Description
Many economists have offered theories about how financial crises develop and how they could be prevented.
ABSTRACT
Title of dissertation: ESSAYS ON FINANCIAL CRISES AND
FINANCIAL REGULATION
Javier Ignacio Bianchi Caporale,
Doctor of Philosophy, 2011
Dissertation directed by: Enrique G. Mendoza
Department of Economics
This dissertation studies the optimal regulatory response to ?nancial crises.
The ?rst two chapters focus on prevention of ?nancial crises, and the third chapter
focuses on resolution of ?nancial crises.
Chapter 1 develops a quantitative theory of overborrowing based on a systemic
risk externality in an emerging market economy. In the model, debt denominated
in foreign currency and balance sheet constraints cause depreciations of the real
exchange rate to be contractionary. The externality arises because when private
agents take debt in good times, they do not internalize that during bad times, the
reduction in demand for consumption causes a higher depreciation of the real ex-
change rate and a further tightening of balance sheet constraints across the economy.
The quantitative analysis suggests that there is an important role for policies that
“throw sand in the wheels of international ?nance.”
Chapter 2 analyzes an externality that arises because of a feedback loop be-
tween asset prices and collateral constraints in a dynamic stochastic general equilib-
rium model calibrated to US data. In the model, a collateral constraint limits private
agents not to borrow more than a fraction of the market value of their collateral
assets, which take the form of an asset in ?xed aggregate supply (e.g. land). When
the collateral constraint binds, ?re-sales of assets cause a Fisherian debt-de?ation
spiral that causes asset prices to decline and the economy’s borrowing ability to
shrink in an endogenous feedback loop. The externality produces deeper recessions
and a larger collapse in asset prices compared to the constrained e?cient allocations.
Chapter 3 studies the macroeconomic and welfare e?ects of government in-
tervention in credit markets during ?nancial crises. A DSGE model to assess the
interaction between ex-post interventions in credit markets and the build-up of risk
ex ante is developed. During a systemic crisis, the central bank ?nds it bene?cial to
bail out the ?nancial sector to relax balance sheet constraints across the economy.
Ex ante, this leads to an increase in risk-taking, making the economy more vulnera-
ble to a ?nancial crisis. We ask whether the central bank should commit to avoiding
a bailout of the ?nancial sector during a systemic crisis. We ?nd that bailouts can
improve welfare by providing insurance against systemic ?nancial crises.
Essays on Financial Crises and Financial Regulation
by
Javier Ignacio Bianchi Caporale
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial ful?llment
of the requirements for the degree of
Doctor of Philosophy
2011
Advisory Committee:
Professor Enrique G. Mendoza (Chair)
Professor Anton Korinek
Professor Lemma Senbet
Professor John Shea
Professor Carlos Vegh
c ? Copyright by
Javier Ignacio Bianchi Caporale
2011
Dedication
To Anita and Francesca.
ii
Acknowledgments
This dissertation could not have been possible without the advice and encour-
agement of my advisors at the University of Maryland.
First and foremost I’d like to thank my advisor, Enrique Mendoza, for his
endless support, encouragement and guidance. He has always made himself available
for help and advice and there has never been an occasion when I’ve knocked on his
door and he hasn’t given me time. He has shown me, by way of example, what an
outstanding academic (and person) should be. The second chapter of my dissertation
emanates from a joint project. It has been a pleasure to work with him and I hope
we continue collaborating on future research.
My other advisors also went beyond the call of their duty. Anton Korinek
deserves special mention. He pushed me to delve deeper in studying the question at
hand. Without his continual support and con?dence, I would have not written my
dissertation. I would like to thank Carlos Vegh. His invaluable advice has shaped
my work and thoughts and will continue to in?uence my future research. I would
like to thank John Shea for generously dedicating time to reading my drafts and for
excellent comments. I am also grateful to Lemma Senbet for kindly accepting to
serve as an external advisor on my committee.
My doctoral research bene?ted greatly from support from a number of institu-
tions. The Department of Economics and the Graduate School at the University of
Maryland and Fulbright provided ?nancial support. I would also like to thank the
Atlanta Fed and the Board of Governors of the Federal Reserve System for ?nancial
iii
support and for hosting me during part of my dissertation. The time I spent at these
two institutions were very fruitful and very special for me. Steve Kay and Federico
Mandelmann were excellent hosts in Atlanta and I owe them special gratitude. At
the Board, I would particularly like to thank John Rogers and Chris Erceg for their
immense support.
I would like to thank my professors at the Universidad de Montevideo for
encouraging me to pursue graduate studies, Fernando Barran, Alejandro Cid, Andres
Jali? and especially Juan Dubra. I have been very fortunate to meet several people
who have helped me and inspired me during my doctoral study. I am especially
grateful to Pablo D’Erasmo, as well as Emine Boz, Lorenzo Caliendo, Bora Durdu,
Bertrand Gruss, Ethan Ilzetski, Tim Kehoe, Leonardo Martinez, Ricardo Nunes,
Carmen Reinhart, and Horacio Saprissa. Discussions in the Liquidity-Lunch with
my colleagues Julien Bengui and Sushant Achyara have been very stimulating as
well. I am grateful to Vickie Fletcher for all her help.
I would like to thank my friends Eduardo, Gabriel, Giorgo, Jose, and Mar-
tin. Thanks for your friendship and support. I am grateful to Alejandro, Daniel,
Cristobal, Kizito and Sebastian for enormous support during my time in Maryland.
I am extremely indebted to my parents Alfredo and Graciela for their uncon-
ditional love, understanding and untold sacri?ce. I would also like to thank my dear
sisters Adriana and Patricia. I treasure moments we have shared.
I am also indebted to Jorge and Cecilia. Thank you for making me feel like a
son. I would also like thank my dear brothers and sisters in-law Ceci, Diego, So?a,
Marcelo, Victoria, Carlos, Fede, Agus, Gonchi and Felipe for their love and support.
iv
Above all I want to thank my wife Anita for her love, care, and encouragement
and our daughter Francesca. Fran you are the joy of our lives. Anita, I could not
imagine my life without you.
College Park, Maryland
May 20, 2011
v
Table of Contents
1 Overborrowing and Systemic Externalities in the Business Cycle 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Equilibrium De?nition . . . . . . . . . . . . . . . . . . . . . . 11
1.4 E?ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Social Planner’s Problem . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Decentralization . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 Borrowing Decisions . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.3 Policy Instruments . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.4 Financial Crises: Incidence and Severity . . . . . . . . . . . . 28
1.5.5 Second Moments . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.6 Welfare E?ects . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.7 Simple forms of intervention: . . . . . . . . . . . . . . . . . . . 34
1.5.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6 Policy Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Overborrowing, Financial Crises and Macro-Prudential Policy (coauthored
with Enrique G. Mendoza) 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.1 Private Optimality Conditions . . . . . . . . . . . . . . . . . 52
2.2.2 Recursive Competitive Equilibrium . . . . . . . . . . . . . . . 55
2.3 Constrained-E?cient Equilibrium . . . . . . . . . . . . . . . . . . . . 57
2.3.1 Equilibrium without collateral constraint . . . . . . . . . . . 57
2.3.2 Recursive Constrained-E?cient Equilibrium . . . . . . . . . . 59
2.3.3 Comparison of Equilibria & ‘Macro-prudential’ Policy . . . . 61
2.4 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.2 Borrowing decisions . . . . . . . . . . . . . . . . . . . . . . . 68
2.4.3 Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4.4 Incidence and Magnitude of Financial Crises . . . . . . . . . . 80
2.4.5 Long-Run Business Cycles . . . . . . . . . . . . . . . . . . . . 86
2.4.6 Properties of Macro-prudential Policies . . . . . . . . . . . . . 89
2.4.7 Welfare E?ects . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 97
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
vi
3 E?cient Bailouts? 108
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2.1 Corporate entities . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2.3 Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.4 Competitive equilibrium . . . . . . . . . . . . . . . . . . . . . 118
3.2.5 Some characterization . . . . . . . . . . . . . . . . . . . . . . 118
3.3 Bailout Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3.1 A second best benchmark . . . . . . . . . . . . . . . . . . . . 120
3.3.2 Policy Experiment . . . . . . . . . . . . . . . . . . . . . . . . 121
3.4 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.1 Calibration and Functional Forms . . . . . . . . . . . . . . . . 124
3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A Proofs (Chapter 1) 132
B An Equivalence Result (Chapter 1) 133
C Sensitivity Analysis (Chapter 1) 135
D Numerical Solution Method (chapter 1) 142
E Numerical Solution Method (chapter 2) 144
vii
Chapter 1
Overborrowing and Systemic Externalities in the Business Cycle
1.1 Introduction
In the wake of the 2008 international ?nancial crisis, there have been intense
debates about reform of the international ?nancial system that emphasize the need
to address the problem of “overborrowing.” The argument typically relies on the
observation that periods of sustained increases in borrowing are often followed by a
devastating disruption in ?nancial markets. This raises the question of why the pri-
vate sector becomes exposed to the dire consequences of ?nancial crises and what the
appropriate policy response should be to reduce the vulnerability to these episodes.
Without a thorough understanding of the underlying ine?ciencies that arise in the
?nancial sector, it seems di?cult to evaluate the merit of proposals that aim to
reform the current international ?nancial architecture.
This paper presents a formal welfare-based analysis of how optimal borrowing
decisions at the individual level can lead to overborrowing at the social level in a
dynamic stochastic general equilibrium (DSGE) model, where ?nancial constraints
give rise to ampli?cation e?ects. As in the theoretical literature (e.g. Lorenzoni,
2008), we analyze constrained e?ciency by considering a social planner that faces the
same ?nancial constraints as the private economy, but internalizes the price e?ects
of its borrowing decisions. Unlike the existing literature, we conduct a quantitative
1
analysis to evaluate the macroeconomic and welfare e?ects of overborrowing. We
study how overborrowing a?ects the incidence and severity of ?nancial crises, the
magnitude of welfare losses, and the features of policy measures that aim to correct
the externality. In a nutshell, we investigate whether overborrowing is in fact a
macroeconomic problem and what should be the optimal policy response.
Our model’s key feature is an occasionally binding credit constraint that limits
borrowing, denominated in the international unit of account (i.e., tradable goods), to
the value of collateral in the form of output from the tradable and nontradable sector,
as in Mendoza (2002). Because debt is partially leveraged in income generated in the
nontradable sector, changes in the relative price of nontradable goods can induce
sharp and sudden adjustments in access to foreign ?nancing. Due to incomplete
markets, agents can only imperfectly insure against adverse shocks. As a result,
when agents have accumulated a large amount of debt and a typical adverse shock
hits, the economy su?ers the typical dislocation associated with an emerging market
crisis. Demand for consumption goods falls, putting downward pressure on the price
of nontradables, which drags down the real exchange rate. This leads to a further
tightening of the credit constraint, setting in motion Fisher’s debt de?ation channel
by which declines in consumption, the real exchange rate, and access to foreign
?nancing mutually reinforce one another, as in Mendoza’s work.
In the model, private agents form rational expectations about the evolution
of macroeconomic variables—in particular the real exchange rate—and correctly
perceive the risks and bene?ts of their borrowing decisions. Nevertheless, they fail
to internalize the general equilibrium e?ects of their borrowing decisions on prices.
2
This is a pecuniary externality that would not impede market e?ciency in the
absence of the credit constraint linked to market prices. However, by reducing the
amount of borrowing ex-ante, a social planner mitigates the decrease in demand for
consumption during crises. This mitigates the real exchange rate depreciation and
prevents a further tightening of ?nancial constraints, making everyone better o?.
Our quantitative analysis shows that the macroeconomic e?ects of the systemic
credit externality are signi?cant. The externality increases the long-run probability
of a ?nancial crisis from 0.4 percent to 5.5 percent and has important e?ects on the
severity of these episodes. In the decentralized equilibrium, consumption drops 17
percent, capital in?ows fall 8 percent, and the real exchange rate drops by 19 percent
in a typical crisis. In the constrained-e?cient allocations, by contrast, consumption
drops 10 percent, capital in?ows barely fall, and the real exchange rate drops by
1 percent. Moreover, the externality allows the model to account for two salient
features of the data: procyclicality of capital in?ows and the high variability of
consumption.
We study a variety of policy measures that can restore constrained e?ciency,
all of which involve restricting the amount of credit in the economy: taxes on debt,
tightening of margins, and capital and liquidity requirements. These measures are
imposed before a crisis hits so that private agents internalize the external costs of
borrowing and the economy becomes less vulnerable to future adverse shocks. In
the calibrated version of our model, the increase in the e?ective cost of borrowing
necessary to implement the constrained-e?cient allocations is about 5 percent on
average, increasing with the level of debt and with the probability of a future ?nan-
3
cial crisis. We also study simple forms of interventions and ascertain that a ?xed
tax on debt can also achieve sizable welfare gains.
Our paper is related to the large literature on the macroeconomic role of ?nan-
cial frictions. Following the work of Bernanke and Gertler (1989) and Kiyotaki and Moore
(1997), various studies have presented dynamic models where ?nancial frictions can
amplify macroeconomic shocks compared to a ?rst-best benchmark where these fric-
tions are absent.
1
Our contribution to this literature is twofold. First, we study the
volatility and the level of ampli?cation of the competitive equilibrium relative to a
second-best benchmark where these frictions are also present. Second, we investigate
several policy measures that can signi?cantly reduce the level of ?nancial instability
and improve welfare by making agents internalize an externality due to ?nancial
accelerator e?ects.
Our paper is related to the theoretical literature that investigates the role of
pecuniary externalities in generating excessive ?nancial fragility, and we borrow ex-
tensively from their insights (see for example Auernheimer and Garcia-Saltos (2000),
Caballero and Krishnamurthy (2001,2003), Lorenzoni (2008), Farhi, Golosov, and Tsyvinski
(2009), and Korinek (2009ab)).
2
In all of these studies, however, the analysis is
qualitative in nature. Our contribution to this literature is to provide a quantitative
assessment of the macroeconomic, policy and welfare implications of overborrowing.
This is an important ?rst step in the evaluation of the potential bene?ts from reg-
1
See for example Aiyagari and Gertler (1999), Bernanke, Gertler, and Gilchrist (1999),
Iacoviello (2005), Gertler, Gilchrist, and Natalucci (2007), Mendoza (2010) (2002,2010).
2
The ine?ciency result of these studies is related to the idea that economies with endoge-
nous borrowing constraint and multiple goods are constrained ine?cient (Kehoe and Levine,
1993) and to the generic ine?ciency result in economies with incomplete markets
(Geneakoplos and Polemarchakis, 1986; Stiglitz, 1982).
4
ulatory measures to correct these externalities and in the study of their practical
implementation.
There is a growing macroeconomic literature that studies optimal policy in
a ?nancial crisis.
3
This literature typically takes as given that the economy is in
a high leverage situation and analyzes the role of policies that can moderate the
impact of a large adverse shock. While this literature provides important insights
on how to respond to crises once they erupt, it does not study how the economy
experiences the surge in debt that leads to the crisis in the ?rst place. This paper
complements this literature by studying how an economy can become vulnerable to
a ?nancial crisis due to excessive borrowing during normal times. We model crises
as infrequent episodes nested within regular business cycles and analyze the role of
policies in reducing an economy’s vulnerability to ?nancial crises, therefore placing
macroprudential policy at the center of the stage. We acknowledge, however, that
because our analysis requires global non-linear solution methods, we abstract from
important real-world features present in larger scale DSGE models.
A related paper that allows for policy intervention during normal times and
crisis times is Benigno et al. (2009). They consider the role of a subsidy on nontrad-
able goods, which the Ramsey planner uses ex-post to mitigate the real exchange
depreciation during crises, but not ex-ante since it is not e?ective to make agents
internalize the full social costs of borrowing. We focus instead on a constrained
planner who directly makes borrowing decisions and show that the decentralization
3
Notable contributions include Christiano, Gust, and Roldos (2004), Kiyotaki and Moore
(2008), Gertler and Karadi (2009), and Gertler and Kiyotaki (2010).
5
requires ex-ante intervention to prevent excessive risk exposure.
Finally, there are a number of other theories of overborrowing that have been
investigated. One theory is moral hazard: banks may lend excessively to take ad-
vantage of some form of government bailout.
4
Uribe (2006) has also studied whether
an economy with an aggregate debt limit tends to overborrow relative to an econ-
omy with debt limits imposed at the level of each individual agent, and found that
borrowing decisions coincide. Our focus is on the comparison between competitive
equilibrium and constrained-e?cient equilibrium when ?nancial constraints that are
linked to market prices generate ampli?cation e?ects.
1.2 Analytical Framework
Consider a representative-agent DSGE model of a small open economy (SOE)
with a tradable goods sector and a nontradable goods sector. Tradable goods can
be used for consumption, external borrowing and lending transactions; nontradable
goods have to be consumed in the domestic economy. The economy is populated by a
continuum of identical, in?nitely-lived households of measure unity with preferences
given by:
E
0
_
?
?
t=0
?
t
u(c
t
)
_
(1.1)
In this expression, E
t
(·) is the time t expectation operator, and ? is the discount fac-
tor. The period utility function u(·) has the constant-relative-risk-aversion (CRRA)
form. The consumption basket c
t
is an Armington-type CES aggregator with elas-
4
See e.g. McKinnon and Pill (1996), Corsetti, Pesenti, and Roubini (1999),
Schneider and Tornell (2004), and Farhi and Tirole (2010).
6
ticity of substitution 1/(? +1) between tradable c
T
and nontradable goods c
N
given
by:
c
t
=
_
?
_
c
T
t
_
??
+ (1 ??)
_
c
N
t
_
??
_
?
1
?
, ? > ?1, ? ? (0, 1)
In each period t, households receive an endowment of tradable goods y
T
t
and an
endowment of nontradable goods y
N
t
. We assume that the vector of endowments
given by y ? (y
T
, y
N
) ? Y ? R
2
++
follows a ?rst-order Markov process. These
endowment shocks are the only source of uncertainty in the model.
The menu of foreign assets available is restricted to a one period, non-state-
contingent bond denominated in units of tradables that pays a ?xed interest rate r,
determined exogenously in the world market.
5
Normalizing the price of tradables
to 1 and denoting the price of nontradable goods by p
N
the budget constraint is:
b
t+1
+ c
T
t
+ p
N
t
c
N
t
= b
t
(1 +r) + y
T
t
+ p
N
t
y
N
t
(1.2)
where b
t+1
denotes bond holdings that households choose at the beginning of time
t. We maintain the convention that positive values of b denote assets. As there is
only one asset, gross and net bond holdings (NFA) coincide.
We assume that creditors restrict loans so that the amount of debt does not
exceed a fraction ?
T
of tradable income and a fraction ?
N
of nontradable income.
5
To have a well-de?ned stochastic steady state, we assume that the discount factor and the
world interest rate are such that ?(1 + r) < 1. If ?(1 + r) ? 1, assets will diverge to in?nity
in equilibrium by the supermartingale convergence theorem (see Chamberlain and Wilson (2000),
2000). See Schmitt-Grohe and Uribe (2003) for other methods to induce stationarity.
7
Speci?cally, the credit constraint is given by:
b
t+1
? ?
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
(1.3)
This credit constraint can be seen as arising from informational and institutional
frictions a?ecting credit relationships (such as monitoring costs, limited enforcement,
asymmetric information, and imperfections in the judicial system), but we do not
model these frictions explicitly. Our focus is on how ?nancial policies can be welfare
improving, taking as given the frictions that lead to these debt contracts, i.e., we will
assume that the social planner is a constrained social planner that is also subject to
this credit constraint.
Discussion of market incompleteness.— A few comments are in order about
the two deviations from complete markets that we introduce here. First, we have
assumed that assets are restricted to a one-period non-state-contingent bond de-
nominated in tradable goods. While agents typically have a richer set of assets
available, this assumption is made for numerical tractability and is meant to cap-
ture the observation that debt in emerging markets is generally short term and
denominated in foreign currency. In turn, these features of debt contracts are gen-
erally seen as an important source of vulnerability in emerging markets (see e.g.
Calvo, Izquierdo, and Loo-Kung, 2006).
The second form of market incompleteness is given by the credit constraint. In
the absence of a credit constraint, households will increase borrowing in bad times
to smooth consumption. This will imply a counterfactual reaction of the current
8
account, which is well-known to rise during recessions in emerging markets. The
credit constraint we have speci?ed has two main features. One crucial feature is
that nontradable goods are part of the collateral. At the empirical level, this is
consistent with evidence that credit booms in the nontradable sector are fueled
by external credit (see e.g. Tornell and Westermann, 2005). At the theoretical
level, this could result because foreign borrowers can seize nontradable goods from
a defaulting borrower, sell them in the domestic market, and repatriate the funds
abroad. A positive gap between ?
T
and ?
N
would re?ect an environment where
creditors have a higher preference for tradable income as collateral. A case where
?
T
= ?
N
would re?ect an environment where creditors request and aim to verify
information on total income of individual borrowers, i.e., they do not document the
sectoral sources of their income.
The second feature is that the collateral is given by current income. At the
empirical level, this assumption is supported by evidence that current income is
a major determinant of credit market access (see e.g., Jappelli, 1990).
6
At the
theoretical level, borrowing limits could depend on current income if households can
engage in fraud in the period they contract debt obligations and prevent creditors
from seizing any future income. If creditors detect the fraud, however, they could
seize household’s current income (see Korinek, 2009a).
An additional argument that our formulation of the credit constraint is suit-
able for a quantitative assessment of the externality is that our model can account
6
For more evidence on credit constraints on households, see Jappelli and Pagano (1989), Zeldes
(1989).
9
reasonably well for the main macro features of emerging market crises, as shown by
Mendoza (2002).
1.3 Equilibrium
1.3.1 Optimality Conditions
The household’s problem is to choose stochastic processes {c
T
t
, c
N
t
, b
t+1
}
t?0
to
maximize the expected present discounted value of utility (2.1) subject to (1.2) and
(1.3), taking b
0
and {p
N
t
}
t?0
as given. The household’s ?rst-order conditions require:
?
t
= u
T
(t) (1.4)
p
N
t
=
_
1 ??
?
__
c
T
t
c
N
t
_
?+1
(1.5)
?
t
= ?(1 +r)E
t
?
t+1
+ µ
t
(1.6)
b
t+1
+
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
? 0, with equality if µ
t
> 0 (1.7)
where ? is the non-negative multiplier associated with the budget constraint and µ
is the non-negative multiplier associated with the credit constraint. The optimality
condition (1.4) equates the marginal utility of tradable consumption to the shadow
value of current wealth. Condition (1.5) equates the marginal rate of substitution of
the two goods, tradables and nontradables, to their relative price. Equation (1.6) is
the Euler equation for bonds. When the credit constraint is binding, there is a wedge
between the current shadow value of wealth and the expected value of reallocating
10
wealth to the next period, given by the shadow price of relaxing the credit constraint
µ
t
. Equation (1.7) is the complementary slackness condition.
Since households are identical, market clearing conditions are given by:
c
N
t
= y
N
t
(1.8)
c
T
t
= y
T
t
+ b
t
(1 +r) ?b
t+1
(1.9)
Notice that equation (1.5) implies that a reduction in c
T
t
generates in equilibrium
a reduction in p
N
t
, which by equation (1.3) reduces the collateral value. Besides
ampli?cation, the credit constraint produces asymmetric responses in the economy:
a binding credit constraint ampli?es the consumption drop in response to a negative
income shock, but no ampli?cation e?ects occur when the credit constraint is slack.
Because of consumption-smoothing e?ects, the demand for borrowing generally de-
creases with current income, and when current income is su?ciently low, the credit
constraint becomes binding.
1.3.2 Equilibrium De?nition
We consider the optimization problem of a representative household in recur-
sive form, which includes, as a crucial state variable, the aggregate bond hold-
ings of the economy. Households need to forecast future aggregate bond hold-
ings that are beyond their control to form expectations of the price of nontrad-
ables. We denote by ?(·) the forecast of aggregate bond holdings for every cur-
11
rent aggregate state (B, y), i.e., B
?
= ?(B, y) . Combining equilibrium conditions
(1.5),(1.8), and (1.9), the forecast price function for nontradables can be expressed
as p
N
(B, y) = (1 ? ?)/(?)
_
(y
T
+ B(1 +r) ??(B, y))/y
N
_
?+1
. The other relevant
state variables for the individual household are its bond holdings and the vector of
endowment shocks. The problem of a representative household can then be written
as:
V (b, B, y) = max
b
?
,c
T
,c
N
u(c(c
T
, c
N
)) +?E
y
?
| y
V (b
?
, B
?
, y
?
) (1.10)
subject to
b
?
+ p
N
(B, y)c
N
+ c
T
= y
T
+ b(1 +r) + p
N
(B, y)y
N
b
?
? ?
_
?
N
p
N
(B, y)y
N
+ ?
T
y
T
_
B
?
= ?(B, y)
where we have followed the convention of denoting current variables without sub-
script and denoting next period variables with the prime superscript. The solution
to the household problem yields decision rules for individual bond holdings
ˆ
b(b, B, y),
tradable consumption ˆ c
T
(b, B, y) and nontradable consumption ˆ c
N
(b, B, y). The
household optimization problem induces a mapping from the perceived law of motion
for aggregate bond holdings to an actual law of motion, given by the representative
agent’s choice
ˆ
b(B, B, y). In a rational expectations equilibrium, as de?ned below,
these two laws of motion must coincide.
De?nition 1 (Decentralized Recursive Competitive Equilibrium)
12
A decentralized recursive competitive equilibrium for our SOE is de?ned by a pricing
function p
N
(B, y), a perceived law of motion ?(B, y) and decision rules
_
ˆ
b(b, B, y), ˆ c
T
(b, B, y), ˆ c
N
(b, B, y)
_
with associated value function V (b, B, y) such that the following conditions hold:
1. Household optimization:
_
ˆ
b(b, B, y), ˆ c
N
(b, B, y), ˆ c
N
(b, B, y), V (b, B, y)
_
solve
the recursive optimization problem of the household for given p
N
(B, y) and
?(B, y).
2. Rational expectation condition: the perceived law of motion is consistent with
the actual law of motion: ?(B, y) =
ˆ
b(B, B, y).
3. Markets clear: y
N
= ˆ c
N
(B, B, y) and ?(B, y) + ˆ c
T
(B, B, y) = y
T
+ B(1 + r).
1.4 E?ciency
1.4.1 Social Planner’s Problem
We previously described the equilibrium achieved when agents take aggregate
variables as given, particularly the price of nontradables. Consider now a benevolent
social planner with restricted planning abilities. We assume that the social planner
can directly choose the level of debt subject to the credit constraint, but allows goods
markets to clear competitively. That is, the planner (a) performs credit operations
and rebates back to households all the proceeds in a lump sum fashion, and (b)
lets households choose their allocation of consumption between tradable goods and
nontradable goods in a competitive way.
As opposed to the representative agent, a social planner internalizes the e?ects
13
of borrowing decisions on the price of nontradables. Critically, the social planner
realizes that a lower debt level mitigates the reduction in the price of nontradables
and prevents a larger drop in borrowing ability when the credit constraint binds.
As a result, we will show that the decentralized equilibrium allocation is not a
constrained Pareto optimum, as de?ned below.
De?nition 2 (Constrained E?ciency)
Let
_
c
T
t
, c
N
t
, b
t+1
_
t?0
be the allocations of the competitive equilibrium yielding util-
ity
ˆ
V . The competitive equilibrium is constrained e?cient if a social planner that
chooses directly {b
t+1
}
t?0
subject to the credit constraint, but lets the goods markets
clear competitively, cannot improve the welfare of households above
ˆ
V .
The social planner’s optimization problem consists of maximizing (2.1) subject
to (1.3),(1.5),(1.8), and (1.9). Substituting for the equilibrium price in (1.3), we can
express the social planner’s optimization problem in recursive form as:
V (b, y) = max
b
?
,c
T
u(c(c
T
, y
N
)) + ?E
y
?
| y
V (b
?
, y
?
) (1.11)
subject to
b
?
+ c
T
= y
T
+ b(1 + r)
b
?
? ?
_
?
N
1 ??
?
_
c
T
y
N
_
?+1
y
N
+ ?
T
y
T
_
Using sequential notation and the superscript “sp” to distinguish the Lagrange mul-
tipliers of the social planner’s problem from the decentralized equilibrium, the ?rst-
14
order conditions for the social planner require:
?
sp
t
= u
T
(t) + µ
sp
t
?
t
(1.12)
?
sp
t
= ?(1 + r)E
t
?
sp
t+1
+ µ
sp
t
(1.13)
b
t+1
+
_
?
N
1 ??
?
_
c
T
t
y
N
t
_
?+1
y
N
t
+ ?
T
y
T
t
_
? 0, with equality if µ
sp
t
> 0
(1.14)
where ?
t
? ?
N
(p
N
t
c
N
t
)/(c
T
t
) (1 +?) > 0 indicates how much the collateral value
changes at equilibrium when there is a change in tradable consumption. Notice that
this term is directly proportional to the fraction of nontradable output that agents
can pledge as collateral, the relative size of the nontradable sector and the inverse
of the elasticity of substitution between tradables and nontradables. We will return
to this expression in the sensitivity analysis.
The key di?erence between the optimization problem of the social planner
relative to households follows from examining (1.12) compared with the correspond-
ing equation for the decentralized equilibrium (1.4). The social planner’s marginal
bene?ts from tradable consumption include the direct increase in utility u
T
(t) and
also the indirect increase in utility µ
sp
t
?
t
. This indirect bene?t, not considered by
private agents, represents how an increase in tradable consumption increases the
price of nontradables and relaxes the credit constraint of all agents by ?
t
, which has
a shadow value of µ
sp
t
. Thus, (1.4) and (1.12) yield the key result that, for given ini-
tial states and allocations at which the credit constraint binds, private agents value
15
wealth less than the social planner, which we highlight in the following remark.
Remark 1 When the credit constraint binds, private agents undervalue wealth.
To see more clearly why this di?erent ex-post valuation generates overborrow-
ing ex-ante, suppose that at time t the constraint is not currently binding. Using
(1.4) and (1.6), the Euler equation for consumption in the decentralized equilibrium
becomes:
u
T
(t) = ?(1 + r)E
t
u
T
(t + 1) (1.15)
Using (1.12) and (1.13), the Euler equation for consumption for the social planner
becomes:
u
T
(t) = ?(1 + r)E
t
_
u
T
(t + 1) + µ
sp
t+1
?
t+1
¸
(1.16)
Consider now a reallocation of wealth by the social planner starting from the
privately optimal allocations in the decentralized equilibrium. In particular, consider
the welfare e?ects of a reduction of one unit of borrowing. Because decentralized
agents are at the optimum, (1.15) shows that the ?rst-order private welfare bene?ts
?(1 + r)E
t
u
T
(t + 1) are equal to the ?rst-order private welfare costs u
T
(t). Using
(1.16), the social planner has a marginal cost of reducing borrowing equal to the
private marginal cost, but faces higher marginal bene?ts: a one unit decrease in bor-
rowing relaxes next-period ability to borrow by (1 + r)?
t+1
, which has a marginal
utility bene?t of µ
sp
t+1
. The uninternalized external bene?ts from savings, or equiv-
alently the uninternalized external marginal cost of borrowing, is then given by the
discounted expected marginal utility cost of the resulting tightening of the credit
16
constraint ?(1 +r)E
t
µ
sp
t+1
?
t+1
. Notice that if the credit constraint does not bind for
any pair (b, y) in the two equilibria, the conditions characterizing both environments
are identical and therefore the allocations coincide.
Proposition 1 (Constrained Ine?ciency) The decentralized equilibrium is not,
in general, constrained e?cient.
Proof : See Appendix A
1.4.2 Decentralization
We study the use of various ?nancial policies in the implementation of the
constrained-e?cient allocations. We start by showing how a tax on debt can restore
constrained e?ciency and then show the equivalence between the tax on debt and
more standard forms of intervention in the ?nancial sector (e.g., capital require-
ments).
Letting ?
t
be the tax charged on debt issued at time t, the Euler equation for
bonds in the regulated decentralized equilibrium (1.6) becomes:
u
T
(t) = ?(1 + r)(1 + ?
t
)E
t
u
T
(t + 1) +µ
t
(1.17)
Proposition 2 (Optimal tax on debt) The constrained-e?cient allocations can
be implemented with an appropriate state contingent tax on debt, with tax revenue
rebated as a lump sum transfer.
Proof : See Appendix A
17
When the credit constraint is not binding in the constrained-e?cient allo-
cations, the tax must be set to ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1)) (variables are
evaluated at the constrained-e?cient allocations). This expression represents the
uninternalized marginal cost of borrowing analyzed above, normalized by the ex-
pected marginal utility. As we will see in the quantitative analysis, this tax increases
with the current level of debt, since a higher current level of debt implies a higher
choice of debt, which increases the probability and the marginal utility cost of a
binding constraint next period. Notice also that if the credit constraint has a zero
probability of binding in the next period, the tax is set to zero.
When the credit constraint is binding, the tax does not generally in?uence the
level of borrowing since the choice of debt is given by the credit constraint (1.3) and
not by the Euler equation (1.17). Setting the tax to ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1))?
(µ
sp
t
?
t
) / (?(1 + r)E
t
u
T
(t + 1)) achieves constrained e?ciency and equalizes the pri-
vate and social shadow values from relaxing the constraint. Notice that an extra
term arises because the social planner internalizes that relaxing the credit constraint
today would have positive e?ects on the current price of nontradables. This term is
negative so that the tax causes the private shadow value of relaxing the constraint
to rise to the social value. As we will show in the quantitative analysis, when the
planner is borrowing up to the limit, the level of borrowing desired by private agents
is also the maximum available. As a result, we ?nd that setting ?
?
t
= 0 when the
constraint binds also implements the constrained-e?cient allocations, and since this
results in a simpler policy we set this tax to zero when we turn to describe its
18
quantitative features.
7
In practice, much of prudential ?nancial regulation is implemented through
banks. To take this into consideration, we develop in Appendix B a simple model
of ?nancial intermediaries, and show that our benchmark economy, in which the
planner sets a tax on debt on borrowers, is equivalent to a economy where the
planner sets capital requirements or reserve requirements on ?nancial institutions.
Throughout the paper, we will refer to the implied tax on debt as the increase in
the cost of debt induced by the use of any of these equivalent policy measures.
Alternatively, the planner could implement the constrained-e?cient allocations
using margin requirements by choosing an adjustment ?
t
? 0 such that the credit
constraint becomes b
t+1
? ?(1 ? ?
t
)
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
. If the socially optimal
amount of borrowing is b
sp
t
, by setting ?
?
t
= 1 ?(b
sp
t
)/
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
the social
planner can restrict the quantity of borrowing and restore constrained e?ciency.
8
Remark 2 (Decentralization) The constrained-e?cient allocations can be im-
plemented with appropriate capital requirements, reserve requirements, or margin
requirements.
7
This result held for the wide range of parameters explored in our quantitative analysis.
8
For this policy to restore constrained e?ciency, it must be the case that
ˆ
b
de
(B, B, y) ?
ˆ
b
sp
(B, y) ?(B, y), which we will show is the case in the numerical analysis. We assume for simplic-
ity that ?
?
is such that the credit constraint always holds with equality in the regulated economy.
In the regulated economy where the adjustment of margins is given by ?
?
, the constraint only
binds in the constrained region and in the tax-region; hence setting ?
t
= 0 in the no-tax region
would deliver the same allocations.
19
1.5 Quantitative Analysis
In this section, we describe the calibration of the model and evaluate the quan-
titative implications of the externality. We solve the competitive equilibrium and the
social planner’s problems numerically using global non-linear methods (described in
detail in the online Appendix).
1.5.1 Calibration
A period in the model represents a year. The baseline calibration uses data
from Argentina, an example of an emerging market with a business cycle that has
been studied extensively. The risk aversion is set at ? = 2, a standard value. The
interest rate is set at r = 4 percent, which is a standard value for the world risk-free
interest rate in the DSGE-SOE literature
We model endowment shocks as a ?rst-order bivariate autoregressive process:
log y
t
= ? log y
t?1
+ ?
t
where y = [y
T
y
N
]
?
, ? is a 2x2 matrix of autocorrelation
coe?cients, and ?
t
= [?
T
t
?
N
t
]
?
follows a bivariate normal distribution with zero
mean and contemporaneous variance-covariance matrix V . This process is estimated
with the HP-?ltered cyclical components of tradables and nontradables GDP from
the World Development Indicators (WDI) for the 1965-2007 period, the longest
time series available from o?cial sources. Following the standard methodology, we
classify manufacturing and primary products as tradables and classify the rest of
the components of GDP as nontradables. The estimates of ? and V are:
20
? =
_
¸
¸
_
0.901 0.495
?0.453 0.225
_
¸
¸
_
V =
_
¸
¸
_
0.00219 0.00162
0.00162 0.00167
_
¸
¸
_
The standard deviations of tradable and nontradable output in the data are
?
y
T = 0.058 and ?
y
N = 0.057, the ?rst-order autocorrelations are ?
y
T = 0.53 and
?
y
N = 0.61, and the correlation between the two is ?
y
T
,y
N = 0.81. Thus cyclical
?uctuations in the two sectors have similar volatility and persistence, and are posi-
tively correlated with each other. We discretize the vector of shocks into a ?rst-order
Markov process, with four grid points for each shock, using the quadrature-based
procedure of Tauchen and Hussey (1991). The mean of the endowments are set to
one without loss of generality.
Table 1.1: Calibration
Value Source/Target
Interest rate r = 0.04 Standard value DSGE-SOE
Risk aversion ? = 2 Standard value DSGE-SOE
Elasticity of substitution 1/(1 +?) = 0.83 Conservative value
Stochastic structure See text Argentina’s economy
Relative credit coe?cients ?
N
/?
T
= 1 Baseline Value
Weight on tradables in CES ? = 0.31 Share of tradable output=32%
Discount factor ? = 0.91 Average NFA-GDP ratio = ?29%
Credit coe?cient ?
T
= 0.32 Frequency of crisis = 5.5%
The intratemporal elasticity of substitution 1/(? + 1) is a crucial parameter
because it a?ects the magnitudes of the price adjustment. For a given reduction in
tradable consumption, a higher elasticity implies a smaller change in the price of
nontradables, and therefore we should expect weaker e?ects from the externality.
The range of estimates for the elasticity of substitution is between 0.40 and 0.83.
9
9
See Mendoza (2006), Gonzalez-Rozada and Neumeyer (2003) and Stockman and Tesar
(1995).
21
As a conservative benchmark, we set ? such that the elasticity of substitution equals
the upper bound of this range and then show how the externality changes with this
parameter.
The ratio ?
N
/?
T
determines the relative quality of nontradable and tradable
output as collateral. It is di?cult, however, to derive a direct mapping from the
data to this ratio. We therefore take a pragmatic approach: we begin by setting
?
N
= ?
T
and then perform extensive sensitivity analysis.
The three remaining parameters are
_
?, ?, ?
T
_
, which are set so that the long-
run moments of the decentralized equilibrium match three historical moments of the
data. The parameter ? governs the tradable share in the CES aggregator and is
calibrated to match a 32 percent share of tradable production.
10
This approach is a
reasonable one to calibrate ? since given the relative endowment and consumption
ratios, ? determines the equilibrium price of nontradables by (1.5) and the share of
tradables in the total value of production. This calibration results in a value of ?
of 0.32.
The discount factor ? is set so that the average net foreign asset position-to-
GDP ratio in the model equals its historical average in Argentina, which is equal
to -29 percent in the dataset constructed by Lane and Milesi-Ferretti (2001). This
calibration results in a value of ? = 0.91, a relatively standard value for annual
frequency in the literature.
The parameter ?
T
is calibrated to match the observed frequency of “Sudden
10
Garcia (2008) reports an average share of tradables of 32 percent using almost a century of
data from Argentina. The average share of tradables is also 32 percent in the data from WDI for
the period 1980-2007.
22
Stops”, which is about 5.5 percent in the cross-country data set of Eichengreen, Gupta, and Mody
(2006).
11
To be consistent with their de?nition of Sudden Stops, we de?ne Sudden
Stops in our model as events where the credit constraint binds and where this leads
to an increase in net capital out?ows that exceeds one standard deviation. This
calibration results in a value of ?
T
equal to 0.32, which is in the range of those used
in the literature (see Mendoza, 2002).
1.5.2 Borrowing Decisions
We ?rst show how the bond accumulation decisions of the social planner dif-
fer from those of private agents and then simulate the model to analyze how this
di?erence a?ects the long-run distribution of debt, the crisis dynamics, and the
unconditional second moments.
Figure 1 shows the bond decision rules in the decentralized equilibrium and
in the constrained-e?cient equilibrium as a function of current bond holdings when
both tradable and nontradable shocks are one-standard-deviation below trend. Since
the mean value of tradable output is 1, we can interpret all results as ratios with
respect to the average output of tradables.
Without the endogenous borrowing constraint, the policy function for next
period’s bond holdings would be monotonically increasing in current bond holdings.
Instead, the policy functions are non-monotonic. The change in the sign of the slope
of the policy function indicates the point at which the credit constraint is satis?ed
11
For the case of Argentina, Eichengreen et al. includes 1989 and 2001 as Sudden Stop events,
yielding a similar frequency to the cross-country average over the sample period.
23
with equality, but is not binding. To the right of this point, the credit constraint is
slack, and bond decision rules display the usual upward-sloping shape. To the left of
this point, next-period bond holdings decrease in current bond holdings. To see why,
notice that a decrease in the current bond position implies a reduction in tradable
consumption for a given choice of next-period bond holdings by equation (1.9). This
in turn lowers the price of nontradables by equation (1.5), which means that the level
of borrowing must be reduced further to satisfy the credit constraint. Comparing
the policy functions against the 45-degree line also shows that for relatively low
current levels of bond holdings, the economy reduces the level of borrowing, which
results in capital out?ows.
We distinguish three regions for all pairs of (b, y) according to the actions taken
by the planner in the regulated economy: a “constrained region,” a “tax region”,
and a “no-tax region.” The constrained region in Figure 1 is given by the range
of b with su?ciently high initial debt such that the credit constraint binds in the
constrained-e?cient equilibrium. In this region, both private agents and the social
planner borrow up to the limit, and decision rules coincide. The long-run probability
of this region is 6.2 percent in the decentralized equilibrium, about twice as much
as for the social planner.
The tax region appears shaded in Figure 1 and corresponds to the pairs of (b, y)
where the social planner would impose a tax on debt in the regulated economy. As
explained above, this is the region where households borrow enough so that the credit
constraint will bind with a strictly positive probability in the next period. Here, the
social planner accumulates uniformly higher bond holdings than households. In fact,
24
?1 ?0.9 ?0.8 ?0.7 ?0.6
?1
?0.9
?0.8
?0.7
?0.6
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
Tax Region
No?Tax Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Social Planner
Decentralized Equilibrium
Figure 1.1: Bond decision rules for negative one-standard-deviation shocks
households continue to borrow up to the limit, over some range of current bonds for
which the social planner would choose a lower borrowing level that monotonically
increases in current bond holdings. The economy spends about 80 percent of its
time in the tax region in both equilibria.
The no-tax region is located to the right of the tax region and corresponds
to the pairs of (b, y) where the credit constraint is slack and the social planner
would not impose a tax on debt. Intuitively, the economy is relatively well-insured
in this region, and the amount of borrowing chosen does not make the economy
vulnerable to a binding constraint. Here, the di?erences in the bond decision rules
become quantitatively smaller, but are non-zero since di?erent future choices of
bond holdings a?ect current optimal choices. The social planner spends 16 percent
of the time in this region, 2 percent more than the decentralized equilibrium.
25
While both the social planner and private households self-insure against the
risk of ?nancial crises, the social planner accumulates extra precautionary savings
above and beyond what households consider privately optimal. As Figure 2 shows,
this implies that the ergodic distribution of bond holdings in the decentralized equi-
librium assigns a higher probability to higher levels of debt. In fact, the decentralized
equilibrium has a 15 percent chance of carrying a larger amount of debt than the
maximum held by the social planner, illustrated by the shaded region in Figure 2.
?1 ?0.95 ?0.9 ?0.85 ?0.8 ?0.75 ?0.7
0
0.35
0.7
1.05
x 10
?3
Bond Holdings
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Figure 1.2: Ergodic Distribution of Bond Holdings
Notice that the large di?erences in the left tail distribution of debt are not
translated into large di?erences in average debt levels: the average debt-to-GDP
ratio is 29.2 percent for the private economy and 28.6 percent for the social planner.
What is crucial is that the social planner reduces the exposure to debt levels that
makes the economy vulnerable to a severe ?nancial crisis when the economy is hit
by an adverse shock.
26
1.5.3 Policy Instruments
Figure 1.3 shows the two types of policy measures that achieve the constrained-
e?cient allocations when shocks are one standard deviation below trend for di?erent
levels of current bond holdings: the left panel shows the e?ective increase in the
cost of borrowing from tax-like measures (taxes, reserve or capital requirements);
the rigth panel shows the adjustment in margin requirements ?.
As explained above, for su?ciently low values of debt, the implied tax on
borrowing is zero. The tax then increases with the level of debt in the tax region,
until the credit constraint becomes binding for the social planner and the tax is set
to zero. On average, the implied tax on debt is 5.2 percent.
?1 ?0.9 ?0.8 ?0.7 ?0.6
0
5
10
15
20
25
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
Implied Tax on Debt
?1 ?0.9 ?0.8 ?0.7 ?0.6
0
5
10
15
20
25
30
35
40
Tightening of Margins
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
Figure 1.3: Policy Instruments for negative one-standard-deviation shocks
The adjustment to the margin requirement is also zero when the constraint
is already binding, but unlike the tax-like measures the adjustment decreases with
the level of debt outside the constrained region. This arises because, as the level
of debt increases, the excess debt capacity is reduced, thereby requiring a smaller
adjustment in margin requirements to reduce the gap and socially desired amount
27
of borrowing. On average, margins are tightened by 9 percent in the tax-region,
which implies that the e?ective fraction that agents can borrow from their income
is reduced from 0.32 to 0.29 in the regulated economy.
1.5.4 Financial Crises: Incidence and Severity
In this section, we establish that overborrowing in the decentralized equilib-
rium leaves the economy vulnerable to more frequent and more severe ?nancial crises.
Using the policy functions of the model, we perform an 80,000-period stochastic time
series simulation of the decentralized and constrained-e?cient equilibrium and use
the resulting data to study the incidence and severity of ?nancial crises. A ?nancial
crisis event is de?ned as a period in which the credit constraint binds, and in which
this leads to an increase in net capital out?ows that exceeds one standard devia-
tion of net capital out?ows in the ergodic distribution of the decentralized economy.
(Results are similar with alternative de?nitions of a crisis event.)
Two important results emerge from the event analysis. First, crises in the
decentralized equilibrium are much more likely: the long-run probability of crises
is 5.5 percent (versus 0.4 percent for the social planner). Thus, by reducing the
amount of debt, the social planner cuts the long-run probability of a ?nancial crisis
more than tenfold.
28
?40 ?35 ?30 ?25 ?20 ?15 ?10 ?5 0 5
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
?3
Percentage change in consumption
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Figure 1.4: Conditional distribution of impact e?ect of ?nancial crises on consump-
tion
Second, the magnitudes of ?nancial crises are substantially more severe be-
cause of the externality. Figure 4 shows the distribution of the response of the
consumption basket on impact during ?nancial crises for the two equilibria, ex-
pressed as a percentage deviation from the average long-run value of consumption.
This ?gure shows that the decentralized equilibrium assigns non-trivial probabilities
to consumption drops of more than 22 percent, while such a fall in consumption is
a zero probability event in the social planner’s allocations.
Drops in the real exchange rate and capital in?ows are also more pronounced
because of the externality. Table 2 compares the responses of these variables during
the median ?nancial crisis under the decentralized equilibrium with the economy’s
response under the constrained-e?cient allocations, conditional on the social planner
having the same level of debt two periods before such crisis and receiving the same
29
sequence of shocks.
12
In this experiment, we ?nd that while the credit constraint also becomes bind-
ing for the social planner at time t, the impact of the adverse shocks is less severe:
consumption falls 10 percent (versus 17 percent in the decentralized equilibrium),
the current account does not increase (versus an 8 percent increase in the decen-
tralized equilibrium), and the real exchange rate depreciates 1 percent (versus 19
percent in the decentralized equilibrium).
13
Notice that since the initial level of debt
and the sequence of shocks are the same for the two equilibria, the di?erence in the
impact of crises is entirely due to the more prudent behavior of the social plan-
ner during the periods preceding the crisis, which makes the required adjustment
following an adverse shock less severe.
Table 1.2: Severity of Financial Crises
Decentralized Equilibrium Social Planner
Consumption -16.7 -10.1
Current Account-GDP 7.8 0.0
Real Exchange Rate 19.2 1.1
Note: Consumption and the depreciation of the real ex-
change rate represent responses on impact expressed as
percentage deviations from averages in the corresponding
ergodic distribution.
12
More precisely, we ?rst de?ne the median ?nancial crisis in the decentralized as the period
with the median current account reversal considering all the crisis episodes in the decentralized
equilibrium. Then we backtrack the initial level of debt two periods before this episode and the
sequence of shocks that hit the economy at time t ? 2, t ? 1, and t. Given this initial level of
debt at t ?2 and the sequence of shocks, we then use the decision rules of the constrained-e?cient
allocations to simulate the economy and compare the impact e?ects at time t with the impact
e?ects in the decentralized equilibrium median crisis.
13
We de?ne the real exchange rate as
_
?
1/(1+?)
+ (1 ??)
1/(1+?)
_
p
N
_
?/(1+?)
_
?(1+?)/?
implying
a one-to-one negative relationship between the price of nontradables and the real exchange rate.
30
1.5.5 Second Moments
Table 3 compares the unconditional second moments for decentralized and
constrained-e?cient equilibria, which are computed using each economy’s ergodic
distribution, and for the Argentinian data. It is apparent that the externality pro-
duces non-trivial e?ects on the volatility of consumption, capital ?ows, and espe-
cially the real exchange rate. Two reasons for this are that, ?rst, the economy
spends most of the time in the tax-region where the bond accumulation decisions
di?er signi?cantly; and second, the decentralized equilibrium experiences ?nancial
crises with a 5.5 percent probability, which is more than 10 times larger than that
of the social planner.
Table 1.3: Second Moments
Decentralized Social
Equilibrium Planner Data
Standard Deviations
Consumption 5.9 5.3 6.2
Real Exchange Rate 9.2 3.5 8.2
Current Account-GDP 2.8 0.6 3.6
Trade Balance-GDP 2.9 0.7 2.4
Correlation with GDP in units of tradables
Consumption 0.84 0.85 0.88
Real Exchange Rate -0.79 -0.43 -0.41
Current Account-GDP -0.76 -0.05 -0.63
Trade Balance-GDP -0.77 -0.22 -0.84
Note: Data is annual from WDI for Argentina from 1965-2007. The real ex-
change rate is calculated as
_
?
1/(1+?)
+ (1 ??)
1/(1+?)
_
p
N
_
?/(1+?)
_
?(1+?)/?
and is measured empirically using value added de?ators.
Table 3 also shows that the model accounts reasonably well for observed busi-
ness cycle moments for Argentina, in line with previous studies. Moreover, it is ap-
parent that the externality is important in accounting for two key regularities in the
31
emerging market business cycle: the high volatility of consumption and the strong
procyclicality of capital ?ows (see e.g. Kaminsky, Reinhart, and Vegh, 2004). The
constrained-e?cient equilibrium cannot account for these two stylized facts. This
occurs because the social planner accumulates su?ciently large precautionary sav-
ings to make large reversals in capital ?ows a much lower probability event compared
to the decentralized equilibrium.
1.5.6 Welfare E?ects
We compute the welfare gains from correcting the externality as the propor-
tional increase in consumption for all possible future histories in the decentralized
equilibrium that would make households indi?erent between remaining in the de-
centralized equilibrium (without government intervention) and correcting the exter-
nality. These calculations explicitly consider the costs of a lower consumption in
the transition to the constrained-e?cient allocations. Because of the homotheticity
of the utility function, the welfare gain ? at a state (b, y) is given by:
(1 + ? (b, y))
1??
V
de
(b, y) = V
sp
(b, y) (1.18)
The welfare gains from correcting the externality are shown in Figure 1.5 as
a function of current bond holdings, for negative, one- standard-deviation endow-
ment shocks. Notice the parallel between the welfare e?ects and the three regions
described in Figure 1, which gives the welfare gains from correcting the externality
a hump shape. In the constrained region, the borrowing decisions coincide; there-
32
fore, the welfare gains only arise from how future allocations will di?er. In the tax
region, because the social planner acts in a signi?cantly more precautionary way,
the welfare gains increase. In the no-tax region, where ?nancial crises are less likely,
borrowing decisions are similar in the two equilibria and welfare gains are smaller.
On average, the welfare gains from correcting the externality are 0.135 percent
of permanent consumption, consistent with the well-known result that the welfare
cost of business cycles is typically small. Even if the planner does not introduce
additional securities that partially complete the market, welfare gains are still larger
than the bene?ts from introducing asset price guarantees (Durdu and Mendoza,
2006) or the bene?ts from introducing indexed bonds (Durdu, 2009), often suggested
as policies to address Sudden Stops (Caballero, 2002).
?1 ?0.9 ?0.8 ?0.7 ?0.6
0.05
0.1
0.15
0.2
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
o
f
P
e
r
m
a
n
e
n
t
C
o
n
s
u
m
p
t
i
o
n
Figure 1.5: Welfare gains from correcting externality for negative one-standard-
deviation shocks
We see these welfare gains of correcting the externality as a lower bound.
