Economics Study on Macroeconomic Volatility and Monetary Economics

Description
While macroeconomics is a broad field of study, there are two areas of research that are emblematic of the discipline: the attempt to understand the causes and consequences of short-run fluctuations in national income (the business cycle), and the attempt to understand the determinants of long-run economic growth (increases in national income).

ABSTRACT
Title of dissertation: ESSAYS ON MACROECONOMIC VOLATILITY
AND MONETARY ECONOMICS
Jeta Menkulasi, Doctor of Philosophy, 2010
Dissertation directed by: Professor Boragan Aruoba
Professor John Haltiwanger
Department of Economics
My dissertation consists of two independent essays on macroeconomic volatility
and monetary economics respectively. The …rst essay explores the implications of
imperfect information on macroeconomic volatility. It o¤ers a micro-founded theory
of time variation in the volatility of aggregate economic activity based on rational inat-
tention. I consider a dynamic general equilibrium model in which …rms are limited in
their ability to process information and allocate their limited attention across aggre-
gate and idiosyncratic states. According to the model, a decrease in the volatility of
aggregate shocks causes the …rms optimally to allocate less attention to the aggregate
environment. As a result, the …rms’ responses, and therefore the aggregate response,
becomes less sensitive to aggregate shocks, amplifying the e¤ect of the initial change
in aggregate shock volatility. As an application, I use the model to explain the Great
Moderation, the well-documented signi…cant decline in aggregate volatility in the U.S.
between 1984 and 2006. The exercise is disciplined by measurements of the changes
in aggregate and idiosyncratic volatilities. The model can account for 90% of the
observed decline in aggregate output volatility. 67% of the decline is due to the direct
e¤ect of the drop in the volatility of aggregate technology shocks and the other 23%
captures the volatility ampli…cation e¤ect due to the optimal attention reallocation
from aggregate to idiosyncratic shocks. A version of the model without rational inat-
tention can capture the former e¤ect but not the latter.
The second essay examines the redistributive e¤ects of monetary policy using a
dynamic general equilibrium model with heterogenous agents. I study the long-run
e¤ects of in‡ation on output, consumption and welfare, as well as the distribution
of wealth in the economy. Unlike in representative agent models, heterogeneity can
potentially allow for bene…cial e¤ects of in‡ation. Increases in the growth rate of
money supply can reduce wealth dispersion, increasing output and welfare.
ESSAYS ON MACROECONOMIC VOLATILITY
AND MONETARY ECONOMICS
by
Jeta Menkulasi
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
2010
Advisory Committee:
Professor Boragan Aruoba (co-chair)
Professor John Haltiwanger (co-chair)
Professor Curt Grimm
Professor Anton Korinek
Professor John Shea
c _ 2010
Jeta Menkulasi
All Rights Reserved
Dedication
To my mother, Kozeta.
ii
Acknowledgements
This dissertation is the end result of a great support from a number of people. Fore-
most, I would like to thank my advisors, Professor Boragan Aruoba and Professor
John Haltiwanger, the co-chairs to this dissertation committee, for their guidance and
extensive time devoted to discussing my research.
I am highly indebted to Prof. Boragan Aruoba for his support over the years. He
has been constantly available to discuss any questions arising in my research in great
details. He has provided invaluable help with conceptual as well as technical obstacles
I have encountered. Most important, he has encouraged me when I have been most
critical of my work. I can never thank him enough for his patience and his persistence.
I am very grateful to Professor John Haltiwanger for his insights and help with
data central to my research. I also thank Professor John Shea for taking his time to
read in great detail my drafts and provide excellent technical comments.
My work has bene…ted extensively from the seminars in the Department of Eco-
nomics. I would like to thank Prof. Anton Korinek, Prof. Carlos Vegh and Prof.
Allan Drazen for their challenging questions, which have improved the quality of my
research. I am particularly grateful to Professor Enrique Mendoza for taking his time
to discuss my work.
iii
I would also like to thank Professor Curt Grimm for accepting to serve as an
external advisor to my committee.