First, the supply side of the economy is the same for both equilibria. If ?nancial
crises distort the e?cient use of production resources, correcting the externality
33
could deliver higher welfare gains. Second, the risk we have considered is only
aggregate; Chatterjee and Corbae (2007) shows that the welfare gains of eliminating
the possibility of a crisis state can be as large as 7 percent of permanent consumption
when considering idiosyncratic risk.
1.5.7 Simple forms of intervention:
Decentralizing the social planner’s allocations requires a state-contingent pol-
icy that might be challenging to implement in practice. Therefore, we also in-
vestigate if more simple forms of intervention can take the economy close to the
second-best. In particular, we ?nd that a ?xed tax on debt induces welfare gains
that are quite close to the second-best solution. The optimal ?xed tax is 3.6 percent,
which is about 70 percent of the average of the state-contingent tax, and achieves 62
percent of the welfare gains from implementing the constrained-e?cient allocations.
This ?xed tax cuts by more than half the probability of a crisis. Allowing the tax
to drop to zero when the credit constraint binds in the regulated economy or in the
constrained region, delivers about the same welfare gains as the ?xed tax.
By contrast, a ?xed tightening in margins across all states of nature delivers
welfare losses. This is intuitive given that tightening margins when the constraint is
already binding delivers signi?cant welfare losses, which outweigh the bene?ts from
a lower average amount of debt. Tightening margins outside the constrained region
only, however, does generate welfare gains, albeit smaller gains than a ?xed tax on
borrowing.
34
1.5.8 Sensitivity Analysis
In this section, we examine the sensitivity of our results to alternative calibra-
tions. Figure 1.6 shows how the average welfare e?ects and the optimal average tax
on debt vary with some key parameters. The online Appendix includes results from
changing all parameter values and details of the sensitivity analysis.
We can gain a better understanding of the results by analyzing the externality
term µ
sp
t
?
t
, which is the wedge between the social shadow value of wealth and the
private shadow value of wealth. Recalling that ?
t
? ?
N
(p
N
t
c
N
t
)/(c
T
t
) (1 +?) , we
have that the fraction of nontradable output that can be collateralized, the size
of the nontradable sector and the elasticity of substitution are the key parameters
determining the price e?ects in the externality term.
To arrive to a unit free measure of how the di?erent parameters a?ect the
price responses, we decompose the e?ects of the di?erent parameters in terms of
elasticities. Two elasticities are crucial for determining the price e?ects on the
borrowing capacity. First is the elasticity that measures how much the depreciation
of nontradables tightens the borrowing constraint. Second is the elasticity that
measures how much nontradables depreciate as a result of an increase in debt service.
Denoting the borrowing limit by ? ? ?
N
p
N
y
N
+ ?
T
y
T
, debt service by DS ?
b(1 +r) ?b
?
, and the elasticity of y with respect to x by ?
y,x
, we can decompose the
elasticity of the borrowing limit with respect to debt service as the product of these
35
two elasticities:
14
?
?,DS
=
_
p
N
y
N
y
T
+ p
N
y
N
_
. ¸¸ .
?
?,p
N
_
(? + 1)
DS
y
T
?DS
_
. ¸¸ .
?
p
N
,DS
(1.19)
In the baseline calibration,?
?,p
N = 0.65 and ?
p
N
,DS
= 0.40 at the median crisis in
the decentralized equilibrium. That is, a reduction in the debt service of 1 percent
would mitigate the drop in the price of nontradables during a median crisis by 0.40
percent (and the real exchange rate by 0.22 percent) and in turn this would increase
the borrowing capacity by 0.40
?
0.65 = 0.26 percent.
It is clear from (1.19) that the elasticity of substitution 1/(1 + ?) between
tradables and nontradables is a key determinant of how much the real exchange
rate depreciates as a result of an increase in debt service: a lower elasticity implies
that a given decrease in tradable consumption requires a greater adjustment in the
real exchange rate to equilibrate the market. Moreover, increases in the level of debt
service lead to a larger depreciation in the real exchange rate all else equal. While
the elasticity of substitution in our baseline calibration is at the high end of the
range of existing estimates, Panel A of Figure 1.6 shows that a smaller elasticity of
substitution increases the optimal average tax on debt and the welfare bene?ts of
correcting the externality. While in theory the overborrowing distortion is present for
any ?nite value of the elasticity, Panel A also shows that in practice, the allocations
of the social planner and the decentralized equilibrium become almost identical for
14
This formula yields as a result of applying the de?nition of ? and DS, the three elasticities
and using ? = ?(??)/(?DS).
36
values of the elasticity greater than 4.
The elasticity decomposition also shows that for low shares of the nontradable
sector, a depreciation of the real exchange rate has smaller e?ects on the borrowing
capacity. By adjusting ? to reduce the equilibrium share of the nontradable sector,
Panel B in Figure 1.6 shows that this is re?ected in lower e?ects from the externality.
We next study changes in ?
T
and ?
N
. It is reasonable to argue that in the
presence of debt denominated in units of tradable, creditors may be less willing
to accept nontradable income as collateral. We explore this idea by varying the
values of ?
N
and ?
T
such that ?
N
= c?
T
for c < 1 and such that the long-run
average of ?
N
p
N
y
N
+ ?
T
y
T
remains as in the baseline calibration. As Panel (D)
in Figure 1.6 shows, the e?ects of the externality remain signi?cant even if the
quality of nontradable collateral is half of the quality of tradable collateral, i.e.
when c = 0.5. In fact, crises in the decentralized equilibrium are 3.5 times more
likely and signi?cant di?erences remain in the severity of these episodes.
In Panel (C) we set ?
T
= ?
N
= ? and vary the value of ?. Notice that there
are two opposing e?ects from an increase in ?. On one hand, since ? scales up
the price e?ects in the externality term, we should expect higher e?ects from the
externality. On the other hand, increasing ? makes the credit constraint less likely
to bind, thereby reducing the externality. As Panel (C) shows, a relatively higher ?
increases the externality.
The other component of the externality term is the shadow value from relaxing
the credit constraint µ. While it is not possible to derive an analytical expression
for µ, for a given state µ should be positively related to the household’s share of
37
tradables in the utility function, and the inverse of the intratemporal and intertem-
poral elasticity of substitution, because these parameters a?ect the utility cost from
a large drop in tradable consumption; µ should also depend on the discount factor
and the interest rate, because these parameters a?ect the household’s impatience
and its willingness to borrow.
0 5 10 15 20
0
1
2
3
4
5
6
0 5 10 15 20
0
0.05
0.1
0.15
0.2
0.25
(A) Elasticity of Substitution
Average Tax (left axis) Welfare(right axis)
0.55 0.6 0.65 0.7 0.75
0
1
2
3
4
5
6
0.55 0.6 0.65 0.7 0.75
0
0.25
0.5
(B) Share of Nontradables in GDP
0.3 0.35 0.4
0
1
2
3
4
5
6
0.3 0.35 0.4
0
0.1
0.2
0.3
(C) Collateral Coefficient
0.4 0.6 0.8 1
0
1
2
3
4
5
6
0.4 0.6 0.8 1
0
0.04
0.08
0.12
0.16
(D) Relative Quality of Nontradable Collateral (c)
Figure 1.6: Sensitivity Analysis: Welfare and Implied Tax (in Percentage)
We extend the model by allowing for production in the nontradable sector with
intermediate inputs, as in Durdu, Mendoza, and Terrones (2009). Speci?cally, ?rms
use intermediate inputs m to produce nontradables with a technology such that y
t
=
A
t
m
?
t
. Firms maximize p
N
t
Am
?
t
?p
m
m
t
and redistribute pro?ts to households, whose
income is now given by the endowment of tradables plus pro?ts. Since a binding
credit constraint induces a depreciation of nontradables, this feature generates a drop
in the value of the marginal product of imported inputs, and therefore a drop in the
38
production of nontradables during ?nancial crises. As a result, since crises in the
decentralized equilibrium generate a larger depreciation, the externality generates a
larger drop in both tradable and nontradable consumption. Setting ? = 0.10 in line
with Goldberg and Campa (2010) and re-calibrating the rest of the parameters, we
?nd that the e?ects of the externality remain very similar overall.
15
Overall, the sensitivity analysis suggests that overborrowing creates signi?cant
distortions for plausible parameterizations. Only when the probability of a binding
credit constraint becomes negligible or when debt de?ation e?ects are very weak do
we ?nd that the e?ects of overborrowing are insigni?cant.
1.6 Policy Remarks
The new paradigm in ?nancial regulation stresses the need for a macropruden-
tial approach to consider how actions of individual market participants can destabi-
lize macroeconomic conditions with adverse e?ects over the whole economy (see e.g.
Borio, 2003). The analysis presented here suggests that overborrowing externalities
have a large enough quantitative impact on welfare to justify macroprudential reg-
ulation. It is worth noting that correcting these externalities does not eliminate the
possibility of ?nancial crises in our simulations, but the incidence and severity of
crises are considerably reduced under regulation. This is consistent with the con-
strained notion of e?ciency that we consider in our analysis: the social planner is
15
We continue to assume that in the constrained-e?cient allocations the social planner makes
borrowing decisions, while households choose consumption allocations and ?rms choose interme-
diate inputs. This yields an identical decentralization to the endowment economy model where a
tax on debt is su?cient to achieve constrained e?ciency.
39
subject to the same ?nancial frictions as the decentralized economy, so that reg-
ulation does not fully eliminate the ?nancial accelerator e?ects that arise when a
negative shock triggers the credit constraint.
In the context of the debate on ?nancial globalization, there is a view that a
Tobin-style tax can help smooth the boom-bust cycle caused by sharp changes in
access to credit in emerging markets. A recent “IMF Sta? Position Note” by Ostry et
al. (2010) emphasizes the bene?ts experienced by emerging markets from the recent
use of reserve requirements, although some controversy remains in the literature.
Our paper contributes to this debate by undertaking a quantitative investigation of
how curbing external ?nance can deliver a reduction in the vulnerability to ?nancial
crises while still allowing an economy to reap the bene?ts of access to global capital
markets. At the same time, fostering the development of ?nancial markets could
also generate signi?cant welfare gains by improving risk sharing and addressing the
root of the externality, i.e., the credit constraint. To the extent that the degree of
?nancial development remains incomplete, our results suggest that there is a scope
for “throwing sand in the wheels of international ?nance.”
1.7 Conclusions
This paper investigates a systemic credit externality that magni?es the inci-
dence and severity of ?nancial crises. Households accumulate precautionary savings
to smooth consumption during the cycle, but they fail to internalize the systemic
feedback e?ects between borrowing decisions, the real exchange rate, and ?nancial
40
constraints. By reducing the amount of debt ex-ante, a social planner mitigates the
downward spiral in the exchange rate and in the borrowing capacity during a crisis,
thereby improving social welfare.
The key contribution of this paper is its quantitative analysis of this external-
ity: we analyze the e?ects on ?nancial crisis dynamics and welfare, and the policy
measures needed to correct this externality. Our main conclusion is that there is
much to gain from introducing macroprudential regulation. Correcting the credit
externality reduces the long-run probability of a ?nancial crisis more than ten times
(from 5.5 percent to 0.4 percent) and reduces the consumption drop during a typical
crisis by 7 percentage points (from 17 percent to 10 percent).
On the policy side, we show that several regulatory measures commonly used to
maintain ?nancial stability can achieve the constrained-e?cient allocations. These
measures e?ectively impose an increase in the cost of borrowing whenever there is a
positive probability of a crisis, but before the crisis materializes so that the economy
becomes less vulnerable to future adverse shocks. While these policies are equivalent
in the model, in practice there are di?erent costs and bene?ts associated with their
actual implementation. We also acknowledge that the actual implementation of
these policies is a challenging task, but we also show that simple interventions such
as a ?xed tax on debt yields sizable welfare gains.
Within our framework, incorporating capital accumulation and specifying a
richer supply side of the economy would be important to extend the quantitative
analysis. There are also other natural extensions of our work. While our externality
stems from a feedback loop between the real exchange rate and ?nancial constraints,
41
our results suggest that pecuniary externalities resulting from a similar mechanism
involving other relative prices might play a quantitatively important role as well.
For example, it would be interesting to study a similar externality involving asset
prices and economic activity. Another direction for future research would be to
study the role for macroprudential regulation in a setup with an explicit role for
?nancial intermediation, as in for example Gertler and Karadi (2009). These issues
remain for future research.
42
Chapter 2
Overborrowing, Financial Crises and Macro-Prudential Policy
(coauthored with Enrique G. Mendoza)
2.1 Introduction
A common argument in narratives of the causes of the 2008 global ?nancial
crisis is that economic agents “borrowed too much.” The notion of “overborrowing,”
however, is often vaguely de?ned or presented as a value judgment on borrowing
decisions, in light of the obvious fact that a prolonged credit boom ended in collapse.
This lack of clarity makes it di?cult to answer two key questions: First, is over-
borrowing a signi?cant macroeconomic problem, in terms of causing ?nancial crises
and playing a central role in driving macro dynamics during both ordinary business
cycles and crises episodes? Second, are the the so-called “macro-prudential” policy
instruments e?ective to contain overborrowing and reduce ?nancial fragility, and if
so what are their main quantitative features?
In this paper, we answer these questions using a dynamic stochastic general
equilibrium model of asset prices and business cycles with credit frictions. We pro-
vide a formal de?nition of overborrowing and use quantitative methods to determine
how much overborrowing the model predicts and how it a?ects business cycles, ?-
nancial crises, and social welfare. We also compute a state-contingent schedule of
43
taxes on debt and dividends that can solve the overborrowing problem.
Our de?nition of overborrowing is in line with the one used in the academic lit-
erature (e.g. Lorenzoni, 2008, Korinek, 2009, Bianchi, 2010): The di?erence between
the amount of credit that an agent obtains acting atomistically in an environment
with a given set of credit frictions, and the amount obtained by a social planner
who faces the same frictions but internalizes the general-equilibrium e?ects of its
borrowing decisions. In the model, the credit friction is in the form of a collateral
constraint on debt that has two important features. First, it drives a wedge between
the marginal costs and bene?ts of borrowing considered by individual agents and
those faced by a social planner. Second, when the constraint binds, it triggers Irv-
ing Fisher’s classic debt-de?ation ?nancial ampli?cation mechanism, which causes
a ?nancial crisis.
This paper contributes to the literature by providing a quantitative assessment
of overborrowing in an equilibrium model of business cycles and asset prices. The
model is similar to those examined by Mendoza and Smith (2006) and Mendoza
(2010). These studies showed that cyclical dynamics in a competitive equilibrium
lead to periods of expansion in which leverage ratios raise enough so that the collat-
eral constraint becomes binding, triggering a Fisherian de?ation that causes sharp
declines in credit, asset prices, and macroeconomic aggregates.
1
In this paper, we
study instead the e?ciency properties of the competitive equilibrium, by comparing
its allocations with those attained by a benevolent social planner subject to the same
1
This is also related to the classic work on ?nancial accelerators by Bernanke and Gertler
(1989) and Kiyotaki and Moore (1997) and the more recent quantitative literature on this topic
as in the work of Jermann and Quadrini (2010).
44
credit frictions as agents in the competitive equilibrium. Thus, while those previ-
ous studies focused on the ampli?cation and asymmetry of the responses of macro
variables to aggregate shocks, we focus here on the di?erences between competitive
equilibria and constrained social optima.
In the model, the collateral constraint limits private agents not to borrow more
than a fraction of the market value of their collateral assets, which take the form
of an asset in ?xed aggregate supply (e.g. land). Private agents take the price of
this asset as given, and hence a “systemic credit externality” arises, because they do
not internalize that, when the collateral constraint binds, ?re-sales of assets cause a
Fisherian debt-de?ation spiral that causes asset prices to decline and the economy’s
borrowing ability to shrink in an endogenous feedback loop. Moreover, when the
constraint binds, production plans are also a?ected, because working capital ?nanc-
ing is needed in order to pay for a fraction of labor costs, and working capital loans
are also subject to the collateral constraint. As a result, when the credit constraint
binds output falls because of a sudden increase in the e?ective cost of labor. This
a?ects dividend streams and therefore equilibrium asset prices, and introduces an
additional vehicle for the credit externality to operate, because private agents do
not internalize the supply-side e?ects of their borrowing decisions.
We conduct a quantitative analysis in a version of the model calibrated to
U.S. data. The results show that ?nancial crises in the competitive equilibrium
are signi?cantly more frequent and more severe than in the constrained-e?cient
equilibrium. The incidence of ?nancial crises is about three times larger. Asset
prices drop about 25 percent in a typical crisis in the decentralized equilibrium,
45
versus 5 percent in the constrained-e?cient equilibrium. Output drops about 50
percent more, because the fall in asset prices reduces access to working capital
?nancing. The more severe asset price collapses also generate a “fat tail” in the
distribution of asset returns in the decentralized equilibrium, which causes the price
of risk to rise 1.5 times and excess returns to rise by 5 times, in both tranquil times
and crisis times. The social planner can replicate exactly the constrained-e?cient
allocations in a decentralized equilibrium by imposing taxes on debt and dividends
of about 1 and -0.5 percent on average respectively
The existing macro literature on credit externalities provides important back-
ground for our work. The externality we study is a pecuniary externality similar to
those examined in the theoretical studies of Caballero and Krishnamurthy (2001),
Lorenzoni (2008), and Korinek (2009). The pecuniary externality in these papers
arises because ?nancial constraints that depend on market prices generate ampli?-
cation e?ects, which are not internalized by private agents. The literature on par-
ticipation constraints in credit markets initiated by the work of Kehoe and Levine
(1993) has also examined the role of ine?ciencies that arise because of endogenous
borrowing constraints. In particular, Jeske (2006) showed that if there is discrimi-
nation against foreign creditors, private agents have a stronger incentive to default
than a planner who internalizes the e?ects of borrowing decisions on the domestic
interest rate, which a?ects the tightness of the participation constraint. Wright
(2006) then showed that as a consequence of this externality, subsidies on capital
?ows restore constrained e?ciency.
Our work is also related to the quantitative studies of macro-prudential policy
46
by Bianchi (2010) and Benigno, Chen, Otrok, Rebucci, and Young (2009). These
authors studied a credit externality at work in the model of emerging markets crises
of Mendoza (2002), in which agents do not internalize the e?ect of their individual
debt plans on the market price of nontradable goods relative to tradables, which
in?uences their ability to borrow from abroad. Bianchi examined how this external-
ity leads to excessive debt accumulation and showed that a tax on debt can restore
constrained e?ciency and reduce the vulnerability to ?nancial crises. Benigno et
al. studied how the e?ects of the overborrowing externality are reduced when the
planner has access to instruments that can a?ect directly labor allocations during
crises.
2
Our analysis di?ers from the above quantitative studies in that we focus on
asset prices as a key factor driving debt dynamics and the credit externality, instead
of the relative price of nontradables. This is important because private debt con-
tracts, particularly mortgage loans like those that drove the high household leverage
ratios of many industrial countries in the years leading to the 2008 crisis, use assets
as collateral. Moreover, from a theoretical standpoint, a collateral constraint linked
to asset prices introduces forward-looking e?ects that are absent when using a credit
constraint linked to goods prices. In particular, expectations of a future ?nancial
crisis a?ect the discount rates applied to future dividends and distort asset prices
even in periods of ?nancial tranquility. In addition, our model also di?ers in that
we study a production economy in which working capital ?nancing is subject to the
2
In a related paper Benigno et al. (2009) found that intervening during ?nancial crisis by
subsidizing nontradable goods leads to large welfare gains.
47
collateral constraint. As a result, the credit externality distorts production plans
and dividend rates, and thus again asset prices.
More recently, the quantitative studies by Nikolov (2009) and Jeanne and Korinek
(2010) examine other models of macro-prudential policy in which assets serve as col-
lateral.
3
Nikolov found that simple rules that impose tighter collateral requirements
may not be welfare-improving in a setup in which consumption is a linear function
that is not in?uenced by precautionary savings. In contrast, precautionary savings
are critical determinants of optimal borrowing decisions in our model, because of
the strong non-linear ampli?cation e?ects caused by the Fisherian debt-de?ation
dynamics, and for the same reason we ?nd that debt taxes are welfare improving.
Jeanne and Korinek construct estimates of a Pigouvian debt tax in a model in which
output follows an exogenous Markov-switching process and individual credit is lim-
ited to the sum of a fraction of aggregate, rather than individual, asset holdings plus
a constant term. In their calibration analysis, this second term dominates and the
probability of crises matches the exogenous probability of a low-output regime, and
as result the tax cannot alter the frequency of crises and has small e?ects on their
magnitude.
4
In contrast, in our model the probability of crises and their output dy-
namics are endogenous, and macro-prudential policy reduces sharply the incidence
and magnitude of crises.
Our results also contrast with the ?ndings of Uribe (2006). He found that
an environment in which agents do not internalize an aggregate borrowing limit
3
Galati and Moessner (2010) conduct an exhaustive survey of the growing literature in research
and policy circles on macro-prudential policy.
4
They also examined the existence of deterministic cycles in a non-stochastic version of the
model.
48
yields identical borrowing decisions to an environment in which the borrowing limit
is internalized.
5
An essential di?erence in our analysis is that the social planner
internalizes not only the borrowing limit but also the price e?ects that arise from
borrowing decisions. Still, our results showing small di?erences in average debt
ratios across competitive and constrained-e?cient equilibria are in line with his
?ndings.
The rest of the paper is organized as follows: Section 2 presents the analyt-
ical framework. Section 3 analyzes constrained e?ciency. Section 4 presents the
quantitative analysis. Section 5 provides conclusions.
2.2 Competitive Equilibrium
We follow Mendoza (2010) in specifying the economic environment in terms of
?rm-household units who make production and consumption decisions. Preferences
are given by:
E
0
_
?
?
t=0
?
t
u(c
t
?G(n
t
))
_
(2.1)
In this expression, E(·) is the expectations operator, ? is the subjective discount
factor, n
t
is labor supply and c
t
is consumption. The period utility function u(·) is
assumed to have the constant-relative-risk-aversion (CRRA) form. The argument of
u(·) is the composite commodity c
t
?G(n
t
) de?ned by Greenwood, Hercowitz, and Hu?man
(1988). G
is a convex, strictly increasing and continuously di?erentiable function
5
He provided analytical results for a canonical endowment economy model with a credit con-
straint where there is an exact equivalence between the two sets of allocations. In addition, he
examined a model in which the exact equivalence of his ?rst example does not hold, but still
overborrowing is negligible.
49
that measures the disutility of labor supply. This formulation of preferences removes
the wealth e?ect on labor supply by making the marginal rate of substitution be-
tween consumption and labor depend on labor only.
Each household can combine land and labor services purchased from other
households to produce ?nal goods using a production technology such that y =
?
t
F(k
t
, h
t
), where F is a decreasing-returns-to-scale production function, k
t
repre-
sents individual land holdings, h
t
represents labor demand and ?
t
is a productivity
shock, which has compact support and follows a ?nite-state, stationary Markov
process. Individual pro?ts from this production activity are therefore given by
?
t
F(k
t
, h
t
) ?w
t
h
t
.
The budget constraint faced by the representative ?rm-household is:
q
t
k
t+1
+ c
t
+
b
t+1
R
t
= q
t
k
t
+ b
t
+ w
t
n
t
+ [?
t
F(k
t
, h
t
) ?w
t
h
t
] (2.2)
where b
t
denotes holdings of one-period, non-state-contingent discount bonds at the
beginning of date t, q
t
is the market price of land, R
t
is the real interest rate, and
w
t
is the wage rate.
The interest rate is assumed to be exogenous. This is equivalent to assuming
that the economy is a price-taker in world credit markets, as in other studies of
the U.S. ?nancial crisis like those of Boz and Mendoza (2010), Corbae and Quintin
(2009) and Howitt (2010), or alternatively it implies that the model can be in-
terpreted as a partial-equilibrium model of the household sector. This assump-
tion is adopted for simplicity, but is also in line with the evidence indicating that
50
in the era of ?nancial globalization even the U.S. risk-free rate has been signif-
icantly in?uenced by outside factors, such as the surge in reserves in emerging
economies and the persistent collapse of investment rates in South East Asia af-
ter 1998. Warnock and Warnock (2009) provide econometric evidence of the sig-
ni?cant e?ect of foreign capital in?ows on U.S. T-bill rates since the mid 1980s.
Mendoza and Quadrini (2009) document that about 1/2 of the surge in net credit
in the U.S. economy since then was ?nanced by foreign capital in?ows, and more
than half of the stock of U.S. treasury bills is now owned by foreign agents.
Household-?rms are subject to a working capital constraint. In particular,
they are required to borrow a fraction ? of the wages bill w
t
h
t
at the beginning
of the period and have to repay at the end of the period. In the conventional
working capital setup, a cash-in-advance-like motive for holding funds to pay for
inputs implies that the wages bill carries a ?nancing cost determined by the inter-
period interest rate. In contrast, here we simply assume that working capital funds
are within-period loans. Hence, the interest rate on working capital is zero, as
in some recent studies on the business cycle implications of working capital and
credit frictions (e.g. Chen and Song (2009)). We follow this approach so as to show
that the e?ects of working capital in our model hinge only on the need to provide
collateral for working capital loans, as explained below, and not on the e?ect of
interest rate ?uctuations on e?ective labor costs.
6
As in Mendoza (2010), agents face a collateral constraint that limits total
6
We could also change to the standard setup, but in our calibration, ? = 0.14 and R = 1.028,
and hence working capital loans would add 0.4 percent to the cost of labor implying that our
?ndings would remain largely unchanged.
51
debt, including both intertemporal debt and atemporal working capital loans, not
to exceed a fraction ? of the market value of asset holdings (i.e. ? imposes a ceiling
on the leverage ratio):
?
b
t+1
R
t
+ ?w
t
h
t
? ?q
t
k
t+1
(2.3)
Following Kiyotaki and Moore (1997) and Aiyagari and Gertler (1999), we interpret
this constraint as resulting from an environment where limited enforcement prevents
lenders to collect more than a fraction ? of the value of a defaulting debtor’s assets,
but we abstract from modeling the contractual relationship explicitly.
2.2.1 Private Optimality Conditions
In the competitive equilibrium, agents maximize (2.1) subject to (2.2) and
(2.3) taking land prices and wages as given. This maximization problem yields the
following optimality conditions for each date t:
w
t
= G
?
(n
t
) (2.4)
?
t
F
h
(k
t
, h
t
) = w
t
[1 +?µ
t
/u
?
(t)] (2.5)
u
?
(t) = ?RE
t
[u
?
(t + 1)] + µ
t
(2.6)
q
t
(u
?
(t) ?µ
t
?) = ?E
t
[u
?
(t + 1) (?
t+1
F
k
(k
t+1
, h
t+1
) + q
t+1
)] (2.7)
where µ
t
? 0 is the Lagrange multiplier on the collateral constraint.
Condition (2.4) is the individual’s labor supply condition, which equates the
marginal disutility of labor with the wage rate. Condition (2.5) is the labor de-
52
mand condition, which equates the marginal productivity of labor with the e?ective
marginal cost of hiring labor. The latter includes the extra ?nancing cost ?µ
t
/u
?
(t)
in the states of nature in which the collateral constraint on working capital binds.
The last two conditions are the Euler equations for bonds and land respectively.
When the collateral constraint binds, condition (2.6) implies that the marginal util-
ity of reallocating consumption to the present exceeds the expected marginal utility
cost of borrowing in the bond market by an amount equal to the shadow price of
relaxing the credit constraint. Condition (2.7) equates the marginal cost of an extra
unit of land investment with its marginal gain. The marginal cost nets out from the
marginal utility of foregone current consumption a fraction ? of the shadow value
of the credit constraint, because the additional unit of land holdings contributes to
relax the borrowing limit.
Condition (2.7) yields the following forward solution for land prices:
q
t
= E
t
_
?
?
j=0
_
j
?
i=0
m
t+1+i
_
d
t+j+1
_
, m
t+1+i
?
?u
?
(t + 1 +i)
u
?
(t + i) ?µ
t+i
?
, d
t
? ?
t
F
k
(k
t
, h
t
)
(2.8)
Thus, we obtain what seems a standard asset pricing condition stating that, at
equilibrium, the date-t price of land is equal to the expected present value of the
future stream of dividends discounted using the stochastic discount factors m
t+1+i
,
for i = 0, ..., ?. The key di?erence with the standard asset pricing condition,
however, is that the discount factors are adjusted to account for the shadow value
of relaxing the credit constraint by purchasing an extra unit of land whenever the
collateral constraint binds (at any date t + i for i = 0, ..., ? ).
53
Combining (2.6), (2.7) and the de?nition of asset returns (R
q
t+1
?
d
t+1
+q
t+1
qt
),
it follows that the expected excess return on land relative to bonds (i.e. the equity
premium), R
ep
t
? E
t
(R
q
t+1
?R), satis?es the following condition:
R
ep
t
=
µ
t
(1 ??)
(u
?
(t) ?µ
t
?)E
t
[m
t+1
]
?
cov
t
(m
t+1
, R
q
t+1
)
E
t
[m
t+1
]
, (2.9)
where cov
t
(m
t+1
, R
q
t+1
) is the date-t conditional covariance between m
t+1
and R
q
t+1
.
Following Mendoza and Smith (2006), we characterize the ?rst term in the
right-hand-side of (2.9) as the direct (?rst-order) e?ect of the collateral constraint
on the equity premium, which re?ects the fact that a binding collateral constraint
exerts pressure to ?re-sell land, depressing its current price.
7
There is also an indirect
(second-order) e?ect given by the fact that cov
t
(m
t+1
, R
q
t+1
) is likely to become
more negative when there is a possibility of a binding credit constraint, because the
collateral constraint makes it harder for agents to smooth consumption.
Given the de?nitions of the Sharpe ratio (S
t
?
R
ep
t
?t(R
q
t+1
)
) and the price of risk
(s
t
? ?
t
(m
t+1
)/E
t
m
t+1
), we can rewrite the expected excess return and the Sharpe
ratio as:
R
ep
t
= S
t
?
t
(R
q
t+1
), S
t
=
µ
t
(1 ??)
(u
?
(t) ?µ
t
?)E
t
[m
t+1
] ?
t
(R
q
t+1
)
??
t
(R
q
t+1
, m
t+1
)s
t
(2.10)
where ?
t
(R
q
t+1
) is the date-t conditional standard deviation of land returns and
?
t
(R
q
t+1
, m
t+1
) is the conditional correlation between R
q
t+1
and m
t+1
. Thus, the col-
7
Notice that this e?ect vanishes when ? = 1, because when 100 percent of the value of land
can be collateralized, the shadow value of relaxing the constraint by acquiring an extra unit of
land equals the shadow value of relaxing it by reducing the debt by one unit.
54
lateral constraint has direct and indirect e?ects on the Sharpe ratio analogous to
those it has on the equity premium. The indirect e?ect reduces to the usual ex-
pression in terms of the product of the price of risk and the correlation between
asset returns and the stochastic discount factor. The direct e?ect is normalized by
the variance of land returns. These relationships will be useful later to study the
quantitative e?ects of the credit externality on asset pricing.
Since q
t
E
t
[R
q
t+1
] ? E
t
[d
t+1
+ q
t+1
], we can rewrite the asset pricing condition
in this way:
q
t
= E
t
?
?
j=0
_
j
?
i=0
E
t+i
R
q
t+1+i
_
?1
d
t+j+1
, (2.11)
Notice that (2.9) and (2.11) imply that a binding collateral constraint at date t
implies an increase in expected excess land returns and a drop in asset prices at t.
Moreover, since expected returns exceed the risk free rate whenever the collateral
constraint is expected to bind at any future date, asset prices at t are a?ected by
collateral constraint not just when the constraints binds at t, but whenever it is
expected to bind at any future date.
2.2.2 Recursive Competitive Equilibrium
The competitive equilibrium is de?ned by stochastic sequences of allocations
{c
t
, k
t+1
, b
t+1
, h
t
, n
t
}
?
t=0
and prices {q
t
, w
t
}
?
t=0
such that: (A) agents maximize utility
(2.1) subject to the sequence of budget and credit constraints given by (2.2) and
(2.3) for t = 0, ..., ?, taking as given {q
t
, w
t
}
t=?
t=0
; (B) the markets of goods, labor
and land clear at each date t. Since land is in ?xed supply
¯
K, the market-clearing
55
condition for land is k
t
=
¯
K. The market clearing condition in the goods and labor
markets are c
t
+
b
t+1
R
= ?
t
F(
¯
K, n
t
) + b
t
and h
t
= n
t
respectively.
We now characterize the competitive equilibrium in recursive form. The state
variables for a particular individual’s optimization problem at time t are the indi-
vidual bond holdings (b), aggregate bond holdings (B), individual land holdings (k),
and the TFP realization (? ). Aggregate land holdings are not carried as a state
variable because land is in ?xed supply. Denoting by ?(B, ?) the agents’ perceived
law of motion of aggregate bonds and q(B, ?) and w(B, ?) the pricing functions for
land and labor respectively, the agents’ recursive optimization problem is:
V (b, k, B, ?) = max
b
?
,k
?
,c,n,h
u(c ?G) + ?E
?
?
|?
[V (b
?
, k
?
, B
?
, ?
?
)] (2.12)
s.t. q(B, ?)k
?
+ c +
b
?
R
= q(B, ?)k + b + w(B, ?)n + [?F(k, h) ?w(B, ?)h]
B
?
= ?(B, ?)
?
b
?
R
+ ?w(B, ?)h ? ?q(B, ?)k
?
The solution to this problem is characterized by the decision rules
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?) and
ˆ
h(b, k, B, ?). The decision rule for bond
holdings induces an actual law of motion for aggregate bonds, which is given by
ˆ
b
?
(B,
¯
K, B, ?). In a recursive rational expectations equilibrium, as de?ned below,
the actual and perceived laws of motion must coincide.
De?nition 3 (Recursive Competitive Equilibrium)
A recursive competitive equilibrium is de?ned by an asset pricing func-
56
tion q(B, ?), a pricing function for labor w(B, ?), a perceived law of mo-
tion for aggregate bond holdings ?(B, ?), and a set of decision rules
_
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?),
ˆ
h(b, k, B, ?)
_
with associated
value function V (b, k, B, ?) such that:
1.
_
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?),
ˆ
h(b, k, B, ?)
_
and V (b, k, B, ?)
solve the agents’ recursive optimization problem, taking as given q(B, ?), w(B, ?)
and ?(B, ?).
2. The perceived law of motion for aggregate bonds is consistent with the actual
law of motion: ?(B, ?) =
ˆ
b
?
(B,
¯
K, B, ?).
3. Wages satisfy w(B, ?) = G
?
(ˆ n(B,
¯
K, B, ?)) and land prices satisfy q(B, ?) =
E
?
?
|?
_
?u
?
(ˆ c(?(B,?),
¯
K,?(B,?),?
?
)) [?
?
F
k(
¯
K,ˆ n(?(B,?)
¯
K,?(B,?),?
?
))+q(?(B,?),?
?
)]
u
?
(ˆ c(B,
¯
K,B,?))??max[0,u
?
(ˆ c(B,
¯
K,B,?))??RE
?
?
|?
u
?
(ˆ c(?(B,?),
¯
K,?(B,?),?
?
)]
_
4. Goods, labor and asset markets clear:
ˆ
b
?
(B,
¯
K,B,?)
R
+ˆ c(B,
¯
K, B, ?) = ?F(
¯
K, ˆ n(B,
¯
K, B, ?))+
B , ˆ n(B,
¯
K, B, ?) =
ˆ
h(B,
¯
K, B, ?) and
ˆ
k(B,
¯
K, B, ?) =
¯
K
2.3 Constrained-E?cient Equilibrium
2.3.1 Equilibrium without collateral constraint
We start studying the e?ciency properties of the competitive equilibrium by
brie?y characterizing an e?cient equilibrium in the absence of the collateral con-
straint (2.3). The allocations of this equilibrium can be represented as the solution
57
to the following standard planning problem:
H(B, ?) = max
b
?
,c,n
u(c ?G) + ?E
?
?
|?
[H(B
?
, ?
?
)] (2.13)
s.t. c +
B
?
R
= ?F(
¯
K, n) + B
and subject also to either this problem’s natural debt limit, which is de?ned by B
?
>
?
min
F(
¯
K,n
?
(?
min
))
R?1
, where ?
min
is the lowest possible realization of TFP and n
?
(?
min
) is
the optimal labor allocation that solves ?
min
F
n
(
¯
K, n) = G
?
, or to a tighter ad-hoc
time- and state-invariant borrowing limit.
The common strategy followed in quantitative studies of the macro e?ects of
collateral constraints (see, for example, Mendoza and Smith, 2006 and Mendoza,
2010) is to compare the allocations of the competitive equilibrium with the Fishe-
rian collateral constraint with those arising from the above benchmark case. The
competitive equilibria with and without the collateral constraint di?er in that in the
former private agents borrow less (since the collateral constraint limits the amount
they can borrow and also because they build precautionary savings to self-insure
against the risk of the occasionally binding credit constraint), and there is ?nancial
ampli?cation of the e?ects of the underlying exogenous shocks (since binding col-
lateral constraints produce large recessions and drops in asset prices). Compared
with the constrained-e?cient equilibrium we de?ne next, however, we will show that
the competitive equilibrium with collateral constraints displays overborrowing (i.e.
agents borrow more than in the constrained-e?cient equilibrium ).
58
2.3.2 Recursive Constrained-E?cient Equilibrium
We study a benevolent social planner who maximizes the agents’ utility subject
to the resource constraint, the collateral constraint and the same menu of assets of
the competitive equilibrium.
8
In particular, we consider a social planner that is con-
strained to have the same “borrowing ability” (the same market-determined value
of collateral assets ?q(B, ?)
¯
K) at every given state as agents in the decentralized
equilibrium, but with the key di?erence that the planner internalizes the e?ects of
its borrowing decisions on the market prices of assets and labor.
9
The recursive problem of the social planner is de?ned as follows:
W(B, ?) = max
B
?
,c,n
u(c ?G) + ?E
?
?
|?
[W(B
?
, ?
?
)] (2.14)
s.t. c +
B
?
R
= ?F(
¯
K, n) + B
?
B
?
R
+ ?w(B, ?)n ? ?q(B, ?)
¯
K
where q(B, ?) is the equilibrium pricing function obtained in the competitive equi-
librium. Wages can be treated in a similar fashion, but it is easier to decentralize
the planner’s allocations as as competitive equilibrium if we assume that the plan-
8
We refer to the social planner’s equilibrium and constrained-e?cient equilibrium interchange-
ably. Our focus is on second-best allocations, so when we refer to the social planner’s choices it
should be understood that we mean the constrained social planner.
9
We could also allow the social planner to manipulate the borrowing ability state by state (i.e.,
by allowing the planner to alter ?q(B, ?)
¯
K). Allowing for this possibility can potentially increase
the welfare losses resulting from the externality but the macroeconomic e?ects are similar. In
addition, since asset prices are forward-looking, this would create a time-inconsistency problem in
the planner’s problem. Allowing the planner to commit to future actions would lead the planner
to internalize not only how today’s choice of debt a?ects tomorrow’s asset prices but also how it
a?ects asset prices and the tightness of collateral constraints in previous periods.
59
ner takes wages as given and wages need to satisfy w(B, ?) = G
?
.
10
Under this
assumption, we impose the optimality condition of labor supply as a condition that
the constrained-e?cient equilibrium must satisfy, in addition to solving problem
(2.14) for given wages.
Using the envelope theorem on the ?rst-order conditions of problem (2.14) and
imposing the labor supply optimality condition, we obtain the following optimality
conditions for the constrained-e?cient equilibrium:
u
?
(t) = ?RE
t
[u
?
(t + 1) + µ
t+1
?
t+1
] +µ
t
, ?
t+1
? ?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
(2.15)
?
t
F
n
(
¯
K, n
t
) = G
?
(n
t
) [1 +?µ
t
/u
?
(t)] (2.16)
The key di?erence between the competitive equilibrium and the constrained-e?cient
allocations follows from examining the Euler equations for bond holdings in both
problems. In particular, the term µ
t+1
?
t+1
in condition (2.15) represents the addi-
tional marginal bene?t of savings considered by the social planner at date t, because
the planner takes into account how an extra unit of bond holdings alters the tight-
ness of the credit constraint through its e?ects on the prices of land and labor at
t+1. Note that, since
?q
t+1
?b
t+1
> 0 and
?w
t+1
?b
t+1
? 0, ?
t+1
is the di?erence of two opposing
e?ects and hence its sign is in principle ambiguous. The term
?q
t+1
?b
t+1
is positive, be-
cause an increase in net worth increases demand for land and land is in ?xed supply.
10
This implies that the social planner does not internalize the direct e?ects of choosing the
contemporaneous labor allocation on contemporaneous wages. We have also investigated the pos-
sibility of having the planner internalize these e?ects but results are very similar. This occurs
again because our calibrated interest rate and working capital requirement are very small.
60
The term
?w
t+1
?b
t+1
is positive, because the e?ective cost of hiring labor increases when
the collateral constraint binds, reducing labor demand and pushing wages down. We
found, however, that the value of ?
t+1
is positive in all our quantitative experiments
with baseline parameter values and variations around them, and this is because
?q
t+1
?b
t+1
is large and positive when the credit constraint binds due the e?ects of the Fisherian
debt-de?ation mechanism.
De?nition 4 (Recursive Constrained-E?cient Equilibrium)
The recursive constrained-e?cient equilibrium is given by a set of decision rules
_
ˆ
B
?
(B, ?), ˆ c(B, ?), ˆ n(B, ?)
_
with associated value function W(B, ?), and wages
w(B, ?) such that:
1.
_
ˆ
B
?
(B, ?), ˆ c(B, ?), ˆ n(B, ?)
_
and W(B, ?) solve the planner’s recursive opti-
mization problem, taking as given w(B, ?) and the competitive equilibrium’s
asset pricing function q(B, ?).
2. Wages satisfy w(B, ?) = G
?
(ˆ n(B, ?)).
2.3.3 Comparison of Equilibria & ‘Macro-prudential’ Policy
Using a simple variational argument, we can show that the allocations of the
competitive equilibrium are ine?cient, in the sense that they violate the conditions
that support the constrained-e?cient equilibrium. In particular, private agents un-
dervalue net worth in periods during which the collateral constraint binds. To see
this, consider ?rst the marginal utility of an increase in individual bond holdings.
By the envelope theorem, in the competitive equilibrium this can be written as
61
?V
?b
= u
?
(t). For the constrained-e?cient economy, however, the marginal bene?t of
an increase in bond holdings takes into account the fact that prices are a?ected by
the increase in bond holdings, and is therefore given by
?W
?b
= u
?
(t) + ?
t
µ
t
. If the
collateral constraint does not bind, µ
t
= 0 and the two expressions coincide. If the
collateral constraint binds, the social bene?ts of a higher level of bonds include the
extra term given by ?
t
µ
t
, because one more unit of aggregate bonds increases the
inter-period ability to borrow by ?
t
which has a marginal value of µ
t
.
The above argument explains why bond holdings are valued di?erently by the
planner and the private agents “ex post,” when the collateral constraint binds. Since
both the planner and the agents are forward looking, however, it follows that those
di?erences in valuation lead to di?erences in the private and social bene?ts of debt
accumulation “ex ante,” when the constraint is not binding. Consider the marginal
cost of increasing the level of debt at date t evaluated at the competitive equilibrium
in a state in which the constraint is not binding. This cost is given by the discounted
expected marginal utility from the implied reduction in consumption next period
?RE [u
?
(t + 1)] . In contrast, the social planner internalizes the e?ect by which the
larger debt reduces tomorrow’s borrowing ability by ?
t+1
, and hence the marginal
cost of borrowing at period t that is not internalized by private agents is given by
?RE
t
_
µ
t+1
_
?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
__
.
We now show that the planner can implement the constrained-e?cient allo-
cations as a competitive equilibrium in the decentralized economy by introducing a
macro-prudential policy that taxes debt and dividends (the latter can turn into a
62
subsidy too, as we show in the next Section, but we refer to it generically as a tax).
11
In particular, the planner can do this by constructing state-contingent schedules of
taxes on bond purchases (?
t
) and on land dividends (?
t
), with the total cost (rev-
enues) ?nanced (rebated) as lump-sum taxes (transfers). The tax on bonds ensures
that the planner’s optimal plans for consumption and bond holdings are consistent
with the Euler equation for bonds in the competitive equilibrium. This requires
setting the tax to ?
t
= E
t
µ
t+1
?
t+1
/E
t
u
?
(t + 1). The tax on land dividends ensures
that these optimal plans and the pricing function q(B, ?) are consistent with the
private agents’ Euler equation for land holdings.
The Euler equations of the competitive equilibrium with the macro-prudential
policy in place become:
u
?
(t) = ?R(1 +?
t
)E
t
[u
?
(t + 1)] +µ
t
(2.17)
q
t
(u
?
(t) ?µ
t
?) = ?E
t
[u
?
(t + 1) (?
t+1
F
k
(k
t+1
, n
t+1
)(1 + ?
t+1
) + q
t+1
)] (2.18)
By combining these two Euler equations we can derive the expected excess return
on land paid in the land market under the macro-prudential policy. In this case,
after-tax returns on land and bonds are de?ned as
˜
R
q
t+1
?
d
t+1
(1+?
t+1
)+q
t+1
qt
and
˜
R
t+1
? R(1 + ?
t
) respectively, and the after-tax expected equity premium reduces
to an expression analogous to that of the decentralized equilibrium:
˜
R
ep
t
=
µ
t
(1 ??)
E
t
[(u
?
(t) ?µ
t
?)m
t+1
]
?
Cov
t
(m
t+1
,
˜
R
q
t+1
)
E
t
[m
t+1
]
(2.19)
11
See Bianchi (2010) for other decentralizations using capital and liquidity requirements and
loan-to-value ratios.
63
This excess return also has a corresponding interpretation in terms of the Sharpe
ratio, the price of risk, and the correlation between land returns and the pricing
kernel as in the case of the competitive equilibrium without macro-prudential policy.
It follows from comparing the expressions for R
ep
t
and
˜
R
ep
t
that di?erences in
the after-tax expected equity premia of the competitive equilibria with and without
macro-prudential policy are determined by di?erences in the direct and indirect
e?ects of the credit constraint in the two environments. As shown in the next
Section, these e?ects are stronger in the decentralized equilibrium without policy
intervention, in which the ine?ciencies of the credit externality are not addressed.
Intuitively, higher leverage and debt in this environment imply that the constraint
binds more often, which strengthens the direct e?ect. In addition, lower net worth
implies that the stochastic discount factor covaries more strongly with the excess
return on land, which strengthens the indirect e?ect. Notice also that dividends in
the constrained-e?cient allocations are discounted at a rate which depends positively
on the tax on debt. This premium is required by the social planner so that the excess
returns re?ect the social costs of borrowing.
2.4 Quantitative Analysis
2.4.1 Calibration
We calibrate the model to annual frequency using data from the U.S. economy.
The functional forms for preferences and technology are the following:
64
u(c ?G) =
_
c ??
n
1+?
1+?
_
1??
?1
1 ??
? > 0, ? > 1 (2.20)
F(k, h) = ?k
?
K
h
?
h
, ?
K
, ?
h
? 0 ?
K
+ ?
h
< 1 (2.21)
The real interest rate is set to R ? 1 = 0.028 per year, which is the ex-
post average real interest rate on U.S. three-month T-bills during the period 1980-
2005. We set ? = 2, which is a standard value in quantitative DSGE models. The
parameter ? is inessential and is set so that mean hours are equal to 1, which requires
? = 0.64. Aggregate land is normalized to
¯
K = 1 without loss of generality and the
share of labor in output ?
h
is equal to 0.64, the standard value. The Frisch elasticity
of labor supply (1/?) is set equal to 1, in line with evidence by Kimball and Shapiro
(2008).
We follow Schmitt-Grohe and Uribe (2007) in taking M1 money balances in
possession of ?rms as a proxy for working capital. Based on the observations that
about two-thirds of M1 are held by ?rms (Mulligan, 1997) and that M1 was on
average about 14 percent of annual GDP over the period 1980 to 2009, we calibrate
the working capital-GDP ratio to be (2/3)0.14 = 0.093. Given the 64 percent labor
share in production, and assuming the collateral constraint does not bind, we obtain
? = 0.093/0.64 = 0.146.
The value of ? is set to 0.96, which is also a standard value but in addition it
supports an average household debt-income ratio in a range that is in line with U.S.
data from the Federal Reserve’s Flow of Funds database. Before the mid-1990s this
65
ratio was stable at about 30 percent. Since then and until just before the 2008 crisis,
it rose steadily to a peak of almost 70 percent. By comparison, the average debt-
income ratio in the stochastic steady-state of our baseline calibration is 38 percent.
A mean debt ratio of 38 percent is sensible because 70 percent was an extreme at
the peak of a credit boom and 30 percent is an average from a period before the
substantial ?nancial innovation of recent years.
Table 2.1: Calibration
Source / target
Interest rate R ?1 = 0.028 U.S. data
Risk aversion ? = 2 Standard DSGE value
Share of labor ?
n
= 0.64 U.S. data
Labor disutility coe?cient ? = 0.64 Normalization
Frisch elasticity parameter ? = 1 Kimball and Shapiro (2008)
Supply of land
¯
K = 1 Normalization
Working capital coe?cient ? = 0.14 Working Capital-GDP=9%
Discount factor ? = 0.96 Debt-GDP ratio= 38%
Collateral coe?cient ? = 0.36 Frequency of Crisis = 3%
Share of land ?
K
= 0.05 Housing-GDP ratio = 1.35
TFP process ?
?
= 0.014, ?
?
= 0.53 Std. dev. and autoc. of U.S. GDP
The values of ?, ?
K
and the TFP process are calibrated to match targets from
U.S. data by simulating the model. We set ?
K
so as to match the average ratio of
housing to GDP at current prices, which is equal to 1.35. The value of housing is
taken from the Flow of Funds, and is measured as real state tangible assets owned
by households (reported in Table B.100, row 4). The model matches the 1.35 ratio
when we set ?
K
= 0.05.
12
12
?
K
represents the share of ?xed assets in GDP, and not the standard share of capital income in
GDP. There is little empirical evidence about the value of this parameter, with estimates that vary
depending, for example, on whether we consider land used for residential or commercial purposes,
or owned by government at di?erent levels. We could also calibrate ?
K
using the fact that, in
66
TFP shocks follow a log-normal AR(1) process log(?
t
) = ? log(?
t?1
) + ?
t
. We
construct a discrete approximation to this process using the quadrature procedure
of Tauchen and Hussey (1991) using 15 nodes. The values of ?
?
and ? are set so that
the standard deviation and ?rst-order autocorrelation of the output series produced
by the model match the corresponding moments for the cyclical component of U.S.
GDP in the sample period 1947-2007 (which are 2.1 percent and 0.5 respectively).
This procedure yields ?
?
= 0.014 and ? = 0.53.
Finally, we set the value of ? so as to match the frequency of ?nancial crises in
U.S. data. We de?ne a ?nancial crisis as an event in which both the credit constraint
binds and there is a decrease in credit of more than one standard deviation. Then,
we set ? so that ?nancial crises in the baseline model simulation occur about 3
percent of the time, which is consistent with the fact that the U.S. has experienced
three major ?nancial crises in the last hundred years.
13
This yields the value of ? =
0.36.
We recognize that several of the parameter values are subject of debate (e.g.
there is a fair amount of disagreement about the Frisch elasticity of labor supply),
or relate to variables that do not have a clear analog in the data (as is the case with
? or ?). Hence, we will perform sensitivity analysis to examine the robustness of
our results to changes in the model’s key parameters.
a deterministic steady state where the collateral constraint does not bind, the value-of-land-GDP
ratio is equal to ?
K
/(R?1), which would imply ?
K
= 1.35(0.028) = 0.038. This yields very similar
results as ?
K
= 0.05.
13
The three crises correspond to the Great Depression, the Savings and Loans Crisis and the
Great Recession (see Reinhart and Rogo? (2008)). While a century may be a short sample for
estimating accurately the probability of a rare event in one country, Mendoza (2010) estimates
a probability of about 3.6 percent for ?nancial crises using a similar de?nition but applied to all
emerging economies using data since 1980.
67
2.4.2 Borrowing decisions
We start the quantitative analysis by exploring the e?ects of the credit ex-
ternality on optimal borrowing plans. The solution method is described in the
Appendix. Since mean output is normalized to 1, all quantities can be interpreted
as fractions of mean output.
The two panels of Figure 1 show the bond decision rules (b
?
) of private agents
and the social planner as a function of b (left panel) as well as the pricing function
for land (right panel), both for a negative two-standard-deviations TFP shock. The
key point is to note that the Fisherian de?ation mechanism generates non-monotonic
bond decision rules, instead of the typical monotonically increasing decision rules.
The point at which bond decision rules switch slope corresponds to the value of b at
which the collateral constraint holds with equality but does not bind. To the right
of this point, the collateral constraint does not bind and the bond decision rules
are upward-sloping. To the left of this point, the bond decision rules are decreasing
in b, because a reduction in current bond holdings results in a sharp reduction in
the price of land, as can be seen in the right panel, and tightens the borrowing
constraint, thus increasing b
?
.
As in Bianchi (2010), we can separate the bond decision rules in the left panel
of Figure 1 into three regions: a “constrained region,” a “high-externality region”
and a “low-externality region.” The “constrained region” is given by the range of
b in the horizontal axis with su?ciently high initial debt (i.e. low b) such that the
collateral constraint binds in the constrained-e?cient equilibrium. This is the range
68
with b ? ?0.385. In this region, the collateral constraint binds in both constrained-
e?cient and competitive equilibria, because the credit externality implies that the
constraint starts binding at higher values of b in the latter than in the former, as
we show below.
By construction, the total amount of debt (i.e. the sum of bond holdings and
working capital) in the constrained region is the same under the constrained-e?cient
allocations and the competitive equilibrium. If working capital were not subject
to the collateral constraint, the two bond decision rules would also be identical.
But with working capital in the constraint the two can di?er. This is because the
e?ective cost of labor di?ers between the two equilibria, since the increase in the
marginal ?nancing cost of labor when the constraint binds, ?µ
t
/u
?
(t), is di?erent.
These di?erences, however, are very small in the numerical experiments, and thus
the bond decision rules are approximately the same in the constrained region.
14
The high-externality region is located to the right of the constrained region,
and it includes the interval ?0.385 < b < ?0.363. Here, the social planner chooses
uniformly higher bond positions (lower debt) than private agents, because of the
di?erent incentives driving the decisions of the two when the constrained region
is near. In fact, private agents hit the credit constraint at b = ?0.383, while at
that initial b the social planner still retains some borrowing capacity. Moreover,
this region is characterized by “?nancial instability,” in the sense that the levels of
14
The choice of b
?
becomes slightly higher for the social planner as b gets closer to the upper
bound of the constrained region, because the deleveraging that occurs around this point is small
enough for the probability of a binding credit constraint next period to be strictly positive. As a
result, for given allocations, conditions (2.15) and (2.6) imply that µ is lower in the constrained-
e?cient allocations.
69
?0.4 ?0.38 ?0.36 ?0.34 ?0.32 ?0.3
?0.4
?0.39
?0.38
?0.37
?0.36
?0.35
?0.34
?0.33
?0.32
?0.31
?0.3
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
H
i
g
h
E
x
t
e
r
n
a
l
i
t
y
R
e
g
i
o
n
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
?0.4 ?0.38 ?0.36 ?0.34 ?0.32 ?0.3
1.1
1.15
1.2
1.25
1.3
1.35
H
i
g
h
E
x
t
e
r
n
a
l
i
t
y
R
e
g
i
o
n
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Current Bond Holdings
L
a
n
d
P
r
i
c
e
Decentralized Equilibrium
Constrained Efficient
Figure 2.1: Bond Decision Rules (left panel) and Land Pricing Function (right panel)
for a Negative Two-standard-deviations TFP Shock
debt chosen for t + 1 are high enough so that a negative TFP shock of standard
magnitude in that period can lead to a binding credit constraint that leads to large
falls in consumption, output, land prices and credit. We will show later that this
is also the region of the state space in which the planner uses actively its macro-
prudential policy to manage the ine?ciencies of the competitive equilibrium.
The low-externality region is the interval for which b ? ?0.363. In this region,
the probability of a binding constraint next period is zero for both the social planner
and the competitive equilibrium. The bond decision rules still di?er, however, be-
70
cause expected marginal utilities di?er for the two equilibria. But the social planner
does not set a tax on debt, because negative shocks cannot lead to a binding credit
constraint in the following period.
The long-run probabilities with which the constrained-e?cient (competitive)
economy visits the three regions of the bond decision rules are 2 (4) percent for
the constrained region, 69 (70) percent for the high-externality region, and 29 (27)
percent for the low-externality region. Both economies spend more than 2/3rds
of the time in the high-externality region, but the prudential actions of the social
planner reduce the probability of entering in the constrained region by a half. Later
we will show that this is re?ected also in ?nancial crises that are much less frequent
and less severe than in the competitive equilibrium.
The larger debt (i.e. lower bond) choices of private agents relative to the social
planner, particularly in the high-externality region, constitute our ?rst measure of
the overborrowing e?ect at work in the competitive equilibrium. The social planner
accumulates extra precautionary savings above and beyond what private individuals
consider optimal in order to self-insure against the risk of ?nancial crises. This
e?ect is quantitatively small in terms of the di?erence between the two decision
rules, but this does not mean that its macroeconomic e?ects are negligible. Later in
this Section we illustrate this point by comparing ?nancial crises events in the two
economies. In addition, the fact that small di?erences in borrowing decisions lead
to major di?erences when a crisis hits can be illustrated using Figure 2 to study
further the dynamics implicit in the bond decision rules.
Figure 2.2 shows bond decision rules for the social planner and the competitive
71
?0.39 ?0.385 ?0.38 ?0.375 ?0.37 ?0.365 ?0.36
?0.39
?0.38
?0.37
?0.36
?0.35
?0.34
?0.33
?0.32
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
Social Planner
Decentralized Equilibrium
b‘=b line
Average TFP
Low TFP
B’
A’
B
A
Figure 2.2: Comparison of Debt Dynamics
equilibrium over the range (-0.39,-0.36) for two TFP scenarios: average TFP and
TFP two-standard-deviations below the mean. The ray from the origin is the b
?
= b
line. We use a narrower range than in Figure 1 to “zoom in” and highlight the
di?erences in decision rules. Assume both economies start at a value of b such that
at average TFP the debt of agents in the competitive equilibrium would remain
unchanged (this is point A with b = ?0.389). If the TFP realization is indeed the
average, private agents in the decentralized equilibrium keep that level of debt. On
72
the other hand, the social planner builds precautionary savings and reduces its debt
to point B with b = ?0.386. Hence, the next period the two economies start at
the debt levels in A and B respectively. Assume now that at this time TFP falls
by two standard deviations. Now we can see the large dynamic implications of the
small di?erences in the bond decision rules of the two economies: The competitive
equilibrium su?ers a major correction caused by the Fisherian de?ation mechanism.
The collateral constraint becomes binding and the economy is forced to a large
deleveraging that results in a sharp reduction in debt (an increase in b to -0.347
at point A
?
). Consumption falls leading to a drop in the the stochastic discount
factor and a drop in asset prices. In contrast, the social planner, while also facing
a binding credit constraint, adjusts it debt marginally to just about b = ?0.379 at
point B
?
. This was possible for the social planner because, taking into account the
risk of a Fisherian de?ation and internalizing its price dynamics, the planner chose
to borrow less than agents in the decentralized equilibrium a period earlier.
Overborrowing can also be assessed by comparing the long-run distributions
of debt and leverage across the competitive and constrained-e?cient equilibria. The
fact that the planner accumulates more precautionary savings implies that its er-
godic distribution concentrates less probability at higher leverage ratios than in
the competitive equilibrium. Figure 2.3 shows the ergodic distributions of leverage
ratios (measured as
?b
t+1
+?wtnt
qt
¯
K
) in the two economies. The maximum leverage ra-
tio in both economies is given by ? but notice that the decentralized equilibrium
concentrates higher probabilities in higher levels of leverage. Comparing averages
across these ergodic distributions, however, mean leverage ratios di?er by less than
73
0.26 0.28 0.3 0.32 0.34 0.36
0
1
2
3
4
5
x 10
?4
Leverage
P
r
o
b
a
b
i
l
i
t
y
Decentralized Equilibrium
Social Planner
Figure 2.3: Ergodic Distribution of Leverage (
?b
t+1
+?wtht
qt
¯
K
)
1 prcent. Hence, overborrowing is relatively small again if measured by comparing
di?erences in unconditional long-run averages of leverage ratios.
15
2.4.3 Asset Returns
Overborrowing has important quantitative implications for asset returns and
their determinants. Figure 2.4 shows the long-run distributions of land returns for
the competitive equilibrium and the social planner. The key di?erence in these
distributions is that the one for the competitive equilibrium features fatter tails.
In particular, there is a sharply fatter left tail in the competitive equilibrium, for
15
Measuring “ex ante” leverage as
?bt+?wtht
qt
¯
K
, we ?nd that leverage ratios in the competitive
equilibrium can exceed the maximum of those for the planner 3 percent of the time and by up to
12 percentage points.
74
which the 99th percentile of returns is about -17.5 percent, v. -1.6 percent in the
constrained-e?cient equilibrium. The fatter left tail in the competitive equilib-
rium corresponds to states in which a negative TFP shock hits when agents have
a relatively high level of debt. Intuitively, as a negative TFP shock hits, expected
dividends decrease and this puts downward pressure on asset returns.
16
In addition,
if the collateral constraint becomes binding, asset ?re-sales lead to a further drop in
asset prices.
We show below that the fatter tails of the distribution of asset returns, and the
associated time-varying risk of ?nancial crises, have substantial e?ects on the risk
premium. These features of our model are similar to those examined in the literature
on asset pricing and “disasters” (see Barro, 2009). Note, however, that this literature
generally treats ?nancial disasters as resulting from exogenous stochastic processes
with fat tails and time-varying volatility, whereas in our setup ?nancial crises and
their time-varying risk are both endogenous.
17
The underlying shocks driving the
model are standard TFP shocks, even in periods of ?nancial crises. In our model, as
in Mendoza (2010), ?nancial crises are endogenous outcomes that occur when shocks
of standard magnitudes trigger a Fisherian de?ation. Table 2.2 reports statistics
that characterize the main properties of asset returns in the constrained-e?cient
and competitive equilibria. We also report statistics for a competitive equilibrium
16
Similarly, the fatter right-tail in the distribution of returns of the competitive equilibrium
corresponds to periods with positive TFP shocks, which were preceded by low asset prices due to
?re sales.
17
The literature on disasters typically uses Epstein-Zin preferences so as to be able to match
the large observed equity premia. Here we use standard CRRA preferences with a risk aversion
coe?cient of 2, and as we show later, we can obtain larger risk premia than in the typical CRRA
setup without credit frictions. Moreover, we obtain realistically large risk premia when the credit
constraint binds.
75
?25 ?20 ?15 ?10 ?5 0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
Return on Land (in percentage))
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Decentralized Equilibrium
Social Planner
Figure 2.4: Ergodic Cumulative Distribution of Land Returns
in which land in the collateral constraint is valued at a ?xed price equal to the
average price across the ergodic distribution ¯ q (i.e. the credit constraint becomes
?
b
t+1
Rt
+ ?w
t
n
t
? ?¯ qk
t+1
).
18
This ?xed-price scenario allows us to compare the
properties of asset returns in the competitive and social planner equilibria with a
setup in which a collateral constraint exists but the Fisherian de?ation channel and
the credit externality are removed.
Table 2.2 lists expected excess returns (i.e. the equity risk premia), the direct
and indirect (covariance) e?ects of the credit constraint on excess returns, the (log)
standard deviation of returns, the price of risk, and the Sharpe ratio. These moments
are reported for the unconditional long-run distributions of each model economy, as
18
Because the asset is in ?xed supply, these allocations would be the same if we use instead
an ad-hoc borrowing limit such that ?
bt+1
Rt
+ ?w
t
n
t
? ?¯ q
¯
K. The price of land, however, would be
lower since with the ad-hoc borrowing constraint land does not have collateral value.
76
well as for distributions conditional on the collateral constraint being binding and
not binding.
The mean unconditional excess return is 1.09 percent in the competitive equi-
librium v. only 0.17 percent in the constrained-e?cient equilibrium and 0.86 percent
in the ?xed-price economy. The risk premium in the competitive equilibrium is large,
about half as large as the risk-free rate. The fact that the other two economies pro-
duce lower premia indicates that the high premium of the competitive equilibrium
is the combined result of the Fisherian de?ation mechanism and the ine?ciencies
induced by the credit externality. Note also that the high premium produced by our
model contrasts sharply with the ?ndings of Heaton and Lucas (1996), who found
that credit frictions without the Fisherian de?ation mechanism do not produce large
premia, unless transactions costs are very large.
19
The excess returns conditional on the collateral constraint not binding in the
constrained-e?cient and ?xed-price economies are in line with those obtained in
classic asset pricing models that display the “equity premium puzzle.” The equity
premia we obtained in these two scenarios are driven only by the covariance e?ect,
as in the classic models, and they are negligible: 0.03 percent in the ?xed-price
economy and 0.06 percent in the constrained-e?cient economy. This is natural
because, without the constraint binding and with the e?ects of the credit externality
and the Fisherian de?ation removed or weakened, the model is in the same class as
those that display the equity premium puzzle. In contrast, our baseline competitive
19
The unconditional premium in the ?xed price economy, at 0.86 percent, is not trivial, but
note that it results from the fact that the constraint binds with very high probability, given the
smaller incentives to accumulate precautionary savings. The risk premium in the unconstrained
region of the ?xed-price model is only 0.03 percent, v. 0.23 in our baseline model.
77
economy yields a 0.23 percent premium conditional on the constraint not binding,
which is small relative to data estimates that range from 6 to 18 percent, but 4 to
8 times larger than in the other two economies.
Conditional on the collateral constraint being binding, mean excess returns
in the competitive equilibrium are nearly 14 percent, 4.86 percent for the social
planner, and 1.29 percent in the ?xed-price economy. Interestingly, the lowest un-
conditional premium is the one for the constrained-e?cient economy (0.17 percent),
but conditional on the constraint binding, the lowest premium is the one for the
?xed-price economy (1.29 percent).This is because on one hand the Fisherian de?a-
tion e?ect is still at work when the collateral constraint binds in the constrained-
e?cient economy, but not in the ?xed-price economy, while on the other hand the
constrained-e?cient economy has a lower probability of hitting the collateral con-
straint (so that the higher premium when the constraint binds does not weigh heavily
when computing the unconditional average). In turn, the probability of hitting the
collateral constraint is higher for the ?xed-price economy, because the incentive to
build precautionary savings is weaker when there is no Fisherian ampli?cation.
The unconditional direct and covariance e?ects of the collateral constraint on
excess returns are signi?cantly stronger in the competitive equilibrium than in the
constrained-e?cient and ?xed-price economies, and even more so if we compare them
conditional on the constraint being binding. Again, the direct and covariance e?ects
are larger in the competitive equilibrium because of the e?ects of the overborrowing
externality and the Fisherian de?ation mechanism.
In terms of the decomposition of excess returns based on condition (2.10),
78
Table 2.2: Asset Pricing Moments
Excess Direct Covariance
Return E?ect E?ect s
t
?
t
(R
q
t+1
) S
t
Decentralized Equilibrium
Unconditional 1.09 0.87 0.22 5.22 3.05 0.79
Constrained 13.94 13.78 0.16 4.05 2.71 11.75
Unconstrained 0.23 0.00 0.23 5.3 3.08 0.05
Constrained-E?cient Equilibrium
Unconditional 0.17 0.11 0.06 2.88 1.85 0.08
Constrained 4.86 4.80 0.06 3.02 2.07 2.38
Unconstrained 0.06 0.00 0.06 2.86 1.84 0.03
Fixed Price Equilibrium
Unconditional 0.86 0.82 0.04 2.59 1.69 0.46
Constrained 1.29 1.23 0.05 2.81 1.84 0.69
Unconstrained 0.03 0.00 0.03 2.16 1.39 0.02
Note: The table reports averages of the conditional excess return after taxes, the direct e?ect, the
covariance e?ect, the price of risk s
t
, the (log) volatility of the return of land denoted ?
t
(R
q
t+1
),
and the Sharpe ratio. All numbers except the Sharpe ratios are in percentage
Table 2.2 shows that the unconditional average of the price of risk is about twice as
large in the decentralized equilibrium than in the constrained-e?cient and ?xed-price
economies. This re?ects the fact that consumption, and therefore the pricing kernel,
?uctuate signi?cantly more in the decentralized equilibrium. The Sharpe ratio and
the variability of land returns are also much larger in the competitive equilibrium.
The increase in the former indicates, however, that the mean excess return rises
signi?cantly more than the variability of returns, which indicates that risk-taking is
“overcompensated” in the competitive equilibrium (relative to the compensation it
receives when the social planner internalizes the credit externality). Note also that
the correlations between land returns and the stochastic discount factor, not shown
in the Table, are very similar under the three equilibria and very close to 1. This is
79
important because it implies that the di?erences in excess returns and Sharpe ratios
cannot be attributed to di?erences in this correlation.
2.4.4 Incidence and Magnitude of Financial Crises
We show now that overborrowing in the competitive equilibrium increases the
incidence and severity of ?nancial crises. To demonstrate this result we construct an
event analysis of ?nancial crises with simulated data obtained by performing long
(100,000-period) stochastic time-series simulations of the competitive, constrained-
e?cient and ?xed-price economies, removing the ?rst 1,000 periods. A ?nancial
crisis episode is de?ned as a period in which the credit constraint binds and this
causes a decrease in credit that exceeds one standard deviation of the ?rst-di?erence
of credit in the corresponding ergodic distribution.
The event analysis exercise is important also because it sheds light on whether
the model can produce ?nancial crises with realistic features, which is a key ?rst step
in order to make the case for treating the normative implications of the model as rel-
evant. We show here that, while we did not aim to build a rich equilibrium business
cycle model so we could keep the analysis of the externality tractable, and hence
our match to the data is not perfect, the model does produce ?nancial crises with
realistic features in terms of abrupt, large declines in allocations, credit, and land
prices, and it supports non-crisis output ?uctuations in line with observed U.S. busi-
ness cycles. Moreover, studies more focused on matching data from ?nancial crisis
events have shown that the Fisherian de?ation mechanism can do well at explaining
80
crisis dynamics nested within realistic long-run business cycle co-movements (see
Mendoza (2010)).
The ?rst important result of the event analysis is that the incidence of ?nancial
crises is signi?cantly higher in the competitive equilibrium. We calibrated ? so
that he competitive economy experiences ?nancial crises with a long-run probability
of 3.0 percent. But with the same ? , ?nancial crises occur in the constrained-
e?cient economy only with 0.9 percent probability in the long run. Thus, the credit
externality increases the frequency of ?nancial crises by a factor of 3.33.
20
The second important result is that ?nancial crises are more severe in the com-
petitive equilibrium. This is illustrated in the event analysis plots shown in Figure
2.5. The event windows are for total credit, consumption, labor, output,TFP and
land prices, all expressed as deviations from long-run averages. These event dynam-
ics are shown for the decentralized, constrained-e?cient, and ?xed-price economies.
We construct comparable event windows for the three scenarios following this
procedure: First we identify ?nancial crisis events in the competitive equilibrium,
and isolate ?ve-year event windows centered in the period in which the crisis takes
place. That is, each event window includes ?ve years, the two years before the
crisis, the year of the crisis, and the two years after. Second, we calculate the
median TFP shock across all of these event windows in each year t ?2 to t +2, and
the median initial debt at t ? 2. This determines an initial value for bonds and a
20
We could also de?ne crises in the constrained-e?cient equilibrium by using the value of the
credit threshold obtained in the competitive equilibrium. However, with this criterion we would
obtain an even lower probability of crises, because credit declines equal to at least one standard
deviation of the ?rst-di?erence of credit in the decentralized equilibrium are zero-probability events
in the constrained e?cient equilibrium.
81
?ve-year sequence of TFP realizations. Third, we feed this sequence of shocks and
initial value of bonds to the decisions rules of each model economy and compute
the corresponding endogenous variables plotted in Figure 2.5. By proceeding in this
way, we ensure that the event dynamics for the three equilibria are simulated using
the same initial state and the same sequence of shocks.
21
The features of ?nancial crises at date t in the competitive economy are in
line with the results in Mendoza (2010): The debt-de?ation mechanism produces
?nancial crises characterized by sharp declines in credit, consumption, asset prices
and output.
The ?ve macro variables illustrated in the event windows show similar dynam-
ics across the three economies in the two years before the ?nancial crisis. When the
crisis hits, however, the collapses observed in the competitive equilibrium are much
larger. Credit falls about 20 percentage points more, and two years after the crisis
the credit stock of the competitive equilibrium remains 10 percentage points below
that of the social planner.
22
Consumption, asset prices, and output also fall much
more sharply in the competitive equilibrium than in the planner’s equilibrium. The
declines in consumption and asset prices are particularly larger (-16 percent v. -5
percent for consumption and -24 percent v. -7 percent for land prices). The asset
price collapse also plays an important role in explaining the more pronounced de-
21
The sequence of TFP shocks is 0.9960, 0.9881, 0.9724, 0.9841, 0.9920 and the initial level of
debt is 1.6 percent above the average.
22
The model overestimates the drop in credit relative to what we have observed so far in the
U.S. crisis (which as of the third quarter of 2010 reached about 7 percent of GDP) . One reason
for this is that in the model, credit is in the form of one-period bonds, whereas in the data, loans
have on average a much larger maturity. In addition, our model does not take into account the
strong policy intervention that took place with the aim to prevent what would have been a larger
credit crunch.
82
cline in credit in the competitive equilibrium, because it re?ects the outcome of the
Fisherian de?ation mechanism. Output falls by 2 percentage points more, and labor
falls almost 3 percentage points more, because of the higher shadow cost of hiring
labor due to the e?ect of the tighter binding credit constraint on access to working
capital.
t?2 t?1 t t+1 t+2
?20
?10
0
Credit
%
Decentralized Equilibrium Social Planner Fixed Price Equilibrium
t?2 t?1 t t+1 t+2
?15
?10
?5
0
5
Consumption
%
t?2 t?1 t t+1 t+2
?25
?15
?5
5
Asset Price
%
t?2 t?1 t t+1 t+2
?8
?6
?4
?2
0
Output
%
t?2 t?1 t t+1 t+2
?6
?4
?2
0
Employment
%
t?2 t?1 t t+1 t+2
?3
?2
?1
0
TFP shock
%
Figure 2.5: Event Analysis: percentage di?erences relative to unconditional averages
83
The event analysis results can also be used to illustrate the relative signi?cance
of the wage and land price components in the externality term ?
t
? ?
¯
K
?qt
?bt
??n
t
?wt
?bt
identi?ed in condition (2.15). Given the unitary Frisch elasticity of labor supply,
wages decrease one-to-one with labor (and hence the event plot for wages would
be identical to the one shown for employment in Figure 2.5). As a result, the
extent to which the drop in wages can help relax the collateral constraint is very
limited. Wages and employment fall about 6 percent at date t , and with a working
capital coe?cient of ? = 0.14, this means that the e?ect of the drop in wages in
the borrowing capacity is 0.14(1 ? 0.06)0.06 = 0.79 percent. On the other hand,
given that
¯
K = 1 and that asset prices fall about 25 percent below trend at date
t, and since ? = 0.36, the e?ect of land ?re sales on the collateral constraint is
0.36(0.25) = 9 percent. Thus, the land price e?ect of the externality is about 10
times bigger than the wage e?ect. This ?nding will play an important role in our
quantitative analysis of the features of the macro-prudential policy later in this
section.
The ?xed-price economy displays very little ampli?cation given that the econ-
omy is free from the Fisherian de?ation mechanism. Credit increases slightly at date
t in order to smooth consumption and remains steady in the following periods. The
fact that land is valued at the average price, and not the market price, contributes
to mitigate the drop in the price of land, since it remains relatively more attractive
as a source of collateral.
To gain more intuition on why land prices drop more because of the credit
externality, we plot in Figure 2.6 the projected conditional sequences of future divi-
84
dends and land returns on land up to 30 periods ahead of a ?nancial crisis that occurs
at date t = 0 (conditional on information available on that date). These are the
sequences of dividends and returns used to compute the present values of dividends
that determine the equilibrium land price at t in the event analysis of Figure 2.5.
The expected land returns start very high when the crisis hits in both competitive
and constrained-e?cient equilibria, but signi?cantly more for the former (at about
40 percent) than the latter (at 10 percent). On the other hand, expected dividends
do not di?er signi?cantly, and therefore we conclude that the sharp change in the
pricing kernel re?ected in the surge in projected land returns when the crisis hits is
what drives the large di?erences in the drop of asset prices.
0 10 20 30
0.0488
0.049
0.0492
0.0494
0.0496
0.0498
0.05
0.0502
Expected Dividend
DE
SP
0 10 20 30
0
10
20
30
40
P
e
r
c
e
n
t
a
g
e
Discount Rate for Dividends
DE
SP
Figure 2.6: Forecast of expected dividends and land returns.
The large deleveraging that takes places when a ?nancial crisis occurs in the
competitive equilibrium implies that projected land returns for the immediate future
(i.e. the ?rst 6 periods after the crisis) drop signi?cantly. Returns are also projected
to fall for the social planner, but at a lower pace, so that in fact the planner projects
85
higher land returns than agents in the competitive equilibrium for a few periods.
Projected dividends for the same immediate future after the crisis are slightly smaller
than the long-run average of 0.05 in both economies because of the persistence of
the TFP shock. In the long-run, expected dividends are slightly higher for the social
planner, because the marginal productivity of land drops less during the ?nancial
crisis as a result of the lower amount of debt. Notice also that the planner projects
to discount dividends with a slightly higher land return in the long run, because the
tax on debt more than o?sets the fact that the risk premium of the planner is lower
(recall that we are comparing after-tax returns as de?ned in Section 2). This arises
because the tax on debt makes bonds relatively more attractive and this leads in
equilibrium to a higher required return on land.
2.4.5 Long-Run Business Cycles
Table 2.3 reports the long-run business cycle moments of the competitive,
constrained-e?cient and ?xed-price equilibria, which are computed using each econ-
omy’s ergodic distribution. The credit externality at work in the competitive equi-
librium produces higher business cycle variability in output and labor, and especially
in consumption, compared with the constrained-e?cient and ?xed-price economies.
The high variability of consumption and credit are consistent with the results in
Bianchi (2010), but we ?nd in addition that the credit externality produces a mod-
erate increase in the variability of labor and a substantial increase in the variability
of land prices and leverage. Notice that the variability in consumption is higher
86
than the variability of output in the decentralized equilibrium which is not the case
in U.S. data. However, if we exclude the crisis periods, the ratio of the variability of
consumption to the variability of GDP would be 0.87 (compared with 0.88 in annual
U.S. data from 1960 to 2007).
It may seem puzzling that we can obtain non-trivial di?erences in long-run
business cycle moments even though ?nancial crises are a low probability event in
the competitive equilibrium. To explain this result, it is useful to go back to Figure
1. This plot shows that even during normal business cycles the optimal plans of the
competitive and constrained-e?cient equilibria di?er, and this is particularly the
case in the high-externality region. Because the economy spends about 70 percent
of the time in this region, where private agents borrow more and are more exposed to
the risk of ?nancial crises, long-run business cycle moments di?er. In addition, the
larger e?ects that occur during crises have a non-trivial e?ect on long-run moments.
This is particularly noticeable in the case of consumption where the variability drops
from 2.7 to 1.7 in the decentralized equilibrium when we exclude the crises episodes.
The business cycle moments for consumption, output and labor in the constrained-
e?cient economy are about the same as those of the ?xed-price economy. This occurs
even though the constrained-e?cient economy is subject to the Fisherian de?ation
mechanism and the ?xed-price economy is not. The reason for this is because the
social planner accumulates extra precautionary savings, which compensate for the
sudden change in the borrowing ability when the credit constraint binds. The con-
straint binds less often and when it does it has weak e?ects on macro variables. On
the other hand, the constrained-e?cient economy does display lower variability in
87
leverage and land prices that the ?xed-price economy, and this occurs because the
social planner internalizes how a drop in the price tightens the collateral constraint.
The output correlations of leverage, credit, and land prices also di?er signif-
icantly across the model economies. The GDP correlations of leverage and credit
are signi?cantly higher in the competitive equilibrium, while the correlation between
the price of land and GDP is lower. The model without credit frictions would have a
natural tendency to produce countercyclical credit because consumption-smoothing
agents want to save in good times and borrow in bad times. This e?ect still domi-
nates in the constrained-e?cient and ?xed-price economies, but in the competitive
equilibrium the collateral constraint and the Fisherian de?ation hamper consump-
tion smoothing enough to produce procyclical credit and a higher GDP-leverage
correlation. Similarly, the GDP-land price correlation is nearly perfect when the
Fisherian de?ation mechanism is weakened (constrained-e?cient case) or removed
(?xed-price case), but falls to about 0.8 in the competitive equilibrium. Because
of the strong procyclicality of land prices, leverage is countercyclical with a GDP
correlation of -0.57. This is in line with the countercyclicality of household leverage
in U.S. data, although the correlation is lower than in the data (the correlation
between the ratio of net household debt to the value of residential land and GDP is
-0.25 at the business cycle frequency).
23
In terms of the ?rst-order autocorrelations, the competitive equilibrium dis-
23
Two caveats on this point. First, at lower frequencies the correlation is positive. As
Boz and Mendoza (2010) report, the household leverage ratio rose together with GDP, land prices
and debt between 1997 and 2007. Second, the countercyclicality of leverage for the household sector
di?ers sharply from the strong procyclicality of leverage in the ?nancial sector (see Adrian and Shin
(2010)).
88
plays lower autocorrelations in all its variables compared to both constrained-e?cient
and ?xed-price equilibria. This occurs because crises in the competitive equilibrium
are characterized by deep but not very prolonged recessions.
Table 2.3: Long Run Moments
Standard Correlation Autocorrelation
Deviation with GDP
DE SP FP DE SP FP DE SP FP
Output 2.10 1.98 1.97 1.00 1.00 1.00 0.50 0.51 0.51
Consumption 2.71 1.87 1.85 0.86 0.99 0.99 0.23 0.56 0.57
Employment 1.25 1.02 0.98 0.97 1.00 1.00 0.42 0.50 0.51
Leverage 3.92 2.72 3.80 -0.57 -0.93 -0.95 0.59 0.69 0.71
Total Credit 3.55 0.95 0.76 0.27 -0.35 -0.42 0.58 0.77 0.81
Land Price 3.95 2.24 3.48 0.79 0.97 0.97 0.16 0.56 0.60
Working capital 2.48 2.04 1.97 0.97 1.00 1.00 0.42 0.50 0.51
Note: ‘DE’ represents the decentralized equilibrium,‘SP’ represents the social planner, ’FP’
represents an economy with land valued at a ?xed price equal to the average of the price of
land in the competitive equilibrium.
2.4.6 Properties of Macro-prudential Policies
Table 4 shows the statistical moments that characterize the state-contingent
schedules of taxes on debt and dividends by which the social planner decentralizes
the constrained-e?cient allocations as a competitive equilibrium. To make the two
comparable, we express the dividend tax as a percent of the price of land.
The unconditional average of the debt tax is 1.07 percent, v. 0.09 when the
constraint binds and 1.09 when it does not. The tax remains positive, albeit small,
on average when the collateral constraint binds, because in some these states the
social planner wants to allocate borrowing ability across bonds and working cap-
ital in a way that di?ers from the competitive equilibrium. If there is a positive
89
probability that the credit constraint will bind again next period, the social planner
allocates less debt capacity to bonds and more to working capital. As a result, a tax
on debt remains necessary in a subset of the constrained region. Note, however, that
these states are not associated with ?nancial crisis events in our simulations. They
correspond to events in which the collateral constraint binds but the deleveraging
that occurs is not strong enough for a crisis to occur.
The debt tax ?uctuates about 2/3rds as much as GDP and is positively cor-
related with leverage, i.e.
?b
t+1
+?wtnt
qt
¯
K
. This is consistent with the macro-prudential
rational behind the tax: The tax is high when leverage is building up and low when
the economy is deleveraging. Note, however, that since leverage itself is negatively
correlated with GDP, the tax also has a negative GDP correlation. When the con-
straint binds, the correlation between the tax and leverage is zero by construction,
because leverage remains constant at the value of ?.
Table 2.4: Long Run Moments of Macro-prudential Policies
Average Standard Correlation
Deviation with Leverage
Debt Dividend Debt Dividend Debt Dividend
Tax Tax Tax Tax Tax Tax
Unconditional 1.07 -0.46 1.41 0.62 0.73 -0.64
Constrained 0.09 0.52 0.41 0.04 0.0 0.0
Unconstrained 1.09 -0.49 1.40 0.61 0.81 -0.79
The unconditional average of the dividend tax is negative (i.e. it is a subsidy),
and it is very small at about -0.46 percent: when the constraint binds it is on average
about 0.52 percent (v. a -0.49 percent average when the constraint does not bind).
90
The fact that on average the planner requires a subsidy on dividends may seem
puzzling, given that land is less risky in the regulated decentralized equilibrium,
as we have shown above. There is another e?ect at work, however, because the
debt tax puts downward pressure on land prices by making bonds relatively more
attractive than land, and this e?ect turns out to be quantitatively larger. Thus,
since by de?nition the constrained-e?cient allocations are required to support the
same land pricing function of the competitive economy without policy intervention,
the planner calls for a dividend subsidy on average in order to o?set the e?ect of the
debt tax on land prices. The variability of the tax on dividends is 0.62 percentage
points, less than 1/3rds the variability of GDP. The correlation between this tax on
dividends and leverage is negative in the unconstrained region re?ecting the negative
correlation between the tax on debt and the tax on land explained above.
The dynamics of the debt and dividend taxes around crisis events are shown
in Figure 2.7. The debt tax is high relative to its average, at about 2.7 percent, at
t ? 2 and t ? 1, and this again re?ects the macro-prudential nature of these taxes:
Their goal is to reduce borrowing so as to mitigate the magnitude of the ?nancial
crisis if bad shocks occur. At date t the debt tax falls to zero, and it rises again at
t +1 and t +2 to about 2 percent. The latter occurs because this close to the crisis
the economy still remains ?nancially fragile (i.e. there is still a non-zero probability
of agents becoming credit constrained next period). The tax on dividends follows a
similar pattern. Dividends are subsidized at a similar rate before and after ?nancial
crises events, but they are actually taxed when crises occur. The reason is again
that the social planner needs to support the same pricing function of the competitive
91
equilibrium that would arise without policy intervention. Hence, with the tax on
debt falling to almost zero, there is pressure for land prices to be higher than what
that pricing function calls for, and hence dividends need to be taxed to o?set this
e?ect.
The macro-prudential behavior of the debt tax is very intuitive and follows
easily from the precautionary behavior of the planner we have described. On the
other hand, the tax on dividends and its dynamic behavior seem less intuitive and
harder to sell as a policy rule (i.e. the notion of proposing to tax dividends at the
through of a ?nancial crisis is bound to be unpopular). The two policy instruments
are required, however, in order to implement exactly the allocations of the con-
strained social planner as a decentralized competitive equilibrium. Moreover, the
planner’s allocations are guaranteed to attain a level of welfare at least as high as
that of the competitive equilibrium without macro-prudential policy, since this equi-
librium remains feasible to the social planner. If one takes the debt tax and not the
tax on dividends, one cannot guarantee this Pareto improvement. Indeed, we solved
a variant of the model in which we introduced the optimal schedule of debt taxes
but left the tax on dividends out, and found that average welfare is actually lower
than without policy intervention by -0.02 percent. This occurs because welfare in
the states of nature in which the constraint is already binding is lower than without
policy intervention.
24
Hence, while our results may provide a justi?cation for the
use of macro-prudential policies, they also provide a warning because selective use
24
If we reduce the debt tax we can obtain again average welfare gains, which again illustrates
the interdependence of macroprudential policies.
92
of macro-prudential policies (i.e. partial implementation of the policy instruments
indicated by the model) can reduce welfare in some states of nature. In this ex-
periment this happens because the selective use of the debt tax without the tax on
dividends lowers asset prices in some states of nature, and reduces welfare in those
states by reducing the value of collateral.
t?2 t?1 t t+1 t+2
0
1
2
3
Tax on Debt
P
e
r
c
e
n
t
a
g
e
t?2 t?1 t t+1 t+2
?1
?0.5
0
0.5
1
Tax on Dividends
P
e
r
c
e
n
t
a
g
e
Figure 2.7: Event Analysis: Macroprudential Policies
Jeanne and Korinek (2010) also compute a schedule of macroprudential taxes
on debt to correct a similar externality that arises because of a collateral constraint
that depends on asset prices. In their constraint, however, the agents’ borrowing ca-
pacity is determined by the aggregate level of assets and by a linear state- and time-
invariant term (i.e. their borrowing constraint is de?ned as
b
t+1
R
? ??q
t
¯
K??). The
fact that their constraint depends on aggregate rather than individual asset hold-
ings, as in our model, matters because it implies that agents do not value additional
asset holdings as a mechanism to manage their borrowing ability.
25
But more im-
25
To illustrate this point, we recomputed our model assuming that the borrowing constraint
depends on the aggregate value of assets, as in their setup. Because assets do not have individual
93
portantly, leaving aside this di?erence, they calibrate parameter values to ? = 0.046,
? = 3.07 and q
t
¯
K = 4.8, which imply that the e?ects of the credit constraint are
driven mainly by ?, and only less than 7 percent (0.07=0.046*4.8/(0.046*4.8+3.07))
of the borrowing ability depends on the value of asset holdings. As a result, the
Fisherian de?ation e?ect and the credit externality are weak, and hence they ?nd
that macroprudential policy lessens the macro e?ects of ?nancial crises much less
than in our setup. The asset price drop is reduced from 12.3 to 10.3 percent, and
the consumption drop is reduced from 6.2 to 5.2 percent (compared with declines
from 24 to 7 and 16 to 5 percent respectively in our model). Moreover, they model
the stochastic process of dividends as an exogenous, regime-switching Markov chain
such that the probability of a crisis (i.e. binding credit constraint) coincides with the
probability of a bad realization of dividends, implying that the probability of busts
is una?ected by macroprudential regulation. Thus, in their setup macro-prudential
policy is much less e?ective at reducing the magnitude of ?nancial crises and has no
impact on their incidence.
2.4.7 Welfare E?ects
We move next to explore the welfare implications of the credit externality.
To this end, we calculate welfare costs as compensating consumption variations for
each state of nature that make agents indi?erent between the allocations of the
competitive equilibrium and the constrained-e?cient allocations. Formally, for a
value as collateral, asset prices drop even more during crises, and this leads private agents to
accumulate more precautionary savings, which results in crises having zero-probability in the long-
run under both competitive and constrained-e?cient equilibria for our baseline calibration.
94
given initial state (B, ?) at date 0, the welfare cost is computed as the value of ?
such that the following condition holds:
E
0
?
?
t=0
?
t
u(c
DE
t
(1 +?) ?G(n
DE
t
)) = E
0
?
?
t=0
?
t
u(c
SP
t
?G(n
SP
t
)) (2.22)
where the superscript DE denotes allocations in the decentralized competitive equi-
librium and the superscript SP denotes the social planner’s allocations. Note that
these welfare costs re?ect also the welfare gains that would be obtained by introduc-
ing the social planner’s optimal debt and dividend tax policies, which by construc-
tion implement the constrained-e?cient allocations as a competitive equilibrium.
The welfare losses of the DE arise from two sources. The ?rst source is the
higher variability of consumption, due to the fact that the credit constraint binds
more often in the DE, and when it binds it induces a larger adjustment in asset
prices and consumption. The second is the e?ciency loss in production that occurs
due to the e?ect of the credit friction on working capital. Without the working
capital constraint, the marginal disutility of labor equals the marginal product of
labor. With the working capital constraint, however, the shadow cost of employing
labor rises when the constraint binds, and this drives a wedge between the marginal
product of labor and its marginal disutility. Again, since the collateral constraint
binds more often in the DE than in the SP, this implies a larger e?ciency loss.
Figure 6 plots the welfare costs of the credit externality as a function of b for
a negative, two-standard-deviations TFP shock. These welfare costs approximate a
bell shape skewed to the left. This is due to the di?erences in the optimal plans of the
95
social planner vis-a-vis private agents in the decentralized equilibrium. Recall than
in the constrained region, the current allocations of the decentralized equilibrium
essentially coincide with those of the constrained-e?cient economy, as described
in Figure 1. Therefore, in this region the welfare gains from implementing the
constrained-e?cient allocations only arise from how future allocations will di?er. On
the other hand, in the high-externality region, the constrained- e?cient allocations
di?er sharply from those of the decentralized equilibrium, and this generally enlarges
the welfare losses caused by the credit externality. Notice that, since the constrained-
e?cient allocations involve more savings and less current consumption, there are
welfare losses in terms of current utility for the social planner, but these are far
outweighed by less vulnerability to sharp decreases in future consumption during
?nancial crises. Finally, as the level of debt is decreased further and the economy
enters the low-externality region, ?nancial crises are unlikely and the welfare costs
of the ine?ciency decrease.
The unconditional average welfare cost over the decentralized equilibrium’s
ergodic distribution of bonds and TFP is 0.046 percentage points of permanent
consumption. This contrasts with Bianchi (2010) who found welfare costs about
3 times larger. Note, however, that our results are in line with his if we express
the welfare costs as a fraction of the variability of consumption. Consumption was
more volatile in his setup because he examined a calibration to data for emerging
economies, which are more volatile than the United States.
The fact that welfare losses from the externality are small although the di?er-
ences in consumption variability are large is related to the well-known Lucas result
96
?0.4 ?0.39 ?0.38 ?0.37 ?0.36 ?0.35 ?0.34 ?0.33 ?0.32 ?0.31 ?0.3
0
0.01
0.02
0.03
0.04
0.05
0.06
Current Bond Holdings
P
e
r
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n
t
a
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o
f
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t
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o
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s
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p
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i
o
n
High Externality Region
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Figure 2.8: Welfare Costs of the Credit Externality for a two-standard-deviations
TFP Shock
that models with CRRA utility, trend-stationary income, and no idiosyncratic un-
certainty produce low welfare costs from consumption ?uctuations. Moreover, the
e?ciency loss in the supply-side when the constraint binds produces low welfare costs
on average because those losses have a low probability in the ergodic distribution.
2.4.8 Sensitivity Analysis
We examine now how the quantitative e?ects of the credit externality change as
we vary the values of the model’s key parameters. Table C.1 shows the main model
statistics for di?erent values of ?, ?, ? and ?. The Table shows the unconditional
averages of the tax on debt and the welfare loss, the covariance e?ect on excess
97
returns, the probability of ?nancial crises, and the impact e?ects of a ?nancial crisis
on key macroeconomic variables. In all of these experiments, only the parameter
listed in the ?rst column changes and the rest of the parameters remain at their
baseline calibration values.
The results of the sensitivity analysis reported in Table C.1 can be understood
more easily by referring to the externality term derived in Section 2: The wedge
between the social and private marginal costs of debt that separate competitive and
constrained-e?cient equilibria, ?RE
t
_
µ
t+1
_
?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
__
. For given ?
and R, the magnitude of the externality is given by the expected product of two
terms: the shadow value of relaxing the credit constraint, µ
t+1
, and the associated
price e?ects ?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
that determine the e?ects of the externality on the
ability to borrow when the constraint binds. As explained earlier, the price e?ects are
driven mostly by
?q
t+1
?b
t+1
, because of the documented large asset price declines when the
collateral constraint binds. It follows therefore, that the quantitative implications
of the credit externality must depend mainly on the parameters that a?ect µ
t+1
and
?q
t+1
?b
t+1
, as well as those that a?ect the probability of hitting the constraint.
The coe?cient of relative risk aversion ? plays a key role because it a?ects
both µ
t+1
and
?q
t+1
?b
t+1
. A high ? implies a low intertemporal elasticity of substitution in
consumption, and therefore a high value from relaxing the constraint since a binding
constraint hinders the ability to smooth consumption across time. A high ? also
makes the stochastic discount factors more sensitive to changes in consumption,
and therefore makes the price of land react more to changes in bond holdings.
Accordingly, rising ? from 2 to 2.5 rises the welfare costs of the credit externality
98
by a factor of 5, and widens the di?erences in the covariance e?ects across the
competitive and constrained-e?cient equilibria. In fact, the covariance e?ect in the
decentralized equilibrium increases from 0.22 to 0.37 whereas for the constrained
e?cient allocations the increase is from 0.06 to 0.08. Stronger precautionary savings
reduce the probability of crises in the competitive equilibrium, and ?nancial crises
become a zero-probability event in the constrained-e?cient equilibrium. Conversely,
reducing ? to 1 makes the externality extremely small, measured either by di?erences
in the incidence or severity of ?nancial crises.
26
The collateral coe?cient ? also plays an important role because it alters the
e?ect of land price changes on the borrowing ability. A higher ? implies that, for a
given price response, the change in the collateral value becomes larger. Thus, this
e?ect makes the externality term larger. On the other hand, a higher ? has two
additional e?ects that go in the opposite direction. First, a higher ? implies that
the direct e?ect of the collateral constraint on the land price is weaker, leading to
a lower fall in the price of land during ?re sales. Second, a higher ? makes the
constraint less likely to bind, reducing the externality. The e?ects of changes in ?
are clearly non-monotonic. If ? is equal to zero, there is no e?ect of prices on the
borrowing-ability. At the same time, for high enough values of ?, the constraint
never binds. In both cases, the externality does not play any role. Quantitatively,
Table C.1 shows that small changes in ? are positively associated with the size of
the ine?ciency. In particular, an increase in ? from the baseline value of 0.36 to
26
Notice that the probability of a crisis in the competitive equilibrium becomes 10 percent, more
than three times larger than the target employed in the baseline calibration due to the reduction
in the level of precautionary savings.
99
0.40 increases the welfare cost of the ine?ciency by a factor of 6 and ?nancial crises
again become a zero-probability event in the constrained-e?cient equilibrium.
The above results have interesting policy implications. In particular, they
suggest that while increasing credit access by rising ? may increase welfare relative to
a more ?nancially constrained environment, rising ? can also strengthen the e?ects of
credit externalities and hence make macro-prudential policies more desirable (since
the welfare cost of the externality also rises).
A high Frisch elasticity of labor supply ((1/?) = 1.2) implies that output
drops more when a negative shock hits. If the credit constraint binds, this implies
that consumption falls more, which increases the marginal utility of consumption
and raises the return rate at which future dividends are discounted.
27
Moreover,
everything else constant, a higher elasticity makes the externality term higher by
weakening the e?ects of wages on the borrowing capacity. Hence, a higher elasticity
of labor supply is associated with higher e?ects from the credit externality, captured
especially by larger di?erences in the severity of ?nancial crises, a higher probability
of crises, and a larger welfare cost of the credit externality.
The fraction of wages that have to be paid in advance ? plays a subtle role. On
one hand, a larger ? increases the shadow value of relaxing the credit constraint, since
this implies a larger rise in the e?ective cost of hiring labor when the constraint binds.
On the other hand, a larger ? implies, ceteris paribus, a weaker e?ect on borrowing
ability, since the reduction of wages that occurs when the collateral constraint binds
27
The increase in leisure mitigates the decrease in the stochastic discount factor but does not
compensate for the fall in consumption
100
has a positive e?ect on the ability to borrow. Quantitatively, increasing (decreasing)
? by 5 percent increases (decreases) slightly the e?ects that re?ect the size of the
externality.
Changes in the volatility and autocorrelation of TFP do not have signi?cant
e?ects. Increasing the variability of TFP implies that ?nancial crises are more
likely to be triggered by a large shock. This results in larger ampli?cation and
a higher bene?t from internalizing price e?ects. In general equilibrium, however,
precautionary savings increase too, resulting in a lower probability of ?nancial crises
for both equilibria. Therefore, the overall e?ects on the externality of a change in
the variability of TFP depend on the relative change in the probability of ?nancial
crisis in both equilibria and the change in the severity of these episodes. An increase
in the autocorrelation of TFP leads to more frequent ?nancial crises for given bond
decision rules. Again, in general equilibrium, precautionary savings increase making
ambiguous the e?ect on the externality.
T
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f
t
h
e
e
v
e
n
t
a
n
a
l
y
s
i
s
)
.
T
h
e
b
a
s
e
l
i
n
e
p
a
r
a
m
e
t
e
r
v
a
l
u
e
s
a
r
e
:
R
?
1
=
0
.
0
2
8
,
?
=
0
.
9
6
,
?
=
2
,
?
h
=
0
.
6
4
,
?
=
0
.
6
4
,
?
=
1
¯
K
=
1
,
?
=
0
.
1
4
,
?
=
0
.
3
6
,
?
K
=
0
.
0
5
,
?
?
=
0
.
0
1
4
,
?
?
=
0
.
5
3
102
In terms of the optimal debt on tax, the results of the sensitivity analysis
produce an important ?nding: The average debt tax of about 1.1 percent is largely
robust to the parameter variations we considered. Except for the scenario that
approximates logarithmic utility (? = 1), in all other scenarios included in Table
C.1 the mean tax ranges between 1.01 and 1.2 percent.
We consider now shocks that a?ect directly the collateral constraint by af-
fecting the extent to which agents can pledge assets as collateral. We consider a
stochastic process for ? that follows a symmetric two-state Markov chain indepen-
dent from shocks to TFP. In line with evidence from Mendoza and Terrones (2008)
on the mean duration of credit booms in industrial countries, we calibrate the proba-
bilities of the Markov chain so that the average duration of each state is 6 years. We
keep the average value of ? as in our benchmark model and consider ?uctuations of
? of 10 percent which is meant to be suggestive. As shown in Table C.1, the e?ects
of the externality remain largely unchanged to this modi?cation.
Overall, the results of the sensitivity analysis show that parameter changes
that weaken the model’s ?nancial ampli?cation mechanism also weaken the magni-
tude of the externality. This results in smaller average taxes, smaller welfare costs
and smaller di?erences in the incidence and severity of ?nancial crises. The coef-
?cient of risk aversion is particularly important also because it in?uences directly
the price elasticity of asset demand, and hence it determines how much asset prices
can be a?ected by the credit externality. This parameter plays a role akin to that
of to the elasticity of substitution in consumption of tradables and non-tradables in
Bianchi (2010), because in his model this elasticity drives the response of the price
103
at which the collateral is valued. Accordingly, he found that the credit externality
has signi?cant e?ects only if the elasticity is su?ciently low.
2.5 Conclusion
This paper examined the positive and normative e?ects of a credit externality
in a dynamic stochastic general equilibrium model in which a collateral constraint
limits access to debt and working capital loans to a fraction of the market value
of an asset in ?xed supply (e.g. land). We compared the allocations and welfare
attained by private agents in a competitive equilibrium in which agents face this
constraint taking prices as given, with those attained by a constrained social planner
that faces the same borrowing limits but takes into account how current borrowing
choices a?ect future asset prices and wages. This planner internalizes the debt-
de?ation process that drives macroeconomic dynamics during ?nancial crises, and
hence borrows less in periods in which the collateral constraint does not bind, so as
to weaken the debt-de?ation process in the states in which the constraint becomes
binding. Conversely, private agents overborrow in periods in which the constraint
does not bind, and hence are exposed to the stronger adverse e?ects of the debt-
de?ation mechanism when a ?nancial crisis occurs.
The novelty of our analysis is in that it quanti?es the e?ects of the credit
externality in a setup in which the credit friction has e?ects on both aggregate
demand and supply. The e?ects on demand are well-known from models with credit
constraints: consumption drops as access to debt becomes constrained, and this
104
induces an endogenous increase in excess returns that leads to a decline in asset
prices. Because collateral is valued at market prices, the drop in asset prices tightens
the collateral constraint further and leads to ?re-sales of assets and a spiraling
decline in asset prices, consumption and debt. On the supply side, production and
labor demand are a?ected by the collateral constraint because ?rms buy labor using
working capital loans that are limited by the collateral constraint, and hence when
the constraint binds the e?ective cost of labor rises, so the demand for labor and
output drops. This a?ects dividend rates and hence feeds back into asset prices.
Previous studies in the macro/?nance literature have shown how these mechanisms
can produce ?nancial crises with features similar to actual ?nancial crises, but the
literature had not conducted a quantitative analysis comparing constrained-e?cient
v. competitive equilibria in an equilibrium model of business cycles and asset prices.
We conducted a quantitative analysis in a version of the model calibrated to
U.S. data. This analysis showed that, even though the credit externality results in
only slightly larger average ratios of debt and leverage to output compared with the
constrained-e?cient allocations (i.e. overborrowing is not large), the credit external-
ity does produce ?nancial crises that are signi?cantly more severe and more frequent
than in the constrained-e?cient equilibrium, and produces higher long-run business
cycle variability. There are also important asset pricing implications. In particular,
the credit externality and its associated higher macroeconomic volatility in the com-
petitive equilibrium produce equity premia, Sharpe ratios, and market price of risk
that are much larger than in the constrained-e?cient equilibrium. We also found
that the degree of risk aversion plays a key role in our results, because it is a key
105
determinant of the response of asset prices to volatility in dividends and stochastic
discount factors. For the credit externality to be important, these price responses
need to be nontrivial, and we found that they are nontrivial already at commonly
used risk aversion parameters, and larger at larger risk aversion coe?cients that are
still in the range of existing estimates.
This analysis has important policy implications. In particular, the social plan-
ner can decentralize the constrained-e?cient allocations as a competitive equilibrium
by introducing an optimal schedule of state-contingent taxes on debt and dividends.
By doing so, it can neutralize the adverse e?ects of the credit externality and pro-
duce an increase in social welfare. In our calibrated model, the tax on debt necessary
to attain this outcome is about 1 percent on average. The tax is higher when the
economy is building up leverage and becoming vulnerable to a ?nancial crisis, but
before a crisis actually occurs, so as to induce private agents to value more the
accumulation of precautionary savings than they do in the competitive equilibrium
without taxes.
These ?ndings are relevant for the ongoing debate on the design of new ?nan-
cial regulation to prevent ?nancial crises, which emphasizes the need for “macro-
prudential” regulation. Our results lend support to this approach by showing that
credit externalities associated with ?re-sales of assets have large adverse macroeco-
nomic e?ects. At the same time, however, we acknowledge that actual implementa-
tion of macro-prudential policies in ?nancial markets remains a challenging task. In
particular, the optimal design of these policies requires detailed information on a va-
riety of credit constraints that private agents and the ?nancial sector face, real-time
106
data on their leverage positions, and access to a rich set of state-contingent policy
instruments. Moreover, as we showed in this paper, implementing only a subset
of the optimal policies because of these limitations (or limitations of the political
process) can reduce welfare in some states.