This dissertation comes at the cost of being away from my family for years. My
family members, my father, Gazmen, my mother, Kozeta and my brother, Fatmir have
been a constant moral support throughout this period. I am lucky to be the daughter
of Kozeta Menkulasi and the granddaughter of Gjystina Dishnica, who have been my
source of strength whenever I doubted myself.
iv
TABLE OF CONTENTS
Page
Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . iii
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Chapter
1 Rational Inattention and Changes in Macroeconomic
Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Information Processing Constraints . . . . . . . . . 8
1.3 The Model Economy . . . . . . . . . . . . . . . . . 11
1.4 Special Case: No Capital and White Noise Distur-
bances . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.5 Numerical Solution of the Benchmark Model . . . . 32
1.6 Shutting Down the Idiosyncratic Channel: Rational
Inattention versus Attention Allocation . . . . . . . 51
1.7 Can Changes in the Volatility of the Idiosyncratic
Environment Cause Changes in the Macroeconomic
Environment ? . . . . . . . . . . . . . . . . . . . . . 53
1.8 Sensitivity Analysis . . . . . . . . . . . . . . . . . . 58
1.9 Endogenous Information Processing Capacity (i) . 69
1.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 78
2 Welfare Cost of Anticipated In‡ation in a Heterogeneous
Agent Model . . . . . . . . . . . . . . . . . . . . . . . . . . 81
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . 81
2.2 A Search Economy: Money is the Only Asset in the
Economy . . . . . . . . . . . . . . . . . . . . . . . . 88
v
2.3 Numerical Solution of the First Model . . . . . . . 98
2.4 An Augmented Search economy: Money and Human
Capital . . . . . . . . . . . . . . . . . . . . . . . . . 102
2.5 Numerical Solution of the Second Model . . . . . . 113
2.6 Welfare Analysis . . . . . . . . . . . . . . . . . . . 128
2.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . 133
Appendix
A Endogeneizing Information Processing Capacity (i) . . . . 137
B Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
C Non-stochastic steady state . . . . . . . . . . . . . . . . . 139
D Why volatility ampli…cation is stronger for aggregate hours
of work than aggregate output . . . . . . . . . . . . . . . . 140
E Derivation of the information ‡ow constraint . . . . . . . . 142
E.1 Information rate of discrete parameter one-dimensional
Gaussian processes . . . . . . . . . . . . . . . . . . 142
E.2 Information rate of discrete parameter multi-
dimensional Gaussian processes . . . . . . . . . . . 144
F Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
G Perfect Information Case . . . . . . . . . . . . . . . . . . . 155
H Nash Bargaining Solution and Seller Heterogeneity . . . . 158
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
vi
List of Tables
1.1 Implied standard deviation for the Idiosyncratic TFP shock . . . . . 38
1.2 Implied standard deviation for the idiosyncratic TFP process -
changing returns to scale parameters . . . . . . . . . . . . . . . . . . 40
1.3 Benchmark Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 42
1.4 Great Moderation: Data versus RBC and Rational Inattention (RI) . 50
1.5 Rational inattention (RI) without the attention allocation problem . . 54
1.6 25%increase in idiosyncratic TFP volatility and no change in aggregate
TFP volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
1.7 Robustness check - changing Labor Supply Elasticity . . . . . . . . . 61
1.8 Robustness check - changing the upper bound of Information Processing
Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
1.9 Robustness check - Persistence of the idiosyncratic TFP process . . . 65
1.10 GHH and Benchmark Preferences - Parameters . . . . . . . . . . . . 68
1.11 GHH vs Benchmark Preferences - Rational inattention (RI) versus
standard RBC model . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
2.1 Benchmark Parameter Values . . . . . . . . . . . . . . . . . . . . . . 100
vii