107
Chapter 3
E?cient Bailouts?
3.1 Introduction
The recent ?nancial crisis has led to massive government intervention in credit
markets. The initial Troubled Assets Relief Program (TARP), for example, required
700 billion dollars to provide credit assistance to ?nancial and non-?nancial institu-
tions. These measures have triggered an intense debate on the desirability of such
interventions. Supporters argue that these bailouts were necessary to avoid a com-
plete meltdown of the ?nancial sector that would have brought an extraordinary
contraction in output and employment. Critics argue that the presence of bailouts
generate incentives for investors to take even more risk ex ante, sowing the seeds
of future crises. Such critics, therefore, propose regulations to limit the ability of
central banks to conduct bailouts.
In this paper, we seek an answer to the following questions: What are the
implications of bailout expectations for the stability of the ?nancial sector? Is it
desirable to prohibit the use of public funds to conduct bailouts? If the central
bank could commit to a bailout policy, under what states of nature is it optimal to
conduct a bailout? How large should these bailouts be?
This paper answers these questions based on a non-linear DSGE model where
credit frictions generate scope for government credit intervention during a ?nancial
108
crisis, but where such e?ects generate more risk-taking before the crisis actually hits.
Recent research (see e.g. Gertler and Kiyotaki, 2010) has developed a quantitative
framework to analyze how governmental intervention can mitigate the credit crunch
and moderate the recession ex post. At the same time, a growing theoretical liter-
ature has emphasized the moral hazard implications of such interventions (see e.g.
Farhi and Tirole, 2010), but little is known about their quantitative implications.
In this paper, we develop a quantitative DSGE model to assess within a uni?ed
framework the interaction between ex-post interventions in credit markets and the
build-up of risk ex ante.
The model features a representative corporate entity that ?nances investment
using equity and debt. There are two key frictions a?ecting the capacity of ?rms to
?nance investment. First, debt contracts are not fully enforceable, giving rise to a
collateral constraint that limits the amount that ?rms can borrow. Second, there is
a constraint on minimum dividends that ?rms must make each period. Therefore,
?rms balance the desire to increase borrowing and investment now with the risk
of becoming ?nancially constrained in the future. When leverage is su?ciently
high and an adverse ?nancial shock hits the economy, ?rms hit their balance sheet
constraints, generating a fall in investment and a protracted recession.
In this environment, we consider a social planner that engages in direct credit
policy with the purpose of relaxing the balance sheet constraints of ?rms, but faces
two types of costs in this intervention. First, there is the static cost of transferring
resources from workers to ?rms. Second, there is the dynamic e?ect (or a moral
hazard e?ect) that arises because ?rms anticipate the planner’s intervention and
109
therefore take more risk. As a result, there is a trade o? involved in the design of a
bailout policy. On the one hand, a commitment to bailing out the entire corporate
sector in some adverse states of nature, i.e. a systemic ?nancial crisis, provides a
form of insurance against such future episodes, which the market would otherwise
not provide. On the other hand, as ?rms adjust their leverage choices, the economy
becomes more exposed to the deadweight losses produced by aggregate ?nancial
distress.
Our answer to the question of whether it is desirable or undesirable to prohibit
the use of public funds to conduct bailouts is that public funds can in fact be used
e?ectively to conduct bailouts. The quantitative analysis shows that is possible to
design a realistic and simple bailout policy so that the welfare bene?ts of such an
intervention outweigh its distortionary e?ects. In particular, we ?nd that in the
presence of a severe systemic ?nancial crisis, it is optimal to engineer a transfer of
funds from workers to ?rms.
This paper relates to di?erent strands of literature. First, there is a growing
literature that studies credit policy in quantitative DSGE models, building on the
work of Bernanke and Gertler (1989) and Kiyotaki and Moore (1997).
1
As in these
papers, we study how government intervention can mitigate ?nancial frictions that
are activated during ?nancial crises. For reasons of tractability, however, this liter-
ature mostly studies the optimal response to unanticipated crises or focuses on the
log-linear dynamics around the deterministic steady state, thereby abstracting from
1
Contributions include Kiyotaki and Moore (2008), Gertler and Kiyotaki (2010),
Gertler and Karadi (2009), Del Negro, Eggertsson, Ferrero, and Kiyotaki (2010),
Guerrieri and Lorenzoni (2011), Gertler, Kiyotaki, and Queralto (2010).
110
risk considerations and the moral hazard e?ects of government intervention. Instead,
a distinctive feature of this paper is the focus on how expectations of future bailouts
a?ect ex-ante risk-taking within a non-linear DSGE model. This is crucial in order
to assess the dynamic implications of credit intervention on ?nancial stability and
on welfare.
2
This paper is also closely related to the theoretical literature that analyzes the
incentive e?ects of bailouts on ?nancial stability. Farhi and Tirole (2010) analyze
how time-consistent systemic bailouts can generate strategic complementarities in
private leverage choices, causing excessive ?nancial fragility. Chari and Kehoe (2010)
show that ?re sale e?ects provide governments with stronger incentives to renegotiate
contracts than private agents, making the time-inconsistency problem more severe
for the government. Diamond and Rajan (2009) show that raising interest rates may
be optimal to penalize excessive risk taking. There are two key di?erences between
our paper and this literature: ?rst, we develop a quantitative framework to assess
the e?ects of bailouts on risk-taking; second, we emphasize the idea that systemic
bailouts can be bene?cial ex-ante as a way to provide insurance. In this aspect,
this paper is related to Schneider and Tornell (2004) and Keister (2010) who also
emphasize the insurance bene?ts of bailouts, but they focus on self-ful?lling crises.
This paper is also related to a growing literature that studies the normative
implications of ?nancial frictions and the role of macroprudential regulation. This
2
Recently, Gertler, Kiyotaki, and Queralto (2010) develop a model where banks face a portfolio
choice between short-term debt and equity and also consider moral hazard e?ects of credit policy
in this portfolio choice. Gertler et al. consider a log-linear approximation around the risk-adjusted
steady state in an equilibrium where ?nancial constraints are always binding. An key structural
di?erence in our model is that we disentangle the role of policy during normal times from the role
of policy during times of systemic crisis by solving the model using global methods.
111
literature emphasizes the role of ex-ante prudential measures to correct pecuniary
externalities due to ?nancial accelerator e?ects.
3
Our paper focuses instead on ex-
post policy measures and their e?ects on ex-ante risk taking decisions.
4
3.2 Analytical Framework
Our model economy is a small open economy populated by ?rms and workers
that are also the shareholders of the ?rms. This model shares with Jermann and Quadrini
(2010) the consideration of dividend policy and with Mendoza (2010) the analysis
of non-linear dynamics. We start by ?rst describing the decisions of the di?erent
agents in the economy, and then we describe the general equilibrium.
3.2.1 Corporate entities
There is a measure one of identical ?rms with technology given by the produc-
tion function F(z
t
, k
t
, n
t
) that combines capital denoted by k, and labor denoted by
h to produce a ?nal good. TFP denoted by z
t
follows a ?rst-order Markov process.
Consistent with the typical timing convention, k
t
is chosen at time t ?1, and there-
fore, they are predetermined at time t. Instead, the input of labor h
t
can be ?exibly
changed in period t.
3
See for example Caballero and Krishnamurthy (2003), Lorenzoni (2008), Korinek (2009),
Bianchi (2010), Bianchi and Mendoza (2010) and Jeanne and Korinek (2010).
4
Benigno, Chen, Otrok, Rebucci, and Young (2010) also consider ex-post policy measures in
response to a pecuniary externality, but focus on policies that a?ect labor allocations as opposed
to policies that a?ect the availability of credit.
112
Capital evolves according to:
k
t+1
= k
t
(1 ??) + i
t
(3.1)
where i
t
is the level of investment and ?
t
is the depreciation rate. Capital accumu-
lation is subject to adjustment costs, given by ?(·).
Firms pay dividends, denoted by d
t
, and issue non-state contingent debt, de-
noted by b
t+1
. The ?ow of funds constraint for ?rms is then given by:
b
t
+ d
t
+ i
t
+ ?(k
t
, k
t+1
) ? F(z
t
, k
t
, n
t
) ?w
t
n
t
+
b
t+1
R
t
+ ?
t
(3.2)
where w
t
is the wage rate, R
t
is the gross interest rate determined in international
markets, and ?
t
is a transfer chosen by the government that will be speci?ed below.
Firms face two types of constraints on their ability to ?nance investment. First,
they are subject to a collateral constraint that limits the amount of borrowing to a
fraction of the value of their assets such that:
b
t+1
? ?
t
k
t+1
(3.3)
. This constraint is similar to those used in existing literature (see Kiyotaki and
Moore, 1997), and we interpret it as arising in an environment with limited en-
forcement between creditors and ?rms; ?
t
is a ?nancial shock and we interpret it,
following Jermann and Quadrini (2010), as a shock originating in the ?nancial sys-
tem.
113
In addition, at each period ?rms are required to pay a minimum amount of
dividends d ?
¯
d. A value of
¯
d ? 0 implies that the issuance of new shares is not
available. A special case is the restriction that dividends need to be non-negative.
This constraint captures the notion that dividend payments are required in order to
reduce agency frictions between shareholders and managers. In addition, it produces
dynamics in dividend payouts which are in line with empirical evidence (see Jermann
and Quadrini (2010)). We abstract, however, from an explicit microfoundation and
instead focus on its implications.
5
Denoting by s the vector of aggregate states, i.e. s = {K, B, ?, z}, the opti-
mization problem for ?rms can be written as:
V (k, b, s) = max
d,h,k
?
,b
?
d +Em
?
(s, s
?
)V (k
?
, b
?
, s
?
) (3.4)
s.t. b + d + k
?
+ ?(k, k
?
) ? (1 ??)k + F(z, k, h) ?wn +
b
?
R
+ ?
b
?
? ?k
?
d ?
¯
d
.
The function V (k, b, s) is the cum-dividend market value of the ?rm and m
?
is the stochastic discount factor, which will be equal in equilibrium to the ratio of
5
Endogeneizing such a constraint would require to model the divergence of interests between
shareholder and corporate manager in an environment with asymmetric information (see e.g. Mayer
1986).
114
marginal utility of household consumption.
The optimality condition for labor demand yields:
F
h
(z
t
, k
t
, h
t
) = w
t
(3.5)
.
There are also two Euler intertemporal conditions that relate the marginal
bene?t from distributing one unit of dividends today with the marginal bene?t of
investing in the available assets and distributing the resulting dividends in the next
period. Denoting by µ the multiplier associated with the borrowing constraint, ?
the multiplier associated with the dividend payout constraint, these Euler equations
are given by:
1 + ?
t
= R
t
E
t
m
t+1
(1 + ?
t+1
) + µ
t
(3.6)
(1+?
t
)(1+?
2
(t, t+1)) = E
t
m
t+1
[1 ?? + F
k
(z
t+1
, k
t+1
, n
t+1
) ??
1
(t + 1, t + 2)] (1+?
t+1
)+?
t
µ
t
(3.7)
.
In the absence of the ?nancial constraints on borrowing and dividend pay-
ments, the cost of raising equity (by reducing dividends), i.e. 1/E
t
m
t+1
, would be
equal to the cost of debt R
t
, and ?rms would be indi?erent at the margin between
equity and debt ?nancing. When the collateral constraint binds, there is a wedge
between the marginal bene?t of borrowing one more unit and distributing it as div-
115
idends in the current period and between the marginal cost of cutting dividends in
the next period to repay the debt increase. In addition, when the dividend payout
constraint binds, a positive wedge arises between the marginal bene?t from invest-
ing one more unit in capital or bonds relative to the marginal cost of cutting one
more unit of dividends. Condition (3.6) suggests also that a binding collateral con-
straint is associated with a binding dividend payout constraint. Intuitively, both
constraints impose a limit on a ?rm’s funding ability, which implies that a tighter
constraint on borrowing imposes pressure on the need to ?nance with equity, which
given that households have concave utility functions, this increases the cost of equity.
Similarly, a binding dividend payout constraint forces higher levels of borrowing for
given investment choices.
3.2.2 Households
There is a continuum of identical households of measure one that maximize:
E
0
?
?
t=0
?
t
u(c
t
?G(n
t
)) (3.8)
where c
t
is consumption, n
t
is labor supply, ? is the discount factor and G(·) is a
twice-continuously di?erentiable, increasing and convex function. The utility func-
tion u(·) has the constant-relative-risk-aversion (CRRA) form; the composite of the
utility function has the GHH form, eliminating wealth e?ects on the labor supply.
Households do not have access to bond markets and are the ?rms’ shareholders.
116
This yields the following budget constraint:
w
t
h
t
+ s
t
(d
t
+ p
t
) = s
t+1
p
t
+ c
t
+ T
t
(3.9)
wheres
t
represents the holdings of ?rm-shares, p
t
represents the price of ?rm-shares,
and T
t
is a lump sum tax to ?nance the cost of the bailout policy to be speci?ed
below.
The ?rst order conditions are given by:
w
t
= G
?
(n
t
) (3.10)
p
t
u
c
(t) = ?E
t
u
c
(t + 1)(1 +p
t+1
) (3.11)
.
The second condition determines the price of shares. Iterating forward on
(3.11) yields that for the ?rm optimization problem to be consistent with the house-
hold optimization problem, it must be the case that the stochastic discount factor
of ?rms is equal to m
t+j
= (?u
?
(c
t+j
?G
?
(n
t+j
)))/(u
?
(c
t+j?1
?G
?
(n
t+j?1
))).
3.2.3 Government
The function of the government is to set a lump sum tax on workers and to
transfer the proceeds to ?rms. A crucial element of the analysis is that transferring
resources to ?rms entails an e?ciency cost ?. Alternatively, this can be interpreted
as a distortionary e?ect of taxation. We assume that the government follows a
117
balanced budget such that:
T
t
= ?
t
(1 +?) (3.12)
3.2.4 Competitive equilibrium
We consider a competitive equilibrium for a small open economy where ?rms
borrow directly from abroad.
De?nition 5 A recursive competitive equilibrium is given by (i) ?rms’ policies
ˆ
d(k, b, s),
ˆ
h(k, b, s),
ˆ
k(k, b, s) and
ˆ
b(k, b, s); households’s policies ˆ s(s, s), ˆ n(s, s); ?rm’s
value V (k, b, s); prices w(s), p(s), m(s, s
?
); government policies ?(s), T(s); and a law
of motion of aggregate variables s
?
= ?(k, b, s) such that: (i) households solve their
optimization problem characterized by (3.9)-(3.11), (ii) ?rms’ policies and ?rms’
value solve (3.4), (iii) prices clear labor market (h
t
= n
t
), equity market (s
t
= 1),
and goods market ((1??)k
t
+F(z
t
, k
t
, n
t
)+b
t
= b
t+1
/R+?(k
t
, k
t+1
)+k
t+1
+c
t
), (iv)
the law of motion is ?(·) is consistent with individual policy functions and stochastic
processes for ? and z.
3.2.5 Some characterization
To illustrate the properties of the model and the e?ects of the bailout policy,
it is useful to consider a few special cases.
Proposition 3 If ?R < 1, the collateral constraint is binding and the dividend
payout constraint may or may not bind.
118
Proof : In a deterministic steady state, m
t
= 1 and (3.6) is simpli?ed to 1 = ?R+µ.
Since ?R < 1, this implies that µ > 0. Substituting the collateral constraint with
equality in the ?rm’s ?ow of funds constraint yields that the dividend constraint
binds if and only if
¯
d > F(¯ z,
¯
k,
¯
h) ?
¯
k(? + ¯ ?(R ? 1)/R) ? G
?
(
¯
h)
¯
h + ? where
¯
k
and
¯
h are the steady state values of capital and labor given by (3.7), (3.10), and
market clearing in labor markets. In general, in a stochastic steady state, these
?nancial constraints may or may not bind depending primarily on the magnitude of
the shocks, ?R, and the level of risk aversion.
Proposition 4 Suppose (i)
¯
d = ??, and (ii) households have unrestricted access
to international credit markets by borrowing and saving at the interest rate R (unlike
the baseline model). Then the competitive equilibrium is una?ected by ?nancial
shocks.
Proof : From a household’s ?rst order condition, it must be the case that REm
?
= 1.
Using the ?rm’s ?rst order condition REm
?
+µ = 1 yields µ = 0. Since the collateral
constraint never binds, the model becomes a standard (RBC) model where the
?nancial structure is independent on the real side.
Remark 6 Suppose (i)
¯
d = ??, and (ii) households do not have access to inter-
national credit markets. Then the competitive equilibrium is a?ected by ?nancial
shocks.
This remark becomes obvious once we observe that if output is su?ciently low at
a given period, households will demand positive dividends to smooth consumption,
119
which implies that ?rms would need to increase their debt position. Therefore, for
low values of ?, the collateral constraint becomes binding.
3.3 Bailout Policy
3.3.1 A second best benchmark
To illustrate the role of bailouts, we start by considering a social planner that
(i) directly chooses all allocations and (ii) is subject to the collateral constraint, but
not the dividend payout constraint.
This problem can be written recursively as:
V (s) = max
d,n,k
?
,b
?
u(c ?G(h)) + ?EV (s
?
) (3.13)
s.t. (1 ??)k + F(z, k, n) ?wn +
b
?
R
? b + d + k
?
+ ?(k, k
?
)
b
?
? ?k
?
.
We now highlight the main results regarding the implementability of this con-
strained social planner’s problem employing bailouts.
Remark 7 If ? = 0, the second best allocations can be implemented by an appro-
priate state contingent bailout policy.
Intuitively, the planner can use cost-free transfers as a substitute for lower
120
dividend payments when the dividend payout constraint becomes binding.
Proposition 5 If
¯
d = ?? and ? = 0, the competitive equilibrium is una?ected by
the bailout policy.
Proof The proof follows simply by noting that the transfers cancel out within the
conditions characterizing the competitive equilibrium.
3.3.2 Policy Experiment
We now consider the general case where the government faces strictly positive
e?ciency costs from transferring resources from ?rms and households. Under these
conditions, the second best allocations cannot be achieved: bailouts introduce a
trade o? between relaxing balance sheet constraints of ?rms and e?ciency costs
associated with the transfer.
The government function is limited to taxing workers and transferring those
resources to ?rms in the same period.
6
Speci?cally, we assume that the government
can commit to following a bailout policy rule ? such that ? = ?(
.
) where ?(
.
) follows
a parametric function that depends on the relevant state variables.
The objective of the government/planner is to choose ?(
.
) to maximize
?
s
0
_
E
?
?
t=0
?
t
u(c
bp
t
?G(n
bp
t
)) |s
0
_
?(s
0
) (3.14)
subject to all allocations and prices being a competitive equilibrium for the speci?ed
6
That is, we rule out the use of the government as a substitute for private credit. Allowing for
government debt is likely to take the economy closer to a ?rst best if there are no other frictions
(e.g. sovereign default risk).
121
bailout policy, here ? represents the joint ergodic distribution of all aggregate states
in the competitive equilibrium without bailout. That is, the policy rule is chosen to
maximize the expected life-time utility from switching to the competitive equilibrium
without intervention to the competitive equilibrium with a bailout policy. This
welfare measure considers explicitly the e?ects associated with the transition from
the stochastic steady state without bailout policy to the stochastic steady state with
bailout policy.
This policy can be interpreted as a form of unconventional monetary policy
(see e.g. Gertler and Karadi, 2010), as one can interpret the recent intervention of
the Federal Reserve in credit markets as a way to facilitate credit to the corporate
sector, given the strains in ?nancial intermediaries.
7
In the quantitative analysis we
search for parametrization of these rules that are simple and realistic such and that
the rules maximize expected life-time utility.
A key feature of this bailout policy is that bailouts are non-targeted, i.e. they
apply to all market participants. That is, even if an individual agent is not under
?nancial distress, it receives a bailout when the overall economy is under distress.
This is akin to an interest rate policy (see Farhi and Tirole, 2010) in which even
?rms that are not under distress can re?nance at a low interest rate when the
7
In practice, the Federal Reserve and the Treasury implemented a variety of policies with
the aim of facilitating credit to the corporate sector including direct lending, debt guarantees,
and equity injections in the banking sector. To simplify the analysis, we do not model ?nancial
intermediaries and capture this class of interventions in a crude way by modeling a direct transfer
from workers to ?rms. What is crucial for our analysis is that this intervention relaxes balance
sheets across the economy and mitigates the fall in credit and investment that occurs during crises.
In our setup, absent of information asymmetries, these policies are likely to yield similar outcomes.
See Philippon and Schnabl (2009) for an evaluation of optimal rescue packages in the context of a
debt overhang problem using a mechanism design approach.
122
central bank conducts an expansionary monetary policy. This introduces strategic
complementarities in ?rms’ ?nancing decisions, as individual agents have more of an
incentive to take a signi?cant amount of risk when other ?rms in the economy are also
taking a large amount of risk. On the other hand, conditioning bailouts on aggregate
variables may also work to mitigate the amount of risk taking, as individual agents
obtain a bailout only when the whole economy comes under ?nancial distress.
8
3.4 Quantitative Analysis
The numerical solution to the model involves several challenges. First, there
are the well-known complications of non-linearities introduced by the absence of
complete markets and in particular the occasionally binding collateral constraint.
Moreover, the state variables in the model are not con?ned to a narrow region of
the state space. We approximate the equilibrium functions over the entire state
space and check that equilibrium conditions are satis?ed at all grid points, allowing
for the two ?nancial constraints to bind only in some states of nature.
9
The
introduction of bailouts introduces an additional complication, which we handle
using a nested ?xed-point algorithm. First, for a given bailout policy, we compute
the implied competitive equilibrium (inner loop). Second, we update the bailout
policy accordingly (outer loop). These two procedures are followed until the two
8
O? equilibrium, while all ?rms receive the same amount of funds in case of a systemic crisis,
the marginal value of those funds depends on how leveraged they are; i.e., those with a tighter
?nancial constraint assign a higher value to the bailout.
9
An additional complication in our setup is that variations in consumption can lead to large
changes in the value of the ?rm, which requires a slow adjustment in the update of the consumption
function along the iterations. Further details are provided in an appendix upon request.
123
loops converge.
3.4.1 Calibration and Functional Forms
We calibrate the model to an annual frequency using data from the U.S. econ-
omy. The functional forms for preferences and technology are the following:
u(c ?G) =
_
c ??
n
1+?
1+?
_
1??
?1
1 ??
? > 0, ? > 1
F(z, k, h) = zk
?
h
1??
?(k
t
, k
t+1
) =
?
k
2
_
k
t+1
?k
t
k
t
_
2
k
t
For a ?rst group of parameters given by ?, ?, ?, ?, we choose values that we
see as reasonably conventional in the literature. The capital share ? is set to 0.32;
the depreciation rate is set at 8 percent; the risk aversion ? is set to 2; and ? is set
so that the Frisch elasticity of labor supply is equal to 1. We normalize the labor
disutility coe?cient ? and the average value of the TFP shock so that employment
and output are equal to one in the deterministic steady state.
The TFP shock and the ?nancial shock are modeled as independent stochastic
processes. Each of these shocks is discretized using a two-state Markov chain. The
TFP shock is constructed using a symmetric simple persistence rule. The realization
of the shock and the persistence of the TFP shock are chosen so that in the model,
?uctuations in GDP are roughly consistent with those in the data. This yields TFP
realizations of plus/minus 1 percent and a probability of remaining in the same state
124
of 78 percent. The calibration of the ?nancial shock is meant to be suggestive. This
shock follows an asymmetric process such that during ”bad times”, the pledgability
of capital falls by 20 percent; this event occurs 3 percent of the time and lasts on
average for two years.
The capital adjustment cost parameter ?
k
is set so that the standard deviation
of investment in the competitive equilibrium without bailouts roughly matches the
standard deviation of investment in the data. This yields ?
k
= 7.8. The remaining
parameters are the discount factor ?, the mean value of ?, and the minimum div-
idend payment
¯
d. These parameters govern the values of leverage as well as how
frequent the constraints on borrowing and dividend bind. For now, we set ? = 0.92,
the mean value of ? = 0.32 and
¯
d = 0.04.
10
With these values, the mean value of
leverage-de?ned as the ratio of debt to market value of the ?rm-equals 0.29, which
is in the upper range of corporate leverage documented by Masulis (1988); the crisis
dynamics are also roughly in line with US data.
3.4.2 Results
Figures 1 and 2 show the laws of motion for capital and debt in the competitive
equilibrium with and without bailout policy. The x-axis in the two ?gures is given
by the current level of debt. (The level of capital is approximately the average value
of capital, and the shocks are given by adverse TFP and adverse ?nancial shocks).
Let us ?rst describe the behavior of the competitive equilibrium without bailouts
10
To facilitate convergence, for the time being, we use an endogenous discount factor without
internalization (see Schmitt-Grohe and Uribe (2003)), such that on average ? = ln(1 +c)
?
= 0.96,
yielding ? = 0.10.
125
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
2.8
2.85
2.9
2.95
3
3.05
3.1
3.15
3.2
Current Debt
F
u
t
u
r
e
C
a
p
i
t
a
l
No Bailout Policy
Bailout Policy
Collateral constraint binding Dividend constraint binding
Figure 3.1: Law of Motion for Capital
before analyzing the e?ects of credit intervention.
As Figure 1 shows, an increase in debt holdings reduces the demand for capital,
since capital is risky and the valuation of future dividends becomes more sensitive
to adverse shocks. Since borrowing ability depends on the next period capital, this
produces a non-monotonic law of motion for debt, as shown in Figure 2. For low
values of current debt, the collateral constraint is not binding. As the value of current
debt increases, the demand for debt increases; but as investment is reduced, the
ability to borrow shrinks. When the current level of debt is about 0.84, the collateral
constraint becomes binding. This is indicated in the ?rst vertical in Figures 1 and 2.
126
For b > 0.84, the next period debt holdings decrease in current debt holdings as the
level of capital and debt are reduced in an endogenous feedback loop. The second
vertical line indicates when the dividend payout constraint becomes binding. Notice
that both investment and the level of future debt are reduced more sharply once the
dividend payout constraint binds. This is intuitive, since a binding dividend payout
constraint limits even more the access to ?nancing.
0.8 0.85 0.9 0.95
0.8
0.81
0.82
0.83
0.84
0.85
0.86
0.87
Current Debt
F
u
t
u
r
e
D
e
b
t
No Bailout Policy
Bailout Policy
Collateral constraint binding Dividend constraint binding
Figure 3.2: Law of Motion for Debt
For now, we consider bailouts that are strictly positive if and only if the
dividend payout constraint is binding and it is increasing in the shadow interest
rate, i.e. the interest rate that will make the collateral constraint hold with equality
127
but not bind at the margin, given prices and allocations.
11
. This makes the bailout
increase in the level of debt and decrease in the level of capital.
To understand how bailouts a?ect the competitive equilibrium, it is useful
to analyze ?rst its e?ects during periods in which the dividend payout constraint
becomes binding. As Figure 2 shows, bailouts allow ?rms to borrow more during
these periods. This occurs because as ?rms receive the bailout transfer, they can
allocate these funds to invest more in capital, which boosts the ?rms’ capacity to
borrow. In the region where the dividend constraint is not binding, ?rms also
borrow more in the competitive equilibrium with bailouts. This occurs because
given that crises become less severe as a result of bailouts, there is a lower incentive
to accumulate precautionary savings during normal times. This leads in turn to a
higher probability of the economy becoming ?nancially constrained in the future.
12
3.4.3 Discussion
In our benchmark model, a bailout from workers to ?rms does not involve
any wealth redistribution since workers are the owners of ?rms. This allows us to
focus on the bailout’s consequences for e?ciency. We can also extend our analysis
to allow bailouts to have wealth redistribution e?ects. In particular, consider a
model where ?rms are owned by entrepreneurs and workers do not hold any shares
of the ?rms. In this context, while workers face a negative wealth e?ect as a result
of carrying the burden of the bailout, there are labor-market spillovers that might
11
Formally, the shadow interest rate is de?ned as R
ef
= (1 + ?
t
)/E
t
m
t+1
(1 + ?
t+1
)
12
The speci?ed bailout policy increases welfare by about 0.1 percentage point of permanent
consumption (please check future updates of the paper for a complete welfare analysis).
128
raise their welfare. In particular, as ?rms do not reduce investment as much during a
?nancial crisis, this leads to higher wages in the recovery of the crisis. For plausible
calibrations, however, the welfare of workers is reduced by bailouts. Intuitively, this
results from capital not increasing enough to make the increase in wages compensate
for the direct cost of ?nancing the bailout.
In our setup, there is also scope for ex-ante prudential measures. The reason is
a form of overborrowing externality. While committing to a bailout in some states of
nature is desirable, these interventions also impose costs that are paid by all workers.
Not internalizing that becoming ?nancially constrained during a systemic ?nancial
crisis triggers a costly intervention by the planner, ?rms borrow too much. Notice
that while ?rms bene?t from other ?rms being constrained due to the systemic
nature of the bailout, this constitutes only a private gain, as the bailout imposes
costs at the social level. A similar point is also made by Chari and Kehoe (2009)
and Farhi and Tirole (2010). In their setup, regulation is designed to reduce the
temptation to conduct bailouts. In our setup, however, regulation also improves the
commitment solution.
We should point out that the possibility to improve welfare in our setup de-
pends on the ability of the government to redirect funds from workers to ?nancially
constrained ?rms. Workers do not have the incentive to transfer these funds to
?rms because this entails only costs at the individual level. Ex ante, the ratio-
nale for committing to such interventions is to provide a form of systemic insurance
against ?nancial crises. This result also points towards the desirability of enhancing
the development of private insurance markets. Clearly, however, there are reasons
129
why the availability of private insurance against systemic episodes is limited by a
host of credit market imperfections (e.g., insurers may go bankrupt in crises). Under
these conditions, our analysis suggests that the government should have a direct role
in providing insurance against systemic ?nancial crises.
Another point that we should emphasize is that we have assumed that the
planner can commit to a speci?c bailout policy. This is motivated by debates about
how to specify a legal framework to put limits on the ability of central banks to bail
out the ?nancial sector (see e.g. the Dodd-Frank Act). Under some states of nature,
however, it might still be feasible for central banks to evade the legal framework. It
would be interesting in our framework to study how the belief that central banks
would deviate from previous commitments could generate an incentive for private
agents to take even more risk as compared to an environment where the planner can
commit to future policies.
3.5 Conclusion
The quantitative analysis shows that it is possible to design a realistic and
simple bailout policy so that the welfare bene?ts of such an intervention outweigh
its distortionary e?ects. A key feature of this intervention is that it is reserved for
extreme episodes of ?nancial distress.
Our results are relevant for ongoing debates about the appropriate role of
central banks during ?nancial crises. Our analysis points towards giving a speci?c
mandate of intervention that supports credit ?ows, albeit in a strictly limited way.
130
Re?ning our analysis would require us to consider explicitly the temptation of central
banks to renege on policy commitments. Within our framework, it would also be
interesting to study how certain policies like capital requirements can help to o?set
these problems of credibility
131
Appendix A
Proofs (Chapter 1)
A.1. Proof of Proposition 1 (Constrained Ine?ciency)
This is a proof by contradiction. Suppose the decentralized equilibrium yields
the same allocations as the constrained-e?cient allocations. Then, we can combine
(1.4) and (1.12), yielding:
?
de
t
= ?
sp
t
+ µ
sp
t
?
t
(A.1)
where we denote with superscript ‘sp’ the Lagrange multipliers of the social
planner and with ‘de’ those of the decentralized equilibrium. Updating this equation
one period forward and taking conditional expectations at time t:
E
t
?
de
t+1
= E
t
?
sp
t+1
+E
t
µ
sp
t+1
?
t+1
(A.2)
Suppose that at time
˜
t, b
˜
t+1
> ?
_
?
N
p
˜
t
y
N
˜
t
+ ?
T
y
T
˜
t
_
. Combining (1.6),(1.7),(1.13),
and (1.14) we obtain:
E
˜
t
?
de
˜
t+1
= E
˜
t
?
sp
˜
t+1
(A.3)
If at time
˜
t+1 the credit constraint binds with positive probability, comparing
(A.3) and (A.2) yields a contradiction.
132
A.2. Proof of Proposition 2 (Optimal tax on debt)
This a proof by construction. Combining the optimality conditions for the social
planner (1.12) and (1.13) yields:
u
T
(t) = ?(1 + r)E
t
(u
T
(t + 1) + µ
sp
t+1
?
t+1
) + µ
sp
t
(1 ??
t
) (A.4)
First, notice that the constrained-e?cient allocations are characterized by
stochastic sequences
_
c
T
t
, c
N
t
, b
t+1
, p
N
t
, µ
sp
t
_
t?0
such that the following conditions
hold: (1.5),(1.8),(1.9),(1.14), (A.4) and µ
sp
t
? 0.
Second, the decentralized equilibrium allocations with taxes on debt are char-
acterized by stochastic sequences
_
c
T
t
, c
N
t
, b
t+1
, p
N
t
, µ
t
, ?
t
, T
t
_
t?0
such that the follow-
ing conditions hold: (1.5),(1.7),(1.8),(1.9),(1.17), T
t
= b
t
(1 +r)?
t?1
and µ
t
? 0.
De?ning the tax as ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1))?(µ
sp
t
?
t
) / (?(1 + r)E
t
u
T
(t + 1))
and redistributing the proceeds lump sum yields that the conditions characterizing
the decentralized equilibrium with the speci?ed tax on debt are identical to those
characterizing the constrained-e?cient allocations.
Appendix B
An Equivalence Result (Chapter 1)
We show in this appendix that the constrained-e?cient allocations can be de-
centralized with regulatory measures directed to the banking sector. Consider the
following simple model. Banks make loans to households at rate r
L
and impose the
constraint (1.2) to guarantee repayment. Banks ?nance these loans by accepting
133
deposits from the rest of the world at rate r and issuing equity in the domestic
markets. We assume that the required return on equity r
e
is higher than the rate
on deposits, i.e., r
e
> r. This could be the outcome of moral hazard or tax dis-
advantages on equity, but we abstract from explicitly modeling this relationship.
Financial intermediation is costless. Banks last for one period, and every period
new banks are set up with free entry into banking.
Without any regulation or any other frictions, banks would ?nance loans only
with deposits, and the resulting equilibrium would be equivalent to the decentralized
equilibrium. We introduce two regulatory measures. First, the planner imposes
capital requirements: banks are required to ?nance a fraction ? of their assets with
equity. Second, the planner imposes reserve requirements: banks are required to
hold a fraction ? of deposits in the form of unremunerated reserve. Thus the banks’
balance sheets become:
Assets Liabilities
b Loans d Deposits
f Reserve requirements e Equity
The objective of the bank is static and consists of maximizing shareholder
value, net of the initial equity investment:
max
b,f,e,d
b(1 +r
L
) + f ?d(1 + r) ?e(1 + r
e
)
134
subject to
f ? d + e
f ? ?d
e ? ?(b + f)
Given that holding reserves and capital is privately costly, banks do not hold excess
reserves or excess capital. In equilibrium, the return from assets must be equal to
the return on liabilities, i.e., r
L
(1 ??(1 ??)) = ?r
e
+(1 ??)r. Therefore, by setting
(?
t
, ?
t
) such that (1 +r
L
t
) = (1 +r)(1 +?
?
t
), the social planner can raise the cost of
borrowing and induce agents to hold the socially optimal amount of debt. Assuming
only capital requirements are used yields: ?
?
t
= (?
?
t
(1 + r))/(r
e
? r). When only
reserve requirements are used, this yields: ?
?
t
= (?
?
t
(1 + r))/(r(1 + ?
?
t
) + ?
?
t
).
Appendix C
Sensitivity Analysis (Chapter 1)
We continue here the sensitivity analysis presented in the body of the paper.
We discuss separately the e?ects of varying each of the parameter values of the
model, using the analysis of the externality term and the elasticity decomposition
studied in the body of the paper. The main quantitative results of all experiments
are shown in Table 4. The table shows for each experiment the average welfare
loss, the average implied tax on debt, the relative volatility of consumption, the
probability of a ?nancial crisis for the decentralized equilibrium and constrained-
e?cient allocations, and the e?ects of a median crisis in consumption, the real
135
exchange rate and the current account for the two equilibria.
Discount Factor (?).— An increase in the discount factor leads to a shift of
the distribution of bond holdings towards a lower amount of debt, leading to less
frequent binding constraints and causing the distribution of the externality term to
concentrate higher probability in a region where its value is zero. This e?ect leads
to smaller e?ects from the externality. There is an opposite e?ect from an increase
in the discount factor. Recall that the maximum welfare gains from the externality
arise in relatively tranquil times because of the reduction in future vulnerability to
?nancial crises. Hence, a higher discount factor makes the economy value relatively
more the bene?ts from a reduction in future variability, which should lead to higher
welfare e?ects from correcting the externality. Quantitatively, we ?nd that the
?rst e?ect is more important. Increasing the discount factor by 0.02 reduces the
average implied tax on debt to 3.3 percent, although large di?erences remain in the
probability of ?nancial crises: the probability of a crisis is 0.2 percent for the social
planner and 4.1 percent for the decentralized equilibrium.
Interest Rate (r).— An increase in the interest rate has e?ects similar to an
increase in the discount factor since both reduce the willingness to borrow and shifts
the economy away from binding constraints. For a given amount of debt, however,
a higher interest rate implies an increase in the debt service, which causes a larger
depreciation of the real exchange rate. Quantitatively, we ?nd that increasing the
interest rate 100bps reduces the implied tax on debt from 5.2 percent to 4.4 percent,
but the e?ects on the incidence and severity of ?nancial crises remain very similar.
136
Risk Aversion (?).— An increase in the risk aversion implies a higher disu-
tility from consumption variability. This implies that a large drop in consumption
generates a higher shadow value from relaxing the credit constraint at a given state
where the constraint binds; therefore, this yields a higher externality term. At the
same time, an increase in risk aversion makes both the social planner and private
agents accumulate more precautionary savings making the constraint less likely to
bind and shifting away the distribution of the externality term towards zero. Quan-
titatively, as shown in Table 4, we ?nd that the e?ects of the externality decreases
(increases) modestly when we consider ? = 5 (? = 1).
Independent shocks.— We model tradable and nontradable endowment shocks
as independent AR(1) processes and analyze the e?ects over the externality. When
shocks are correlated, both tradable and nontradable shocks typically fall during ?-
nancial crises. The fact that nontradables fall, however, mitigates the fall in the price
of nontradables and the tightening of ?nancial constraints. This channel suggests
that making the two shocks independent should reduce the e?ects of the externality.
There is another channel, however, by which making the shocks independent causes
the externality to have higher e?ects. For the baseline calibration, the risk aver-
sion and the elasticity of substitution between tradables and nontradables are such
that tradable and nontradable goods are Edgeworth substitutes. As a result, a fall
in the endowment of nontradables when the credit constraint binds, increases the
marginal utility from tradable consumption, which increases the desire to borrow
and increases the shadow value from relaxing the credit constraint. Quantitatively,
137
we ?nd that the e?ects over the shadow value from relaxing the credit constraint
are stronger than those a?ecting the price e?ects, so that the di?erences in severity
of ?nancial crises become even stronger.
Volatility and Persistance (Cov(?)).— An increase in the volatility of endow-
ment shocks increases the severity of ?nancial crises, in terms of the ampli?ca-
tion e?ects and the disutility cost from a binding constraint. This e?ect increases
the externality term. At the same time, private agents have an incentive to in-
crease relatively more precautionary savings in response to the increase in volatility.
This occurs because the concavity of the utility function implies that a given in-
crease in variability is more costly in the decentralized equilibrium compared to the
constrained-e?cient allocations. In fact, when we vary simultaneously the volatility
of the shocks to the endowment processes by 15 percent, we ?nd that the externality
decreases modestly with a higher volatility.
An increase in persistence leads to a higher probability of ?nancial crises for a
given level of precautionary savings although it does not alter the size of the shocks
and the severity of ?nancial crises. When we vary the autocorrelation of the endow-
ment shocks by 15 percent, we ?nd that a higher autocorrelation is associated with
larger e?ects from the externality. In fact, the experiment with higher autocorre-
lation yields larger di?erences in the incidence and severity of ?nancial crises, and
this leads to larger welfare e?ects.
Elasticity of Substitution(1/(1+?)).— As explained in the paper, the elasticity
of substitution between tradables and nontradables determines the debt service elas-
138
ticity of the real exchange rate, which is in turn a key component of the externality
term. Moreover, the elasticity of substitution also a?ects the incentive to accumu-
late precautionary savings: the lower the elasticity of substitution the higher the
disutility from drops in consumption during ?nancial crises. This second channel is
similar to the increase in the risk aversion, but we ?nd that the channel a?ecting
directly the price e?ects are quantitatively more important.
Share of tradables (?).— As explained in the paper, the weight of tradables
in the utility function determines the borrowing limit elasticity of the real exchange
rate and is key for the e?ects on the externality. There is another e?ect of this
parameter. A higher share of tradables in the utility function implies that large
drops in tradables consumption during ?nancial crises are more costly, causing an
increase in precautionary savings. As explained before, this second channel becomes
qualitatively ambiguous, but we ?nd that the price e?ects, which unambiguously
increase the externality, are more signi?cant.
Credit Coe?cient (?).— We set ?
T
= ?
N
= ?. An increase in ? has two e?ects.
First, it increases directly the externality term, because for a given drop in the price
of nontradables the e?ects over the borrowing ability are directly proportional to ?.
Second, it makes the constraint less likely to bind, hence reducing the e?ects of the
externality. On one hand, when ? is 0, there is no borrowing; therefore an increase
in ? raises the e?ects of the externality. On the other hand, for a very large ?, the
credit constraint never binds and there are no e?ects from the externality in the long
run. Quantitatively, we ?nd that increasing ? from 0.32 to 0.36 increases the welfare
139
e?ects of the externality to 0.22 percentage points of permanent consumption. In
addition, consumption during a median crisis drops almost three times as much
in the decentralized equilibrium compared to the constrained-e?cient equilibrium.
Reducing ? to 0.28 reduces also slightly the e?ects of the externality but crises in
the decentralized equilibrium remain ten times more likely than in the constrained-
e?cient equilibrium.
140
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D
P
.
Appendix D
Numerical Solution Method (chapter 1)
The computation of the competitive equilibrium requires solving for functions
B(b, y), P
N
(b, y),C
T
(b, y) such that:
P
N
(b, y) =
_
1 ??
?
__
C
T
(b, y)
y
N
_
?+1
(D.1)
u
T
(C
T
(b, y), y
N
) ? ?(1 + r)E
y
?
/y
u
T
(C
T
(B(b, y), y
?
), y
N?
) (D.2)
B(b, y) ? ?
_
?
N
P
N
(b, y)y
N
+ ?
T
y
T
_
with = if (25) holds with strict inequality
(D.3)
B(b, y) +C
T
(b, y) = b(1 + r) + y
T
(D.4)
where u
T
(C
T
(b, y), y
N
) = u
C
(C(b, y))C
T
(b, y), C(b, y) =
_
?
_
C
T
(b, y)
_
??
+ (1 ??)
_
y
N
_
??
_
?
1
?
and y = (y
T
, y
N
).
The algorithm employed to solve for the competitive equilibrium is based on
the time iteration algorithm modi?ed to address the occasionally binding endogenous
constraint. The algorithm follows these steps:
1
1. Generate a discrete grid for the economy’s bond position G
b
= {b
1,
b
2
, ...b
M
}
and the shock state space G
Y
= {y
1,
y
2
, ...y
N
} and choose an interpolation
scheme for evaluating the functions outside the grid of bonds. We use 800
1
For the social planner’s allocations, we use a standard value function iteration algorithm.
142
points in the grid for bonds and interpolate the functions using a piecewise
linear approximation.
2. Conjecture P
N
K
(b, y), B
K
(b, y), C
T
K
(b, y) at time K ?b ? G
b
and ?y ? G
Y
.
3. Set j = 1
4. Solve for the values of P
N
K?j
(b, y), B
K?j
(b, y), C
T
K?j
(b, y) at time K ? j using
(D.1),(D.2),(D.3),(D.4) and B
K?j+1
(b, y), P
N
K?j+1
(b, y), C
T
K?j+1
(b, y), ?b ? G
b
and ?y ? G
Y
:
(a) Set B
K?j
(b, y) = ?
_
?
N
P
N
K?j+1
(b, y)y
N
+ ?
T
y
T
_
and compute C
T
K?j
(b, y)
from (D.4)
(b) Compute
U = u
T
(C
T
K?j
(b, y), y
N
) ??(1 +r)E
y
?
/y
u
T
(C
T
K?j
(B
K?j
(b, y), y
?
), y
N
?
)
(c) If U > 0, the credit constraint binds; move to (e).
(d) Solve for B
K?j
(b, y), C
T
K?j
(b, y) using (D.2) and (D.4) with a root ?nding
algorithm.
(e) Set P
N
K?j
(b, y) =
_
1??
?
_
_
C
T
K?j
(b,y)
y
N
_
?+1
5. Evaluate convergence. If sup
b?G
b
,y?G
Y
?x
K?j
(b, y) ? x
K?j+1
(b, y)? < ? for
x = B, C
T
, P
N
we have found the competitive equilibrium. Otherwise, set
x
K?j
(b, y) = ?x
K?j
(b, y) + (1 ? ?)x
K?j+1
(b, y) and j ; j + 1 and go to step
4. We use values of ? close to 1.
143
Appendix E
Numerical Solution Method (chapter 2)
The computation of the competitive equilibrium requires solving for functions
B(b, ?), q(b, ?), C(b, ?), N(b, ?), µ(b, ?) such that:
C(b, ?) +
B(b, ?)
R
= ?F(K, N(b, ?)) + b (E.1)
?
B(b, ?)
R
+ ?G
?
(N(b, ?))N(b, ?) ? ?q(b, ?)K (E.2)
u
?
(t) = ?RE
?
?
/?
_
u
?
(C(B(b, ?), ?
?
))
_
+ µ(b, ?) (E.3)
?F
n
(K, N(b, ?)) = G
?
(N(b, ?))N(b, ?)(1 +?µ(b, ?)/u
?
(C(b, ?)) (E.4)
q(b, ?) =
?E
?
?
/?
_
u
?
(c(B(b, ?), ?
?
))?
?
F
k
(K, N(B(b, ?), ?
?
)) + q(B(b, ?), ?
?
)
¸
(u
?
(C(b, ?) ) ?µ(b, ?)?)
(E.5)
We solve the model using a time iteration algorithm developed by Coleman
(1990) modi?ed to address the occasionally binding endogenous constraint. The
algorithm follows these steps:
1
1. Generate a discrete grid for the economy’s bond position G
b
= {b
1,
b
2
, ...b
M
}
and the shock state space G
?
= {?
1,
?
2
, ...?
N
} and choose an interpolation
scheme for evaluating the functions outside the grid of bonds. We use 300
1
For the social planner’s allocations, we use the same algorithm operating on the planner’s
optimality conditions.
144
points in the grid for bonds and interpolate the functions using a piecewise
linear approximation.
2. Conjecture B
K
(b, ?), q
K
(b, ?), C
K
(b, ?), N
K
(b, ?), µ
K
(b, ?) at time K ? b ? G
b
and ? ? ? G
?
.
3. Set j = 1
4. Solve for the values of B
K?j
(b, ?), q
K?j
(b, ?), C
K?j
(b, ?), N
K?j
(b, ?), µ
K?j
(b, ?)
at time K?j using (E.1),(E.2),(E.3),(E.4), (E.5) and B
K?j+1
(b, ?), q
K?j+1
(b, ?), C
K?j+1
(b, ?),
N
K?j+1
(b, ?), µ
K?j+1
(b, ?)? b ? G
b
and ? Y ? G
Y
:
(a) Assume collateral constraint (E.2) is not binding. Set µ
K?j
(b, ?) = 0 and
solve for N
K?j
(b, ?) using (E.4). Solve for B
K?j
(b, ?) and C
K?j
(b, ?) using
(E.1) and(E.3) and a root ?nding algorithm.
(b) Check whether ?
B
K?j
(b,?)
R
+ ?G
?
(N(b, ?))N
K?j
(b, ?) ? ?q
K?J+1
(b, ?)K
holds.
(c) If constraint is satis?ed, move to next grid point.
(d) Otherwise, solve for µ(b, ?), N
K?J
(b, ?), B
K?j
(b, ?) using (E.2), (E.3) and(E.4)
with equality.
(e) Solve for q
N
K?j
(b, ?) using (E.5)
5. Evaluate convergence. If sup
B,?
?x
K?j
(B, ?)?x
K?j+1
(B, ?)? < ? for x =B, C, q, µ, N
we have found the competitive equilibrium. Otherwise, set x
K?j
(B, ?) =
x
K?j+1
(B, ?) and j ;j + 1 and go to step 4.
145
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doc_519888650.pdf
Many economists have offered theories about how financial crises develop and how they could be prevented.
ABSTRACT
Title of dissertation: ESSAYS ON FINANCIAL CRISES AND
FINANCIAL REGULATION
Javier Ignacio Bianchi Caporale,
Doctor of Philosophy, 2011
Dissertation directed by: Enrique G. Mendoza
Department of Economics
This dissertation studies the optimal regulatory response to ?nancial crises.
The ?rst two chapters focus on prevention of ?nancial crises, and the third chapter
focuses on resolution of ?nancial crises.
Chapter 1 develops a quantitative theory of overborrowing based on a systemic
risk externality in an emerging market economy. In the model, debt denominated
in foreign currency and balance sheet constraints cause depreciations of the real
exchange rate to be contractionary. The externality arises because when private
agents take debt in good times, they do not internalize that during bad times, the
reduction in demand for consumption causes a higher depreciation of the real ex-
change rate and a further tightening of balance sheet constraints across the economy.
The quantitative analysis suggests that there is an important role for policies that
“throw sand in the wheels of international ?nance.”
Chapter 2 analyzes an externality that arises because of a feedback loop be-
tween asset prices and collateral constraints in a dynamic stochastic general equilib-
rium model calibrated to US data. In the model, a collateral constraint limits private
agents not to borrow more than a fraction of the market value of their collateral
assets, which take the form of an asset in ?xed aggregate supply (e.g. land). When
the collateral constraint binds, ?re-sales of assets cause a Fisherian debt-de?ation
spiral that causes asset prices to decline and the economy’s borrowing ability to
shrink in an endogenous feedback loop. The externality produces deeper recessions
and a larger collapse in asset prices compared to the constrained e?cient allocations.
Chapter 3 studies the macroeconomic and welfare e?ects of government in-
tervention in credit markets during ?nancial crises. A DSGE model to assess the
interaction between ex-post interventions in credit markets and the build-up of risk
ex ante is developed. During a systemic crisis, the central bank ?nds it bene?cial to
bail out the ?nancial sector to relax balance sheet constraints across the economy.
Ex ante, this leads to an increase in risk-taking, making the economy more vulnera-
ble to a ?nancial crisis. We ask whether the central bank should commit to avoiding
a bailout of the ?nancial sector during a systemic crisis. We ?nd that bailouts can
improve welfare by providing insurance against systemic ?nancial crises.
Essays on Financial Crises and Financial Regulation
by
Javier Ignacio Bianchi Caporale
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial ful?llment
of the requirements for the degree of
Doctor of Philosophy
2011
Advisory Committee:
Professor Enrique G. Mendoza (Chair)
Professor Anton Korinek
Professor Lemma Senbet
Professor John Shea
Professor Carlos Vegh
c ? Copyright by
Javier Ignacio Bianchi Caporale
2011
Dedication
To Anita and Francesca.
ii
Acknowledgments
This dissertation could not have been possible without the advice and encour-
agement of my advisors at the University of Maryland.
First and foremost I’d like to thank my advisor, Enrique Mendoza, for his
endless support, encouragement and guidance. He has always made himself available
for help and advice and there has never been an occasion when I’ve knocked on his
door and he hasn’t given me time. He has shown me, by way of example, what an
outstanding academic (and person) should be. The second chapter of my dissertation
emanates from a joint project. It has been a pleasure to work with him and I hope
we continue collaborating on future research.
My other advisors also went beyond the call of their duty. Anton Korinek
deserves special mention. He pushed me to delve deeper in studying the question at
hand. Without his continual support and con?dence, I would have not written my
dissertation. I would like to thank Carlos Vegh. His invaluable advice has shaped
my work and thoughts and will continue to in?uence my future research. I would
like to thank John Shea for generously dedicating time to reading my drafts and for
excellent comments. I am also grateful to Lemma Senbet for kindly accepting to
serve as an external advisor on my committee.
My doctoral research bene?ted greatly from support from a number of institu-
tions. The Department of Economics and the Graduate School at the University of
Maryland and Fulbright provided ?nancial support. I would also like to thank the
Atlanta Fed and the Board of Governors of the Federal Reserve System for ?nancial
iii
support and for hosting me during part of my dissertation. The time I spent at these
two institutions were very fruitful and very special for me. Steve Kay and Federico
Mandelmann were excellent hosts in Atlanta and I owe them special gratitude. At
the Board, I would particularly like to thank John Rogers and Chris Erceg for their
immense support.
I would like to thank my professors at the Universidad de Montevideo for
encouraging me to pursue graduate studies, Fernando Barran, Alejandro Cid, Andres
Jali? and especially Juan Dubra. I have been very fortunate to meet several people
who have helped me and inspired me during my doctoral study. I am especially
grateful to Pablo D’Erasmo, as well as Emine Boz, Lorenzo Caliendo, Bora Durdu,
Bertrand Gruss, Ethan Ilzetski, Tim Kehoe, Leonardo Martinez, Ricardo Nunes,
Carmen Reinhart, and Horacio Saprissa. Discussions in the Liquidity-Lunch with
my colleagues Julien Bengui and Sushant Achyara have been very stimulating as
well. I am grateful to Vickie Fletcher for all her help.
I would like to thank my friends Eduardo, Gabriel, Giorgo, Jose, and Mar-
tin. Thanks for your friendship and support. I am grateful to Alejandro, Daniel,
Cristobal, Kizito and Sebastian for enormous support during my time in Maryland.
I am extremely indebted to my parents Alfredo and Graciela for their uncon-
ditional love, understanding and untold sacri?ce. I would also like to thank my dear
sisters Adriana and Patricia. I treasure moments we have shared.
I am also indebted to Jorge and Cecilia. Thank you for making me feel like a
son. I would also like thank my dear brothers and sisters in-law Ceci, Diego, So?a,
Marcelo, Victoria, Carlos, Fede, Agus, Gonchi and Felipe for their love and support.
iv
Above all I want to thank my wife Anita for her love, care, and encouragement
and our daughter Francesca. Fran you are the joy of our lives. Anita, I could not
imagine my life without you.
College Park, Maryland
May 20, 2011
v
Table of Contents
1 Overborrowing and Systemic Externalities in the Business Cycle 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.3.1 Optimality Conditions . . . . . . . . . . . . . . . . . . . . . . 10
1.3.2 Equilibrium De?nition . . . . . . . . . . . . . . . . . . . . . . 11
1.4 E?ciency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.4.1 Social Planner’s Problem . . . . . . . . . . . . . . . . . . . . . 13
1.4.2 Decentralization . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.5.2 Borrowing Decisions . . . . . . . . . . . . . . . . . . . . . . . 23
1.5.3 Policy Instruments . . . . . . . . . . . . . . . . . . . . . . . . 27
1.5.4 Financial Crises: Incidence and Severity . . . . . . . . . . . . 28
1.5.5 Second Moments . . . . . . . . . . . . . . . . . . . . . . . . . 31
1.5.6 Welfare E?ects . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.5.7 Simple forms of intervention: . . . . . . . . . . . . . . . . . . . 34
1.5.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 35
1.6 Policy Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
1.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2 Overborrowing, Financial Crises and Macro-Prudential Policy (coauthored
with Enrique G. Mendoza) 43
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.2 Competitive Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . 49
2.2.1 Private Optimality Conditions . . . . . . . . . . . . . . . . . 52
2.2.2 Recursive Competitive Equilibrium . . . . . . . . . . . . . . . 55
2.3 Constrained-E?cient Equilibrium . . . . . . . . . . . . . . . . . . . . 57
2.3.1 Equilibrium without collateral constraint . . . . . . . . . . . 57
2.3.2 Recursive Constrained-E?cient Equilibrium . . . . . . . . . . 59
2.3.3 Comparison of Equilibria & ‘Macro-prudential’ Policy . . . . 61
2.4 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.4.2 Borrowing decisions . . . . . . . . . . . . . . . . . . . . . . . 68
2.4.3 Asset Returns . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.4.4 Incidence and Magnitude of Financial Crises . . . . . . . . . . 80
2.4.5 Long-Run Business Cycles . . . . . . . . . . . . . . . . . . . . 86
2.4.6 Properties of Macro-prudential Policies . . . . . . . . . . . . . 89
2.4.7 Welfare E?ects . . . . . . . . . . . . . . . . . . . . . . . . . . 94
2.4.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . 97
2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
vi
3 E?cient Bailouts? 108
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
3.2 Analytical Framework . . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2.1 Corporate entities . . . . . . . . . . . . . . . . . . . . . . . . . 112
3.2.2 Households . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.2.3 Government . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
3.2.4 Competitive equilibrium . . . . . . . . . . . . . . . . . . . . . 118
3.2.5 Some characterization . . . . . . . . . . . . . . . . . . . . . . 118
3.3 Bailout Policy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.3.1 A second best benchmark . . . . . . . . . . . . . . . . . . . . 120
3.3.2 Policy Experiment . . . . . . . . . . . . . . . . . . . . . . . . 121
3.4 Quantitative Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.4.1 Calibration and Functional Forms . . . . . . . . . . . . . . . . 124
3.4.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
3.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128
3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
A Proofs (Chapter 1) 132
B An Equivalence Result (Chapter 1) 133
C Sensitivity Analysis (Chapter 1) 135
D Numerical Solution Method (chapter 1) 142
E Numerical Solution Method (chapter 2) 144
vii
Chapter 1
Overborrowing and Systemic Externalities in the Business Cycle
1.1 Introduction
In the wake of the 2008 international ?nancial crisis, there have been intense
debates about reform of the international ?nancial system that emphasize the need
to address the problem of “overborrowing.” The argument typically relies on the
observation that periods of sustained increases in borrowing are often followed by a
devastating disruption in ?nancial markets. This raises the question of why the pri-
vate sector becomes exposed to the dire consequences of ?nancial crises and what the
appropriate policy response should be to reduce the vulnerability to these episodes.
Without a thorough understanding of the underlying ine?ciencies that arise in the
?nancial sector, it seems di?cult to evaluate the merit of proposals that aim to
reform the current international ?nancial architecture.
This paper presents a formal welfare-based analysis of how optimal borrowing
decisions at the individual level can lead to overborrowing at the social level in a
dynamic stochastic general equilibrium (DSGE) model, where ?nancial constraints
give rise to ampli?cation e?ects. As in the theoretical literature (e.g. Lorenzoni,
2008), we analyze constrained e?ciency by considering a social planner that faces the
same ?nancial constraints as the private economy, but internalizes the price e?ects
of its borrowing decisions. Unlike the existing literature, we conduct a quantitative
1
analysis to evaluate the macroeconomic and welfare e?ects of overborrowing. We
study how overborrowing a?ects the incidence and severity of ?nancial crises, the
magnitude of welfare losses, and the features of policy measures that aim to correct
the externality. In a nutshell, we investigate whether overborrowing is in fact a
macroeconomic problem and what should be the optimal policy response.
Our model’s key feature is an occasionally binding credit constraint that limits
borrowing, denominated in the international unit of account (i.e., tradable goods), to
the value of collateral in the form of output from the tradable and nontradable sector,
as in Mendoza (2002). Because debt is partially leveraged in income generated in the
nontradable sector, changes in the relative price of nontradable goods can induce
sharp and sudden adjustments in access to foreign ?nancing. Due to incomplete
markets, agents can only imperfectly insure against adverse shocks. As a result,
when agents have accumulated a large amount of debt and a typical adverse shock
hits, the economy su?ers the typical dislocation associated with an emerging market
crisis. Demand for consumption goods falls, putting downward pressure on the price
of nontradables, which drags down the real exchange rate. This leads to a further
tightening of the credit constraint, setting in motion Fisher’s debt de?ation channel
by which declines in consumption, the real exchange rate, and access to foreign
?nancing mutually reinforce one another, as in Mendoza’s work.
In the model, private agents form rational expectations about the evolution
of macroeconomic variables—in particular the real exchange rate—and correctly
perceive the risks and bene?ts of their borrowing decisions. Nevertheless, they fail
to internalize the general equilibrium e?ects of their borrowing decisions on prices.
2
This is a pecuniary externality that would not impede market e?ciency in the
absence of the credit constraint linked to market prices. However, by reducing the
amount of borrowing ex-ante, a social planner mitigates the decrease in demand for
consumption during crises. This mitigates the real exchange rate depreciation and
prevents a further tightening of ?nancial constraints, making everyone better o?.
Our quantitative analysis shows that the macroeconomic e?ects of the systemic
credit externality are signi?cant. The externality increases the long-run probability
of a ?nancial crisis from 0.4 percent to 5.5 percent and has important e?ects on the
severity of these episodes. In the decentralized equilibrium, consumption drops 17
percent, capital in?ows fall 8 percent, and the real exchange rate drops by 19 percent
in a typical crisis. In the constrained-e?cient allocations, by contrast, consumption
drops 10 percent, capital in?ows barely fall, and the real exchange rate drops by
1 percent. Moreover, the externality allows the model to account for two salient
features of the data: procyclicality of capital in?ows and the high variability of
consumption.
We study a variety of policy measures that can restore constrained e?ciency,
all of which involve restricting the amount of credit in the economy: taxes on debt,
tightening of margins, and capital and liquidity requirements. These measures are
imposed before a crisis hits so that private agents internalize the external costs of
borrowing and the economy becomes less vulnerable to future adverse shocks. In
the calibrated version of our model, the increase in the e?ective cost of borrowing
necessary to implement the constrained-e?cient allocations is about 5 percent on
average, increasing with the level of debt and with the probability of a future ?nan-
3
cial crisis. We also study simple forms of interventions and ascertain that a ?xed
tax on debt can also achieve sizable welfare gains.
Our paper is related to the large literature on the macroeconomic role of ?nan-
cial frictions. Following the work of Bernanke and Gertler (1989) and Kiyotaki and Moore
(1997), various studies have presented dynamic models where ?nancial frictions can
amplify macroeconomic shocks compared to a ?rst-best benchmark where these fric-
tions are absent.
1
Our contribution to this literature is twofold. First, we study the
volatility and the level of ampli?cation of the competitive equilibrium relative to a
second-best benchmark where these frictions are also present. Second, we investigate
several policy measures that can signi?cantly reduce the level of ?nancial instability
and improve welfare by making agents internalize an externality due to ?nancial
accelerator e?ects.
Our paper is related to the theoretical literature that investigates the role of
pecuniary externalities in generating excessive ?nancial fragility, and we borrow ex-
tensively from their insights (see for example Auernheimer and Garcia-Saltos (2000),
Caballero and Krishnamurthy (2001,2003), Lorenzoni (2008), Farhi, Golosov, and Tsyvinski
(2009), and Korinek (2009ab)).
2
In all of these studies, however, the analysis is
qualitative in nature. Our contribution to this literature is to provide a quantitative
assessment of the macroeconomic, policy and welfare implications of overborrowing.
This is an important ?rst step in the evaluation of the potential bene?ts from reg-
1
See for example Aiyagari and Gertler (1999), Bernanke, Gertler, and Gilchrist (1999),
Iacoviello (2005), Gertler, Gilchrist, and Natalucci (2007), Mendoza (2010) (2002,2010).
2
The ine?ciency result of these studies is related to the idea that economies with endoge-
nous borrowing constraint and multiple goods are constrained ine?cient (Kehoe and Levine,
1993) and to the generic ine?ciency result in economies with incomplete markets
(Geneakoplos and Polemarchakis, 1986; Stiglitz, 1982).
4
ulatory measures to correct these externalities and in the study of their practical
implementation.
There is a growing macroeconomic literature that studies optimal policy in
a ?nancial crisis.
3
This literature typically takes as given that the economy is in
a high leverage situation and analyzes the role of policies that can moderate the
impact of a large adverse shock. While this literature provides important insights
on how to respond to crises once they erupt, it does not study how the economy
experiences the surge in debt that leads to the crisis in the ?rst place. This paper
complements this literature by studying how an economy can become vulnerable to
a ?nancial crisis due to excessive borrowing during normal times. We model crises
as infrequent episodes nested within regular business cycles and analyze the role of
policies in reducing an economy’s vulnerability to ?nancial crises, therefore placing
macroprudential policy at the center of the stage. We acknowledge, however, that
because our analysis requires global non-linear solution methods, we abstract from
important real-world features present in larger scale DSGE models.
A related paper that allows for policy intervention during normal times and
crisis times is Benigno et al. (2009). They consider the role of a subsidy on nontrad-
able goods, which the Ramsey planner uses ex-post to mitigate the real exchange
depreciation during crises, but not ex-ante since it is not e?ective to make agents
internalize the full social costs of borrowing. We focus instead on a constrained
planner who directly makes borrowing decisions and show that the decentralization
3
Notable contributions include Christiano, Gust, and Roldos (2004), Kiyotaki and Moore
(2008), Gertler and Karadi (2009), and Gertler and Kiyotaki (2010).
5
requires ex-ante intervention to prevent excessive risk exposure.
Finally, there are a number of other theories of overborrowing that have been
investigated. One theory is moral hazard: banks may lend excessively to take ad-
vantage of some form of government bailout.
4
Uribe (2006) has also studied whether
an economy with an aggregate debt limit tends to overborrow relative to an econ-
omy with debt limits imposed at the level of each individual agent, and found that
borrowing decisions coincide. Our focus is on the comparison between competitive
equilibrium and constrained-e?cient equilibrium when ?nancial constraints that are
linked to market prices generate ampli?cation e?ects.
1.2 Analytical Framework
Consider a representative-agent DSGE model of a small open economy (SOE)
with a tradable goods sector and a nontradable goods sector. Tradable goods can
be used for consumption, external borrowing and lending transactions; nontradable
goods have to be consumed in the domestic economy. The economy is populated by a
continuum of identical, in?nitely-lived households of measure unity with preferences
given by:
E
0
_
?
?
t=0
?
t
u(c
t
)
_
(1.1)
In this expression, E
t
(·) is the time t expectation operator, and ? is the discount fac-
tor. The period utility function u(·) has the constant-relative-risk-aversion (CRRA)
form. The consumption basket c
t
is an Armington-type CES aggregator with elas-
4
See e.g. McKinnon and Pill (1996), Corsetti, Pesenti, and Roubini (1999),
Schneider and Tornell (2004), and Farhi and Tirole (2010).
6
ticity of substitution 1/(? +1) between tradable c
T
and nontradable goods c
N
given
by:
c
t
=
_
?
_
c
T
t
_
??
+ (1 ??)
_
c
N
t
_
??
_
?
1
?
, ? > ?1, ? ? (0, 1)
In each period t, households receive an endowment of tradable goods y
T
t
and an
endowment of nontradable goods y
N
t
. We assume that the vector of endowments
given by y ? (y
T
, y
N
) ? Y ? R
2
++
follows a ?rst-order Markov process. These
endowment shocks are the only source of uncertainty in the model.
The menu of foreign assets available is restricted to a one period, non-state-
contingent bond denominated in units of tradables that pays a ?xed interest rate r,
determined exogenously in the world market.
5
Normalizing the price of tradables
to 1 and denoting the price of nontradable goods by p
N
the budget constraint is:
b
t+1
+ c
T
t
+ p
N
t
c
N
t
= b
t
(1 +r) + y
T
t
+ p
N
t
y
N
t
(1.2)
where b
t+1
denotes bond holdings that households choose at the beginning of time
t. We maintain the convention that positive values of b denote assets. As there is
only one asset, gross and net bond holdings (NFA) coincide.
We assume that creditors restrict loans so that the amount of debt does not
exceed a fraction ?
T
of tradable income and a fraction ?
N
of nontradable income.
5
To have a well-de?ned stochastic steady state, we assume that the discount factor and the
world interest rate are such that ?(1 + r) < 1. If ?(1 + r) ? 1, assets will diverge to in?nity
in equilibrium by the supermartingale convergence theorem (see Chamberlain and Wilson (2000),
2000). See Schmitt-Grohe and Uribe (2003) for other methods to induce stationarity.
7
Speci?cally, the credit constraint is given by:
b
t+1
? ?
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
(1.3)
This credit constraint can be seen as arising from informational and institutional
frictions a?ecting credit relationships (such as monitoring costs, limited enforcement,
asymmetric information, and imperfections in the judicial system), but we do not
model these frictions explicitly. Our focus is on how ?nancial policies can be welfare
improving, taking as given the frictions that lead to these debt contracts, i.e., we will
assume that the social planner is a constrained social planner that is also subject to
this credit constraint.
Discussion of market incompleteness.— A few comments are in order about
the two deviations from complete markets that we introduce here. First, we have
assumed that assets are restricted to a one-period non-state-contingent bond de-
nominated in tradable goods. While agents typically have a richer set of assets
available, this assumption is made for numerical tractability and is meant to cap-
ture the observation that debt in emerging markets is generally short term and
denominated in foreign currency. In turn, these features of debt contracts are gen-
erally seen as an important source of vulnerability in emerging markets (see e.g.
Calvo, Izquierdo, and Loo-Kung, 2006).
The second form of market incompleteness is given by the credit constraint. In
the absence of a credit constraint, households will increase borrowing in bad times
to smooth consumption. This will imply a counterfactual reaction of the current
8
account, which is well-known to rise during recessions in emerging markets. The
credit constraint we have speci?ed has two main features. One crucial feature is
that nontradable goods are part of the collateral. At the empirical level, this is
consistent with evidence that credit booms in the nontradable sector are fueled
by external credit (see e.g. Tornell and Westermann, 2005). At the theoretical
level, this could result because foreign borrowers can seize nontradable goods from
a defaulting borrower, sell them in the domestic market, and repatriate the funds
abroad. A positive gap between ?
T
and ?
N
would re?ect an environment where
creditors have a higher preference for tradable income as collateral. A case where
?
T
= ?
N
would re?ect an environment where creditors request and aim to verify
information on total income of individual borrowers, i.e., they do not document the
sectoral sources of their income.
The second feature is that the collateral is given by current income. At the
empirical level, this assumption is supported by evidence that current income is
a major determinant of credit market access (see e.g., Jappelli, 1990).
6
At the
theoretical level, borrowing limits could depend on current income if households can
engage in fraud in the period they contract debt obligations and prevent creditors
from seizing any future income. If creditors detect the fraud, however, they could
seize household’s current income (see Korinek, 2009a).
An additional argument that our formulation of the credit constraint is suit-
able for a quantitative assessment of the externality is that our model can account
6
For more evidence on credit constraints on households, see Jappelli and Pagano (1989), Zeldes
(1989).
9
reasonably well for the main macro features of emerging market crises, as shown by
Mendoza (2002).
1.3 Equilibrium
1.3.1 Optimality Conditions
The household’s problem is to choose stochastic processes {c
T
t
, c
N
t
, b
t+1
}
t?0
to
maximize the expected present discounted value of utility (2.1) subject to (1.2) and
(1.3), taking b
0
and {p
N
t
}
t?0
as given. The household’s ?rst-order conditions require:
?
t
= u
T
(t) (1.4)
p
N
t
=
_
1 ??
?
__
c
T
t
c
N
t
_
?+1
(1.5)
?
t
= ?(1 +r)E
t
?
t+1
+ µ
t
(1.6)
b
t+1
+
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
? 0, with equality if µ
t
> 0 (1.7)
where ? is the non-negative multiplier associated with the budget constraint and µ
is the non-negative multiplier associated with the credit constraint. The optimality
condition (1.4) equates the marginal utility of tradable consumption to the shadow
value of current wealth. Condition (1.5) equates the marginal rate of substitution of
the two goods, tradables and nontradables, to their relative price. Equation (1.6) is
the Euler equation for bonds. When the credit constraint is binding, there is a wedge
between the current shadow value of wealth and the expected value of reallocating
10
wealth to the next period, given by the shadow price of relaxing the credit constraint
µ
t
. Equation (1.7) is the complementary slackness condition.
Since households are identical, market clearing conditions are given by:
c
N
t
= y
N
t
(1.8)
c
T
t
= y
T
t
+ b
t
(1 +r) ?b
t+1
(1.9)
Notice that equation (1.5) implies that a reduction in c
T
t
generates in equilibrium
a reduction in p
N
t
, which by equation (1.3) reduces the collateral value. Besides
ampli?cation, the credit constraint produces asymmetric responses in the economy:
a binding credit constraint ampli?es the consumption drop in response to a negative
income shock, but no ampli?cation e?ects occur when the credit constraint is slack.
Because of consumption-smoothing e?ects, the demand for borrowing generally de-
creases with current income, and when current income is su?ciently low, the credit
constraint becomes binding.
1.3.2 Equilibrium De?nition
We consider the optimization problem of a representative household in recur-
sive form, which includes, as a crucial state variable, the aggregate bond hold-
ings of the economy. Households need to forecast future aggregate bond hold-
ings that are beyond their control to form expectations of the price of nontrad-
ables. We denote by ?(·) the forecast of aggregate bond holdings for every cur-
11
rent aggregate state (B, y), i.e., B
?
= ?(B, y) . Combining equilibrium conditions
(1.5),(1.8), and (1.9), the forecast price function for nontradables can be expressed
as p
N
(B, y) = (1 ? ?)/(?)
_
(y
T
+ B(1 +r) ??(B, y))/y
N
_
?+1
. The other relevant
state variables for the individual household are its bond holdings and the vector of
endowment shocks. The problem of a representative household can then be written
as:
V (b, B, y) = max
b
?
,c
T
,c
N
u(c(c
T
, c
N
)) +?E
y
?
| y
V (b
?
, B
?
, y
?
) (1.10)
subject to
b
?
+ p
N
(B, y)c
N
+ c
T
= y
T
+ b(1 +r) + p
N
(B, y)y
N
b
?
? ?
_
?
N
p
N
(B, y)y
N
+ ?
T
y
T
_
B
?
= ?(B, y)
where we have followed the convention of denoting current variables without sub-
script and denoting next period variables with the prime superscript. The solution
to the household problem yields decision rules for individual bond holdings
ˆ
b(b, B, y),
tradable consumption ˆ c
T
(b, B, y) and nontradable consumption ˆ c
N
(b, B, y). The
household optimization problem induces a mapping from the perceived law of motion
for aggregate bond holdings to an actual law of motion, given by the representative
agent’s choice
ˆ
b(B, B, y). In a rational expectations equilibrium, as de?ned below,
these two laws of motion must coincide.
De?nition 1 (Decentralized Recursive Competitive Equilibrium)
12
A decentralized recursive competitive equilibrium for our SOE is de?ned by a pricing
function p
N
(B, y), a perceived law of motion ?(B, y) and decision rules
_
ˆ
b(b, B, y), ˆ c
T
(b, B, y), ˆ c
N
(b, B, y)
_
with associated value function V (b, B, y) such that the following conditions hold:
1. Household optimization:
_
ˆ
b(b, B, y), ˆ c
N
(b, B, y), ˆ c
N
(b, B, y), V (b, B, y)
_
solve
the recursive optimization problem of the household for given p
N
(B, y) and
?(B, y).
2. Rational expectation condition: the perceived law of motion is consistent with
the actual law of motion: ?(B, y) =
ˆ
b(B, B, y).
3. Markets clear: y
N
= ˆ c
N
(B, B, y) and ?(B, y) + ˆ c
T
(B, B, y) = y
T
+ B(1 + r).
1.4 E?ciency
1.4.1 Social Planner’s Problem
We previously described the equilibrium achieved when agents take aggregate
variables as given, particularly the price of nontradables. Consider now a benevolent
social planner with restricted planning abilities. We assume that the social planner
can directly choose the level of debt subject to the credit constraint, but allows goods
markets to clear competitively. That is, the planner (a) performs credit operations
and rebates back to households all the proceeds in a lump sum fashion, and (b)
lets households choose their allocation of consumption between tradable goods and
nontradable goods in a competitive way.
As opposed to the representative agent, a social planner internalizes the e?ects
13
of borrowing decisions on the price of nontradables. Critically, the social planner
realizes that a lower debt level mitigates the reduction in the price of nontradables
and prevents a larger drop in borrowing ability when the credit constraint binds.
As a result, we will show that the decentralized equilibrium allocation is not a
constrained Pareto optimum, as de?ned below.
De?nition 2 (Constrained E?ciency)
Let
_
c
T
t
, c
N
t
, b
t+1
_
t?0
be the allocations of the competitive equilibrium yielding util-
ity
ˆ
V . The competitive equilibrium is constrained e?cient if a social planner that
chooses directly {b
t+1
}
t?0
subject to the credit constraint, but lets the goods markets
clear competitively, cannot improve the welfare of households above
ˆ
V .
The social planner’s optimization problem consists of maximizing (2.1) subject
to (1.3),(1.5),(1.8), and (1.9). Substituting for the equilibrium price in (1.3), we can
express the social planner’s optimization problem in recursive form as:
V (b, y) = max
b
?
,c
T
u(c(c
T
, y
N
)) + ?E
y
?
| y
V (b
?
, y
?
) (1.11)
subject to
b
?
+ c
T
= y
T
+ b(1 + r)
b
?
? ?
_
?
N
1 ??
?
_
c
T
y
N
_
?+1
y
N
+ ?
T
y
T
_
Using sequential notation and the superscript “sp” to distinguish the Lagrange mul-
tipliers of the social planner’s problem from the decentralized equilibrium, the ?rst-
14
order conditions for the social planner require:
?
sp
t
= u
T
(t) + µ
sp
t
?
t
(1.12)
?
sp
t
= ?(1 + r)E
t
?
sp
t+1
+ µ
sp
t
(1.13)
b
t+1
+
_
?
N
1 ??
?
_
c
T
t
y
N
t
_
?+1
y
N
t
+ ?
T
y
T
t
_
? 0, with equality if µ
sp
t
> 0
(1.14)
where ?
t
? ?
N
(p
N
t
c
N
t
)/(c
T
t
) (1 +?) > 0 indicates how much the collateral value
changes at equilibrium when there is a change in tradable consumption. Notice that
this term is directly proportional to the fraction of nontradable output that agents
can pledge as collateral, the relative size of the nontradable sector and the inverse
of the elasticity of substitution between tradables and nontradables. We will return
to this expression in the sensitivity analysis.
The key di?erence between the optimization problem of the social planner
relative to households follows from examining (1.12) compared with the correspond-
ing equation for the decentralized equilibrium (1.4). The social planner’s marginal
bene?ts from tradable consumption include the direct increase in utility u
T
(t) and
also the indirect increase in utility µ
sp
t
?
t
. This indirect bene?t, not considered by
private agents, represents how an increase in tradable consumption increases the
price of nontradables and relaxes the credit constraint of all agents by ?
t
, which has
a shadow value of µ
sp
t
. Thus, (1.4) and (1.12) yield the key result that, for given ini-
tial states and allocations at which the credit constraint binds, private agents value
15
wealth less than the social planner, which we highlight in the following remark.
Remark 1 When the credit constraint binds, private agents undervalue wealth.
To see more clearly why this di?erent ex-post valuation generates overborrow-
ing ex-ante, suppose that at time t the constraint is not currently binding. Using
(1.4) and (1.6), the Euler equation for consumption in the decentralized equilibrium
becomes:
u
T
(t) = ?(1 + r)E
t
u
T
(t + 1) (1.15)
Using (1.12) and (1.13), the Euler equation for consumption for the social planner
becomes:
u
T
(t) = ?(1 + r)E
t
_
u
T
(t + 1) + µ
sp
t+1
?
t+1
¸
(1.16)
Consider now a reallocation of wealth by the social planner starting from the
privately optimal allocations in the decentralized equilibrium. In particular, consider
the welfare e?ects of a reduction of one unit of borrowing. Because decentralized
agents are at the optimum, (1.15) shows that the ?rst-order private welfare bene?ts
?(1 + r)E
t
u
T
(t + 1) are equal to the ?rst-order private welfare costs u
T
(t). Using
(1.16), the social planner has a marginal cost of reducing borrowing equal to the
private marginal cost, but faces higher marginal bene?ts: a one unit decrease in bor-
rowing relaxes next-period ability to borrow by (1 + r)?
t+1
, which has a marginal
utility bene?t of µ
sp
t+1
. The uninternalized external bene?ts from savings, or equiv-
alently the uninternalized external marginal cost of borrowing, is then given by the
discounted expected marginal utility cost of the resulting tightening of the credit
16
constraint ?(1 +r)E
t
µ
sp
t+1
?
t+1
. Notice that if the credit constraint does not bind for
any pair (b, y) in the two equilibria, the conditions characterizing both environments
are identical and therefore the allocations coincide.
Proposition 1 (Constrained Ine?ciency) The decentralized equilibrium is not,
in general, constrained e?cient.
Proof : See Appendix A
1.4.2 Decentralization
We study the use of various ?nancial policies in the implementation of the
constrained-e?cient allocations. We start by showing how a tax on debt can restore
constrained e?ciency and then show the equivalence between the tax on debt and
more standard forms of intervention in the ?nancial sector (e.g., capital require-
ments).
Letting ?
t
be the tax charged on debt issued at time t, the Euler equation for
bonds in the regulated decentralized equilibrium (1.6) becomes:
u
T
(t) = ?(1 + r)(1 + ?
t
)E
t
u
T
(t + 1) +µ
t
(1.17)
Proposition 2 (Optimal tax on debt) The constrained-e?cient allocations can
be implemented with an appropriate state contingent tax on debt, with tax revenue
rebated as a lump sum transfer.
Proof : See Appendix A
17
When the credit constraint is not binding in the constrained-e?cient allo-
cations, the tax must be set to ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1)) (variables are
evaluated at the constrained-e?cient allocations). This expression represents the
uninternalized marginal cost of borrowing analyzed above, normalized by the ex-
pected marginal utility. As we will see in the quantitative analysis, this tax increases
with the current level of debt, since a higher current level of debt implies a higher
choice of debt, which increases the probability and the marginal utility cost of a
binding constraint next period. Notice also that if the credit constraint has a zero
probability of binding in the next period, the tax is set to zero.
When the credit constraint is binding, the tax does not generally in?uence the
level of borrowing since the choice of debt is given by the credit constraint (1.3) and
not by the Euler equation (1.17). Setting the tax to ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1))?
(µ
sp
t
?
t
) / (?(1 + r)E
t
u
T
(t + 1)) achieves constrained e?ciency and equalizes the pri-
vate and social shadow values from relaxing the constraint. Notice that an extra
term arises because the social planner internalizes that relaxing the credit constraint
today would have positive e?ects on the current price of nontradables. This term is
negative so that the tax causes the private shadow value of relaxing the constraint
to rise to the social value. As we will show in the quantitative analysis, when the
planner is borrowing up to the limit, the level of borrowing desired by private agents
is also the maximum available. As a result, we ?nd that setting ?
?
t
= 0 when the
constraint binds also implements the constrained-e?cient allocations, and since this
results in a simpler policy we set this tax to zero when we turn to describe its
18
quantitative features.
7
In practice, much of prudential ?nancial regulation is implemented through
banks. To take this into consideration, we develop in Appendix B a simple model
of ?nancial intermediaries, and show that our benchmark economy, in which the
planner sets a tax on debt on borrowers, is equivalent to a economy where the
planner sets capital requirements or reserve requirements on ?nancial institutions.
Throughout the paper, we will refer to the implied tax on debt as the increase in
the cost of debt induced by the use of any of these equivalent policy measures.
Alternatively, the planner could implement the constrained-e?cient allocations
using margin requirements by choosing an adjustment ?
t
? 0 such that the credit
constraint becomes b
t+1
? ?(1 ? ?
t
)
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
. If the socially optimal
amount of borrowing is b
sp
t
, by setting ?
?
t
= 1 ?(b
sp
t
)/
_
?
N
p
N
t
y
N
t
+ ?
T
y
T
t
_
the social
planner can restrict the quantity of borrowing and restore constrained e?ciency.
8
Remark 2 (Decentralization) The constrained-e?cient allocations can be im-
plemented with appropriate capital requirements, reserve requirements, or margin
requirements.
7
This result held for the wide range of parameters explored in our quantitative analysis.
8
For this policy to restore constrained e?ciency, it must be the case that
ˆ
b
de
(B, B, y) ?
ˆ
b
sp
(B, y) ?(B, y), which we will show is the case in the numerical analysis. We assume for simplic-
ity that ?
?
is such that the credit constraint always holds with equality in the regulated economy.
In the regulated economy where the adjustment of margins is given by ?
?
, the constraint only
binds in the constrained region and in the tax-region; hence setting ?
t
= 0 in the no-tax region
would deliver the same allocations.
19
1.5 Quantitative Analysis
In this section, we describe the calibration of the model and evaluate the quan-
titative implications of the externality. We solve the competitive equilibrium and the
social planner’s problems numerically using global non-linear methods (described in
detail in the online Appendix).
1.5.1 Calibration
A period in the model represents a year. The baseline calibration uses data
from Argentina, an example of an emerging market with a business cycle that has
been studied extensively. The risk aversion is set at ? = 2, a standard value. The
interest rate is set at r = 4 percent, which is a standard value for the world risk-free
interest rate in the DSGE-SOE literature
We model endowment shocks as a ?rst-order bivariate autoregressive process:
log y
t
= ? log y
t?1
+ ?
t
where y = [y
T
y
N
]
?
, ? is a 2x2 matrix of autocorrelation
coe?cients, and ?
t
= [?
T
t
?
N
t
]
?
follows a bivariate normal distribution with zero
mean and contemporaneous variance-covariance matrix V . This process is estimated
with the HP-?ltered cyclical components of tradables and nontradables GDP from
the World Development Indicators (WDI) for the 1965-2007 period, the longest
time series available from o?cial sources. Following the standard methodology, we
classify manufacturing and primary products as tradables and classify the rest of
the components of GDP as nontradables. The estimates of ? and V are:
20
? =
_
¸
¸
_
0.901 0.495
?0.453 0.225
_
¸
¸
_
V =
_
¸
¸
_
0.00219 0.00162
0.00162 0.00167
_
¸
¸
_
The standard deviations of tradable and nontradable output in the data are
?
y
T = 0.058 and ?
y
N = 0.057, the ?rst-order autocorrelations are ?
y
T = 0.53 and
?
y
N = 0.61, and the correlation between the two is ?
y
T
,y
N = 0.81. Thus cyclical
?uctuations in the two sectors have similar volatility and persistence, and are posi-
tively correlated with each other. We discretize the vector of shocks into a ?rst-order
Markov process, with four grid points for each shock, using the quadrature-based
procedure of Tauchen and Hussey (1991). The mean of the endowments are set to
one without loss of generality.
Table 1.1: Calibration
Value Source/Target
Interest rate r = 0.04 Standard value DSGE-SOE
Risk aversion ? = 2 Standard value DSGE-SOE
Elasticity of substitution 1/(1 +?) = 0.83 Conservative value
Stochastic structure See text Argentina’s economy
Relative credit coe?cients ?
N
/?
T
= 1 Baseline Value
Weight on tradables in CES ? = 0.31 Share of tradable output=32%
Discount factor ? = 0.91 Average NFA-GDP ratio = ?29%
Credit coe?cient ?
T
= 0.32 Frequency of crisis = 5.5%
The intratemporal elasticity of substitution 1/(? + 1) is a crucial parameter
because it a?ects the magnitudes of the price adjustment. For a given reduction in
tradable consumption, a higher elasticity implies a smaller change in the price of
nontradables, and therefore we should expect weaker e?ects from the externality.
The range of estimates for the elasticity of substitution is between 0.40 and 0.83.
9
9
See Mendoza (2006), Gonzalez-Rozada and Neumeyer (2003) and Stockman and Tesar
(1995).
21
As a conservative benchmark, we set ? such that the elasticity of substitution equals
the upper bound of this range and then show how the externality changes with this
parameter.
The ratio ?
N
/?
T
determines the relative quality of nontradable and tradable
output as collateral. It is di?cult, however, to derive a direct mapping from the
data to this ratio. We therefore take a pragmatic approach: we begin by setting
?
N
= ?
T
and then perform extensive sensitivity analysis.
The three remaining parameters are
_
?, ?, ?
T
_
, which are set so that the long-
run moments of the decentralized equilibrium match three historical moments of the
data. The parameter ? governs the tradable share in the CES aggregator and is
calibrated to match a 32 percent share of tradable production.
10
This approach is a
reasonable one to calibrate ? since given the relative endowment and consumption
ratios, ? determines the equilibrium price of nontradables by (1.5) and the share of
tradables in the total value of production. This calibration results in a value of ?
of 0.32.
The discount factor ? is set so that the average net foreign asset position-to-
GDP ratio in the model equals its historical average in Argentina, which is equal
to -29 percent in the dataset constructed by Lane and Milesi-Ferretti (2001). This
calibration results in a value of ? = 0.91, a relatively standard value for annual
frequency in the literature.
The parameter ?
T
is calibrated to match the observed frequency of “Sudden
10
Garcia (2008) reports an average share of tradables of 32 percent using almost a century of
data from Argentina. The average share of tradables is also 32 percent in the data from WDI for
the period 1980-2007.
22
Stops”, which is about 5.5 percent in the cross-country data set of Eichengreen, Gupta, and Mody
(2006).
11
To be consistent with their de?nition of Sudden Stops, we de?ne Sudden
Stops in our model as events where the credit constraint binds and where this leads
to an increase in net capital out?ows that exceeds one standard deviation. This
calibration results in a value of ?
T
equal to 0.32, which is in the range of those used
in the literature (see Mendoza, 2002).
1.5.2 Borrowing Decisions
We ?rst show how the bond accumulation decisions of the social planner dif-
fer from those of private agents and then simulate the model to analyze how this
di?erence a?ects the long-run distribution of debt, the crisis dynamics, and the
unconditional second moments.
Figure 1 shows the bond decision rules in the decentralized equilibrium and
in the constrained-e?cient equilibrium as a function of current bond holdings when
both tradable and nontradable shocks are one-standard-deviation below trend. Since
the mean value of tradable output is 1, we can interpret all results as ratios with
respect to the average output of tradables.
Without the endogenous borrowing constraint, the policy function for next
period’s bond holdings would be monotonically increasing in current bond holdings.
Instead, the policy functions are non-monotonic. The change in the sign of the slope
of the policy function indicates the point at which the credit constraint is satis?ed
11
For the case of Argentina, Eichengreen et al. includes 1989 and 2001 as Sudden Stop events,
yielding a similar frequency to the cross-country average over the sample period.
23
with equality, but is not binding. To the right of this point, the credit constraint is
slack, and bond decision rules display the usual upward-sloping shape. To the left of
this point, next-period bond holdings decrease in current bond holdings. To see why,
notice that a decrease in the current bond position implies a reduction in tradable
consumption for a given choice of next-period bond holdings by equation (1.9). This
in turn lowers the price of nontradables by equation (1.5), which means that the level
of borrowing must be reduced further to satisfy the credit constraint. Comparing
the policy functions against the 45-degree line also shows that for relatively low
current levels of bond holdings, the economy reduces the level of borrowing, which
results in capital out?ows.
We distinguish three regions for all pairs of (b, y) according to the actions taken
by the planner in the regulated economy: a “constrained region,” a “tax region”,
and a “no-tax region.” The constrained region in Figure 1 is given by the range
of b with su?ciently high initial debt such that the credit constraint binds in the
constrained-e?cient equilibrium. In this region, both private agents and the social
planner borrow up to the limit, and decision rules coincide. The long-run probability
of this region is 6.2 percent in the decentralized equilibrium, about twice as much
as for the social planner.
The tax region appears shaded in Figure 1 and corresponds to the pairs of (b, y)
where the social planner would impose a tax on debt in the regulated economy. As
explained above, this is the region where households borrow enough so that the credit
constraint will bind with a strictly positive probability in the next period. Here, the
social planner accumulates uniformly higher bond holdings than households. In fact,
24
?1 ?0.9 ?0.8 ?0.7 ?0.6
?1
?0.9
?0.8
?0.7
?0.6
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
Tax Region
No?Tax Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Social Planner
Decentralized Equilibrium
Figure 1.1: Bond decision rules for negative one-standard-deviation shocks
households continue to borrow up to the limit, over some range of current bonds for
which the social planner would choose a lower borrowing level that monotonically
increases in current bond holdings. The economy spends about 80 percent of its
time in the tax region in both equilibria.
The no-tax region is located to the right of the tax region and corresponds
to the pairs of (b, y) where the credit constraint is slack and the social planner
would not impose a tax on debt. Intuitively, the economy is relatively well-insured
in this region, and the amount of borrowing chosen does not make the economy
vulnerable to a binding constraint. Here, the di?erences in the bond decision rules
become quantitatively smaller, but are non-zero since di?erent future choices of
bond holdings a?ect current optimal choices. The social planner spends 16 percent
of the time in this region, 2 percent more than the decentralized equilibrium.
25
While both the social planner and private households self-insure against the
risk of ?nancial crises, the social planner accumulates extra precautionary savings
above and beyond what households consider privately optimal. As Figure 2 shows,
this implies that the ergodic distribution of bond holdings in the decentralized equi-
librium assigns a higher probability to higher levels of debt. In fact, the decentralized
equilibrium has a 15 percent chance of carrying a larger amount of debt than the
maximum held by the social planner, illustrated by the shaded region in Figure 2.
?1 ?0.95 ?0.9 ?0.85 ?0.8 ?0.75 ?0.7
0
0.35
0.7
1.05
x 10
?3
Bond Holdings
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Figure 1.2: Ergodic Distribution of Bond Holdings
Notice that the large di?erences in the left tail distribution of debt are not
translated into large di?erences in average debt levels: the average debt-to-GDP
ratio is 29.2 percent for the private economy and 28.6 percent for the social planner.
What is crucial is that the social planner reduces the exposure to debt levels that
makes the economy vulnerable to a severe ?nancial crisis when the economy is hit
by an adverse shock.
26
1.5.3 Policy Instruments
Figure 1.3 shows the two types of policy measures that achieve the constrained-
e?cient allocations when shocks are one standard deviation below trend for di?erent
levels of current bond holdings: the left panel shows the e?ective increase in the
cost of borrowing from tax-like measures (taxes, reserve or capital requirements);
the rigth panel shows the adjustment in margin requirements ?.
As explained above, for su?ciently low values of debt, the implied tax on
borrowing is zero. The tax then increases with the level of debt in the tax region,
until the credit constraint becomes binding for the social planner and the tax is set
to zero. On average, the implied tax on debt is 5.2 percent.
?1 ?0.9 ?0.8 ?0.7 ?0.6
0
5
10
15
20
25
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
Implied Tax on Debt
?1 ?0.9 ?0.8 ?0.7 ?0.6
0
5
10
15
20
25
30
35
40
Tightening of Margins
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
Figure 1.3: Policy Instruments for negative one-standard-deviation shocks
The adjustment to the margin requirement is also zero when the constraint
is already binding, but unlike the tax-like measures the adjustment decreases with
the level of debt outside the constrained region. This arises because, as the level
of debt increases, the excess debt capacity is reduced, thereby requiring a smaller
adjustment in margin requirements to reduce the gap and socially desired amount
27
of borrowing. On average, margins are tightened by 9 percent in the tax-region,
which implies that the e?ective fraction that agents can borrow from their income
is reduced from 0.32 to 0.29 in the regulated economy.
1.5.4 Financial Crises: Incidence and Severity
In this section, we establish that overborrowing in the decentralized equilib-
rium leaves the economy vulnerable to more frequent and more severe ?nancial crises.
Using the policy functions of the model, we perform an 80,000-period stochastic time
series simulation of the decentralized and constrained-e?cient equilibrium and use
the resulting data to study the incidence and severity of ?nancial crises. A ?nancial
crisis event is de?ned as a period in which the credit constraint binds, and in which
this leads to an increase in net capital out?ows that exceeds one standard devia-
tion of net capital out?ows in the ergodic distribution of the decentralized economy.
(Results are similar with alternative de?nitions of a crisis event.)
Two important results emerge from the event analysis. First, crises in the
decentralized equilibrium are much more likely: the long-run probability of crises
is 5.5 percent (versus 0.4 percent for the social planner). Thus, by reducing the
amount of debt, the social planner cuts the long-run probability of a ?nancial crisis
more than tenfold.
28
?40 ?35 ?30 ?25 ?20 ?15 ?10 ?5 0 5
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
?3
Percentage change in consumption
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Figure 1.4: Conditional distribution of impact e?ect of ?nancial crises on consump-
tion
Second, the magnitudes of ?nancial crises are substantially more severe be-
cause of the externality. Figure 4 shows the distribution of the response of the
consumption basket on impact during ?nancial crises for the two equilibria, ex-
pressed as a percentage deviation from the average long-run value of consumption.
This ?gure shows that the decentralized equilibrium assigns non-trivial probabilities
to consumption drops of more than 22 percent, while such a fall in consumption is
a zero probability event in the social planner’s allocations.
Drops in the real exchange rate and capital in?ows are also more pronounced
because of the externality. Table 2 compares the responses of these variables during
the median ?nancial crisis under the decentralized equilibrium with the economy’s
response under the constrained-e?cient allocations, conditional on the social planner
having the same level of debt two periods before such crisis and receiving the same
29
sequence of shocks.
12
In this experiment, we ?nd that while the credit constraint also becomes bind-
ing for the social planner at time t, the impact of the adverse shocks is less severe:
consumption falls 10 percent (versus 17 percent in the decentralized equilibrium),
the current account does not increase (versus an 8 percent increase in the decen-
tralized equilibrium), and the real exchange rate depreciates 1 percent (versus 19
percent in the decentralized equilibrium).
13
Notice that since the initial level of debt
and the sequence of shocks are the same for the two equilibria, the di?erence in the
impact of crises is entirely due to the more prudent behavior of the social plan-
ner during the periods preceding the crisis, which makes the required adjustment
following an adverse shock less severe.
Table 1.2: Severity of Financial Crises
Decentralized Equilibrium Social Planner
Consumption -16.7 -10.1
Current Account-GDP 7.8 0.0
Real Exchange Rate 19.2 1.1
Note: Consumption and the depreciation of the real ex-
change rate represent responses on impact expressed as
percentage deviations from averages in the corresponding
ergodic distribution.
12
More precisely, we ?rst de?ne the median ?nancial crisis in the decentralized as the period
with the median current account reversal considering all the crisis episodes in the decentralized
equilibrium. Then we backtrack the initial level of debt two periods before this episode and the
sequence of shocks that hit the economy at time t ? 2, t ? 1, and t. Given this initial level of
debt at t ?2 and the sequence of shocks, we then use the decision rules of the constrained-e?cient
allocations to simulate the economy and compare the impact e?ects at time t with the impact
e?ects in the decentralized equilibrium median crisis.
13
We de?ne the real exchange rate as
_
?
1/(1+?)
+ (1 ??)
1/(1+?)
_
p
N
_
?/(1+?)
_
?(1+?)/?
implying
a one-to-one negative relationship between the price of nontradables and the real exchange rate.
30
1.5.5 Second Moments
Table 3 compares the unconditional second moments for decentralized and
constrained-e?cient equilibria, which are computed using each economy’s ergodic
distribution, and for the Argentinian data. It is apparent that the externality pro-
duces non-trivial e?ects on the volatility of consumption, capital ?ows, and espe-
cially the real exchange rate. Two reasons for this are that, ?rst, the economy
spends most of the time in the tax-region where the bond accumulation decisions
di?er signi?cantly; and second, the decentralized equilibrium experiences ?nancial
crises with a 5.5 percent probability, which is more than 10 times larger than that
of the social planner.
Table 1.3: Second Moments
Decentralized Social
Equilibrium Planner Data
Standard Deviations
Consumption 5.9 5.3 6.2
Real Exchange Rate 9.2 3.5 8.2
Current Account-GDP 2.8 0.6 3.6
Trade Balance-GDP 2.9 0.7 2.4
Correlation with GDP in units of tradables
Consumption 0.84 0.85 0.88
Real Exchange Rate -0.79 -0.43 -0.41
Current Account-GDP -0.76 -0.05 -0.63
Trade Balance-GDP -0.77 -0.22 -0.84
Note: Data is annual from WDI for Argentina from 1965-2007. The real ex-
change rate is calculated as
_
?
1/(1+?)
+ (1 ??)
1/(1+?)
_
p
N
_
?/(1+?)
_
?(1+?)/?
and is measured empirically using value added de?ators.
Table 3 also shows that the model accounts reasonably well for observed busi-
ness cycle moments for Argentina, in line with previous studies. Moreover, it is ap-
parent that the externality is important in accounting for two key regularities in the
31
emerging market business cycle: the high volatility of consumption and the strong
procyclicality of capital ?ows (see e.g. Kaminsky, Reinhart, and Vegh, 2004). The
constrained-e?cient equilibrium cannot account for these two stylized facts. This
occurs because the social planner accumulates su?ciently large precautionary sav-
ings to make large reversals in capital ?ows a much lower probability event compared
to the decentralized equilibrium.
1.5.6 Welfare E?ects
We compute the welfare gains from correcting the externality as the propor-
tional increase in consumption for all possible future histories in the decentralized
equilibrium that would make households indi?erent between remaining in the de-
centralized equilibrium (without government intervention) and correcting the exter-
nality. These calculations explicitly consider the costs of a lower consumption in
the transition to the constrained-e?cient allocations. Because of the homotheticity
of the utility function, the welfare gain ? at a state (b, y) is given by:
(1 + ? (b, y))
1??
V
de
(b, y) = V
sp
(b, y) (1.18)
The welfare gains from correcting the externality are shown in Figure 1.5 as
a function of current bond holdings, for negative, one- standard-deviation endow-
ment shocks. Notice the parallel between the welfare e?ects and the three regions
described in Figure 1, which gives the welfare gains from correcting the externality
a hump shape. In the constrained region, the borrowing decisions coincide; there-
32
fore, the welfare gains only arise from how future allocations will di?er. In the tax
region, because the social planner acts in a signi?cantly more precautionary way,
the welfare gains increase. In the no-tax region, where ?nancial crises are less likely,
borrowing decisions are similar in the two equilibria and welfare gains are smaller.
On average, the welfare gains from correcting the externality are 0.135 percent
of permanent consumption, consistent with the well-known result that the welfare
cost of business cycles is typically small. Even if the planner does not introduce
additional securities that partially complete the market, welfare gains are still larger
than the bene?ts from introducing asset price guarantees (Durdu and Mendoza,
2006) or the bene?ts from introducing indexed bonds (Durdu, 2009), often suggested
as policies to address Sudden Stops (Caballero, 2002).
?1 ?0.9 ?0.8 ?0.7 ?0.6
0.05
0.1
0.15
0.2
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
o
f
P
e
r
m
a
n
e
n
t
C
o
n
s
u
m
p
t
i
o
n
Figure 1.5: Welfare gains from correcting externality for negative one-standard-
deviation shocks
We see these welfare gains of correcting the externality as a lower bound.
First, the supply side of the economy is the same for both equilibria. If ?nancial
crises distort the e?cient use of production resources, correcting the externality
33
could deliver higher welfare gains. Second, the risk we have considered is only
aggregate; Chatterjee and Corbae (2007) shows that the welfare gains of eliminating
the possibility of a crisis state can be as large as 7 percent of permanent consumption
when considering idiosyncratic risk.
1.5.7 Simple forms of intervention:
Decentralizing the social planner’s allocations requires a state-contingent pol-
icy that might be challenging to implement in practice. Therefore, we also in-
vestigate if more simple forms of intervention can take the economy close to the
second-best. In particular, we ?nd that a ?xed tax on debt induces welfare gains
that are quite close to the second-best solution. The optimal ?xed tax is 3.6 percent,
which is about 70 percent of the average of the state-contingent tax, and achieves 62
percent of the welfare gains from implementing the constrained-e?cient allocations.
This ?xed tax cuts by more than half the probability of a crisis. Allowing the tax
to drop to zero when the credit constraint binds in the regulated economy or in the
constrained region, delivers about the same welfare gains as the ?xed tax.
By contrast, a ?xed tightening in margins across all states of nature delivers
welfare losses. This is intuitive given that tightening margins when the constraint is
already binding delivers signi?cant welfare losses, which outweigh the bene?ts from
a lower average amount of debt. Tightening margins outside the constrained region
only, however, does generate welfare gains, albeit smaller gains than a ?xed tax on
borrowing.
34
1.5.8 Sensitivity Analysis
In this section, we examine the sensitivity of our results to alternative calibra-
tions. Figure 1.6 shows how the average welfare e?ects and the optimal average tax
on debt vary with some key parameters. The online Appendix includes results from
changing all parameter values and details of the sensitivity analysis.
We can gain a better understanding of the results by analyzing the externality
term µ
sp
t
?
t
, which is the wedge between the social shadow value of wealth and the
private shadow value of wealth. Recalling that ?
t
? ?
N
(p
N
t
c
N
t
)/(c
T
t
) (1 +?) , we
have that the fraction of nontradable output that can be collateralized, the size
of the nontradable sector and the elasticity of substitution are the key parameters
determining the price e?ects in the externality term.
To arrive to a unit free measure of how the di?erent parameters a?ect the
price responses, we decompose the e?ects of the di?erent parameters in terms of
elasticities. Two elasticities are crucial for determining the price e?ects on the
borrowing capacity. First is the elasticity that measures how much the depreciation
of nontradables tightens the borrowing constraint. Second is the elasticity that
measures how much nontradables depreciate as a result of an increase in debt service.
Denoting the borrowing limit by ? ? ?
N
p
N
y
N
+ ?
T
y
T
, debt service by DS ?
b(1 +r) ?b
?
, and the elasticity of y with respect to x by ?
y,x
, we can decompose the
elasticity of the borrowing limit with respect to debt service as the product of these
35
two elasticities:
14
?
?,DS
=
_
p
N
y
N
y
T
+ p
N
y
N
_
. ¸¸ .
?
?,p
N
_
(? + 1)
DS
y
T
?DS
_
. ¸¸ .
?
p
N
,DS
(1.19)
In the baseline calibration,?
?,p
N = 0.65 and ?
p
N
,DS
= 0.40 at the median crisis in
the decentralized equilibrium. That is, a reduction in the debt service of 1 percent
would mitigate the drop in the price of nontradables during a median crisis by 0.40
percent (and the real exchange rate by 0.22 percent) and in turn this would increase
the borrowing capacity by 0.40
?
0.65 = 0.26 percent.