2.2 Welfare cost of moving from 0% to 10% in‡ation . . . . . . . . . . . . 101
2.3 Benchmark Parameter Values - Human Capital Augmented Model . . 114
2.4 Welfare cost of moving from 0% to 10% in‡ation - Decreasing Returns
to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
2.5 Welfare cost of moving from 0% to 10% in‡ation - Constant Returns
to Scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
viii
List of Figures
1.1 Comparison of Patterns of Firm and Aggregate Volatility using
Employment Growth Rates from LBD . . . . . . . . . . . . . . . . . 9
1.2 Impulse Response to an aggregate TFP shock . . . . . . . . . . . . . 44
1.3 Impulse response of …rm level input (labor and capital) choices to an
innovation in idiosyncratic TFP . . . . . . . . . . . . . . . . . . . . . 45
1.4 Business Cycle Statistics - Perfect Information vs Rational Inattention 46
1.5 Impulse Responses to an aggregate TFP shock across di¤erent TFP
volatility regime and information structures . . . . . . . . . . . . . . 48
1.6 Impulse response of output and hours to an innovation in aggregate
TFP across di¤erent idiosyncratic volatility regimes . . . . . . . . . . 56
1.7 Elasticity of aggregate volatility with respect to aggregate shock
volatility. Linear cost in acquiring new information processing capacity. 75
2.1 Welfare implications of expansionary monetary policy . . . . . . . . . 99
2.2 The long-run e¤ects of expansionary monetary policy - Costant
Returns to Scale technology . . . . . . . . . . . . . . . . . . . . . . . 117
2.3 Type-speci…c long-run e¤ects of expansionary monetary policy - Con-
stant returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
ix
2.4 Distribution of wealth and human capital - Constant returns to scale 120
2.5 Long run e¤ect of in‡ation on aggregate variables - Constant returns
to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.6 Type speci…c long-run e¤ects of monetary expansion. Decreasing
Returns to Scale CM production technology . . . . . . . . . . . . . . 123
2.7 Dispersion in wealth and human capital - Decreasing Returns to Scale
CM production technology . . . . . . . . . . . . . . . . . . . . . . . . 124
2.8 The long run e¤ect of monetary policy on aggregate variables -
Decreasing returns to scale . . . . . . . . . . . . . . . . . . . . . . . . 125
2.9 Welfare e¤ects on in‡ation - Decreasing Returns to Scale . . . . . . . 131
2.10 Welfare consequences of in‡ation - Constant Returns to Scale . . . . 132
x
Chapter 1
Rational Inattention and Changes in Macroeconomic
Volatility
1.1 Introduction
There was a well-documented decline in U.S. macroeconomic volatility lasting from
the mid-1980s until 2006, followed by a renewed high macroeconomic volatility since
2007. This chapter aims to explain the Great Moderation and to help understand the
return to increased macroeconomic volatility.
During the Great Moderation, the volatility of aggregate output in the U.S.
declined by 50%. The leading explanations of the Great Moderation include better
monetary policy, structural changes such as better inventory management, and lower
volatility of shocks hitting the economy. The …rst two explanations have proven to
account only for part of the decline in macroeconomic volatility.
1
As for the ‘good luck’ hypothesis, one can explain a 50%decline in output volatility
in a standard RBC model only to the extent that the volatility of aggregate technology
1
Ahmed, Levin, and Wilson (2004), Arias, Hansen, and Ohanian (2006) and Stock and
Watson (2003) compare hypotheses and conclude that in recent years the U.S. economy has
to a large extent simply been hit by smaller shocks.
1
shocks declines by the same amount.
2
This opens the question of whether aggregate
TFP volatility has in fact experienced such a decline. TFP series compiled by Basu,
Fernald, and Kimball (2006) at an annual frequency covering the period 1949 - 1996
show only a 15% decline in the volatility of TFP innovations during the Great Mod-
eration. Quarterly series by Fernald (2009) covering a longer time period 1949 - 2006
and using a di¤erent methodology exhibit a 34% decline.