It is clear from (1.19) that the elasticity of substitution 1/(1 + ?) between
tradables and nontradables is a key determinant of how much the real exchange
rate depreciates as a result of an increase in debt service: a lower elasticity implies
that a given decrease in tradable consumption requires a greater adjustment in the
real exchange rate to equilibrate the market. Moreover, increases in the level of debt
service lead to a larger depreciation in the real exchange rate all else equal. While
the elasticity of substitution in our baseline calibration is at the high end of the
range of existing estimates, Panel A of Figure 1.6 shows that a smaller elasticity of
substitution increases the optimal average tax on debt and the welfare bene?ts of
correcting the externality. While in theory the overborrowing distortion is present for
any ?nite value of the elasticity, Panel A also shows that in practice, the allocations
of the social planner and the decentralized equilibrium become almost identical for
14
This formula yields as a result of applying the de?nition of ? and DS, the three elasticities
and using ? = ?(??)/(?DS).
36
values of the elasticity greater than 4.
The elasticity decomposition also shows that for low shares of the nontradable
sector, a depreciation of the real exchange rate has smaller e?ects on the borrowing
capacity. By adjusting ? to reduce the equilibrium share of the nontradable sector,
Panel B in Figure 1.6 shows that this is re?ected in lower e?ects from the externality.
We next study changes in ?
T
and ?
N
. It is reasonable to argue that in the
presence of debt denominated in units of tradable, creditors may be less willing
to accept nontradable income as collateral. We explore this idea by varying the
values of ?
N
and ?
T
such that ?
N
= c?
T
for c < 1 and such that the long-run
average of ?
N
p
N
y
N
+ ?
T
y
T
remains as in the baseline calibration. As Panel (D)
in Figure 1.6 shows, the e?ects of the externality remain signi?cant even if the
quality of nontradable collateral is half of the quality of tradable collateral, i.e.
when c = 0.5. In fact, crises in the decentralized equilibrium are 3.5 times more
likely and signi?cant di?erences remain in the severity of these episodes.
In Panel (C) we set ?
T
= ?
N
= ? and vary the value of ?. Notice that there
are two opposing e?ects from an increase in ?. On one hand, since ? scales up
the price e?ects in the externality term, we should expect higher e?ects from the
externality. On the other hand, increasing ? makes the credit constraint less likely
to bind, thereby reducing the externality. As Panel (C) shows, a relatively higher ?
increases the externality.
The other component of the externality term is the shadow value from relaxing
the credit constraint µ. While it is not possible to derive an analytical expression
for µ, for a given state µ should be positively related to the household’s share of
37
tradables in the utility function, and the inverse of the intratemporal and intertem-
poral elasticity of substitution, because these parameters a?ect the utility cost from
a large drop in tradable consumption; µ should also depend on the discount factor
and the interest rate, because these parameters a?ect the household’s impatience
and its willingness to borrow.
0 5 10 15 20
0
1
2
3
4
5
6
0 5 10 15 20
0
0.05
0.1
0.15
0.2
0.25
(A) Elasticity of Substitution
Average Tax (left axis) Welfare(right axis)
0.55 0.6 0.65 0.7 0.75
0
1
2
3
4
5
6
0.55 0.6 0.65 0.7 0.75
0
0.25
0.5
(B) Share of Nontradables in GDP
0.3 0.35 0.4
0
1
2
3
4
5
6
0.3 0.35 0.4
0
0.1
0.2
0.3
(C) Collateral Coefficient
0.4 0.6 0.8 1
0
1
2
3
4
5
6
0.4 0.6 0.8 1
0
0.04
0.08
0.12
0.16
(D) Relative Quality of Nontradable Collateral (c)
Figure 1.6: Sensitivity Analysis: Welfare and Implied Tax (in Percentage)
We extend the model by allowing for production in the nontradable sector with
intermediate inputs, as in Durdu, Mendoza, and Terrones (2009). Speci?cally, ?rms
use intermediate inputs m to produce nontradables with a technology such that y
t
=
A
t
m
?
t
. Firms maximize p
N
t
Am
?
t
?p
m
m
t
and redistribute pro?ts to households, whose
income is now given by the endowment of tradables plus pro?ts. Since a binding
credit constraint induces a depreciation of nontradables, this feature generates a drop
in the value of the marginal product of imported inputs, and therefore a drop in the
38
production of nontradables during ?nancial crises. As a result, since crises in the
decentralized equilibrium generate a larger depreciation, the externality generates a
larger drop in both tradable and nontradable consumption. Setting ? = 0.10 in line
with Goldberg and Campa (2010) and re-calibrating the rest of the parameters, we
?nd that the e?ects of the externality remain very similar overall.
15
Overall, the sensitivity analysis suggests that overborrowing creates signi?cant
distortions for plausible parameterizations. Only when the probability of a binding
credit constraint becomes negligible or when debt de?ation e?ects are very weak do
we ?nd that the e?ects of overborrowing are insigni?cant.
1.6 Policy Remarks
The new paradigm in ?nancial regulation stresses the need for a macropruden-
tial approach to consider how actions of individual market participants can destabi-
lize macroeconomic conditions with adverse e?ects over the whole economy (see e.g.
Borio, 2003). The analysis presented here suggests that overborrowing externalities
have a large enough quantitative impact on welfare to justify macroprudential reg-
ulation. It is worth noting that correcting these externalities does not eliminate the
possibility of ?nancial crises in our simulations, but the incidence and severity of
crises are considerably reduced under regulation. This is consistent with the con-
strained notion of e?ciency that we consider in our analysis: the social planner is
15
We continue to assume that in the constrained-e?cient allocations the social planner makes
borrowing decisions, while households choose consumption allocations and ?rms choose interme-
diate inputs. This yields an identical decentralization to the endowment economy model where a
tax on debt is su?cient to achieve constrained e?ciency.
39
subject to the same ?nancial frictions as the decentralized economy, so that reg-
ulation does not fully eliminate the ?nancial accelerator e?ects that arise when a
negative shock triggers the credit constraint.
In the context of the debate on ?nancial globalization, there is a view that a
Tobin-style tax can help smooth the boom-bust cycle caused by sharp changes in
access to credit in emerging markets. A recent “IMF Sta? Position Note” by Ostry et
al. (2010) emphasizes the bene?ts experienced by emerging markets from the recent
use of reserve requirements, although some controversy remains in the literature.
Our paper contributes to this debate by undertaking a quantitative investigation of
how curbing external ?nance can deliver a reduction in the vulnerability to ?nancial
crises while still allowing an economy to reap the bene?ts of access to global capital
markets. At the same time, fostering the development of ?nancial markets could
also generate signi?cant welfare gains by improving risk sharing and addressing the
root of the externality, i.e., the credit constraint. To the extent that the degree of
?nancial development remains incomplete, our results suggest that there is a scope
for “throwing sand in the wheels of international ?nance.”
1.7 Conclusions
This paper investigates a systemic credit externality that magni?es the inci-
dence and severity of ?nancial crises. Households accumulate precautionary savings
to smooth consumption during the cycle, but they fail to internalize the systemic
feedback e?ects between borrowing decisions, the real exchange rate, and ?nancial
40
constraints. By reducing the amount of debt ex-ante, a social planner mitigates the
downward spiral in the exchange rate and in the borrowing capacity during a crisis,
thereby improving social welfare.
The key contribution of this paper is its quantitative analysis of this external-
ity: we analyze the e?ects on ?nancial crisis dynamics and welfare, and the policy
measures needed to correct this externality. Our main conclusion is that there is
much to gain from introducing macroprudential regulation. Correcting the credit
externality reduces the long-run probability of a ?nancial crisis more than ten times
(from 5.5 percent to 0.4 percent) and reduces the consumption drop during a typical
crisis by 7 percentage points (from 17 percent to 10 percent).
On the policy side, we show that several regulatory measures commonly used to
maintain ?nancial stability can achieve the constrained-e?cient allocations. These
measures e?ectively impose an increase in the cost of borrowing whenever there is a
positive probability of a crisis, but before the crisis materializes so that the economy
becomes less vulnerable to future adverse shocks. While these policies are equivalent
in the model, in practice there are di?erent costs and bene?ts associated with their
actual implementation. We also acknowledge that the actual implementation of
these policies is a challenging task, but we also show that simple interventions such
as a ?xed tax on debt yields sizable welfare gains.
Within our framework, incorporating capital accumulation and specifying a
richer supply side of the economy would be important to extend the quantitative
analysis. There are also other natural extensions of our work. While our externality
stems from a feedback loop between the real exchange rate and ?nancial constraints,
41
our results suggest that pecuniary externalities resulting from a similar mechanism
involving other relative prices might play a quantitatively important role as well.
For example, it would be interesting to study a similar externality involving asset
prices and economic activity. Another direction for future research would be to
study the role for macroprudential regulation in a setup with an explicit role for
?nancial intermediation, as in for example Gertler and Karadi (2009). These issues
remain for future research.
42
Chapter 2
Overborrowing, Financial Crises and Macro-Prudential Policy
(coauthored with Enrique G. Mendoza)
2.1 Introduction
A common argument in narratives of the causes of the 2008 global ?nancial
crisis is that economic agents “borrowed too much.” The notion of “overborrowing,”
however, is often vaguely de?ned or presented as a value judgment on borrowing
decisions, in light of the obvious fact that a prolonged credit boom ended in collapse.
This lack of clarity makes it di?cult to answer two key questions: First, is over-
borrowing a signi?cant macroeconomic problem, in terms of causing ?nancial crises
and playing a central role in driving macro dynamics during both ordinary business
cycles and crises episodes? Second, are the the so-called “macro-prudential” policy
instruments e?ective to contain overborrowing and reduce ?nancial fragility, and if
so what are their main quantitative features?
In this paper, we answer these questions using a dynamic stochastic general
equilibrium model of asset prices and business cycles with credit frictions. We pro-
vide a formal de?nition of overborrowing and use quantitative methods to determine
how much overborrowing the model predicts and how it a?ects business cycles, ?-
nancial crises, and social welfare. We also compute a state-contingent schedule of
43
taxes on debt and dividends that can solve the overborrowing problem.
Our de?nition of overborrowing is in line with the one used in the academic lit-
erature (e.g. Lorenzoni, 2008, Korinek, 2009, Bianchi, 2010): The di?erence between
the amount of credit that an agent obtains acting atomistically in an environment
with a given set of credit frictions, and the amount obtained by a social planner
who faces the same frictions but internalizes the general-equilibrium e?ects of its
borrowing decisions. In the model, the credit friction is in the form of a collateral
constraint on debt that has two important features. First, it drives a wedge between
the marginal costs and bene?ts of borrowing considered by individual agents and
those faced by a social planner. Second, when the constraint binds, it triggers Irv-
ing Fisher’s classic debt-de?ation ?nancial ampli?cation mechanism, which causes
a ?nancial crisis.
This paper contributes to the literature by providing a quantitative assessment
of overborrowing in an equilibrium model of business cycles and asset prices. The
model is similar to those examined by Mendoza and Smith (2006) and Mendoza
(2010). These studies showed that cyclical dynamics in a competitive equilibrium
lead to periods of expansion in which leverage ratios raise enough so that the collat-
eral constraint becomes binding, triggering a Fisherian de?ation that causes sharp
declines in credit, asset prices, and macroeconomic aggregates.
1
In this paper, we
study instead the e?ciency properties of the competitive equilibrium, by comparing
its allocations with those attained by a benevolent social planner subject to the same
1
This is also related to the classic work on ?nancial accelerators by Bernanke and Gertler
(1989) and Kiyotaki and Moore (1997) and the more recent quantitative literature on this topic
as in the work of Jermann and Quadrini (2010).
44
credit frictions as agents in the competitive equilibrium. Thus, while those previ-
ous studies focused on the ampli?cation and asymmetry of the responses of macro
variables to aggregate shocks, we focus here on the di?erences between competitive
equilibria and constrained social optima.
In the model, the collateral constraint limits private agents not to borrow more
than a fraction of the market value of their collateral assets, which take the form
of an asset in ?xed aggregate supply (e.g. land). Private agents take the price of
this asset as given, and hence a “systemic credit externality” arises, because they do
not internalize that, when the collateral constraint binds, ?re-sales of assets cause a
Fisherian debt-de?ation spiral that causes asset prices to decline and the economy’s
borrowing ability to shrink in an endogenous feedback loop. Moreover, when the
constraint binds, production plans are also a?ected, because working capital ?nanc-
ing is needed in order to pay for a fraction of labor costs, and working capital loans
are also subject to the collateral constraint. As a result, when the credit constraint
binds output falls because of a sudden increase in the e?ective cost of labor. This
a?ects dividend streams and therefore equilibrium asset prices, and introduces an
additional vehicle for the credit externality to operate, because private agents do
not internalize the supply-side e?ects of their borrowing decisions.
We conduct a quantitative analysis in a version of the model calibrated to
U.S. data. The results show that ?nancial crises in the competitive equilibrium
are signi?cantly more frequent and more severe than in the constrained-e?cient
equilibrium. The incidence of ?nancial crises is about three times larger. Asset
prices drop about 25 percent in a typical crisis in the decentralized equilibrium,
45
versus 5 percent in the constrained-e?cient equilibrium. Output drops about 50
percent more, because the fall in asset prices reduces access to working capital
?nancing. The more severe asset price collapses also generate a “fat tail” in the
distribution of asset returns in the decentralized equilibrium, which causes the price
of risk to rise 1.5 times and excess returns to rise by 5 times, in both tranquil times
and crisis times. The social planner can replicate exactly the constrained-e?cient
allocations in a decentralized equilibrium by imposing taxes on debt and dividends
of about 1 and -0.5 percent on average respectively
The existing macro literature on credit externalities provides important back-
ground for our work. The externality we study is a pecuniary externality similar to
those examined in the theoretical studies of Caballero and Krishnamurthy (2001),
Lorenzoni (2008), and Korinek (2009). The pecuniary externality in these papers
arises because ?nancial constraints that depend on market prices generate ampli?-
cation e?ects, which are not internalized by private agents. The literature on par-
ticipation constraints in credit markets initiated by the work of Kehoe and Levine
(1993) has also examined the role of ine?ciencies that arise because of endogenous
borrowing constraints. In particular, Jeske (2006) showed that if there is discrimi-
nation against foreign creditors, private agents have a stronger incentive to default
than a planner who internalizes the e?ects of borrowing decisions on the domestic
interest rate, which a?ects the tightness of the participation constraint. Wright
(2006) then showed that as a consequence of this externality, subsidies on capital
?ows restore constrained e?ciency.
Our work is also related to the quantitative studies of macro-prudential policy
46
by Bianchi (2010) and Benigno, Chen, Otrok, Rebucci, and Young (2009). These
authors studied a credit externality at work in the model of emerging markets crises
of Mendoza (2002), in which agents do not internalize the e?ect of their individual
debt plans on the market price of nontradable goods relative to tradables, which
in?uences their ability to borrow from abroad. Bianchi examined how this external-
ity leads to excessive debt accumulation and showed that a tax on debt can restore
constrained e?ciency and reduce the vulnerability to ?nancial crises. Benigno et
al. studied how the e?ects of the overborrowing externality are reduced when the
planner has access to instruments that can a?ect directly labor allocations during
crises.
2
Our analysis di?ers from the above quantitative studies in that we focus on
asset prices as a key factor driving debt dynamics and the credit externality, instead
of the relative price of nontradables. This is important because private debt con-
tracts, particularly mortgage loans like those that drove the high household leverage
ratios of many industrial countries in the years leading to the 2008 crisis, use assets
as collateral. Moreover, from a theoretical standpoint, a collateral constraint linked
to asset prices introduces forward-looking e?ects that are absent when using a credit
constraint linked to goods prices. In particular, expectations of a future ?nancial
crisis a?ect the discount rates applied to future dividends and distort asset prices
even in periods of ?nancial tranquility. In addition, our model also di?ers in that
we study a production economy in which working capital ?nancing is subject to the
2
In a related paper Benigno et al. (2009) found that intervening during ?nancial crisis by
subsidizing nontradable goods leads to large welfare gains.
47
collateral constraint. As a result, the credit externality distorts production plans
and dividend rates, and thus again asset prices.
More recently, the quantitative studies by Nikolov (2009) and Jeanne and Korinek
(2010) examine other models of macro-prudential policy in which assets serve as col-
lateral.
3
Nikolov found that simple rules that impose tighter collateral requirements
may not be welfare-improving in a setup in which consumption is a linear function
that is not in?uenced by precautionary savings. In contrast, precautionary savings
are critical determinants of optimal borrowing decisions in our model, because of
the strong non-linear ampli?cation e?ects caused by the Fisherian debt-de?ation
dynamics, and for the same reason we ?nd that debt taxes are welfare improving.
Jeanne and Korinek construct estimates of a Pigouvian debt tax in a model in which
output follows an exogenous Markov-switching process and individual credit is lim-
ited to the sum of a fraction of aggregate, rather than individual, asset holdings plus
a constant term. In their calibration analysis, this second term dominates and the
probability of crises matches the exogenous probability of a low-output regime, and
as result the tax cannot alter the frequency of crises and has small e?ects on their
magnitude.
4
In contrast, in our model the probability of crises and their output dy-
namics are endogenous, and macro-prudential policy reduces sharply the incidence
and magnitude of crises.
Our results also contrast with the ?ndings of Uribe (2006). He found that
an environment in which agents do not internalize an aggregate borrowing limit
3
Galati and Moessner (2010) conduct an exhaustive survey of the growing literature in research
and policy circles on macro-prudential policy.
4
They also examined the existence of deterministic cycles in a non-stochastic version of the
model.
48
yields identical borrowing decisions to an environment in which the borrowing limit
is internalized.
5
An essential di?erence in our analysis is that the social planner
internalizes not only the borrowing limit but also the price e?ects that arise from
borrowing decisions. Still, our results showing small di?erences in average debt
ratios across competitive and constrained-e?cient equilibria are in line with his
?ndings.
The rest of the paper is organized as follows: Section 2 presents the analyt-
ical framework. Section 3 analyzes constrained e?ciency. Section 4 presents the
quantitative analysis. Section 5 provides conclusions.
2.2 Competitive Equilibrium
We follow Mendoza (2010) in specifying the economic environment in terms of
?rm-household units who make production and consumption decisions. Preferences
are given by:
E
0
_
?
?
t=0
?
t
u(c
t
?G(n
t
))
_
(2.1)
In this expression, E(·) is the expectations operator, ? is the subjective discount
factor, n
t
is labor supply and c
t
is consumption. The period utility function u(·) is
assumed to have the constant-relative-risk-aversion (CRRA) form. The argument of
u(·) is the composite commodity c
t
?G(n
t
) de?ned by Greenwood, Hercowitz, and Hu?man
(1988). G
is a convex, strictly increasing and continuously di?erentiable function5
He provided analytical results for a canonical endowment economy model with a credit con-
straint where there is an exact equivalence between the two sets of allocations. In addition, he
examined a model in which the exact equivalence of his ?rst example does not hold, but still
overborrowing is negligible.
49
that measures the disutility of labor supply. This formulation of preferences removes
the wealth e?ect on labor supply by making the marginal rate of substitution be-
tween consumption and labor depend on labor only.
Each household can combine land and labor services purchased from other
households to produce ?nal goods using a production technology such that y =
?
t
F(k
t
, h
t
), where F is a decreasing-returns-to-scale production function, k
t
repre-
sents individual land holdings, h
t
represents labor demand and ?
t
is a productivity
shock, which has compact support and follows a ?nite-state, stationary Markov
process. Individual pro?ts from this production activity are therefore given by
?
t
F(k
t
, h
t
) ?w
t
h
t
.
The budget constraint faced by the representative ?rm-household is:
q
t
k
t+1
+ c
t
+
b
t+1
R
t
= q
t
k
t
+ b
t
+ w
t
n
t
+ [?
t
F(k
t
, h
t
) ?w
t
h
t
] (2.2)
where b
t
denotes holdings of one-period, non-state-contingent discount bonds at the
beginning of date t, q
t
is the market price of land, R
t
is the real interest rate, and
w
t
is the wage rate.
The interest rate is assumed to be exogenous. This is equivalent to assuming
that the economy is a price-taker in world credit markets, as in other studies of
the U.S. ?nancial crisis like those of Boz and Mendoza (2010), Corbae and Quintin
(2009) and Howitt (2010), or alternatively it implies that the model can be in-
terpreted as a partial-equilibrium model of the household sector. This assump-
tion is adopted for simplicity, but is also in line with the evidence indicating that
50
in the era of ?nancial globalization even the U.S. risk-free rate has been signif-
icantly in?uenced by outside factors, such as the surge in reserves in emerging
economies and the persistent collapse of investment rates in South East Asia af-
ter 1998. Warnock and Warnock (2009) provide econometric evidence of the sig-
ni?cant e?ect of foreign capital in?ows on U.S. T-bill rates since the mid 1980s.
Mendoza and Quadrini (2009) document that about 1/2 of the surge in net credit
in the U.S. economy since then was ?nanced by foreign capital in?ows, and more
than half of the stock of U.S. treasury bills is now owned by foreign agents.
Household-?rms are subject to a working capital constraint. In particular,
they are required to borrow a fraction ? of the wages bill w
t
h
t
at the beginning
of the period and have to repay at the end of the period. In the conventional
working capital setup, a cash-in-advance-like motive for holding funds to pay for
inputs implies that the wages bill carries a ?nancing cost determined by the inter-
period interest rate. In contrast, here we simply assume that working capital funds
are within-period loans. Hence, the interest rate on working capital is zero, as
in some recent studies on the business cycle implications of working capital and
credit frictions (e.g. Chen and Song (2009)). We follow this approach so as to show
that the e?ects of working capital in our model hinge only on the need to provide
collateral for working capital loans, as explained below, and not on the e?ect of
interest rate ?uctuations on e?ective labor costs.
6
As in Mendoza (2010), agents face a collateral constraint that limits total
6
We could also change to the standard setup, but in our calibration, ? = 0.14 and R = 1.028,
and hence working capital loans would add 0.4 percent to the cost of labor implying that our
?ndings would remain largely unchanged.
51
debt, including both intertemporal debt and atemporal working capital loans, not
to exceed a fraction ? of the market value of asset holdings (i.e. ? imposes a ceiling
on the leverage ratio):
?
b
t+1
R
t
+ ?w
t
h
t
? ?q
t
k
t+1
(2.3)
Following Kiyotaki and Moore (1997) and Aiyagari and Gertler (1999), we interpret
this constraint as resulting from an environment where limited enforcement prevents
lenders to collect more than a fraction ? of the value of a defaulting debtor’s assets,
but we abstract from modeling the contractual relationship explicitly.
2.2.1 Private Optimality Conditions
In the competitive equilibrium, agents maximize (2.1) subject to (2.2) and
(2.3) taking land prices and wages as given. This maximization problem yields the
following optimality conditions for each date t:
w
t
= G
?
(n
t
) (2.4)
?
t
F
h
(k
t
, h
t
) = w
t
[1 +?µ
t
/u
?
(t)] (2.5)
u
?
(t) = ?RE
t
[u
?
(t + 1)] + µ
t
(2.6)
q
t
(u
?
(t) ?µ
t
?) = ?E
t
[u
?
(t + 1) (?
t+1
F
k
(k
t+1
, h
t+1
) + q
t+1
)] (2.7)
where µ
t
? 0 is the Lagrange multiplier on the collateral constraint.
Condition (2.4) is the individual’s labor supply condition, which equates the
marginal disutility of labor with the wage rate. Condition (2.5) is the labor de-
52
mand condition, which equates the marginal productivity of labor with the e?ective
marginal cost of hiring labor. The latter includes the extra ?nancing cost ?µ
t
/u
?
(t)
in the states of nature in which the collateral constraint on working capital binds.
The last two conditions are the Euler equations for bonds and land respectively.
When the collateral constraint binds, condition (2.6) implies that the marginal util-
ity of reallocating consumption to the present exceeds the expected marginal utility
cost of borrowing in the bond market by an amount equal to the shadow price of
relaxing the credit constraint. Condition (2.7) equates the marginal cost of an extra
unit of land investment with its marginal gain. The marginal cost nets out from the
marginal utility of foregone current consumption a fraction ? of the shadow value
of the credit constraint, because the additional unit of land holdings contributes to
relax the borrowing limit.
Condition (2.7) yields the following forward solution for land prices:
q
t
= E
t
_
?
?
j=0
_
j
?
i=0
m
t+1+i
_
d
t+j+1
_
, m
t+1+i
?
?u
?
(t + 1 +i)
u
?
(t + i) ?µ
t+i
?
, d
t
? ?
t
F
k
(k
t
, h
t
)
(2.8)
Thus, we obtain what seems a standard asset pricing condition stating that, at
equilibrium, the date-t price of land is equal to the expected present value of the
future stream of dividends discounted using the stochastic discount factors m
t+1+i
,
for i = 0, ..., ?. The key di?erence with the standard asset pricing condition,
however, is that the discount factors are adjusted to account for the shadow value
of relaxing the credit constraint by purchasing an extra unit of land whenever the
collateral constraint binds (at any date t + i for i = 0, ..., ? ).
53
Combining (2.6), (2.7) and the de?nition of asset returns (R
q
t+1
?
d
t+1
+q
t+1
qt
),
it follows that the expected excess return on land relative to bonds (i.e. the equity
premium), R
ep
t
? E
t
(R
q
t+1
?R), satis?es the following condition:
R
ep
t
=
µ
t
(1 ??)
(u
?
(t) ?µ
t
?)E
t
[m
t+1
]
?
cov
t
(m
t+1
, R
q
t+1
)
E
t
[m
t+1
]
, (2.9)
where cov
t
(m
t+1
, R
q
t+1
) is the date-t conditional covariance between m
t+1
and R
q
t+1
.
Following Mendoza and Smith (2006), we characterize the ?rst term in the
right-hand-side of (2.9) as the direct (?rst-order) e?ect of the collateral constraint
on the equity premium, which re?ects the fact that a binding collateral constraint
exerts pressure to ?re-sell land, depressing its current price.
7
There is also an indirect
(second-order) e?ect given by the fact that cov
t
(m
t+1
, R
q
t+1
) is likely to become
more negative when there is a possibility of a binding credit constraint, because the
collateral constraint makes it harder for agents to smooth consumption.
Given the de?nitions of the Sharpe ratio (S
t
?
R
ep
t
?t(R
q
t+1
)
) and the price of risk
(s
t
? ?
t
(m
t+1
)/E
t
m
t+1
), we can rewrite the expected excess return and the Sharpe
ratio as:
R
ep
t
= S
t
?
t
(R
q
t+1
), S
t
=
µ
t
(1 ??)
(u
?
(t) ?µ
t
?)E
t
[m
t+1
] ?
t
(R
q
t+1
)
??
t
(R
q
t+1
, m
t+1
)s
t
(2.10)
where ?
t
(R
q
t+1
) is the date-t conditional standard deviation of land returns and
?
t
(R
q
t+1
, m
t+1
) is the conditional correlation between R
q
t+1
and m
t+1
. Thus, the col-
7
Notice that this e?ect vanishes when ? = 1, because when 100 percent of the value of land
can be collateralized, the shadow value of relaxing the constraint by acquiring an extra unit of
land equals the shadow value of relaxing it by reducing the debt by one unit.
54
lateral constraint has direct and indirect e?ects on the Sharpe ratio analogous to
those it has on the equity premium. The indirect e?ect reduces to the usual ex-
pression in terms of the product of the price of risk and the correlation between
asset returns and the stochastic discount factor. The direct e?ect is normalized by
the variance of land returns. These relationships will be useful later to study the
quantitative e?ects of the credit externality on asset pricing.
Since q
t
E
t
[R
q
t+1
] ? E
t
[d
t+1
+ q
t+1
], we can rewrite the asset pricing condition
in this way:
q
t
= E
t
?
?
j=0
_
j
?
i=0
E
t+i
R
q
t+1+i
_
?1
d
t+j+1
, (2.11)
Notice that (2.9) and (2.11) imply that a binding collateral constraint at date t
implies an increase in expected excess land returns and a drop in asset prices at t.
Moreover, since expected returns exceed the risk free rate whenever the collateral
constraint is expected to bind at any future date, asset prices at t are a?ected by
collateral constraint not just when the constraints binds at t, but whenever it is
expected to bind at any future date.
2.2.2 Recursive Competitive Equilibrium
The competitive equilibrium is de?ned by stochastic sequences of allocations
{c
t
, k
t+1
, b
t+1
, h
t
, n
t
}
?
t=0
and prices {q
t
, w
t
}
?
t=0
such that: (A) agents maximize utility
(2.1) subject to the sequence of budget and credit constraints given by (2.2) and
(2.3) for t = 0, ..., ?, taking as given {q
t
, w
t
}
t=?
t=0
; (B) the markets of goods, labor
and land clear at each date t. Since land is in ?xed supply
¯
K, the market-clearing
55
condition for land is k
t
=
¯
K. The market clearing condition in the goods and labor
markets are c
t
+
b
t+1
R
= ?
t
F(
¯
K, n
t
) + b
t
and h
t
= n
t
respectively.
We now characterize the competitive equilibrium in recursive form. The state
variables for a particular individual’s optimization problem at time t are the indi-
vidual bond holdings (b), aggregate bond holdings (B), individual land holdings (k),
and the TFP realization (? ). Aggregate land holdings are not carried as a state
variable because land is in ?xed supply. Denoting by ?(B, ?) the agents’ perceived
law of motion of aggregate bonds and q(B, ?) and w(B, ?) the pricing functions for
land and labor respectively, the agents’ recursive optimization problem is:
V (b, k, B, ?) = max
b
?
,k
?
,c,n,h
u(c ?G
) + ?E?
?
|?
[V (b
?
, k
?
, B
?
, ?
?
)] (2.12)
s.t. q(B, ?)k
?
+ c +
b
?
R
= q(B, ?)k + b + w(B, ?)n + [?F(k, h) ?w(B, ?)h]
B
?
= ?(B, ?)
?
b
?
R
+ ?w(B, ?)h ? ?q(B, ?)k
?
The solution to this problem is characterized by the decision rules
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?) and
ˆ
h(b, k, B, ?). The decision rule for bond
holdings induces an actual law of motion for aggregate bonds, which is given by
ˆ
b
?
(B,
¯
K, B, ?). In a recursive rational expectations equilibrium, as de?ned below,
the actual and perceived laws of motion must coincide.
De?nition 3 (Recursive Competitive Equilibrium)
A recursive competitive equilibrium is de?ned by an asset pricing func-
56
tion q(B, ?), a pricing function for labor w(B, ?), a perceived law of mo-
tion for aggregate bond holdings ?(B, ?), and a set of decision rules
_
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?),
ˆ
h(b, k, B, ?)
_
with associated
value function V (b, k, B, ?) such that:
1.
_
ˆ
b
?
(b, k, B, ?),
ˆ
k
?
(b, k, B, ?), ˆ c(b, k, B, ?), ˆ n(b, k, B, ?),
ˆ
h(b, k, B, ?)
_
and V (b, k, B, ?)
solve the agents’ recursive optimization problem, taking as given q(B, ?), w(B, ?)
and ?(B, ?).
2. The perceived law of motion for aggregate bonds is consistent with the actual
law of motion: ?(B, ?) =
ˆ
b
?
(B,
¯
K, B, ?).
3. Wages satisfy w(B, ?) = G
?
(ˆ n(B,
¯
K, B, ?)) and land prices satisfy q(B, ?) =
E
?
?
|?
_
?u
?
(ˆ c(?(B,?),
¯
K,?(B,?),?
?
)) [?
?
F
k(
¯
K,ˆ n(?(B,?)
¯
K,?(B,?),?
?
))+q(?(B,?),?
?
)]
u
?
(ˆ c(B,
¯
K,B,?))??max[0,u
?
(ˆ c(B,
¯
K,B,?))??RE
?
?
|?
u
?
(ˆ c(?(B,?),
¯
K,?(B,?),?
?
)]
_
4. Goods, labor and asset markets clear:
ˆ
b
?
(B,
¯
K,B,?)
R
+ˆ c(B,
¯
K, B, ?) = ?F(
¯
K, ˆ n(B,
¯
K, B, ?))+
B , ˆ n(B,
¯
K, B, ?) =
ˆ
h(B,
¯
K, B, ?) and
ˆ
k(B,
¯
K, B, ?) =
¯
K
2.3 Constrained-E?cient Equilibrium
2.3.1 Equilibrium without collateral constraint
We start studying the e?ciency properties of the competitive equilibrium by
brie?y characterizing an e?cient equilibrium in the absence of the collateral con-
straint (2.3). The allocations of this equilibrium can be represented as the solution
57
to the following standard planning problem:
H(B, ?) = max
b
?
,c,n
u(c ?G
) + ?E?
?
|?
[H(B
?
, ?
?
)] (2.13)
s.t. c +
B
?
R
= ?F(
¯
K, n) + B
and subject also to either this problem’s natural debt limit, which is de?ned by B
?
>
?
min
F(
¯
K,n
?
(?
min
))
R?1
, where ?
min
is the lowest possible realization of TFP and n
?
(?
min
) is
the optimal labor allocation that solves ?
min
F
n
(
¯
K, n) = G
?
, or to a tighter ad-hoctime- and state-invariant borrowing limit.
The common strategy followed in quantitative studies of the macro e?ects of
collateral constraints (see, for example, Mendoza and Smith, 2006 and Mendoza,
2010) is to compare the allocations of the competitive equilibrium with the Fishe-
rian collateral constraint with those arising from the above benchmark case. The
competitive equilibria with and without the collateral constraint di?er in that in the
former private agents borrow less (since the collateral constraint limits the amount
they can borrow and also because they build precautionary savings to self-insure
against the risk of the occasionally binding credit constraint), and there is ?nancial
ampli?cation of the e?ects of the underlying exogenous shocks (since binding col-
lateral constraints produce large recessions and drops in asset prices). Compared
with the constrained-e?cient equilibrium we de?ne next, however, we will show that
the competitive equilibrium with collateral constraints displays overborrowing (i.e.
agents borrow more than in the constrained-e?cient equilibrium ).
58
2.3.2 Recursive Constrained-E?cient Equilibrium
We study a benevolent social planner who maximizes the agents’ utility subject
to the resource constraint, the collateral constraint and the same menu of assets of
the competitive equilibrium.
8
In particular, we consider a social planner that is con-
strained to have the same “borrowing ability” (the same market-determined value
of collateral assets ?q(B, ?)
¯
K) at every given state as agents in the decentralized
equilibrium, but with the key di?erence that the planner internalizes the e?ects of
its borrowing decisions on the market prices of assets and labor.
9
The recursive problem of the social planner is de?ned as follows:
W(B, ?) = max
B
?
,c,n
u(c ?G
) + ?E?
?
|?
[W(B
?
, ?
?
)] (2.14)
s.t. c +
B
?
R
= ?F(
¯
K, n) + B
?
B
?
R
+ ?w(B, ?)n ? ?q(B, ?)
¯
K
where q(B, ?) is the equilibrium pricing function obtained in the competitive equi-
librium. Wages can be treated in a similar fashion, but it is easier to decentralize
the planner’s allocations as as competitive equilibrium if we assume that the plan-
8
We refer to the social planner’s equilibrium and constrained-e?cient equilibrium interchange-
ably. Our focus is on second-best allocations, so when we refer to the social planner’s choices it
should be understood that we mean the constrained social planner.
9
We could also allow the social planner to manipulate the borrowing ability state by state (i.e.,
by allowing the planner to alter ?q(B, ?)
¯
K). Allowing for this possibility can potentially increase
the welfare losses resulting from the externality but the macroeconomic e?ects are similar. In
addition, since asset prices are forward-looking, this would create a time-inconsistency problem in
the planner’s problem. Allowing the planner to commit to future actions would lead the planner
to internalize not only how today’s choice of debt a?ects tomorrow’s asset prices but also how it
a?ects asset prices and the tightness of collateral constraints in previous periods.
59
ner takes wages as given and wages need to satisfy w(B, ?) = G
?
.10
Under this
assumption, we impose the optimality condition of labor supply as a condition that
the constrained-e?cient equilibrium must satisfy, in addition to solving problem
(2.14) for given wages.
Using the envelope theorem on the ?rst-order conditions of problem (2.14) and
imposing the labor supply optimality condition, we obtain the following optimality
conditions for the constrained-e?cient equilibrium:
u
?
(t) = ?RE
t
[u
?
(t + 1) + µ
t+1
?
t+1
] +µ
t
, ?
t+1
? ?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
(2.15)
?
t
F
n
(
¯
K, n
t
) = G
?
(n
t
) [1 +?µ
t
/u
?
(t)] (2.16)
The key di?erence between the competitive equilibrium and the constrained-e?cient
allocations follows from examining the Euler equations for bond holdings in both
problems. In particular, the term µ
t+1
?
t+1
in condition (2.15) represents the addi-
tional marginal bene?t of savings considered by the social planner at date t, because
the planner takes into account how an extra unit of bond holdings alters the tight-
ness of the credit constraint through its e?ects on the prices of land and labor at
t+1. Note that, since
?q
t+1
?b
t+1
> 0 and
?w
t+1
?b
t+1
? 0, ?
t+1
is the di?erence of two opposing
e?ects and hence its sign is in principle ambiguous. The term
?q
t+1
?b
t+1
is positive, be-
cause an increase in net worth increases demand for land and land is in ?xed supply.
10
This implies that the social planner does not internalize the direct e?ects of choosing the
contemporaneous labor allocation on contemporaneous wages. We have also investigated the pos-
sibility of having the planner internalize these e?ects but results are very similar. This occurs
again because our calibrated interest rate and working capital requirement are very small.
60
The term
?w
t+1
?b
t+1
is positive, because the e?ective cost of hiring labor increases when
the collateral constraint binds, reducing labor demand and pushing wages down. We
found, however, that the value of ?
t+1
is positive in all our quantitative experiments
with baseline parameter values and variations around them, and this is because
?q
t+1
?b
t+1
is large and positive when the credit constraint binds due the e?ects of the Fisherian
debt-de?ation mechanism.
De?nition 4 (Recursive Constrained-E?cient Equilibrium)
The recursive constrained-e?cient equilibrium is given by a set of decision rules
_
ˆ
B
?
(B, ?), ˆ c(B, ?), ˆ n(B, ?)
_
with associated value function W(B, ?), and wages
w(B, ?) such that:
1.
_
ˆ
B
?
(B, ?), ˆ c(B, ?), ˆ n(B, ?)
_
and W(B, ?) solve the planner’s recursive opti-
mization problem, taking as given w(B, ?) and the competitive equilibrium’s
asset pricing function q(B, ?).
2. Wages satisfy w(B, ?) = G
?
(ˆ n(B, ?)).
2.3.3 Comparison of Equilibria & ‘Macro-prudential’ Policy
Using a simple variational argument, we can show that the allocations of the
competitive equilibrium are ine?cient, in the sense that they violate the conditions
that support the constrained-e?cient equilibrium. In particular, private agents un-
dervalue net worth in periods during which the collateral constraint binds. To see
this, consider ?rst the marginal utility of an increase in individual bond holdings.
By the envelope theorem, in the competitive equilibrium this can be written as
61
?V
?b
= u
?
(t). For the constrained-e?cient economy, however, the marginal bene?t of
an increase in bond holdings takes into account the fact that prices are a?ected by
the increase in bond holdings, and is therefore given by
?W
?b
= u
?
(t) + ?
t
µ
t
. If the
collateral constraint does not bind, µ
t
= 0 and the two expressions coincide. If the
collateral constraint binds, the social bene?ts of a higher level of bonds include the
extra term given by ?
t
µ
t
, because one more unit of aggregate bonds increases the
inter-period ability to borrow by ?
t
which has a marginal value of µ
t
.
The above argument explains why bond holdings are valued di?erently by the
planner and the private agents “ex post,” when the collateral constraint binds. Since
both the planner and the agents are forward looking, however, it follows that those
di?erences in valuation lead to di?erences in the private and social bene?ts of debt
accumulation “ex ante,” when the constraint is not binding. Consider the marginal
cost of increasing the level of debt at date t evaluated at the competitive equilibrium
in a state in which the constraint is not binding. This cost is given by the discounted
expected marginal utility from the implied reduction in consumption next period
?RE [u
?
(t + 1)] . In contrast, the social planner internalizes the e?ect by which the
larger debt reduces tomorrow’s borrowing ability by ?
t+1
, and hence the marginal
cost of borrowing at period t that is not internalized by private agents is given by
?RE
t
_
µ
t+1
_
?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
__
.
We now show that the planner can implement the constrained-e?cient allo-
cations as a competitive equilibrium in the decentralized economy by introducing a
macro-prudential policy that taxes debt and dividends (the latter can turn into a
62
subsidy too, as we show in the next Section, but we refer to it generically as a tax).
11
In particular, the planner can do this by constructing state-contingent schedules of
taxes on bond purchases (?
t
) and on land dividends (?
t
), with the total cost (rev-
enues) ?nanced (rebated) as lump-sum taxes (transfers). The tax on bonds ensures
that the planner’s optimal plans for consumption and bond holdings are consistent
with the Euler equation for bonds in the competitive equilibrium. This requires
setting the tax to ?
t
= E
t
µ
t+1
?
t+1
/E
t
u
?
(t + 1). The tax on land dividends ensures
that these optimal plans and the pricing function q(B, ?) are consistent with the
private agents’ Euler equation for land holdings.
The Euler equations of the competitive equilibrium with the macro-prudential
policy in place become:
u
?
(t) = ?R(1 +?
t
)E
t
[u
?
(t + 1)] +µ
t
(2.17)
q
t
(u
?
(t) ?µ
t
?) = ?E
t
[u
?
(t + 1) (?
t+1
F
k
(k
t+1
, n
t+1
)(1 + ?
t+1
) + q
t+1
)] (2.18)
By combining these two Euler equations we can derive the expected excess return
on land paid in the land market under the macro-prudential policy. In this case,
after-tax returns on land and bonds are de?ned as
˜
R
q
t+1
?
d
t+1
(1+?
t+1
)+q
t+1
qt
and
˜
R
t+1
? R(1 + ?
t
) respectively, and the after-tax expected equity premium reduces
to an expression analogous to that of the decentralized equilibrium:
˜
R
ep
t
=
µ
t
(1 ??)
E
t
[(u
?
(t) ?µ
t
?)m
t+1
]
?
Cov
t
(m
t+1
,
˜
R
q
t+1
)
E
t
[m
t+1
]
(2.19)
11
See Bianchi (2010) for other decentralizations using capital and liquidity requirements and
loan-to-value ratios.
63
This excess return also has a corresponding interpretation in terms of the Sharpe
ratio, the price of risk, and the correlation between land returns and the pricing
kernel as in the case of the competitive equilibrium without macro-prudential policy.
It follows from comparing the expressions for R
ep
t
and
˜
R
ep
t
that di?erences in
the after-tax expected equity premia of the competitive equilibria with and without
macro-prudential policy are determined by di?erences in the direct and indirect
e?ects of the credit constraint in the two environments. As shown in the next
Section, these e?ects are stronger in the decentralized equilibrium without policy
intervention, in which the ine?ciencies of the credit externality are not addressed.
Intuitively, higher leverage and debt in this environment imply that the constraint
binds more often, which strengthens the direct e?ect. In addition, lower net worth
implies that the stochastic discount factor covaries more strongly with the excess
return on land, which strengthens the indirect e?ect. Notice also that dividends in
the constrained-e?cient allocations are discounted at a rate which depends positively
on the tax on debt. This premium is required by the social planner so that the excess
returns re?ect the social costs of borrowing.
2.4 Quantitative Analysis
2.4.1 Calibration
We calibrate the model to annual frequency using data from the U.S. economy.
The functional forms for preferences and technology are the following:
64
u(c ?G
) =_
c ??
n
1+?
1+?
_
1??
?1
1 ??
? > 0, ? > 1 (2.20)
F(k, h) = ?k
?
K
h
?
h
, ?
K
, ?
h
? 0 ?
K
+ ?
h
< 1 (2.21)
The real interest rate is set to R ? 1 = 0.028 per year, which is the ex-
post average real interest rate on U.S. three-month T-bills during the period 1980-
2005. We set ? = 2, which is a standard value in quantitative DSGE models. The
parameter ? is inessential and is set so that mean hours are equal to 1, which requires
? = 0.64. Aggregate land is normalized to
¯
K = 1 without loss of generality and the
share of labor in output ?
h
is equal to 0.64, the standard value. The Frisch elasticity
of labor supply (1/?) is set equal to 1, in line with evidence by Kimball and Shapiro
(2008).
We follow Schmitt-Grohe and Uribe (2007) in taking M1 money balances in
possession of ?rms as a proxy for working capital. Based on the observations that
about two-thirds of M1 are held by ?rms (Mulligan, 1997) and that M1 was on
average about 14 percent of annual GDP over the period 1980 to 2009, we calibrate
the working capital-GDP ratio to be (2/3)0.14 = 0.093. Given the 64 percent labor
share in production, and assuming the collateral constraint does not bind, we obtain
? = 0.093/0.64 = 0.146.
The value of ? is set to 0.96, which is also a standard value but in addition it
supports an average household debt-income ratio in a range that is in line with U.S.
data from the Federal Reserve’s Flow of Funds database. Before the mid-1990s this
65
ratio was stable at about 30 percent. Since then and until just before the 2008 crisis,
it rose steadily to a peak of almost 70 percent. By comparison, the average debt-
income ratio in the stochastic steady-state of our baseline calibration is 38 percent.
A mean debt ratio of 38 percent is sensible because 70 percent was an extreme at
the peak of a credit boom and 30 percent is an average from a period before the
substantial ?nancial innovation of recent years.
Table 2.1: Calibration
Source / target
Interest rate R ?1 = 0.028 U.S. data
Risk aversion ? = 2 Standard DSGE value
Share of labor ?
n
= 0.64 U.S. data
Labor disutility coe?cient ? = 0.64 Normalization
Frisch elasticity parameter ? = 1 Kimball and Shapiro (2008)
Supply of land
¯
K = 1 Normalization
Working capital coe?cient ? = 0.14 Working Capital-GDP=9%
Discount factor ? = 0.96 Debt-GDP ratio= 38%
Collateral coe?cient ? = 0.36 Frequency of Crisis = 3%
Share of land ?
K
= 0.05 Housing-GDP ratio = 1.35
TFP process ?
?
= 0.014, ?
?
= 0.53 Std. dev. and autoc. of U.S. GDP
The values of ?, ?
K
and the TFP process are calibrated to match targets from
U.S. data by simulating the model. We set ?
K
so as to match the average ratio of
housing to GDP at current prices, which is equal to 1.35. The value of housing is
taken from the Flow of Funds, and is measured as real state tangible assets owned
by households (reported in Table B.100, row 4). The model matches the 1.35 ratio
when we set ?
K
= 0.05.
12
12
?
K
represents the share of ?xed assets in GDP, and not the standard share of capital income in
GDP. There is little empirical evidence about the value of this parameter, with estimates that vary
depending, for example, on whether we consider land used for residential or commercial purposes,
or owned by government at di?erent levels. We could also calibrate ?
K
using the fact that, in
66
TFP shocks follow a log-normal AR(1) process log(?
t
) = ? log(?
t?1
) + ?
t
. We
construct a discrete approximation to this process using the quadrature procedure
of Tauchen and Hussey (1991) using 15 nodes. The values of ?
?
and ? are set so that
the standard deviation and ?rst-order autocorrelation of the output series produced
by the model match the corresponding moments for the cyclical component of U.S.
GDP in the sample period 1947-2007 (which are 2.1 percent and 0.5 respectively).
This procedure yields ?
?
= 0.014 and ? = 0.53.
Finally, we set the value of ? so as to match the frequency of ?nancial crises in
U.S. data. We de?ne a ?nancial crisis as an event in which both the credit constraint
binds and there is a decrease in credit of more than one standard deviation. Then,
we set ? so that ?nancial crises in the baseline model simulation occur about 3
percent of the time, which is consistent with the fact that the U.S. has experienced
three major ?nancial crises in the last hundred years.
13
This yields the value of ? =
0.36.
We recognize that several of the parameter values are subject of debate (e.g.
there is a fair amount of disagreement about the Frisch elasticity of labor supply),
or relate to variables that do not have a clear analog in the data (as is the case with
? or ?). Hence, we will perform sensitivity analysis to examine the robustness of
our results to changes in the model’s key parameters.
a deterministic steady state where the collateral constraint does not bind, the value-of-land-GDP
ratio is equal to ?
K
/(R?1), which would imply ?
K
= 1.35(0.028) = 0.038. This yields very similar
results as ?
K
= 0.05.
13
The three crises correspond to the Great Depression, the Savings and Loans Crisis and the
Great Recession (see Reinhart and Rogo? (2008)). While a century may be a short sample for
estimating accurately the probability of a rare event in one country, Mendoza (2010) estimates
a probability of about 3.6 percent for ?nancial crises using a similar de?nition but applied to all
emerging economies using data since 1980.
67
2.4.2 Borrowing decisions
We start the quantitative analysis by exploring the e?ects of the credit ex-
ternality on optimal borrowing plans. The solution method is described in the
Appendix. Since mean output is normalized to 1, all quantities can be interpreted
as fractions of mean output.
The two panels of Figure 1 show the bond decision rules (b
?
) of private agents
and the social planner as a function of b (left panel) as well as the pricing function
for land (right panel), both for a negative two-standard-deviations TFP shock. The
key point is to note that the Fisherian de?ation mechanism generates non-monotonic
bond decision rules, instead of the typical monotonically increasing decision rules.
The point at which bond decision rules switch slope corresponds to the value of b at
which the collateral constraint holds with equality but does not bind. To the right
of this point, the collateral constraint does not bind and the bond decision rules
are upward-sloping. To the left of this point, the bond decision rules are decreasing
in b, because a reduction in current bond holdings results in a sharp reduction in
the price of land, as can be seen in the right panel, and tightens the borrowing
constraint, thus increasing b
?
.
As in Bianchi (2010), we can separate the bond decision rules in the left panel
of Figure 1 into three regions: a “constrained region,” a “high-externality region”
and a “low-externality region.” The “constrained region” is given by the range of
b in the horizontal axis with su?ciently high initial debt (i.e. low b) such that the
collateral constraint binds in the constrained-e?cient equilibrium. This is the range
68
with b ? ?0.385. In this region, the collateral constraint binds in both constrained-
e?cient and competitive equilibria, because the credit externality implies that the
constraint starts binding at higher values of b in the latter than in the former, as
we show below.
By construction, the total amount of debt (i.e. the sum of bond holdings and
working capital) in the constrained region is the same under the constrained-e?cient
allocations and the competitive equilibrium. If working capital were not subject
to the collateral constraint, the two bond decision rules would also be identical.
But with working capital in the constraint the two can di?er. This is because the
e?ective cost of labor di?ers between the two equilibria, since the increase in the
marginal ?nancing cost of labor when the constraint binds, ?µ
t
/u
?
(t), is di?erent.
These di?erences, however, are very small in the numerical experiments, and thus
the bond decision rules are approximately the same in the constrained region.
14
The high-externality region is located to the right of the constrained region,
and it includes the interval ?0.385 < b < ?0.363. Here, the social planner chooses
uniformly higher bond positions (lower debt) than private agents, because of the
di?erent incentives driving the decisions of the two when the constrained region
is near. In fact, private agents hit the credit constraint at b = ?0.383, while at
that initial b the social planner still retains some borrowing capacity. Moreover,
this region is characterized by “?nancial instability,” in the sense that the levels of
14
The choice of b
?
becomes slightly higher for the social planner as b gets closer to the upper
bound of the constrained region, because the deleveraging that occurs around this point is small
enough for the probability of a binding credit constraint next period to be strictly positive. As a
result, for given allocations, conditions (2.15) and (2.6) imply that µ is lower in the constrained-
e?cient allocations.
69
?0.4 ?0.38 ?0.36 ?0.34 ?0.32 ?0.3
?0.4
?0.39
?0.38
?0.37
?0.36
?0.35
?0.34
?0.33
?0.32
?0.31
?0.3
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
H
i
g
h
E
x
t
e
r
n
a
l
i
t
y
R
e
g
i
o
n
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
?0.4 ?0.38 ?0.36 ?0.34 ?0.32 ?0.3
1.1
1.15
1.2
1.25
1.3
1.35
H
i
g
h
E
x
t
e
r
n
a
l
i
t
y
R
e
g
i
o
n
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Current Bond Holdings
L
a
n
d
P
r
i
c
e
Decentralized Equilibrium
Constrained Efficient
Figure 2.1: Bond Decision Rules (left panel) and Land Pricing Function (right panel)
for a Negative Two-standard-deviations TFP Shock
debt chosen for t + 1 are high enough so that a negative TFP shock of standard
magnitude in that period can lead to a binding credit constraint that leads to large
falls in consumption, output, land prices and credit. We will show later that this
is also the region of the state space in which the planner uses actively its macro-
prudential policy to manage the ine?ciencies of the competitive equilibrium.
The low-externality region is the interval for which b ? ?0.363. In this region,
the probability of a binding constraint next period is zero for both the social planner
and the competitive equilibrium. The bond decision rules still di?er, however, be-
70
cause expected marginal utilities di?er for the two equilibria. But the social planner
does not set a tax on debt, because negative shocks cannot lead to a binding credit
constraint in the following period.
The long-run probabilities with which the constrained-e?cient (competitive)
economy visits the three regions of the bond decision rules are 2 (4) percent for
the constrained region, 69 (70) percent for the high-externality region, and 29 (27)
percent for the low-externality region. Both economies spend more than 2/3rds
of the time in the high-externality region, but the prudential actions of the social
planner reduce the probability of entering in the constrained region by a half. Later
we will show that this is re?ected also in ?nancial crises that are much less frequent
and less severe than in the competitive equilibrium.