3
This clearly poses a problem for the ’good luck’ hypothesis using a standard
RBC model. If pure technology shocks have experienced at most a 34% decline in
volatility, a RBC model can explain only a 34% decline in output volatility. This
chapter o¤ers a mechanism that breaks this linear relationship between aggregate
TFP shock volatility and output volatility. I propose an imperfect information setting
in the form of rational inattention, in which changes in the volatility of aggregate
shocks are ampli…ed. Benchmark calibration of the model shows that a 34% decline
in aggregate TFP shock volatility can generate a 46% decline in output volatility.
Rational inattention captures the idea that agents in the economy base their
decisions not on the true state of the economy but on the perceived state, which is
2
See Arias, Hansen and Ohanian (2006) for a discussion of aggregate TFP volatility
changes and the Great Moderation. Standard RBC models are characterized by an almost
linear relationship between the volatility of aggregate technology shock and the volatility of
aggregate output. This relationship is exactly linear up to a …rst order approximation and
very close to linear for higher order appoximations.
3
Basu, Fernald, and Kimball (2006) correct for aggregation issues, variable capacity
utilization, deviations from constant returns to scale and imperfect competition. Fernald
(2009) builds a quarterly series of total factor productivity that corrects only for variable
capacity utilization.
2
conditioned on their information set (Sims, 2003). Limited in their ability to process
information, agents choose the optimal nature and precision of signals to reduce their
uncertainty regarding the true state of the economy. One can think about the problem
as a signal extraction problem, where the signal’s noise properties are endogenously
determined. In other words, the precision of the signals received as well as their
statistical properties are choice variables. The restriction on the ability to process
information limits how precise the signals can be. In the case where there is more
than one state that agents in the economy are interested in tracking, the information
processing problem becomes one of attention allocation: how to allocate information
(attention) across multiple states, or in signal extraction terminology, how to allocate
precision across multiple signals. This allocation will depend on the relevance of each
state in the objective function as well as the properties of their stochastic processes,
such as their relative persistence and volatility. More information will be allocated to
variables with a higher variance or lower persistence for a given variance.
4
This chapter applies this ‘attention allocation’ problem to an otherwise standard
RBC model with heterogenous …rms and explores the transmission mechanism of
shocks in the economy. The focus of the chapter is the relationship between the
volatility of aggregate technology and the volatility of aggregate outcomes such as
output, labor, investment and consumption. Firms’ pro…ts depend on both aggregate
and idiosyncratic state variables. Bounded in their ability to process information,
4
See Ma´ckowiak and Wiederholt (2009a)
3
they have to decide how to allocate the information ‡ow across states. Given a higher
relative volatility of the idiosyncratic state, …rms will allocate more attention to the
idiosyncratic environment and hence be more responsive to idiosyncratic shocks and
less responsive to aggregate shocks. This leads to a dampening and delay in the
response of endogenous variables to an innovation in the aggregate shock.
As the relative volatility of idiosyncratic versus aggregate states changes, so does
the optimal allocation of attention. In the face of a decline in aggregate TFP shock
volatility (’good luck’, in the terminology of the Great Moderation literature), …rms
will reallocate their attention away from the aggregate environment since the relative
volatility of the idiosyncratic environment has increased. This leads to an additional
moderating e¤ect. Hence, the decline in the volatility of aggregate outcomes is bigger
than the decline in the volatility of the aggregate shock. This is in stark contrast with
the full information version of the model, which is the standard rational expectations
RBC model.
Evidence on …rm-level data compiled by Davis, Haltiwanger, Jarmin and Miranda
(2006) show that …rm-level employment growth rate volatility has declined during the
Great Moderation period by 9%, as compared to the 40-50% decline in its aggregate
counterpart (Figure 1.1).
5
Using indirect inference, I estimate a similar (9%) decline in
5
Figure 1.1 reports the 10-year window rolling standard deviations for …rm-level and
aggregate employment growth rates. The rolling standard deviations are normalized to 1
for the baseline year 1980.
4
the volatility of idiosyncratic TFP, which combined with the 34% decline in aggregate
TFP volatility, implies an increase in the idiosyncratic-to-aggregate volatility ratio.
6
In the benchmark calibration this model can account for 90% of the decline in
aggregate output volatility experienced by the U.S. in the past 30 years. 67% of
the decline is due to direct e¤ect of the drop in the volatility of aggregate technology
shocks and the other 23%captures the volatility ampli…cation e¤ect due to the optimal
attention reallocation from aggregate to idiosyncratic shocks. This chapter presents
the idea that the reduction in macroeconomic volatility in the mid-1980s has not been
solely due to smaller aggregate shocks, but also to an increase in the relative volatility
of idiosyncratic shocks as compared to aggregate shocks, which via an attention re-
allocation has altered equilibrium behavior.
While I focus on the Great Moderation as the most obvious case study in the time
variation of aggregate volatility, it is important to note that this mechanism is more
general than the application in this chapter. By allowing the idiosyncractic environ-
ment to play a role for aggregate dynamics, rational inattention in this model o¤ers a
new relationship between microeconomic and macroeconomic volatility. Because the
idiosyncratic environment serves as a diversion of attention, changes in idiosyncratic
volatility can a¤ect aggregate dynamics without any change in the aggregate tech-
nology shock process. In order to expose the role of idiosyncratic shocks for aggregate
dynamics more directly, I ask whether changes in the idiosyncratic state volatility
6
See Section 5.1 for details on the indirect inference exercise.
5
alone can produce changes in aggregate volatility. My calibrated model shows that a
hypothetical 25% increase in the volatility of the idiosyncratic state alone can produce
an 11% decline in the volatility of aggregate output.
Starting with the …nancial crisis of 2007, there has been a renewed high degree
of macroeconomic volatility. To the extent that there has been an increase in the
volatility of the underlying aggregate shocks in the economy, this model predicts
a reallocation of attention towards the aggregate environment by agents in the
economy. This will in turn amplify initial changes in the volatility of aggregate
shocks. Hence, the current increase in macroeconomic volatility might be partially
due to more volatile aggregate shocks and partially due to more attention being
reallocated towards the macroeconomic environment.
There have been several applications of rational inattention in the literature.
Ma´ckowiak and Wiederholt (2009a) study the response of prices to aggregate nom-
inal shocks versus idiosyncratic shocks in a partial equilibrium framework. They show
how the attention allocation mechanism of …rms under rational inattention leads to
prices being more responsive to idiosyncratic shocks and less responsive to aggregate
nominal shocks. This chapter di¤ers from Ma´ckowiak and Wiederholt (2009a) in two
dimensions. First, I apply this mechanism in a general equilibrium real business cycle
framework to study how rational inattention a¤ects the transmission mechanism of
aggregate technology shocks. Second, this chapter discovers a new outcome of rational
inattention, which is a volatility ampli…cation e¤ect. One main contribution of this
6
chapter is that I conduct a disciplined quantitative exercise of whether the Ma´ckowiak
and Wiederholt (2009a) mechanism can explain the Great Moderation.
Applications of rational inattention in a dynamic general equilibrium setting
include Paciello (2008), Luo and Young (2009), and Ma´ckowiak and Wiederholt
(2009b). Paciello (2008) and Ma´ckowiak and Wiederholt (2009b) explore the dif-
ferential response of prices to various aggregate and idiosyncratic shocks.
7
Rational
inattention is shown to account for the sluggish response of prices to monetary shocks
on one hand and their quicker adjustment to neutral technology shocks on the other.
Luo and Young (2009) introduce rational inattention in a stochastic growth model
with permanent technology shocks and explore the extent to which rational inatten-
tion can enrich the weak internal propagation mechanism of shocks in RBC theory.