The larger debt (i.e. lower bond) choices of private agents relative to the social
planner, particularly in the high-externality region, constitute our ?rst measure of
the overborrowing e?ect at work in the competitive equilibrium. The social planner
accumulates extra precautionary savings above and beyond what private individuals
consider optimal in order to self-insure against the risk of ?nancial crises. This
e?ect is quantitatively small in terms of the di?erence between the two decision
rules, but this does not mean that its macroeconomic e?ects are negligible. Later in
this Section we illustrate this point by comparing ?nancial crises events in the two
economies. In addition, the fact that small di?erences in borrowing decisions lead
to major di?erences when a crisis hits can be illustrated using Figure 2 to study
further the dynamics implicit in the bond decision rules.
Figure 2.2 shows bond decision rules for the social planner and the competitive
71
?0.39 ?0.385 ?0.38 ?0.375 ?0.37 ?0.365 ?0.36
?0.39
?0.38
?0.37
?0.36
?0.35
?0.34
?0.33
?0.32
Current Bond Holdings
N
e
x
t
P
e
r
i
o
d
t
B
o
n
d
H
o
l
d
i
n
g
s
Social Planner
Decentralized Equilibrium
b‘=b line
Average TFP
Low TFP
B’
A’
B
A
Figure 2.2: Comparison of Debt Dynamics
equilibrium over the range (-0.39,-0.36) for two TFP scenarios: average TFP and
TFP two-standard-deviations below the mean. The ray from the origin is the b
?
= b
line. We use a narrower range than in Figure 1 to “zoom in” and highlight the
di?erences in decision rules. Assume both economies start at a value of b such that
at average TFP the debt of agents in the competitive equilibrium would remain
unchanged (this is point A with b = ?0.389). If the TFP realization is indeed the
average, private agents in the decentralized equilibrium keep that level of debt. On
72
the other hand, the social planner builds precautionary savings and reduces its debt
to point B with b = ?0.386. Hence, the next period the two economies start at
the debt levels in A and B respectively. Assume now that at this time TFP falls
by two standard deviations. Now we can see the large dynamic implications of the
small di?erences in the bond decision rules of the two economies: The competitive
equilibrium su?ers a major correction caused by the Fisherian de?ation mechanism.
The collateral constraint becomes binding and the economy is forced to a large
deleveraging that results in a sharp reduction in debt (an increase in b to -0.347
at point A
?
). Consumption falls leading to a drop in the the stochastic discount
factor and a drop in asset prices. In contrast, the social planner, while also facing
a binding credit constraint, adjusts it debt marginally to just about b = ?0.379 at
point B
?
. This was possible for the social planner because, taking into account the
risk of a Fisherian de?ation and internalizing its price dynamics, the planner chose
to borrow less than agents in the decentralized equilibrium a period earlier.
Overborrowing can also be assessed by comparing the long-run distributions
of debt and leverage across the competitive and constrained-e?cient equilibria. The
fact that the planner accumulates more precautionary savings implies that its er-
godic distribution concentrates less probability at higher leverage ratios than in
the competitive equilibrium. Figure 2.3 shows the ergodic distributions of leverage
ratios (measured as
?b
t+1
+?wtnt
qt
¯
K
) in the two economies. The maximum leverage ra-
tio in both economies is given by ? but notice that the decentralized equilibrium
concentrates higher probabilities in higher levels of leverage. Comparing averages
across these ergodic distributions, however, mean leverage ratios di?er by less than
73
0.26 0.28 0.3 0.32 0.34 0.36
0
1
2
3
4
5
x 10
?4
Leverage
P
r
o
b
a
b
i
l
i
t
y
Decentralized Equilibrium
Social Planner
Figure 2.3: Ergodic Distribution of Leverage (
?b
t+1
+?wtht
qt
¯
K
)
1 prcent. Hence, overborrowing is relatively small again if measured by comparing
di?erences in unconditional long-run averages of leverage ratios.
15
2.4.3 Asset Returns
Overborrowing has important quantitative implications for asset returns and
their determinants. Figure 2.4 shows the long-run distributions of land returns for
the competitive equilibrium and the social planner. The key di?erence in these
distributions is that the one for the competitive equilibrium features fatter tails.
In particular, there is a sharply fatter left tail in the competitive equilibrium, for
15
Measuring “ex ante” leverage as
?bt+?wtht
qt
¯
K
, we ?nd that leverage ratios in the competitive
equilibrium can exceed the maximum of those for the planner 3 percent of the time and by up to
12 percentage points.
74
which the 99th percentile of returns is about -17.5 percent, v. -1.6 percent in the
constrained-e?cient equilibrium. The fatter left tail in the competitive equilib-
rium corresponds to states in which a negative TFP shock hits when agents have
a relatively high level of debt. Intuitively, as a negative TFP shock hits, expected
dividends decrease and this puts downward pressure on asset returns.
16
In addition,
if the collateral constraint becomes binding, asset ?re-sales lead to a further drop in
asset prices.
We show below that the fatter tails of the distribution of asset returns, and the
associated time-varying risk of ?nancial crises, have substantial e?ects on the risk
premium. These features of our model are similar to those examined in the literature
on asset pricing and “disasters” (see Barro, 2009). Note, however, that this literature
generally treats ?nancial disasters as resulting from exogenous stochastic processes
with fat tails and time-varying volatility, whereas in our setup ?nancial crises and
their time-varying risk are both endogenous.
17
The underlying shocks driving the
model are standard TFP shocks, even in periods of ?nancial crises. In our model, as
in Mendoza (2010), ?nancial crises are endogenous outcomes that occur when shocks
of standard magnitudes trigger a Fisherian de?ation. Table 2.2 reports statistics
that characterize the main properties of asset returns in the constrained-e?cient
and competitive equilibria. We also report statistics for a competitive equilibrium
16
Similarly, the fatter right-tail in the distribution of returns of the competitive equilibrium
corresponds to periods with positive TFP shocks, which were preceded by low asset prices due to
?re sales.
17
The literature on disasters typically uses Epstein-Zin preferences so as to be able to match
the large observed equity premia. Here we use standard CRRA preferences with a risk aversion
coe?cient of 2, and as we show later, we can obtain larger risk premia than in the typical CRRA
setup without credit frictions. Moreover, we obtain realistically large risk premia when the credit
constraint binds.
75
?25 ?20 ?15 ?10 ?5 0 5 10 15 20 25 30 35 40
0
0.2
0.4
0.6
0.8
1
Return on Land (in percentage))
P
r
o
b
a
b
i
l
i
t
y
Social Planner
Decentralized Equilibrium
Decentralized Equilibrium
Social Planner
Figure 2.4: Ergodic Cumulative Distribution of Land Returns
in which land in the collateral constraint is valued at a ?xed price equal to the
average price across the ergodic distribution ¯ q (i.e. the credit constraint becomes
?
b
t+1
Rt
+ ?w
t
n
t
? ?¯ qk
t+1
).
18
This ?xed-price scenario allows us to compare the
properties of asset returns in the competitive and social planner equilibria with a
setup in which a collateral constraint exists but the Fisherian de?ation channel and
the credit externality are removed.
Table 2.2 lists expected excess returns (i.e. the equity risk premia), the direct
and indirect (covariance) e?ects of the credit constraint on excess returns, the (log)
standard deviation of returns, the price of risk, and the Sharpe ratio. These moments
are reported for the unconditional long-run distributions of each model economy, as
18
Because the asset is in ?xed supply, these allocations would be the same if we use instead
an ad-hoc borrowing limit such that ?
bt+1
Rt
+ ?w
t
n
t
? ?¯ q
¯
K. The price of land, however, would be
lower since with the ad-hoc borrowing constraint land does not have collateral value.
76
well as for distributions conditional on the collateral constraint being binding and
not binding.
The mean unconditional excess return is 1.09 percent in the competitive equi-
librium v. only 0.17 percent in the constrained-e?cient equilibrium and 0.86 percent
in the ?xed-price economy. The risk premium in the competitive equilibrium is large,
about half as large as the risk-free rate. The fact that the other two economies pro-
duce lower premia indicates that the high premium of the competitive equilibrium
is the combined result of the Fisherian de?ation mechanism and the ine?ciencies
induced by the credit externality. Note also that the high premium produced by our
model contrasts sharply with the ?ndings of Heaton and Lucas (1996), who found
that credit frictions without the Fisherian de?ation mechanism do not produce large
premia, unless transactions costs are very large.
19
The excess returns conditional on the collateral constraint not binding in the
constrained-e?cient and ?xed-price economies are in line with those obtained in
classic asset pricing models that display the “equity premium puzzle.” The equity
premia we obtained in these two scenarios are driven only by the covariance e?ect,
as in the classic models, and they are negligible: 0.03 percent in the ?xed-price
economy and 0.06 percent in the constrained-e?cient economy. This is natural
because, without the constraint binding and with the e?ects of the credit externality
and the Fisherian de?ation removed or weakened, the model is in the same class as
those that display the equity premium puzzle. In contrast, our baseline competitive
19
The unconditional premium in the ?xed price economy, at 0.86 percent, is not trivial, but
note that it results from the fact that the constraint binds with very high probability, given the
smaller incentives to accumulate precautionary savings. The risk premium in the unconstrained
region of the ?xed-price model is only 0.03 percent, v. 0.23 in our baseline model.
77
economy yields a 0.23 percent premium conditional on the constraint not binding,
which is small relative to data estimates that range from 6 to 18 percent, but 4 to
8 times larger than in the other two economies.
Conditional on the collateral constraint being binding, mean excess returns
in the competitive equilibrium are nearly 14 percent, 4.86 percent for the social
planner, and 1.29 percent in the ?xed-price economy. Interestingly, the lowest un-
conditional premium is the one for the constrained-e?cient economy (0.17 percent),
but conditional on the constraint binding, the lowest premium is the one for the
?xed-price economy (1.29 percent).This is because on one hand the Fisherian de?a-
tion e?ect is still at work when the collateral constraint binds in the constrained-
e?cient economy, but not in the ?xed-price economy, while on the other hand the
constrained-e?cient economy has a lower probability of hitting the collateral con-
straint (so that the higher premium when the constraint binds does not weigh heavily
when computing the unconditional average). In turn, the probability of hitting the
collateral constraint is higher for the ?xed-price economy, because the incentive to
build precautionary savings is weaker when there is no Fisherian ampli?cation.
The unconditional direct and covariance e?ects of the collateral constraint on
excess returns are signi?cantly stronger in the competitive equilibrium than in the
constrained-e?cient and ?xed-price economies, and even more so if we compare them
conditional on the constraint being binding. Again, the direct and covariance e?ects
are larger in the competitive equilibrium because of the e?ects of the overborrowing
externality and the Fisherian de?ation mechanism.
In terms of the decomposition of excess returns based on condition (2.10),
78
Table 2.2: Asset Pricing Moments
Excess Direct Covariance
Return E?ect E?ect s
t
?
t
(R
q
t+1
) S
t
Decentralized Equilibrium
Unconditional 1.09 0.87 0.22 5.22 3.05 0.79
Constrained 13.94 13.78 0.16 4.05 2.71 11.75
Unconstrained 0.23 0.00 0.23 5.3 3.08 0.05
Constrained-E?cient Equilibrium
Unconditional 0.17 0.11 0.06 2.88 1.85 0.08
Constrained 4.86 4.80 0.06 3.02 2.07 2.38
Unconstrained 0.06 0.00 0.06 2.86 1.84 0.03
Fixed Price Equilibrium
Unconditional 0.86 0.82 0.04 2.59 1.69 0.46
Constrained 1.29 1.23 0.05 2.81 1.84 0.69
Unconstrained 0.03 0.00 0.03 2.16 1.39 0.02
Note: The table reports averages of the conditional excess return after taxes, the direct e?ect, the
covariance e?ect, the price of risk s
t
, the (log) volatility of the return of land denoted ?
t
(R
q
t+1
),
and the Sharpe ratio. All numbers except the Sharpe ratios are in percentage
Table 2.2 shows that the unconditional average of the price of risk is about twice as
large in the decentralized equilibrium than in the constrained-e?cient and ?xed-price
economies. This re?ects the fact that consumption, and therefore the pricing kernel,
?uctuate signi?cantly more in the decentralized equilibrium. The Sharpe ratio and
the variability of land returns are also much larger in the competitive equilibrium.
The increase in the former indicates, however, that the mean excess return rises
signi?cantly more than the variability of returns, which indicates that risk-taking is
“overcompensated” in the competitive equilibrium (relative to the compensation it
receives when the social planner internalizes the credit externality). Note also that
the correlations between land returns and the stochastic discount factor, not shown
in the Table, are very similar under the three equilibria and very close to 1. This is
79
important because it implies that the di?erences in excess returns and Sharpe ratios
cannot be attributed to di?erences in this correlation.
2.4.4 Incidence and Magnitude of Financial Crises
We show now that overborrowing in the competitive equilibrium increases the
incidence and severity of ?nancial crises. To demonstrate this result we construct an
event analysis of ?nancial crises with simulated data obtained by performing long
(100,000-period) stochastic time-series simulations of the competitive, constrained-
e?cient and ?xed-price economies, removing the ?rst 1,000 periods. A ?nancial
crisis episode is de?ned as a period in which the credit constraint binds and this
causes a decrease in credit that exceeds one standard deviation of the ?rst-di?erence
of credit in the corresponding ergodic distribution.
The event analysis exercise is important also because it sheds light on whether
the model can produce ?nancial crises with realistic features, which is a key ?rst step
in order to make the case for treating the normative implications of the model as rel-
evant. We show here that, while we did not aim to build a rich equilibrium business
cycle model so we could keep the analysis of the externality tractable, and hence
our match to the data is not perfect, the model does produce ?nancial crises with
realistic features in terms of abrupt, large declines in allocations, credit, and land
prices, and it supports non-crisis output ?uctuations in line with observed U.S. busi-
ness cycles. Moreover, studies more focused on matching data from ?nancial crisis
events have shown that the Fisherian de?ation mechanism can do well at explaining
80
crisis dynamics nested within realistic long-run business cycle co-movements (see
Mendoza (2010)).
The ?rst important result of the event analysis is that the incidence of ?nancial
crises is signi?cantly higher in the competitive equilibrium. We calibrated ? so
that he competitive economy experiences ?nancial crises with a long-run probability
of 3.0 percent. But with the same ? , ?nancial crises occur in the constrained-
e?cient economy only with 0.9 percent probability in the long run. Thus, the credit
externality increases the frequency of ?nancial crises by a factor of 3.33.
20
The second important result is that ?nancial crises are more severe in the com-
petitive equilibrium. This is illustrated in the event analysis plots shown in Figure
2.5. The event windows are for total credit, consumption, labor, output,TFP and
land prices, all expressed as deviations from long-run averages. These event dynam-
ics are shown for the decentralized, constrained-e?cient, and ?xed-price economies.
We construct comparable event windows for the three scenarios following this
procedure: First we identify ?nancial crisis events in the competitive equilibrium,
and isolate ?ve-year event windows centered in the period in which the crisis takes
place. That is, each event window includes ?ve years, the two years before the
crisis, the year of the crisis, and the two years after. Second, we calculate the
median TFP shock across all of these event windows in each year t ?2 to t +2, and
the median initial debt at t ? 2. This determines an initial value for bonds and a
20
We could also de?ne crises in the constrained-e?cient equilibrium by using the value of the
credit threshold obtained in the competitive equilibrium. However, with this criterion we would
obtain an even lower probability of crises, because credit declines equal to at least one standard
deviation of the ?rst-di?erence of credit in the decentralized equilibrium are zero-probability events
in the constrained e?cient equilibrium.
81
?ve-year sequence of TFP realizations. Third, we feed this sequence of shocks and
initial value of bonds to the decisions rules of each model economy and compute
the corresponding endogenous variables plotted in Figure 2.5. By proceeding in this
way, we ensure that the event dynamics for the three equilibria are simulated using
the same initial state and the same sequence of shocks.
21
The features of ?nancial crises at date t in the competitive economy are in
line with the results in Mendoza (2010): The debt-de?ation mechanism produces
?nancial crises characterized by sharp declines in credit, consumption, asset prices
and output.
The ?ve macro variables illustrated in the event windows show similar dynam-
ics across the three economies in the two years before the ?nancial crisis. When the
crisis hits, however, the collapses observed in the competitive equilibrium are much
larger. Credit falls about 20 percentage points more, and two years after the crisis
the credit stock of the competitive equilibrium remains 10 percentage points below
that of the social planner.
22
Consumption, asset prices, and output also fall much
more sharply in the competitive equilibrium than in the planner’s equilibrium. The
declines in consumption and asset prices are particularly larger (-16 percent v. -5
percent for consumption and -24 percent v. -7 percent for land prices). The asset
price collapse also plays an important role in explaining the more pronounced de-
21
The sequence of TFP shocks is 0.9960, 0.9881, 0.9724, 0.9841, 0.9920 and the initial level of
debt is 1.6 percent above the average.
22
The model overestimates the drop in credit relative to what we have observed so far in the
U.S. crisis (which as of the third quarter of 2010 reached about 7 percent of GDP) . One reason
for this is that in the model, credit is in the form of one-period bonds, whereas in the data, loans
have on average a much larger maturity. In addition, our model does not take into account the
strong policy intervention that took place with the aim to prevent what would have been a larger
credit crunch.
82
cline in credit in the competitive equilibrium, because it re?ects the outcome of the
Fisherian de?ation mechanism. Output falls by 2 percentage points more, and labor
falls almost 3 percentage points more, because of the higher shadow cost of hiring
labor due to the e?ect of the tighter binding credit constraint on access to working
capital.
t?2 t?1 t t+1 t+2
?20
?10
0
Credit
%
Decentralized Equilibrium Social Planner Fixed Price Equilibrium
t?2 t?1 t t+1 t+2
?15
?10
?5
0
5
Consumption
%
t?2 t?1 t t+1 t+2
?25
?15
?5
5
Asset Price
%
t?2 t?1 t t+1 t+2
?8
?6
?4
?2
0
Output
%
t?2 t?1 t t+1 t+2
?6
?4
?2
0
Employment
%
t?2 t?1 t t+1 t+2
?3
?2
?1
0
TFP shock
%
Figure 2.5: Event Analysis: percentage di?erences relative to unconditional averages
83
The event analysis results can also be used to illustrate the relative signi?cance
of the wage and land price components in the externality term ?
t
? ?
¯
K
?qt
?bt
??n
t
?wt
?bt
identi?ed in condition (2.15). Given the unitary Frisch elasticity of labor supply,
wages decrease one-to-one with labor (and hence the event plot for wages would
be identical to the one shown for employment in Figure 2.5). As a result, the
extent to which the drop in wages can help relax the collateral constraint is very
limited. Wages and employment fall about 6 percent at date t , and with a working
capital coe?cient of ? = 0.14, this means that the e?ect of the drop in wages in
the borrowing capacity is 0.14(1 ? 0.06)0.06 = 0.79 percent. On the other hand,
given that
¯
K = 1 and that asset prices fall about 25 percent below trend at date
t, and since ? = 0.36, the e?ect of land ?re sales on the collateral constraint is
0.36(0.25) = 9 percent. Thus, the land price e?ect of the externality is about 10
times bigger than the wage e?ect. This ?nding will play an important role in our
quantitative analysis of the features of the macro-prudential policy later in this
section.
The ?xed-price economy displays very little ampli?cation given that the econ-
omy is free from the Fisherian de?ation mechanism. Credit increases slightly at date
t in order to smooth consumption and remains steady in the following periods. The
fact that land is valued at the average price, and not the market price, contributes
to mitigate the drop in the price of land, since it remains relatively more attractive
as a source of collateral.
To gain more intuition on why land prices drop more because of the credit
externality, we plot in Figure 2.6 the projected conditional sequences of future divi-
84
dends and land returns on land up to 30 periods ahead of a ?nancial crisis that occurs
at date t = 0 (conditional on information available on that date). These are the
sequences of dividends and returns used to compute the present values of dividends
that determine the equilibrium land price at t in the event analysis of Figure 2.5.
The expected land returns start very high when the crisis hits in both competitive
and constrained-e?cient equilibria, but signi?cantly more for the former (at about
40 percent) than the latter (at 10 percent). On the other hand, expected dividends
do not di?er signi?cantly, and therefore we conclude that the sharp change in the
pricing kernel re?ected in the surge in projected land returns when the crisis hits is
what drives the large di?erences in the drop of asset prices.
0 10 20 30
0.0488
0.049
0.0492
0.0494
0.0496
0.0498
0.05
0.0502
Expected Dividend
DE
SP
0 10 20 30
0
10
20
30
40
P
e
r
c
e
n
t
a
g
e
Discount Rate for Dividends
DE
SP
Figure 2.6: Forecast of expected dividends and land returns.
The large deleveraging that takes places when a ?nancial crisis occurs in the
competitive equilibrium implies that projected land returns for the immediate future
(i.e. the ?rst 6 periods after the crisis) drop signi?cantly. Returns are also projected
to fall for the social planner, but at a lower pace, so that in fact the planner projects
85
higher land returns than agents in the competitive equilibrium for a few periods.
Projected dividends for the same immediate future after the crisis are slightly smaller
than the long-run average of 0.05 in both economies because of the persistence of
the TFP shock. In the long-run, expected dividends are slightly higher for the social
planner, because the marginal productivity of land drops less during the ?nancial
crisis as a result of the lower amount of debt. Notice also that the planner projects
to discount dividends with a slightly higher land return in the long run, because the
tax on debt more than o?sets the fact that the risk premium of the planner is lower
(recall that we are comparing after-tax returns as de?ned in Section 2). This arises
because the tax on debt makes bonds relatively more attractive and this leads in
equilibrium to a higher required return on land.
2.4.5 Long-Run Business Cycles
Table 2.3 reports the long-run business cycle moments of the competitive,
constrained-e?cient and ?xed-price equilibria, which are computed using each econ-
omy’s ergodic distribution. The credit externality at work in the competitive equi-
librium produces higher business cycle variability in output and labor, and especially
in consumption, compared with the constrained-e?cient and ?xed-price economies.
The high variability of consumption and credit are consistent with the results in
Bianchi (2010), but we ?nd in addition that the credit externality produces a mod-
erate increase in the variability of labor and a substantial increase in the variability
of land prices and leverage. Notice that the variability in consumption is higher
86
than the variability of output in the decentralized equilibrium which is not the case
in U.S. data. However, if we exclude the crisis periods, the ratio of the variability of
consumption to the variability of GDP would be 0.87 (compared with 0.88 in annual
U.S. data from 1960 to 2007).
It may seem puzzling that we can obtain non-trivial di?erences in long-run
business cycle moments even though ?nancial crises are a low probability event in
the competitive equilibrium. To explain this result, it is useful to go back to Figure
1. This plot shows that even during normal business cycles the optimal plans of the
competitive and constrained-e?cient equilibria di?er, and this is particularly the
case in the high-externality region. Because the economy spends about 70 percent
of the time in this region, where private agents borrow more and are more exposed to
the risk of ?nancial crises, long-run business cycle moments di?er. In addition, the
larger e?ects that occur during crises have a non-trivial e?ect on long-run moments.
This is particularly noticeable in the case of consumption where the variability drops
from 2.7 to 1.7 in the decentralized equilibrium when we exclude the crises episodes.
The business cycle moments for consumption, output and labor in the constrained-
e?cient economy are about the same as those of the ?xed-price economy. This occurs
even though the constrained-e?cient economy is subject to the Fisherian de?ation
mechanism and the ?xed-price economy is not. The reason for this is because the
social planner accumulates extra precautionary savings, which compensate for the
sudden change in the borrowing ability when the credit constraint binds. The con-
straint binds less often and when it does it has weak e?ects on macro variables. On
the other hand, the constrained-e?cient economy does display lower variability in
87
leverage and land prices that the ?xed-price economy, and this occurs because the
social planner internalizes how a drop in the price tightens the collateral constraint.
The output correlations of leverage, credit, and land prices also di?er signif-
icantly across the model economies. The GDP correlations of leverage and credit
are signi?cantly higher in the competitive equilibrium, while the correlation between
the price of land and GDP is lower. The model without credit frictions would have a
natural tendency to produce countercyclical credit because consumption-smoothing
agents want to save in good times and borrow in bad times. This e?ect still domi-
nates in the constrained-e?cient and ?xed-price economies, but in the competitive
equilibrium the collateral constraint and the Fisherian de?ation hamper consump-
tion smoothing enough to produce procyclical credit and a higher GDP-leverage
correlation. Similarly, the GDP-land price correlation is nearly perfect when the
Fisherian de?ation mechanism is weakened (constrained-e?cient case) or removed
(?xed-price case), but falls to about 0.8 in the competitive equilibrium. Because
of the strong procyclicality of land prices, leverage is countercyclical with a GDP
correlation of -0.57. This is in line with the countercyclicality of household leverage
in U.S. data, although the correlation is lower than in the data (the correlation
between the ratio of net household debt to the value of residential land and GDP is
-0.25 at the business cycle frequency).
23
In terms of the ?rst-order autocorrelations, the competitive equilibrium dis-
23
Two caveats on this point. First, at lower frequencies the correlation is positive. As
Boz and Mendoza (2010) report, the household leverage ratio rose together with GDP, land prices
and debt between 1997 and 2007. Second, the countercyclicality of leverage for the household sector
di?ers sharply from the strong procyclicality of leverage in the ?nancial sector (see Adrian and Shin
(2010)).
88
plays lower autocorrelations in all its variables compared to both constrained-e?cient
and ?xed-price equilibria. This occurs because crises in the competitive equilibrium
are characterized by deep but not very prolonged recessions.
Table 2.3: Long Run Moments
Standard Correlation Autocorrelation
Deviation with GDP
DE SP FP DE SP FP DE SP FP
Output 2.10 1.98 1.97 1.00 1.00 1.00 0.50 0.51 0.51
Consumption 2.71 1.87 1.85 0.86 0.99 0.99 0.23 0.56 0.57
Employment 1.25 1.02 0.98 0.97 1.00 1.00 0.42 0.50 0.51
Leverage 3.92 2.72 3.80 -0.57 -0.93 -0.95 0.59 0.69 0.71
Total Credit 3.55 0.95 0.76 0.27 -0.35 -0.42 0.58 0.77 0.81
Land Price 3.95 2.24 3.48 0.79 0.97 0.97 0.16 0.56 0.60
Working capital 2.48 2.04 1.97 0.97 1.00 1.00 0.42 0.50 0.51
Note: ‘DE’ represents the decentralized equilibrium,‘SP’ represents the social planner, ’FP’
represents an economy with land valued at a ?xed price equal to the average of the price of
land in the competitive equilibrium.
2.4.6 Properties of Macro-prudential Policies
Table 4 shows the statistical moments that characterize the state-contingent
schedules of taxes on debt and dividends by which the social planner decentralizes
the constrained-e?cient allocations as a competitive equilibrium. To make the two
comparable, we express the dividend tax as a percent of the price of land.
The unconditional average of the debt tax is 1.07 percent, v. 0.09 when the
constraint binds and 1.09 when it does not. The tax remains positive, albeit small,
on average when the collateral constraint binds, because in some these states the
social planner wants to allocate borrowing ability across bonds and working cap-
ital in a way that di?ers from the competitive equilibrium. If there is a positive
89
probability that the credit constraint will bind again next period, the social planner
allocates less debt capacity to bonds and more to working capital. As a result, a tax
on debt remains necessary in a subset of the constrained region. Note, however, that
these states are not associated with ?nancial crisis events in our simulations. They
correspond to events in which the collateral constraint binds but the deleveraging
that occurs is not strong enough for a crisis to occur.
The debt tax ?uctuates about 2/3rds as much as GDP and is positively cor-
related with leverage, i.e.
?b
t+1
+?wtnt
qt
¯
K
. This is consistent with the macro-prudential
rational behind the tax: The tax is high when leverage is building up and low when
the economy is deleveraging. Note, however, that since leverage itself is negatively
correlated with GDP, the tax also has a negative GDP correlation. When the con-
straint binds, the correlation between the tax and leverage is zero by construction,
because leverage remains constant at the value of ?.
Table 2.4: Long Run Moments of Macro-prudential Policies
Average Standard Correlation
Deviation with Leverage
Debt Dividend Debt Dividend Debt Dividend
Tax Tax Tax Tax Tax Tax
Unconditional 1.07 -0.46 1.41 0.62 0.73 -0.64
Constrained 0.09 0.52 0.41 0.04 0.0 0.0
Unconstrained 1.09 -0.49 1.40 0.61 0.81 -0.79
The unconditional average of the dividend tax is negative (i.e. it is a subsidy),
and it is very small at about -0.46 percent: when the constraint binds it is on average
about 0.52 percent (v. a -0.49 percent average when the constraint does not bind).
90
The fact that on average the planner requires a subsidy on dividends may seem
puzzling, given that land is less risky in the regulated decentralized equilibrium,
as we have shown above. There is another e?ect at work, however, because the
debt tax puts downward pressure on land prices by making bonds relatively more
attractive than land, and this e?ect turns out to be quantitatively larger. Thus,
since by de?nition the constrained-e?cient allocations are required to support the
same land pricing function of the competitive economy without policy intervention,
the planner calls for a dividend subsidy on average in order to o?set the e?ect of the
debt tax on land prices. The variability of the tax on dividends is 0.62 percentage
points, less than 1/3rds the variability of GDP. The correlation between this tax on
dividends and leverage is negative in the unconstrained region re?ecting the negative
correlation between the tax on debt and the tax on land explained above.
The dynamics of the debt and dividend taxes around crisis events are shown
in Figure 2.7. The debt tax is high relative to its average, at about 2.7 percent, at
t ? 2 and t ? 1, and this again re?ects the macro-prudential nature of these taxes:
Their goal is to reduce borrowing so as to mitigate the magnitude of the ?nancial
crisis if bad shocks occur. At date t the debt tax falls to zero, and it rises again at
t +1 and t +2 to about 2 percent. The latter occurs because this close to the crisis
the economy still remains ?nancially fragile (i.e. there is still a non-zero probability
of agents becoming credit constrained next period). The tax on dividends follows a
similar pattern. Dividends are subsidized at a similar rate before and after ?nancial
crises events, but they are actually taxed when crises occur. The reason is again
that the social planner needs to support the same pricing function of the competitive
91
equilibrium that would arise without policy intervention. Hence, with the tax on
debt falling to almost zero, there is pressure for land prices to be higher than what
that pricing function calls for, and hence dividends need to be taxed to o?set this
e?ect.
The macro-prudential behavior of the debt tax is very intuitive and follows
easily from the precautionary behavior of the planner we have described. On the
other hand, the tax on dividends and its dynamic behavior seem less intuitive and
harder to sell as a policy rule (i.e. the notion of proposing to tax dividends at the
through of a ?nancial crisis is bound to be unpopular). The two policy instruments
are required, however, in order to implement exactly the allocations of the con-
strained social planner as a decentralized competitive equilibrium. Moreover, the
planner’s allocations are guaranteed to attain a level of welfare at least as high as
that of the competitive equilibrium without macro-prudential policy, since this equi-
librium remains feasible to the social planner. If one takes the debt tax and not the
tax on dividends, one cannot guarantee this Pareto improvement. Indeed, we solved
a variant of the model in which we introduced the optimal schedule of debt taxes
but left the tax on dividends out, and found that average welfare is actually lower
than without policy intervention by -0.02 percent. This occurs because welfare in
the states of nature in which the constraint is already binding is lower than without
policy intervention.
24
Hence, while our results may provide a justi?cation for the
use of macro-prudential policies, they also provide a warning because selective use
24
If we reduce the debt tax we can obtain again average welfare gains, which again illustrates
the interdependence of macroprudential policies.
92
of macro-prudential policies (i.e. partial implementation of the policy instruments
indicated by the model) can reduce welfare in some states of nature. In this ex-
periment this happens because the selective use of the debt tax without the tax on
dividends lowers asset prices in some states of nature, and reduces welfare in those
states by reducing the value of collateral.
t?2 t?1 t t+1 t+2
0
1
2
3
Tax on Debt
P
e
r
c
e
n
t
a
g
e
t?2 t?1 t t+1 t+2
?1
?0.5
0
0.5
1
Tax on Dividends
P
e
r
c
e
n
t
a
g
e
Figure 2.7: Event Analysis: Macroprudential Policies
Jeanne and Korinek (2010) also compute a schedule of macroprudential taxes
on debt to correct a similar externality that arises because of a collateral constraint
that depends on asset prices. In their constraint, however, the agents’ borrowing ca-
pacity is determined by the aggregate level of assets and by a linear state- and time-
invariant term (i.e. their borrowing constraint is de?ned as
b
t+1
R
? ??q
t
¯
K??). The
fact that their constraint depends on aggregate rather than individual asset hold-
ings, as in our model, matters because it implies that agents do not value additional
asset holdings as a mechanism to manage their borrowing ability.
25
But more im-
25
To illustrate this point, we recomputed our model assuming that the borrowing constraint
depends on the aggregate value of assets, as in their setup. Because assets do not have individual
93
portantly, leaving aside this di?erence, they calibrate parameter values to ? = 0.046,
? = 3.07 and q
t
¯
K = 4.8, which imply that the e?ects of the credit constraint are
driven mainly by ?, and only less than 7 percent (0.07=0.046*4.8/(0.046*4.8+3.07))
of the borrowing ability depends on the value of asset holdings. As a result, the
Fisherian de?ation e?ect and the credit externality are weak, and hence they ?nd
that macroprudential policy lessens the macro e?ects of ?nancial crises much less
than in our setup. The asset price drop is reduced from 12.3 to 10.3 percent, and
the consumption drop is reduced from 6.2 to 5.2 percent (compared with declines
from 24 to 7 and 16 to 5 percent respectively in our model). Moreover, they model
the stochastic process of dividends as an exogenous, regime-switching Markov chain
such that the probability of a crisis (i.e. binding credit constraint) coincides with the
probability of a bad realization of dividends, implying that the probability of busts
is una?ected by macroprudential regulation. Thus, in their setup macro-prudential
policy is much less e?ective at reducing the magnitude of ?nancial crises and has no
impact on their incidence.
2.4.7 Welfare E?ects
We move next to explore the welfare implications of the credit externality.
To this end, we calculate welfare costs as compensating consumption variations for
each state of nature that make agents indi?erent between the allocations of the
competitive equilibrium and the constrained-e?cient allocations. Formally, for a
value as collateral, asset prices drop even more during crises, and this leads private agents to
accumulate more precautionary savings, which results in crises having zero-probability in the long-
run under both competitive and constrained-e?cient equilibria for our baseline calibration.
94
given initial state (B, ?) at date 0, the welfare cost is computed as the value of ?
such that the following condition holds:
E
0
?
?
t=0
?
t
u(c
DE
t
(1 +?) ?G(n
DE
t
)) = E
0
?
?
t=0
?
t
u(c
SP
t
?G(n
SP
t
)) (2.22)
where the superscript DE denotes allocations in the decentralized competitive equi-
librium and the superscript SP denotes the social planner’s allocations. Note that
these welfare costs re?ect also the welfare gains that would be obtained by introduc-
ing the social planner’s optimal debt and dividend tax policies, which by construc-
tion implement the constrained-e?cient allocations as a competitive equilibrium.
The welfare losses of the DE arise from two sources. The ?rst source is the
higher variability of consumption, due to the fact that the credit constraint binds
more often in the DE, and when it binds it induces a larger adjustment in asset
prices and consumption. The second is the e?ciency loss in production that occurs
due to the e?ect of the credit friction on working capital. Without the working
capital constraint, the marginal disutility of labor equals the marginal product of
labor. With the working capital constraint, however, the shadow cost of employing
labor rises when the constraint binds, and this drives a wedge between the marginal
product of labor and its marginal disutility. Again, since the collateral constraint
binds more often in the DE than in the SP, this implies a larger e?ciency loss.
Figure 6 plots the welfare costs of the credit externality as a function of b for
a negative, two-standard-deviations TFP shock. These welfare costs approximate a
bell shape skewed to the left. This is due to the di?erences in the optimal plans of the
95
social planner vis-a-vis private agents in the decentralized equilibrium. Recall than
in the constrained region, the current allocations of the decentralized equilibrium
essentially coincide with those of the constrained-e?cient economy, as described
in Figure 1. Therefore, in this region the welfare gains from implementing the
constrained-e?cient allocations only arise from how future allocations will di?er. On
the other hand, in the high-externality region, the constrained- e?cient allocations
di?er sharply from those of the decentralized equilibrium, and this generally enlarges
the welfare losses caused by the credit externality. Notice that, since the constrained-
e?cient allocations involve more savings and less current consumption, there are
welfare losses in terms of current utility for the social planner, but these are far
outweighed by less vulnerability to sharp decreases in future consumption during
?nancial crises. Finally, as the level of debt is decreased further and the economy
enters the low-externality region, ?nancial crises are unlikely and the welfare costs
of the ine?ciency decrease.
The unconditional average welfare cost over the decentralized equilibrium’s
ergodic distribution of bonds and TFP is 0.046 percentage points of permanent
consumption. This contrasts with Bianchi (2010) who found welfare costs about
3 times larger. Note, however, that our results are in line with his if we express
the welfare costs as a fraction of the variability of consumption. Consumption was
more volatile in his setup because he examined a calibration to data for emerging
economies, which are more volatile than the United States.
The fact that welfare losses from the externality are small although the di?er-
ences in consumption variability are large is related to the well-known Lucas result
96
?0.4 ?0.39 ?0.38 ?0.37 ?0.36 ?0.35 ?0.34 ?0.33 ?0.32 ?0.31 ?0.3
0
0.01
0.02
0.03
0.04
0.05
0.06
Current Bond Holdings
P
e
r
c
e
n
t
a
g
e
o
f
P
e
r
m
a
n
e
n
t
C
o
n
s
u
m
p
t
i
o
n
High Externality Region
Low Externality Region
C
o
n
s
t
r
a
i
n
e
d
R
e
g
i
o
n
Figure 2.8: Welfare Costs of the Credit Externality for a two-standard-deviations
TFP Shock
that models with CRRA utility, trend-stationary income, and no idiosyncratic un-
certainty produce low welfare costs from consumption ?uctuations. Moreover, the
e?ciency loss in the supply-side when the constraint binds produces low welfare costs
on average because those losses have a low probability in the ergodic distribution.
2.4.8 Sensitivity Analysis
We examine now how the quantitative e?ects of the credit externality change as
we vary the values of the model’s key parameters. Table C.1 shows the main model
statistics for di?erent values of ?, ?, ? and ?. The Table shows the unconditional
averages of the tax on debt and the welfare loss, the covariance e?ect on excess
97
returns, the probability of ?nancial crises, and the impact e?ects of a ?nancial crisis
on key macroeconomic variables. In all of these experiments, only the parameter
listed in the ?rst column changes and the rest of the parameters remain at their
baseline calibration values.
The results of the sensitivity analysis reported in Table C.1 can be understood
more easily by referring to the externality term derived in Section 2: The wedge
between the social and private marginal costs of debt that separate competitive and
constrained-e?cient equilibria, ?RE
t
_
µ
t+1
_
?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
__
. For given ?
and R, the magnitude of the externality is given by the expected product of two
terms: the shadow value of relaxing the credit constraint, µ
t+1
, and the associated
price e?ects ?
¯
K
?q
t+1
?b
t+1
??n
t+1
?w
t+1
?b
t+1
that determine the e?ects of the externality on the
ability to borrow when the constraint binds. As explained earlier, the price e?ects are
driven mostly by
?q
t+1
?b
t+1
, because of the documented large asset price declines when the
collateral constraint binds. It follows therefore, that the quantitative implications
of the credit externality must depend mainly on the parameters that a?ect µ
t+1
and
?q
t+1
?b
t+1
, as well as those that a?ect the probability of hitting the constraint.
The coe?cient of relative risk aversion ? plays a key role because it a?ects
both µ
t+1
and
?q
t+1
?b
t+1
. A high ? implies a low intertemporal elasticity of substitution in
consumption, and therefore a high value from relaxing the constraint since a binding
constraint hinders the ability to smooth consumption across time. A high ? also
makes the stochastic discount factors more sensitive to changes in consumption,
and therefore makes the price of land react more to changes in bond holdings.
Accordingly, rising ? from 2 to 2.5 rises the welfare costs of the credit externality
98
by a factor of 5, and widens the di?erences in the covariance e?ects across the
competitive and constrained-e?cient equilibria. In fact, the covariance e?ect in the
decentralized equilibrium increases from 0.22 to 0.37 whereas for the constrained
e?cient allocations the increase is from 0.06 to 0.08. Stronger precautionary savings
reduce the probability of crises in the competitive equilibrium, and ?nancial crises
become a zero-probability event in the constrained-e?cient equilibrium. Conversely,
reducing ? to 1 makes the externality extremely small, measured either by di?erences
in the incidence or severity of ?nancial crises.
26
The collateral coe?cient ? also plays an important role because it alters the
e?ect of land price changes on the borrowing ability. A higher ? implies that, for a
given price response, the change in the collateral value becomes larger. Thus, this
e?ect makes the externality term larger. On the other hand, a higher ? has two
additional e?ects that go in the opposite direction. First, a higher ? implies that
the direct e?ect of the collateral constraint on the land price is weaker, leading to
a lower fall in the price of land during ?re sales. Second, a higher ? makes the
constraint less likely to bind, reducing the externality. The e?ects of changes in ?
are clearly non-monotonic. If ? is equal to zero, there is no e?ect of prices on the
borrowing-ability. At the same time, for high enough values of ?, the constraint
never binds. In both cases, the externality does not play any role. Quantitatively,
Table C.1 shows that small changes in ? are positively associated with the size of
the ine?ciency. In particular, an increase in ? from the baseline value of 0.36 to
26
Notice that the probability of a crisis in the competitive equilibrium becomes 10 percent, more
than three times larger than the target employed in the baseline calibration due to the reduction
in the level of precautionary savings.
99
0.40 increases the welfare cost of the ine?ciency by a factor of 6 and ?nancial crises
again become a zero-probability event in the constrained-e?cient equilibrium.
The above results have interesting policy implications. In particular, they
suggest that while increasing credit access by rising ? may increase welfare relative to
a more ?nancially constrained environment, rising ? can also strengthen the e?ects of
credit externalities and hence make macro-prudential policies more desirable (since
the welfare cost of the externality also rises).
A high Frisch elasticity of labor supply ((1/?) = 1.2) implies that output
drops more when a negative shock hits. If the credit constraint binds, this implies
that consumption falls more, which increases the marginal utility of consumption
and raises the return rate at which future dividends are discounted.
27
Moreover,
everything else constant, a higher elasticity makes the externality term higher by
weakening the e?ects of wages on the borrowing capacity. Hence, a higher elasticity
of labor supply is associated with higher e?ects from the credit externality, captured
especially by larger di?erences in the severity of ?nancial crises, a higher probability
of crises, and a larger welfare cost of the credit externality.
The fraction of wages that have to be paid in advance ? plays a subtle role. On
one hand, a larger ? increases the shadow value of relaxing the credit constraint, since
this implies a larger rise in the e?ective cost of hiring labor when the constraint binds.
On the other hand, a larger ? implies, ceteris paribus, a weaker e?ect on borrowing
ability, since the reduction of wages that occurs when the collateral constraint binds
27
The increase in leisure mitigates the decrease in the stochastic discount factor but does not
compensate for the fall in consumption
100
has a positive e?ect on the ability to borrow. Quantitatively, increasing (decreasing)
? by 5 percent increases (decreases) slightly the e?ects that re?ect the size of the
externality.
Changes in the volatility and autocorrelation of TFP do not have signi?cant
e?ects. Increasing the variability of TFP implies that ?nancial crises are more
likely to be triggered by a large shock. This results in larger ampli?cation and
a higher bene?t from internalizing price e?ects. In general equilibrium, however,
precautionary savings increase too, resulting in a lower probability of ?nancial crises
for both equilibria. Therefore, the overall e?ects on the externality of a change in
the variability of TFP depend on the relative change in the probability of ?nancial
crisis in both equilibria and the change in the severity of these episodes. An increase
in the autocorrelation of TFP leads to more frequent ?nancial crises for given bond
decision rules. Again, in general equilibrium, precautionary savings increase making
ambiguous the e?ect on the externality.
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6
N
o
t
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3
102
In terms of the optimal debt on tax, the results of the sensitivity analysis
produce an important ?nding: The average debt tax of about 1.1 percent is largely
robust to the parameter variations we considered. Except for the scenario that
approximates logarithmic utility (? = 1), in all other scenarios included in Table
C.1 the mean tax ranges between 1.01 and 1.2 percent.
We consider now shocks that a?ect directly the collateral constraint by af-
fecting the extent to which agents can pledge assets as collateral. We consider a
stochastic process for ? that follows a symmetric two-state Markov chain indepen-
dent from shocks to TFP. In line with evidence from Mendoza and Terrones (2008)
on the mean duration of credit booms in industrial countries, we calibrate the proba-
bilities of the Markov chain so that the average duration of each state is 6 years. We
keep the average value of ? as in our benchmark model and consider ?uctuations of
? of 10 percent which is meant to be suggestive. As shown in Table C.1, the e?ects
of the externality remain largely unchanged to this modi?cation.
Overall, the results of the sensitivity analysis show that parameter changes
that weaken the model’s ?nancial ampli?cation mechanism also weaken the magni-
tude of the externality. This results in smaller average taxes, smaller welfare costs
and smaller di?erences in the incidence and severity of ?nancial crises. The coef-
?cient of risk aversion is particularly important also because it in?uences directly
the price elasticity of asset demand, and hence it determines how much asset prices
can be a?ected by the credit externality. This parameter plays a role akin to that
of to the elasticity of substitution in consumption of tradables and non-tradables in
Bianchi (2010), because in his model this elasticity drives the response of the price
103
at which the collateral is valued. Accordingly, he found that the credit externality
has signi?cant e?ects only if the elasticity is su?ciently low.
2.5 Conclusion
This paper examined the positive and normative e?ects of a credit externality
in a dynamic stochastic general equilibrium model in which a collateral constraint
limits access to debt and working capital loans to a fraction of the market value
of an asset in ?xed supply (e.g. land). We compared the allocations and welfare
attained by private agents in a competitive equilibrium in which agents face this
constraint taking prices as given, with those attained by a constrained social planner
that faces the same borrowing limits but takes into account how current borrowing
choices a?ect future asset prices and wages. This planner internalizes the debt-
de?ation process that drives macroeconomic dynamics during ?nancial crises, and
hence borrows less in periods in which the collateral constraint does not bind, so as
to weaken the debt-de?ation process in the states in which the constraint becomes
binding. Conversely, private agents overborrow in periods in which the constraint
does not bind, and hence are exposed to the stronger adverse e?ects of the debt-
de?ation mechanism when a ?nancial crisis occurs.
The novelty of our analysis is in that it quanti?es the e?ects of the credit
externality in a setup in which the credit friction has e?ects on both aggregate
demand and supply. The e?ects on demand are well-known from models with credit
constraints: consumption drops as access to debt becomes constrained, and this
104
induces an endogenous increase in excess returns that leads to a decline in asset
prices. Because collateral is valued at market prices, the drop in asset prices tightens
the collateral constraint further and leads to ?re-sales of assets and a spiraling
decline in asset prices, consumption and debt. On the supply side, production and
labor demand are a?ected by the collateral constraint because ?rms buy labor using
working capital loans that are limited by the collateral constraint, and hence when
the constraint binds the e?ective cost of labor rises, so the demand for labor and
output drops. This a?ects dividend rates and hence feeds back into asset prices.
Previous studies in the macro/?nance literature have shown how these mechanisms
can produce ?nancial crises with features similar to actual ?nancial crises, but the
literature had not conducted a quantitative analysis comparing constrained-e?cient
v. competitive equilibria in an equilibrium model of business cycles and asset prices.
We conducted a quantitative analysis in a version of the model calibrated to
U.S. data. This analysis showed that, even though the credit externality results in
only slightly larger average ratios of debt and leverage to output compared with the
constrained-e?cient allocations (i.e. overborrowing is not large), the credit external-
ity does produce ?nancial crises that are signi?cantly more severe and more frequent
than in the constrained-e?cient equilibrium, and produces higher long-run business
cycle variability. There are also important asset pricing implications. In particular,
the credit externality and its associated higher macroeconomic volatility in the com-
petitive equilibrium produce equity premia, Sharpe ratios, and market price of risk
that are much larger than in the constrained-e?cient equilibrium. We also found
that the degree of risk aversion plays a key role in our results, because it is a key
105
determinant of the response of asset prices to volatility in dividends and stochastic
discount factors. For the credit externality to be important, these price responses
need to be nontrivial, and we found that they are nontrivial already at commonly
used risk aversion parameters, and larger at larger risk aversion coe?cients that are
still in the range of existing estimates.
This analysis has important policy implications. In particular, the social plan-
ner can decentralize the constrained-e?cient allocations as a competitive equilibrium
by introducing an optimal schedule of state-contingent taxes on debt and dividends.
By doing so, it can neutralize the adverse e?ects of the credit externality and pro-
duce an increase in social welfare. In our calibrated model, the tax on debt necessary
to attain this outcome is about 1 percent on average. The tax is higher when the
economy is building up leverage and becoming vulnerable to a ?nancial crisis, but
before a crisis actually occurs, so as to induce private agents to value more the
accumulation of precautionary savings than they do in the competitive equilibrium
without taxes.
These ?ndings are relevant for the ongoing debate on the design of new ?nan-
cial regulation to prevent ?nancial crises, which emphasizes the need for “macro-
prudential” regulation. Our results lend support to this approach by showing that
credit externalities associated with ?re-sales of assets have large adverse macroeco-
nomic e?ects. At the same time, however, we acknowledge that actual implementa-
tion of macro-prudential policies in ?nancial markets remains a challenging task. In
particular, the optimal design of these policies requires detailed information on a va-
riety of credit constraints that private agents and the ?nancial sector face, real-time
106
data on their leverage positions, and access to a rich set of state-contingent policy
instruments. Moreover, as we showed in this paper, implementing only a subset
of the optimal policies because of these limitations (or limitations of the political
process) can reduce welfare in some states.
107
Chapter 3
E?cient Bailouts?
3.1 Introduction
The recent ?nancial crisis has led to massive government intervention in credit
markets. The initial Troubled Assets Relief Program (TARP), for example, required
700 billion dollars to provide credit assistance to ?nancial and non-?nancial institu-
tions. These measures have triggered an intense debate on the desirability of such
interventions. Supporters argue that these bailouts were necessary to avoid a com-
plete meltdown of the ?nancial sector that would have brought an extraordinary
contraction in output and employment. Critics argue that the presence of bailouts
generate incentives for investors to take even more risk ex ante, sowing the seeds
of future crises. Such critics, therefore, propose regulations to limit the ability of
central banks to conduct bailouts.
In this paper, we seek an answer to the following questions: What are the
implications of bailout expectations for the stability of the ?nancial sector? Is it
desirable to prohibit the use of public funds to conduct bailouts? If the central
bank could commit to a bailout policy, under what states of nature is it optimal to
conduct a bailout? How large should these bailouts be?
This paper answers these questions based on a non-linear DSGE model where
credit frictions generate scope for government credit intervention during a ?nancial
108
crisis, but where such e?ects generate more risk-taking before the crisis actually hits.
Recent research (see e.g. Gertler and Kiyotaki, 2010) has developed a quantitative
framework to analyze how governmental intervention can mitigate the credit crunch
and moderate the recession ex post. At the same time, a growing theoretical liter-
ature has emphasized the moral hazard implications of such interventions (see e.g.
Farhi and Tirole, 2010), but little is known about their quantitative implications.
In this paper, we develop a quantitative DSGE model to assess within a uni?ed
framework the interaction between ex-post interventions in credit markets and the
build-up of risk ex ante.
The model features a representative corporate entity that ?nances investment
using equity and debt. There are two key frictions a?ecting the capacity of ?rms to
?nance investment. First, debt contracts are not fully enforceable, giving rise to a
collateral constraint that limits the amount that ?rms can borrow. Second, there is
a constraint on minimum dividends that ?rms must make each period. Therefore,
?rms balance the desire to increase borrowing and investment now with the risk
of becoming ?nancially constrained in the future. When leverage is su?ciently
high and an adverse ?nancial shock hits the economy, ?rms hit their balance sheet
constraints, generating a fall in investment and a protracted recession.
In this environment, we consider a social planner that engages in direct credit
policy with the purpose of relaxing the balance sheet constraints of ?rms, but faces
two types of costs in this intervention. First, there is the static cost of transferring
resources from workers to ?rms. Second, there is the dynamic e?ect (or a moral
hazard e?ect) that arises because ?rms anticipate the planner’s intervention and
109
therefore take more risk. As a result, there is a trade o? involved in the design of a
bailout policy. On the one hand, a commitment to bailing out the entire corporate
sector in some adverse states of nature, i.e. a systemic ?nancial crisis, provides a
form of insurance against such future episodes, which the market would otherwise
not provide. On the other hand, as ?rms adjust their leverage choices, the economy
becomes more exposed to the deadweight losses produced by aggregate ?nancial
distress.
Our answer to the question of whether it is desirable or undesirable to prohibit
the use of public funds to conduct bailouts is that public funds can in fact be used
e?ectively to conduct bailouts. The quantitative analysis shows that is possible to
design a realistic and simple bailout policy so that the welfare bene?ts of such an
intervention outweigh its distortionary e?ects. In particular, we ?nd that in the
presence of a severe systemic ?nancial crisis, it is optimal to engineer a transfer of
funds from workers to ?rms.