This chapter overlaps with their paper in that we both study the propagation mecha-
nism of technology shocks in an RBC framework. It di¤ers on the question of interest
as well as in the solution method employed. I explore the second moment e¤ects
of rational inattention in an RBC framework, with the Great Moderation being the
main case study. I also solve for a competitive equilibrium, which allows for a solution
of rational inattention models with multiple state variables and accounts for general
equilibrium e¤ects on the propagation of shocks.
Overall the contribution of this chapter in the literature is twofold. First, it is the
…rst paper to expose a volatility-ampli…cation result in rational inattention models
7
The main di¤erence between Mackowiak and Widerholt (2009b) and Paciello (2008)
and is that the latter considers only two aggregate shocks, whereas the former includes
idiosyncratic shocks as well.
7
with attention allocation. Second, it o¤ers a newapplication of the rational inattention
theory.
This chapter is organized as follows: section 1.2 introduces the tools from informa-
tion theory that are applied in my rational inattention setting. Section 1.3 introduces
the benchmark model. In section 1.4, I study a simple version of the model that
has an analytical solution to illustrate the main mechanism in this chapter. Section
1.5 presents the calibration procedure and the numerical results for the benchmark
model. In section 1.6, I distinguish between the roles of rational inattention (deci-
sion making under information processing constraints and one state variable) and
attention allocation (rational inattention with multiple state variables). I show that
simply restricting the ability to process information without having the problem of
allocating information does not lead to a volatility ampli…cation e¤ect. Section 1.7
examines whether changes in the volatility of the idiosyncratic environment alone can
lead to changes in aggregate volatility. Section 1.8 includes the sensitivity analysis.
Section 1.9 analyses the implications of a model with endogenously determined upper
bound on the ability to process information. Section 1.10 concludes.
1.2 Information Processing Constraints
In this section I introduce concepts from Information Theory that are used to quantify
information ‡ow and discuss how one can model a constraint in processing informa-
tion. The rate of information ‡ow is measured as the rate in uncertainty reduction,
8
Figure 1.1: Comparison of Patterns of Firm and Aggregate Volatility using
Employment Growth Rates from LBD
0.2
0.4
0.6
0.8
1.0
1.2
82 84 86 88 90 92 94 96 98 00
Aggregate Volatility
Firm Level Volatility
s
t
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
(
n
o
r
m
a
l
i
z
e
d
)
Source: Longitudinal Business Database (LBD), Davis,
Haltiwanger, Jarmin and Miranda (2006)
9
where the uncertainty regarding a random variable is measured by its entropy. Con-
sider a random variable A, whose probability density function is ,(A). The entropy
of A equals ÷1 [log(,(A)] . It’s important to note that uncertainty about a random
variable does not depend on its realizations but on the probability distribution of
those realizations. Given the Gaussian setting of the model that will follow, I con-
sider the entropy of a normally distributed variable. If A is normally distributed, then
its entropy equals
H(A) =
1
2
log
2
(2:c\ cA))
Hence, the uncertainty regarding a normally distributed variable is summarized by
its variance. Conditional entropy measures the conditional uncertainty of random
variable A given another random variable 1 . When A and 1 follow a joint normal
distribution, the conditional entropy becomes
H(A[1 ) =
1
2
log
2
(2:c\ cA[1 ))
Having quanti…ed the uncertainty of a random variable, information ‡ow is then
de…ned as the rate at which this uncertainty is reduced. More speci…cally:
1(A; 1 ) = H(A) ÷H(A[1 )
That is, the rate of information ‡ow between two random variables equals the di¤er-
ence between prior uncertainty and the posterior uncertainty. In the case that the two
variables are independent from each other, the reduction in uncertainty will be zero,
since knowing 1 gives no information regarding A and hence the prior and posterior
10
uncertainty will be the same. Constraints in the ability to process information are
modelled as limits in the rate at which uncertainty about a random variable can be
reduced. Formally, an information processing constraint is de…ned as:
1(A; 1 ) _ i
where i is the capacity of the channel through which information is processed, which
places an upper bound on the rate of uncertainty reduction through this channel.