This paper relates to di?erent strands of literature. First, there is a growing
literature that studies credit policy in quantitative DSGE models, building on the
work of Bernanke and Gertler (1989) and Kiyotaki and Moore (1997).
1
As in these
papers, we study how government intervention can mitigate ?nancial frictions that
are activated during ?nancial crises. For reasons of tractability, however, this liter-
ature mostly studies the optimal response to unanticipated crises or focuses on the
log-linear dynamics around the deterministic steady state, thereby abstracting from
1
Contributions include Kiyotaki and Moore (2008), Gertler and Kiyotaki (2010),
Gertler and Karadi (2009), Del Negro, Eggertsson, Ferrero, and Kiyotaki (2010),
Guerrieri and Lorenzoni (2011), Gertler, Kiyotaki, and Queralto (2010).
110
risk considerations and the moral hazard e?ects of government intervention. Instead,
a distinctive feature of this paper is the focus on how expectations of future bailouts
a?ect ex-ante risk-taking within a non-linear DSGE model. This is crucial in order
to assess the dynamic implications of credit intervention on ?nancial stability and
on welfare.
2
This paper is also closely related to the theoretical literature that analyzes the
incentive e?ects of bailouts on ?nancial stability. Farhi and Tirole (2010) analyze
how time-consistent systemic bailouts can generate strategic complementarities in
private leverage choices, causing excessive ?nancial fragility. Chari and Kehoe (2010)
show that ?re sale e?ects provide governments with stronger incentives to renegotiate
contracts than private agents, making the time-inconsistency problem more severe
for the government. Diamond and Rajan (2009) show that raising interest rates may
be optimal to penalize excessive risk taking. There are two key di?erences between
our paper and this literature: ?rst, we develop a quantitative framework to assess
the e?ects of bailouts on risk-taking; second, we emphasize the idea that systemic
bailouts can be bene?cial ex-ante as a way to provide insurance. In this aspect,
this paper is related to Schneider and Tornell (2004) and Keister (2010) who also
emphasize the insurance bene?ts of bailouts, but they focus on self-ful?lling crises.
This paper is also related to a growing literature that studies the normative
implications of ?nancial frictions and the role of macroprudential regulation. This
2
Recently, Gertler, Kiyotaki, and Queralto (2010) develop a model where banks face a portfolio
choice between short-term debt and equity and also consider moral hazard e?ects of credit policy
in this portfolio choice. Gertler et al. consider a log-linear approximation around the risk-adjusted
steady state in an equilibrium where ?nancial constraints are always binding. An key structural
di?erence in our model is that we disentangle the role of policy during normal times from the role
of policy during times of systemic crisis by solving the model using global methods.
111
literature emphasizes the role of ex-ante prudential measures to correct pecuniary
externalities due to ?nancial accelerator e?ects.
3
Our paper focuses instead on ex-
post policy measures and their e?ects on ex-ante risk taking decisions.
4
3.2 Analytical Framework
Our model economy is a small open economy populated by ?rms and workers
that are also the shareholders of the ?rms. This model shares with Jermann and Quadrini
(2010) the consideration of dividend policy and with Mendoza (2010) the analysis
of non-linear dynamics. We start by ?rst describing the decisions of the di?erent
agents in the economy, and then we describe the general equilibrium.
3.2.1 Corporate entities
There is a measure one of identical ?rms with technology given by the produc-
tion function F(z
t
, k
t
, n
t
) that combines capital denoted by k, and labor denoted by
h to produce a ?nal good. TFP denoted by z
t
follows a ?rst-order Markov process.
Consistent with the typical timing convention, k
t
is chosen at time t ?1, and there-
fore, they are predetermined at time t. Instead, the input of labor h
t
can be ?exibly
changed in period t.
3
See for example Caballero and Krishnamurthy (2003), Lorenzoni (2008), Korinek (2009),
Bianchi (2010), Bianchi and Mendoza (2010) and Jeanne and Korinek (2010).
4
Benigno, Chen, Otrok, Rebucci, and Young (2010) also consider ex-post policy measures in
response to a pecuniary externality, but focus on policies that a?ect labor allocations as opposed
to policies that a?ect the availability of credit.
112
Capital evolves according to:
k
t+1
= k
t
(1 ??) + i
t
(3.1)
where i
t
is the level of investment and ?
t
is the depreciation rate. Capital accumu-
lation is subject to adjustment costs, given by ?(·).
Firms pay dividends, denoted by d
t
, and issue non-state contingent debt, de-
noted by b
t+1
. The ?ow of funds constraint for ?rms is then given by:
b
t
+ d
t
+ i
t
+ ?(k
t
, k
t+1
) ? F(z
t
, k
t
, n
t
) ?w
t
n
t
+
b
t+1
R
t
+ ?
t
(3.2)
where w
t
is the wage rate, R
t
is the gross interest rate determined in international
markets, and ?
t
is a transfer chosen by the government that will be speci?ed below.
Firms face two types of constraints on their ability to ?nance investment. First,
they are subject to a collateral constraint that limits the amount of borrowing to a
fraction of the value of their assets such that:
b
t+1
? ?
t
k
t+1
(3.3)
. This constraint is similar to those used in existing literature (see Kiyotaki and
Moore, 1997), and we interpret it as arising in an environment with limited en-
forcement between creditors and ?rms; ?
t
is a ?nancial shock and we interpret it,
following Jermann and Quadrini (2010), as a shock originating in the ?nancial sys-
tem.
113
In addition, at each period ?rms are required to pay a minimum amount of
dividends d ?
¯
d. A value of
¯
d ? 0 implies that the issuance of new shares is not
available. A special case is the restriction that dividends need to be non-negative.
This constraint captures the notion that dividend payments are required in order to
reduce agency frictions between shareholders and managers. In addition, it produces
dynamics in dividend payouts which are in line with empirical evidence (see Jermann
and Quadrini (2010)). We abstract, however, from an explicit microfoundation and
instead focus on its implications.
5
Denoting by s the vector of aggregate states, i.e. s = {K, B, ?, z}, the opti-
mization problem for ?rms can be written as:
V (k, b, s) = max
d,h,k
?
,b
?
d +Em
?
(s, s
?
)V (k
?
, b
?
, s
?
) (3.4)
s.t. b + d + k
?
+ ?(k, k
?
) ? (1 ??)k + F(z, k, h) ?wn +
b
?
R
+ ?
b
?
? ?k
?
d ?
¯
d
.
The function V (k, b, s) is the cum-dividend market value of the ?rm and m
?
is the stochastic discount factor, which will be equal in equilibrium to the ratio of
5
Endogeneizing such a constraint would require to model the divergence of interests between
shareholder and corporate manager in an environment with asymmetric information (see e.g. Mayer
1986).
114
marginal utility of household consumption.
The optimality condition for labor demand yields:
F
h
(z
t
, k
t
, h
t
) = w
t
(3.5)
.
There are also two Euler intertemporal conditions that relate the marginal
bene?t from distributing one unit of dividends today with the marginal bene?t of
investing in the available assets and distributing the resulting dividends in the next
period. Denoting by µ the multiplier associated with the borrowing constraint, ?
the multiplier associated with the dividend payout constraint, these Euler equations
are given by:
1 + ?
t
= R
t
E
t
m
t+1
(1 + ?
t+1
) + µ
t
(3.6)
(1+?
t
)(1+?
2
(t, t+1)) = E
t
m
t+1
[1 ?? + F
k
(z
t+1
, k
t+1
, n
t+1
) ??
1
(t + 1, t + 2)] (1+?
t+1
)+?
t
µ
t
(3.7)
.
In the absence of the ?nancial constraints on borrowing and dividend pay-
ments, the cost of raising equity (by reducing dividends), i.e. 1/E
t
m
t+1
, would be
equal to the cost of debt R
t
, and ?rms would be indi?erent at the margin between
equity and debt ?nancing. When the collateral constraint binds, there is a wedge
between the marginal bene?t of borrowing one more unit and distributing it as div-
115
idends in the current period and between the marginal cost of cutting dividends in
the next period to repay the debt increase. In addition, when the dividend payout
constraint binds, a positive wedge arises between the marginal bene?t from invest-
ing one more unit in capital or bonds relative to the marginal cost of cutting one
more unit of dividends. Condition (3.6) suggests also that a binding collateral con-
straint is associated with a binding dividend payout constraint. Intuitively, both
constraints impose a limit on a ?rm’s funding ability, which implies that a tighter
constraint on borrowing imposes pressure on the need to ?nance with equity, which
given that households have concave utility functions, this increases the cost of equity.
Similarly, a binding dividend payout constraint forces higher levels of borrowing for
given investment choices.
3.2.2 Households
There is a continuum of identical households of measure one that maximize:
E
0
?
?
t=0
?
t
u(c
t
?G(n
t
)) (3.8)
where c
t
is consumption, n
t
is labor supply, ? is the discount factor and G(·) is a
twice-continuously di?erentiable, increasing and convex function. The utility func-
tion u(·) has the constant-relative-risk-aversion (CRRA) form; the composite of the
utility function has the GHH form, eliminating wealth e?ects on the labor supply.
Households do not have access to bond markets and are the ?rms’ shareholders.
116
This yields the following budget constraint:
w
t
h
t
+ s
t
(d
t
+ p
t
) = s
t+1
p
t
+ c
t
+ T
t
(3.9)
wheres
t
represents the holdings of ?rm-shares, p
t
represents the price of ?rm-shares,
and T
t
is a lump sum tax to ?nance the cost of the bailout policy to be speci?ed
below.
The ?rst order conditions are given by:
w
t
= G
?
(n
t
) (3.10)
p
t
u
c
(t) = ?E
t
u
c
(t + 1)(1 +p
t+1
) (3.11)
.
The second condition determines the price of shares. Iterating forward on
(3.11) yields that for the ?rm optimization problem to be consistent with the house-
hold optimization problem, it must be the case that the stochastic discount factor
of ?rms is equal to m
t+j
= (?u
?
(c
t+j
?G
?
(n
t+j
)))/(u
?
(c
t+j?1
?G
?
(n
t+j?1
))).
3.2.3 Government
The function of the government is to set a lump sum tax on workers and to
transfer the proceeds to ?rms. A crucial element of the analysis is that transferring
resources to ?rms entails an e?ciency cost ?. Alternatively, this can be interpreted
as a distortionary e?ect of taxation. We assume that the government follows a
117
balanced budget such that:
T
t
= ?
t
(1 +?) (3.12)
3.2.4 Competitive equilibrium
We consider a competitive equilibrium for a small open economy where ?rms
borrow directly from abroad.
De?nition 5 A recursive competitive equilibrium is given by (i) ?rms’ policies
ˆ
d(k, b, s),
ˆ
h(k, b, s),
ˆ
k(k, b, s) and
ˆ
b(k, b, s); households’s policies ˆ s(s, s), ˆ n(s, s); ?rm’s
value V (k, b, s); prices w(s), p(s), m(s, s
?
); government policies ?(s), T(s); and a law
of motion of aggregate variables s
?
= ?(k, b, s) such that: (i) households solve their
optimization problem characterized by (3.9)-(3.11), (ii) ?rms’ policies and ?rms’
value solve (3.4), (iii) prices clear labor market (h
t
= n
t
), equity market (s
t
= 1),
and goods market ((1??)k
t
+F(z
t
, k
t
, n
t
)+b
t
= b
t+1
/R+?(k
t
, k
t+1
)+k
t+1
+c
t
), (iv)
the law of motion is ?(·) is consistent with individual policy functions and stochastic
processes for ? and z.
3.2.5 Some characterization
To illustrate the properties of the model and the e?ects of the bailout policy,
it is useful to consider a few special cases.
Proposition 3 If ?R < 1, the collateral constraint is binding and the dividend
payout constraint may or may not bind.
118
Proof : In a deterministic steady state, m
t
= 1 and (3.6) is simpli?ed to 1 = ?R+µ.
Since ?R < 1, this implies that µ > 0. Substituting the collateral constraint with
equality in the ?rm’s ?ow of funds constraint yields that the dividend constraint
binds if and only if
¯
d > F(¯ z,
¯
k,
¯
h) ?
¯
k(? + ¯ ?(R ? 1)/R) ? G
?
(
¯
h)
¯
h + ? where
¯
k
and
¯
h are the steady state values of capital and labor given by (3.7), (3.10), and
market clearing in labor markets. In general, in a stochastic steady state, these
?nancial constraints may or may not bind depending primarily on the magnitude of
the shocks, ?R, and the level of risk aversion.
Proposition 4 Suppose (i)
¯
d = ??, and (ii) households have unrestricted access
to international credit markets by borrowing and saving at the interest rate R (unlike
the baseline model). Then the competitive equilibrium is una?ected by ?nancial
shocks.
Proof : From a household’s ?rst order condition, it must be the case that REm
?
= 1.
Using the ?rm’s ?rst order condition REm
?
+µ = 1 yields µ = 0. Since the collateral
constraint never binds, the model becomes a standard (RBC) model where the
?nancial structure is independent on the real side.
Remark 6 Suppose (i)
¯
d = ??, and (ii) households do not have access to inter-
national credit markets. Then the competitive equilibrium is a?ected by ?nancial
shocks.
This remark becomes obvious once we observe that if output is su?ciently low at
a given period, households will demand positive dividends to smooth consumption,
119
which implies that ?rms would need to increase their debt position. Therefore, for
low values of ?, the collateral constraint becomes binding.
3.3 Bailout Policy
3.3.1 A second best benchmark
To illustrate the role of bailouts, we start by considering a social planner that
(i) directly chooses all allocations and (ii) is subject to the collateral constraint, but
not the dividend payout constraint.
This problem can be written recursively as:
V (s) = max
d,n,k
?
,b
?
u(c ?G(h)) + ?EV (s
?
) (3.13)
s.t. (1 ??)k + F(z, k, n) ?wn +
b
?
R
? b + d + k
?
+ ?(k, k
?
)
b
?
? ?k
?
.
We now highlight the main results regarding the implementability of this con-
strained social planner’s problem employing bailouts.
Remark 7 If ? = 0, the second best allocations can be implemented by an appro-
priate state contingent bailout policy.
Intuitively, the planner can use cost-free transfers as a substitute for lower
120
dividend payments when the dividend payout constraint becomes binding.
Proposition 5 If
¯
d = ?? and ? = 0, the competitive equilibrium is una?ected by
the bailout policy.
Proof The proof follows simply by noting that the transfers cancel out within the
conditions characterizing the competitive equilibrium.
3.3.2 Policy Experiment
We now consider the general case where the government faces strictly positive
e?ciency costs from transferring resources from ?rms and households. Under these
conditions, the second best allocations cannot be achieved: bailouts introduce a
trade o? between relaxing balance sheet constraints of ?rms and e?ciency costs
associated with the transfer.
The government function is limited to taxing workers and transferring those
resources to ?rms in the same period.
6
Speci?cally, we assume that the government
can commit to following a bailout policy rule ? such that ? = ?(
.
) where ?(
.
) follows
a parametric function that depends on the relevant state variables.
The objective of the government/planner is to choose ?(
.
) to maximize
?
s
0
_
E
?
?
t=0
?
t
u(c
bp
t
?G(n
bp
t
)) |s
0
_
?(s
0
) (3.14)
subject to all allocations and prices being a competitive equilibrium for the speci?ed
6
That is, we rule out the use of the government as a substitute for private credit. Allowing for
government debt is likely to take the economy closer to a ?rst best if there are no other frictions
(e.g. sovereign default risk).
121
bailout policy, here ? represents the joint ergodic distribution of all aggregate states
in the competitive equilibrium without bailout. That is, the policy rule is chosen to
maximize the expected life-time utility from switching to the competitive equilibrium
without intervention to the competitive equilibrium with a bailout policy. This
welfare measure considers explicitly the e?ects associated with the transition from
the stochastic steady state without bailout policy to the stochastic steady state with
bailout policy.
This policy can be interpreted as a form of unconventional monetary policy
(see e.g. Gertler and Karadi, 2010), as one can interpret the recent intervention of
the Federal Reserve in credit markets as a way to facilitate credit to the corporate
sector, given the strains in ?nancial intermediaries.
7
In the quantitative analysis we
search for parametrization of these rules that are simple and realistic such and that
the rules maximize expected life-time utility.
A key feature of this bailout policy is that bailouts are non-targeted, i.e. they
apply to all market participants. That is, even if an individual agent is not under
?nancial distress, it receives a bailout when the overall economy is under distress.
This is akin to an interest rate policy (see Farhi and Tirole, 2010) in which even
?rms that are not under distress can re?nance at a low interest rate when the
7
In practice, the Federal Reserve and the Treasury implemented a variety of policies with
the aim of facilitating credit to the corporate sector including direct lending, debt guarantees,
and equity injections in the banking sector. To simplify the analysis, we do not model ?nancial
intermediaries and capture this class of interventions in a crude way by modeling a direct transfer
from workers to ?rms. What is crucial for our analysis is that this intervention relaxes balance
sheets across the economy and mitigates the fall in credit and investment that occurs during crises.
In our setup, absent of information asymmetries, these policies are likely to yield similar outcomes.
See Philippon and Schnabl (2009) for an evaluation of optimal rescue packages in the context of a
debt overhang problem using a mechanism design approach.
122
central bank conducts an expansionary monetary policy. This introduces strategic
complementarities in ?rms’ ?nancing decisions, as individual agents have more of an
incentive to take a signi?cant amount of risk when other ?rms in the economy are also
taking a large amount of risk. On the other hand, conditioning bailouts on aggregate
variables may also work to mitigate the amount of risk taking, as individual agents
obtain a bailout only when the whole economy comes under ?nancial distress.
8
3.4 Quantitative Analysis
The numerical solution to the model involves several challenges. First, there
are the well-known complications of non-linearities introduced by the absence of
complete markets and in particular the occasionally binding collateral constraint.
Moreover, the state variables in the model are not con?ned to a narrow region of
the state space. We approximate the equilibrium functions over the entire state
space and check that equilibrium conditions are satis?ed at all grid points, allowing
for the two ?nancial constraints to bind only in some states of nature.
9
The
introduction of bailouts introduces an additional complication, which we handle
using a nested ?xed-point algorithm. First, for a given bailout policy, we compute
the implied competitive equilibrium (inner loop). Second, we update the bailout
policy accordingly (outer loop). These two procedures are followed until the two
8
O? equilibrium, while all ?rms receive the same amount of funds in case of a systemic crisis,
the marginal value of those funds depends on how leveraged they are; i.e., those with a tighter
?nancial constraint assign a higher value to the bailout.
9
An additional complication in our setup is that variations in consumption can lead to large
changes in the value of the ?rm, which requires a slow adjustment in the update of the consumption
function along the iterations. Further details are provided in an appendix upon request.
123
loops converge.
3.4.1 Calibration and Functional Forms
We calibrate the model to an annual frequency using data from the U.S. econ-
omy. The functional forms for preferences and technology are the following:
u(c ?G
) =_
c ??
n
1+?
1+?
_
1??
?1
1 ??
? > 0, ? > 1
F(z, k, h) = zk
?
h
1??
?(k
t
, k
t+1
) =
?
k
2
_
k
t+1
?k
t
k
t
_
2
k
t
For a ?rst group of parameters given by ?, ?, ?, ?, we choose values that we
see as reasonably conventional in the literature. The capital share ? is set to 0.32;
the depreciation rate is set at 8 percent; the risk aversion ? is set to 2; and ? is set
so that the Frisch elasticity of labor supply is equal to 1. We normalize the labor
disutility coe?cient ? and the average value of the TFP shock so that employment
and output are equal to one in the deterministic steady state.
The TFP shock and the ?nancial shock are modeled as independent stochastic
processes. Each of these shocks is discretized using a two-state Markov chain. The
TFP shock is constructed using a symmetric simple persistence rule. The realization
of the shock and the persistence of the TFP shock are chosen so that in the model,
?uctuations in GDP are roughly consistent with those in the data. This yields TFP
realizations of plus/minus 1 percent and a probability of remaining in the same state
124
of 78 percent. The calibration of the ?nancial shock is meant to be suggestive. This
shock follows an asymmetric process such that during ”bad times”, the pledgability
of capital falls by 20 percent; this event occurs 3 percent of the time and lasts on
average for two years.
The capital adjustment cost parameter ?
k
is set so that the standard deviation
of investment in the competitive equilibrium without bailouts roughly matches the
standard deviation of investment in the data. This yields ?
k
= 7.8. The remaining
parameters are the discount factor ?, the mean value of ?, and the minimum div-
idend payment
¯
d. These parameters govern the values of leverage as well as how
frequent the constraints on borrowing and dividend bind. For now, we set ? = 0.92,
the mean value of ? = 0.32 and
¯
d = 0.04.
10
With these values, the mean value of
leverage-de?ned as the ratio of debt to market value of the ?rm-equals 0.29, which
is in the upper range of corporate leverage documented by Masulis (1988); the crisis
dynamics are also roughly in line with US data.
3.4.2 Results
Figures 1 and 2 show the laws of motion for capital and debt in the competitive
equilibrium with and without bailout policy. The x-axis in the two ?gures is given
by the current level of debt. (The level of capital is approximately the average value
of capital, and the shocks are given by adverse TFP and adverse ?nancial shocks).
Let us ?rst describe the behavior of the competitive equilibrium without bailouts
10
To facilitate convergence, for the time being, we use an endogenous discount factor without
internalization (see Schmitt-Grohe and Uribe (2003)), such that on average ? = ln(1 +c)
?
= 0.96,
yielding ? = 0.10.
125
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 1
2.8
2.85
2.9
2.95
3
3.05
3.1
3.15
3.2
Current Debt
F
u
t
u
r
e
C
a
p
i
t
a
l
No Bailout Policy
Bailout Policy
Collateral constraint binding Dividend constraint binding
Figure 3.1: Law of Motion for Capital
before analyzing the e?ects of credit intervention.
As Figure 1 shows, an increase in debt holdings reduces the demand for capital,
since capital is risky and the valuation of future dividends becomes more sensitive
to adverse shocks. Since borrowing ability depends on the next period capital, this
produces a non-monotonic law of motion for debt, as shown in Figure 2. For low
values of current debt, the collateral constraint is not binding. As the value of current
debt increases, the demand for debt increases; but as investment is reduced, the
ability to borrow shrinks. When the current level of debt is about 0.84, the collateral
constraint becomes binding. This is indicated in the ?rst vertical in Figures 1 and 2.
126
For b > 0.84, the next period debt holdings decrease in current debt holdings as the
level of capital and debt are reduced in an endogenous feedback loop. The second
vertical line indicates when the dividend payout constraint becomes binding. Notice
that both investment and the level of future debt are reduced more sharply once the
dividend payout constraint binds. This is intuitive, since a binding dividend payout
constraint limits even more the access to ?nancing.
0.8 0.85 0.9 0.95
0.8
0.81
0.82
0.83
0.84
0.85
0.86
0.87
Current Debt
F
u
t
u
r
e
D
e
b
t
No Bailout Policy
Bailout Policy
Collateral constraint binding Dividend constraint binding
Figure 3.2: Law of Motion for Debt
For now, we consider bailouts that are strictly positive if and only if the
dividend payout constraint is binding and it is increasing in the shadow interest
rate, i.e. the interest rate that will make the collateral constraint hold with equality
127
but not bind at the margin, given prices and allocations.
11
. This makes the bailout
increase in the level of debt and decrease in the level of capital.
To understand how bailouts a?ect the competitive equilibrium, it is useful
to analyze ?rst its e?ects during periods in which the dividend payout constraint
becomes binding. As Figure 2 shows, bailouts allow ?rms to borrow more during
these periods. This occurs because as ?rms receive the bailout transfer, they can
allocate these funds to invest more in capital, which boosts the ?rms’ capacity to
borrow. In the region where the dividend constraint is not binding, ?rms also
borrow more in the competitive equilibrium with bailouts. This occurs because
given that crises become less severe as a result of bailouts, there is a lower incentive
to accumulate precautionary savings during normal times. This leads in turn to a
higher probability of the economy becoming ?nancially constrained in the future.
12
3.4.3 Discussion
In our benchmark model, a bailout from workers to ?rms does not involve
any wealth redistribution since workers are the owners of ?rms. This allows us to
focus on the bailout’s consequences for e?ciency. We can also extend our analysis
to allow bailouts to have wealth redistribution e?ects. In particular, consider a
model where ?rms are owned by entrepreneurs and workers do not hold any shares
of the ?rms. In this context, while workers face a negative wealth e?ect as a result
of carrying the burden of the bailout, there are labor-market spillovers that might
11
Formally, the shadow interest rate is de?ned as R
ef
= (1 + ?
t
)/E
t
m
t+1
(1 + ?
t+1
)
12
The speci?ed bailout policy increases welfare by about 0.1 percentage point of permanent
consumption (please check future updates of the paper for a complete welfare analysis).
128
raise their welfare. In particular, as ?rms do not reduce investment as much during a
?nancial crisis, this leads to higher wages in the recovery of the crisis. For plausible
calibrations, however, the welfare of workers is reduced by bailouts. Intuitively, this
results from capital not increasing enough to make the increase in wages compensate
for the direct cost of ?nancing the bailout.
In our setup, there is also scope for ex-ante prudential measures. The reason is
a form of overborrowing externality. While committing to a bailout in some states of
nature is desirable, these interventions also impose costs that are paid by all workers.
Not internalizing that becoming ?nancially constrained during a systemic ?nancial
crisis triggers a costly intervention by the planner, ?rms borrow too much. Notice
that while ?rms bene?t from other ?rms being constrained due to the systemic
nature of the bailout, this constitutes only a private gain, as the bailout imposes
costs at the social level. A similar point is also made by Chari and Kehoe (2009)
and Farhi and Tirole (2010). In their setup, regulation is designed to reduce the
temptation to conduct bailouts. In our setup, however, regulation also improves the
commitment solution.
We should point out that the possibility to improve welfare in our setup de-
pends on the ability of the government to redirect funds from workers to ?nancially
constrained ?rms. Workers do not have the incentive to transfer these funds to
?rms because this entails only costs at the individual level. Ex ante, the ratio-
nale for committing to such interventions is to provide a form of systemic insurance
against ?nancial crises. This result also points towards the desirability of enhancing
the development of private insurance markets. Clearly, however, there are reasons
129
why the availability of private insurance against systemic episodes is limited by a
host of credit market imperfections (e.g., insurers may go bankrupt in crises). Under
these conditions, our analysis suggests that the government should have a direct role
in providing insurance against systemic ?nancial crises.
Another point that we should emphasize is that we have assumed that the
planner can commit to a speci?c bailout policy. This is motivated by debates about
how to specify a legal framework to put limits on the ability of central banks to bail
out the ?nancial sector (see e.g. the Dodd-Frank Act). Under some states of nature,
however, it might still be feasible for central banks to evade the legal framework. It
would be interesting in our framework to study how the belief that central banks
would deviate from previous commitments could generate an incentive for private
agents to take even more risk as compared to an environment where the planner can
commit to future policies.
3.5 Conclusion
The quantitative analysis shows that it is possible to design a realistic and
simple bailout policy so that the welfare bene?ts of such an intervention outweigh
its distortionary e?ects. A key feature of this intervention is that it is reserved for
extreme episodes of ?nancial distress.
Our results are relevant for ongoing debates about the appropriate role of
central banks during ?nancial crises. Our analysis points towards giving a speci?c
mandate of intervention that supports credit ?ows, albeit in a strictly limited way.
130
Re?ning our analysis would require us to consider explicitly the temptation of central
banks to renege on policy commitments. Within our framework, it would also be
interesting to study how certain policies like capital requirements can help to o?set
these problems of credibility
131
Appendix A
Proofs (Chapter 1)
A.1. Proof of Proposition 1 (Constrained Ine?ciency)
This is a proof by contradiction. Suppose the decentralized equilibrium yields
the same allocations as the constrained-e?cient allocations. Then, we can combine
(1.4) and (1.12), yielding:
?
de
t
= ?
sp
t
+ µ
sp
t
?
t
(A.1)
where we denote with superscript ‘sp’ the Lagrange multipliers of the social
planner and with ‘de’ those of the decentralized equilibrium. Updating this equation
one period forward and taking conditional expectations at time t:
E
t
?
de
t+1
= E
t
?
sp
t+1
+E
t
µ
sp
t+1
?
t+1
(A.2)
Suppose that at time
˜
t, b
˜
t+1
> ?
_
?
N
p
˜
t
y
N
˜
t
+ ?
T
y
T
˜
t
_
. Combining (1.6),(1.7),(1.13),
and (1.14) we obtain:
E
˜
t
?
de
˜
t+1
= E
˜
t
?
sp
˜
t+1
(A.3)
If at time
˜
t+1 the credit constraint binds with positive probability, comparing
(A.3) and (A.2) yields a contradiction.
132
A.2. Proof of Proposition 2 (Optimal tax on debt)
This a proof by construction. Combining the optimality conditions for the social
planner (1.12) and (1.13) yields:
u
T
(t) = ?(1 + r)E
t
(u
T
(t + 1) + µ
sp
t+1
?
t+1
) + µ
sp
t
(1 ??
t
) (A.4)
First, notice that the constrained-e?cient allocations are characterized by
stochastic sequences
_
c
T
t
, c
N
t
, b
t+1
, p
N
t
, µ
sp
t
_
t?0
such that the following conditions
hold: (1.5),(1.8),(1.9),(1.14), (A.4) and µ
sp
t
? 0.
Second, the decentralized equilibrium allocations with taxes on debt are char-
acterized by stochastic sequences
_
c
T
t
, c
N
t
, b
t+1
, p
N
t
, µ
t
, ?
t
, T
t
_
t?0
such that the follow-
ing conditions hold: (1.5),(1.7),(1.8),(1.9),(1.17), T
t
= b
t
(1 +r)?
t?1
and µ
t
? 0.
De?ning the tax as ?
?
t
=
_
E
t
µ
sp
t+1
?
t+1
_
/ (E
t
u
T
(t + 1))?(µ
sp
t
?
t
) / (?(1 + r)E
t
u
T
(t + 1))
and redistributing the proceeds lump sum yields that the conditions characterizing
the decentralized equilibrium with the speci?ed tax on debt are identical to those
characterizing the constrained-e?cient allocations.
Appendix B
An Equivalence Result (Chapter 1)
We show in this appendix that the constrained-e?cient allocations can be de-
centralized with regulatory measures directed to the banking sector. Consider the
following simple model. Banks make loans to households at rate r
L
and impose the
constraint (1.2) to guarantee repayment. Banks ?nance these loans by accepting
133
deposits from the rest of the world at rate r and issuing equity in the domestic
markets. We assume that the required return on equity r
e
is higher than the rate
on deposits, i.e., r
e
> r. This could be the outcome of moral hazard or tax dis-
advantages on equity, but we abstract from explicitly modeling this relationship.
Financial intermediation is costless. Banks last for one period, and every period
new banks are set up with free entry into banking.
Without any regulation or any other frictions, banks would ?nance loans only
with deposits, and the resulting equilibrium would be equivalent to the decentralized
equilibrium. We introduce two regulatory measures. First, the planner imposes
capital requirements: banks are required to ?nance a fraction ? of their assets with
equity. Second, the planner imposes reserve requirements: banks are required to
hold a fraction ? of deposits in the form of unremunerated reserve. Thus the banks’
balance sheets become:
Assets Liabilities
b Loans d Deposits
f Reserve requirements e Equity
The objective of the bank is static and consists of maximizing shareholder
value, net of the initial equity investment:
max
b,f,e,d
b(1 +r
L
) + f ?d(1 + r) ?e(1 + r
e
)
134
subject to
f ? d + e
f ? ?d
e ? ?(b + f)
Given that holding reserves and capital is privately costly, banks do not hold excess
reserves or excess capital. In equilibrium, the return from assets must be equal to
the return on liabilities, i.e., r
L
(1 ??(1 ??)) = ?r
e
+(1 ??)r. Therefore, by setting
(?
t
, ?
t
) such that (1 +r
L
t
) = (1 +r)(1 +?
?
t
), the social planner can raise the cost of
borrowing and induce agents to hold the socially optimal amount of debt. Assuming
only capital requirements are used yields: ?
?
t
= (?
?
t
(1 + r))/(r
e
? r). When only
reserve requirements are used, this yields: ?
?
t
= (?
?
t
(1 + r))/(r(1 + ?
?
t
) + ?
?
t
).
Appendix C
Sensitivity Analysis (Chapter 1)
We continue here the sensitivity analysis presented in the body of the paper.
We discuss separately the e?ects of varying each of the parameter values of the
model, using the analysis of the externality term and the elasticity decomposition
studied in the body of the paper. The main quantitative results of all experiments
are shown in Table 4. The table shows for each experiment the average welfare
loss, the average implied tax on debt, the relative volatility of consumption, the
probability of a ?nancial crisis for the decentralized equilibrium and constrained-
e?cient allocations, and the e?ects of a median crisis in consumption, the real
135
exchange rate and the current account for the two equilibria.
Discount Factor (?).— An increase in the discount factor leads to a shift of
the distribution of bond holdings towards a lower amount of debt, leading to less
frequent binding constraints and causing the distribution of the externality term to
concentrate higher probability in a region where its value is zero. This e?ect leads
to smaller e?ects from the externality. There is an opposite e?ect from an increase
in the discount factor. Recall that the maximum welfare gains from the externality
arise in relatively tranquil times because of the reduction in future vulnerability to
?nancial crises. Hence, a higher discount factor makes the economy value relatively
more the bene?ts from a reduction in future variability, which should lead to higher
welfare e?ects from correcting the externality. Quantitatively, we ?nd that the
?rst e?ect is more important. Increasing the discount factor by 0.02 reduces the
average implied tax on debt to 3.3 percent, although large di?erences remain in the
probability of ?nancial crises: the probability of a crisis is 0.2 percent for the social
planner and 4.1 percent for the decentralized equilibrium.
Interest Rate (r).— An increase in the interest rate has e?ects similar to an
increase in the discount factor since both reduce the willingness to borrow and shifts
the economy away from binding constraints. For a given amount of debt, however,
a higher interest rate implies an increase in the debt service, which causes a larger
depreciation of the real exchange rate. Quantitatively, we ?nd that increasing the
interest rate 100bps reduces the implied tax on debt from 5.2 percent to 4.4 percent,
but the e?ects on the incidence and severity of ?nancial crises remain very similar.
136
Risk Aversion (?).— An increase in the risk aversion implies a higher disu-
tility from consumption variability. This implies that a large drop in consumption
generates a higher shadow value from relaxing the credit constraint at a given state
where the constraint binds; therefore, this yields a higher externality term. At the
same time, an increase in risk aversion makes both the social planner and private
agents accumulate more precautionary savings making the constraint less likely to
bind and shifting away the distribution of the externality term towards zero. Quan-
titatively, as shown in Table 4, we ?nd that the e?ects of the externality decreases
(increases) modestly when we consider ? = 5 (? = 1).
Independent shocks.— We model tradable and nontradable endowment shocks
as independent AR(1) processes and analyze the e?ects over the externality. When
shocks are correlated, both tradable and nontradable shocks typically fall during ?-
nancial crises. The fact that nontradables fall, however, mitigates the fall in the price
of nontradables and the tightening of ?nancial constraints. This channel suggests
that making the two shocks independent should reduce the e?ects of the externality.
There is another channel, however, by which making the shocks independent causes
the externality to have higher e?ects. For the baseline calibration, the risk aver-
sion and the elasticity of substitution between tradables and nontradables are such
that tradable and nontradable goods are Edgeworth substitutes. As a result, a fall
in the endowment of nontradables when the credit constraint binds, increases the
marginal utility from tradable consumption, which increases the desire to borrow
and increases the shadow value from relaxing the credit constraint. Quantitatively,
137
we ?nd that the e?ects over the shadow value from relaxing the credit constraint
are stronger than those a?ecting the price e?ects, so that the di?erences in severity
of ?nancial crises become even stronger.
Volatility and Persistance (Cov(?)).— An increase in the volatility of endow-
ment shocks increases the severity of ?nancial crises, in terms of the ampli?ca-
tion e?ects and the disutility cost from a binding constraint. This e?ect increases
the externality term. At the same time, private agents have an incentive to in-
crease relatively more precautionary savings in response to the increase in volatility.
This occurs because the concavity of the utility function implies that a given in-
crease in variability is more costly in the decentralized equilibrium compared to the
constrained-e?cient allocations. In fact, when we vary simultaneously the volatility
of the shocks to the endowment processes by 15 percent, we ?nd that the externality
decreases modestly with a higher volatility.
An increase in persistence leads to a higher probability of ?nancial crises for a
given level of precautionary savings although it does not alter the size of the shocks
and the severity of ?nancial crises. When we vary the autocorrelation of the endow-
ment shocks by 15 percent, we ?nd that a higher autocorrelation is associated with
larger e?ects from the externality. In fact, the experiment with higher autocorre-
lation yields larger di?erences in the incidence and severity of ?nancial crises, and
this leads to larger welfare e?ects.
Elasticity of Substitution(1/(1+?)).— As explained in the paper, the elasticity
of substitution between tradables and nontradables determines the debt service elas-
138
ticity of the real exchange rate, which is in turn a key component of the externality
term. Moreover, the elasticity of substitution also a?ects the incentive to accumu-
late precautionary savings: the lower the elasticity of substitution the higher the
disutility from drops in consumption during ?nancial crises. This second channel is
similar to the increase in the risk aversion, but we ?nd that the channel a?ecting
directly the price e?ects are quantitatively more important.
Share of tradables (?).— As explained in the paper, the weight of tradables
in the utility function determines the borrowing limit elasticity of the real exchange
rate and is key for the e?ects on the externality. There is another e?ect of this
parameter. A higher share of tradables in the utility function implies that large
drops in tradables consumption during ?nancial crises are more costly, causing an
increase in precautionary savings. As explained before, this second channel becomes
qualitatively ambiguous, but we ?nd that the price e?ects, which unambiguously
increase the externality, are more signi?cant.
Credit Coe?cient (?).— We set ?
T
= ?
N
= ?. An increase in ? has two e?ects.
First, it increases directly the externality term, because for a given drop in the price
of nontradables the e?ects over the borrowing ability are directly proportional to ?.
Second, it makes the constraint less likely to bind, hence reducing the e?ects of the
externality. On one hand, when ? is 0, there is no borrowing; therefore an increase
in ? raises the e?ects of the externality. On the other hand, for a very large ?, the
credit constraint never binds and there are no e?ects from the externality in the long
run. Quantitatively, we ?nd that increasing ? from 0.32 to 0.36 increases the welfare
139
e?ects of the externality to 0.22 percentage points of permanent consumption. In
addition, consumption during a median crisis drops almost three times as much
in the decentralized equilibrium compared to the constrained-e?cient equilibrium.
Reducing ? to 0.28 reduces also slightly the e?ects of the externality but crises in
the decentralized equilibrium remain ten times more likely than in the constrained-
e?cient equilibrium.
140
T
a
b
l
e
C
.
1
:
S
e
n
s
i
t
i
v
i
t
y
A
n
a
l
y
s
i
s
S
e
v
e
r
i
t
y
o
f
F
i
n
a
n
c
i
a
l
C
r
i
s
e
s
P
r
o
b
a
b
.
C
r
i
s
i
s
C
o
n
s
u
m
p
t
i
o
n
R
E
R
C
u
r
r
e
n
t
A
c
c
o
u
n
t
W
e
l
f
a
r
e
T
a
x
o
n
D
e
b
t
?
c
d
e
/
?
c
s
p
D
E
S
P
D
E
S
P
D
E
S
P
D
E
S
P
b
a
s
e
l
i
n
e
0
.
1
3
5
.
2
1
.
1
3
5
.
5
0
.
4
-
1
6
.
7
-
1
0
.
0
1
9
.
2
1
.
1
7
.
9
0
.
0
?
=
0
.
9
3
0
.
0
8
3
.
3
1
.
0
8
4
.
1
0
.
2
-
1
3
.
0
-
9
.
2
1
7
.
6
7
.
7
5
.
8
1
.
3
?
=
0
.
8
9
0
.
1
9
7
.
3
1
.
1
8
6
.
4
0
.
4
-
1
7
.
9
-
1
0
.
0
2
2
.
1
1
.
0
9
.
5
-
0
.
2
r
=
0
.
0
5
0
.
1
1
4
.
4
1
.
1
1
4
.
7
0
.
3
-
1
6
.
1
-
1
0
.
2
1
7
.
6
1
.
3
6
.
9
-
0
.
1
r
=
0
.
0
3
0
.
1
6
6
.
0
1
.
1
6
6
.
0
0
.
4
-
1
7
.
3
-
1
0
.
1
2
0
.
7
1
.
1
8
.
8
-
0
.
1
?
=
5
0
.
1
1
4
.
7
1
.
0
6
2
.
4
0
.
2
-
1
2
.
8
-
7
.
9
1
7
.
2
4
.
3
5
.
5
-
0
.
2
?
=
1
0
.
1
5
5
.
3
1
.
2
1
6
.
4
0
.
4
-
1
9
.
4
-
1
0
.
1
2
5
.
7
1
.
1
1
1
.
6
-
0
.
1
I
n
d
e
p
e
n
d
e
n
t
s
h
o
c
k
s
0
.
1
5
5
.
0
1
.
2
4
.
9
0
.
3
-
1
2
.
4
-
2
.
2
3
5
.
4
1
4
.
7
1
3
.
0
1
.
0
V
o
l
a
t
i
l
i
t
y
?
(
1
5
%
l
e
s
s
)
0
.
1
7
5
.
3
1
.
2
1
4
.
9
0
.
0
-
1
8
.
9
-
1
0
.
1
2
3
.
9
0
.
2
1
0
.
9
-
0
.
1
V
o
l
a
t
i
l
i
t
y
?
(
1
5
%
m
o
r
e
)
0
.
1
3
5
.
0
1
.
1
0
5
.
5
0
.
4
-
1
0
.
5
-
6
.
5
1
5
.
3
5
.
1
6
.
6
2
.
1
A
u
t
o
c
o
r
r
e
l
a
t
i
o
n
?
(
1
5
%
l
e
s
s
)
0
.
1
2
5
.
1
1
.
1
2
6
.
0
0
.
4
-
1
5
.
4
-
9
.
6
1
7
.
4
1
.
6
6
.
7
-
0
.
2
A
u
t
o
c
o
r
r
e
l
a
t
i
o
n
?
(
1
5
%
m
o
r
e
)
0
.
1
6
5
.
2
1
.
1
7
4
.
9
0
.
1
-
1
8
.
6
-
1
0
.
5
2
3
.
0
1
.
2
1
0
.
0
0
.
0
?
=
0
.
3
6
0
.
2
1
5
.
9
1
.
1
5
4
.
3
0
.
4
-
1
4
.
5
-
5
.
6
2
8
.
0
7
.
3
1
0
.
4
0
.
0
?
=
0
.
2
8
0
.
0
9
4
.
7
1
.
1
0
6
.
4
0
.
6
-
1
3
.
5
-
8
.
2
1
8
.
9
5
.
1
6
.
5
0
.
3
?
=
0
.
3
4
0
.
0
8
4
.
4
1
.
1
1
6
.
5
0
.
9
-
1
3
.
8
-
9
.
9
1
7
.
2
8
.
5
6
.
6
2
.
0
?
=
0
.
2
9
0
.
2
2
5
.
8
1
.
1
4
4
.
1
0
.
0
-
2
2
.
3
-
1
0
.
2
3
5
.
6
1
.
6
1
5
.
9
0
.
0
1
/
(
?
+
1
)
=
1
.
0
0
.
0
7
4
.
3
1
.
1
0
7
.
7
1
.
3
-
9
.
8
-
6
.
3
1
2
.
7
4
.
9
6
.
3
2
.
3
1
/
(
?
+
1
)
=
0
.
7
0
.
1
9
5
.
8
1
.
1
5
4
.
5
0
.
5
-
1
7
.
0
-
8
.
0
3
0
.
0
5
.
2
1
1
.
2
0
.
0
I
n
t
e
r
m
e
d
i
a
t
e
i
n
p
u
t
s
0
.
1
4
5
.
4
1
.
1
0
5
.
5
0
.
3
-
1
5
.
4
-
9
.
0
1
9
.
5
3
.
4
6
.
4
0
.
0
?
N
/
?
T
=
0
.
5
0
.
0
4
3
.
8
1
.
0
8
7
.
0
2
.
0
-
1
7
.
1
-
1
3
.
8
1
1
.
4
1
.
9
5
.
3
0
.
0
N
o
t
e
:
‘
D
E
’
r
e
p
r
e
s
e
n
t
s
t
h
e
d
e
c
e
n
t
r
a
l
i
z
e
d
e
q
u
i
l
i
b
r
i
u
m
,
‘
S
P
’
r
e
p
r
e
s
e
n
t
s
t
h
e
s
o
c
i
a
l
p
l
a
n
n
e
r
.
T
a
x
o
n
d
e
b
t
i
s
t
h
e
u
n
c
o
n
d
i
t
i
o
n
a
l
a
v
e
r
a
g
e
o
f
t
h
e
o
p
t
i
m
a
l
t
a
x
o
n
d
e
b
t
.
W
e
l
f
a
r
e
r
e
p
r
e
s
e
n
t
t
h
e
u
n
c
o
n
d
i
t
i
o
n
a
l
a
v
e
r
a
g
e
o
f
t
h
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w
e
l
f
a
r
e
g
a
i
n
s
f
r
o
m
c
o
r
r
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c
t
i
n
g
t
h
e
e
x
t
e
r
n
a
l
i
t
y
.
?
c
d
e
/
?
c
s
p
r
e
p
r
e
s
e
n
t
s
t
h
e
r
e
l
a
t
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e
v
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l
a
t
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i
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c
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s
u
m
p
t
i
o
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.
T
h
e
t
a
x
o
n
d
e
b
t
,
w
e
l
f
a
r
e
a
n
d
t
h
e
p
r
o
b
a
b
i
l
i
t
y
o
f
a
c
r
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s
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s
a
r
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e
x
p
r
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s
s
e
d
i
n
p
e
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c
e
n
t
a
g
e
.
C
o
n
s
u
m
p
t
i
o
n
a
n
d
t
h
e
d
e
p
r
e
c
i
a
t
i
o
n
o
f
t
h
e
r
e
a
l
e
x
c
h
a
n
g
e
r
a
t
e
(
R
E
R
)
a
r
e
e
x
p
r
e
s
s
e
d
a
s
p
e
r
c
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n
t
a
g
e
d
e
v
i
a
t
i
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n
s
f
r
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r
u
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v
a
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e
s
d
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r
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a
m
e
d
i
a
n
c
r
i
s
i
s
(
s
e
e
f
o
o
t
n
o
t
e
1
2
i
n
t
h
e
b
o
d
y
o
f
t
h
e
p
a
p
e
r
)
.
C
u
r
r
e
n
t
a
c
c
o
u
n
t
i
s
e
x
p
r
e
s
s
e
d
a
s
a
p
e
r
c
e
n
t
a
g
e
o
f
G
D
P
.
Appendix D
Numerical Solution Method (chapter 1)
The computation of the competitive equilibrium requires solving for functions
B(b, y), P
N
(b, y),C
T
(b, y) such that:
P
N
(b, y) =
_
1 ??
?
__
C
T
(b, y)
y
N
_
?+1
(D.1)
u
T
(C
T
(b, y), y
N
) ? ?(1 + r)E
y
?
/y
u
T
(C
T
(B(b, y), y
?
), y
N?
) (D.2)
B(b, y) ? ?
_
?
N
P
N
(b, y)y
N
+ ?
T
y
T
_
with = if (25) holds with strict inequality
(D.3)
B(b, y) +C
T
(b, y) = b(1 + r) + y
T
(D.4)
where u
T
(C
T
(b, y), y
N
) = u
C
(C(b, y))C
T
(b, y), C(b, y) =
_
?
_
C
T
(b, y)
_
??
+ (1 ??)
_
y
N
_
??
_
?
1
?
and y = (y
T
, y
N
).
The algorithm employed to solve for the competitive equilibrium is based on
the time iteration algorithm modi?ed to address the occasionally binding endogenous
constraint. The algorithm follows these steps:
1
1. Generate a discrete grid for the economy’s bond position G
b
= {b
1,
b
2
, ...b
M
}
and the shock state space G
Y
= {y
1,
y
2
, ...y
N
} and choose an interpolation
scheme for evaluating the functions outside the grid of bonds. We use 800
1
For the social planner’s allocations, we use a standard value function iteration algorithm.
142
points in the grid for bonds and interpolate the functions using a piecewise
linear approximation.
2. Conjecture P
N
K
(b, y), B
K
(b, y), C
T
K
(b, y) at time K ?b ? G
b
and ?y ? G
Y
.
3. Set j = 1
4. Solve for the values of P
N
K?j
(b, y), B
K?j
(b, y), C
T
K?j
(b, y) at time K ? j using
(D.1),(D.2),(D.3),(D.4) and B
K?j+1
(b, y), P
N
K?j+1
(b, y), C
T
K?j+1
(b, y), ?b ? G
b
and ?y ? G
Y
:
(a) Set B
K?j
(b, y) = ?
_
?
N
P
N
K?j+1
(b, y)y
N
+ ?
T
y
T
_
and compute C
T
K?j
(b, y)
from (D.4)
(b) Compute
U = u
T
(C
T
K?j
(b, y), y
N
) ??(1 +r)E
y
?
/y
u
T
(C
T
K?j
(B
K?j
(b, y), y
?
), y
N
?
)
(c) If U > 0, the credit constraint binds; move to (e).
(d) Solve for B
K?j
(b, y), C
T
K?j
(b, y) using (D.2) and (D.4) with a root ?nding
algorithm.
(e) Set P
N
K?j
(b, y) =
_
1??
?
_
_
C
T
K?j
(b,y)
y
N
_
?+1
5. Evaluate convergence. If sup
b?G
b
,y?G
Y
?x
K?j
(b, y) ? x
K?j+1
(b, y)? < ? for
x = B, C
T
, P
N
we have found the competitive equilibrium. Otherwise, set
x
K?j
(b, y) = ?x
K?j
(b, y) + (1 ? ?)x
K?j+1
(b, y) and j ; j + 1 and go to step
4. We use values of ? close to 1.
143
Appendix E
Numerical Solution Method (chapter 2)
The computation of the competitive equilibrium requires solving for functions
B(b, ?), q(b, ?), C(b, ?), N(b, ?), µ(b, ?) such that:
C(b, ?) +
B(b, ?)
R
= ?F(K, N(b, ?)) + b (E.1)
?
B(b, ?)
R
+ ?G
?
(N(b, ?))N(b, ?) ? ?q(b, ?)K (E.2)
u
?
(t) = ?RE
?
?
/?
_
u
?
(C(B(b, ?), ?
?
))
_
+ µ(b, ?) (E.3)
?F
n
(K, N(b, ?)) = G
?
(N(b, ?))N(b, ?)(1 +?µ(b, ?)/u
?
(C(b, ?)) (E.4)
q(b, ?) =
?E
?
?
/?
_
u
?
(c(B(b, ?), ?
?
))?
?
F
k
(K, N(B(b, ?), ?
?
)) + q(B(b, ?), ?
?
)
¸
(u
?
(C(b, ?) ) ?µ(b, ?)?)
(E.5)
We solve the model using a time iteration algorithm developed by Coleman
(1990) modi?ed to address the occasionally binding endogenous constraint. The
algorithm follows these steps:
1
1. Generate a discrete grid for the economy’s bond position G
b
= {b
1,
b
2
, ...b
M
}
and the shock state space G
?
= {?
1,
?
2
, ...?
N
} and choose an interpolation
scheme for evaluating the functions outside the grid of bonds. We use 300
1
For the social planner’s allocations, we use the same algorithm operating on the planner’s
optimality conditions.
144
points in the grid for bonds and interpolate the functions using a piecewise
linear approximation.
2. Conjecture B
K
(b, ?), q
K
(b, ?), C
K
(b, ?), N
K
(b, ?), µ
K
(b, ?) at time K ? b ? G
b
and ? ? ? G
?
.
3. Set j = 1
4. Solve for the values of B
K?j
(b, ?), q
K?j
(b, ?), C
K?j
(b, ?), N
K?j
(b, ?), µ
K?j
(b, ?)
at time K?j using (E.1),(E.2),(E.3),(E.4), (E.5) and B
K?j+1
(b, ?), q
K?j+1
(b, ?), C
K?j+1
(b, ?),
N
K?j+1
(b, ?), µ
K?j+1
(b, ?)? b ? G
b
and ? Y ? G
Y
:
(a) Assume collateral constraint (E.2) is not binding. Set µ
K?j
(b, ?) = 0 and
solve for N
K?j
(b, ?) using (E.4). Solve for B
K?j
(b, ?) and C
K?j
(b, ?) using
(E.1) and(E.3) and a root ?nding algorithm.
(b) Check whether ?
B
K?j
(b,?)
R
+ ?G
?
(N(b, ?))N
K?j
(b, ?) ? ?q
K?J+1
(b, ?)K
holds.
(c) If constraint is satis?ed, move to next grid point.
(d) Otherwise, solve for µ(b, ?), N
K?J
(b, ?), B
K?j
(b, ?) using (E.2), (E.3) and(E.4)
with equality.
(e) Solve for q
N
K?j
(b, ?) using (E.5)
5. Evaluate convergence. If sup
B,?
?x
K?j
(B, ?)?x
K?j+1
(B, ?)? < ? for x =B, C, q, µ, N
we have found the competitive equilibrium. Otherwise, set x
K?j
(B, ?) =
x
K?j+1
(B, ?) and j ;j + 1 and go to step 4.
145
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