The channel is referred to as the device through which individuals process informa-
tion (e.g. their brain) and the capacity refers to a technological constraint on the
maximum amount of information that can be processed through this channel (Sims,
1988, 2003, 2006). As Sims (2006) notes, it’s important to distinguish between various
economic environments where such a description of uncertainty and limited informa-
tion is logically consistent. Information processing constraints measured as limits to
the capacity of a Shannon channel, as de…ned above, are consistent with an environ-
ment where information is publicly available and the only cost to making use of this
information is the human information-processing capacity cost.
1.3 The Model Economy
In this section I develop a dynamic general equilibrium model representing an
economy populated by households and …rms. Given the availability of data on …rm-
level volatility, I will focus on the decision making process of …rms facing a constraint
11
in their information processing capabilities. There is a continuum of …rms that pro-
duce a homogenous product using labor and capital and face a decreasing returns to
scale production function as well as …rm-speci…c technology shocks. Households are
assumed to make their consumption, labor and investment decisions under perfect
information. That is, they don’t face constraints in their information processing
capacity. This assumption is made for tractability purposes.
1.3.1 Firms
This part of the model is similar to Restuccia and Rogerson (2004) as well as Bar-
telsman, Haltiwanger and Scarpetta (2009) with the main features of the model being
diminishing returns to scale and heterogenous production units as in Hopenhayn
(1992) and Hopenhayn and Rogerson (1993). The main di¤erence between this model
and the above papers is that I abstract from the entry and exit decision of …rms.
The assumption of decreasing returns to scale allows me to pin down …rm-level
employment and capital, which will then form the basis of comparison with the …rm-
level dynamics we see in the data. There are two approaches to obtaining a non-
degenerate distribution of …rm size, the …rst being a single-good model where …rms
operate under decreasing returns to scale and perfect competition, and the second
being a model with di¤erentiated products and imperfect competition, which yields a
non-degenerate distribution in size due to curvature in preferences. To avoid concerns
about price setting and to keep the model as close as possible to the standard RBC
12
model, I use decreasing returns to scale to get a non-degenerate distribution of …rm
size. Obtaining a non-degenerate distribution of …rm size is important in supporting
a distribution of the idiosyncratic productivity in equilibrium, and hence, exploring
the role of the idiosyncratic environment.
The production technology each …rm faces is
¸
it
= c
oI
c
o
.I
/
c
it
|
c
it
. c +o < 1 (1.1)
where c
t
and c
it
are the common and idiosyncratic components of …rm-speci…c TFP
respectively. In an environment of heterogeneous …rms and decreasing returns to
scale there may be a motive for entry and exit of …rms. To avoid keeping track of this
dimension I assume that in equilibrium there is no entry or exit. One can think of
various institutional barriers that could make such movements very costly for a …rm.
In this model …rms are not heterogenous in the products they produce but rather in
the idiosyncratic TFP levels they face. They di¤er in their production levels as well
as in the level of labor and capital they hire. Common and idiosyncratic components
of …rm-level TFP follow exogenous stochastic processes de…ned by
c
t
= j
¹
c
t1
+
t
(1.2)
c
it
= j
1
c
it1
+n
t
(1.3)
where
t
~ `(0. o
2
.
), n
t
~ `(0. o
2
&
). and both variables are iid over time and uncor-
related with each other.
13
Pro…ts in each period are
/
it,
|
it,
n
t,
:
t
) = c
oI
c
o
.I
/
c
it
|
c
it
÷n
t
|
it
÷:
t
/
it
(1.4)
where the wage and rental rate in the economy are taken as given by the …rm.
The …rm has to choose the level of capital and labor inputs that maximizes its
pro…ts subject to the informational constraints it faces. Formally …rm i in period t
chooses /

it
and |

it
to solve the following problem
max
fI
.I
,|
.I
g
_
1
1

t=t
~
,
t
(/
it,
|
it,
n
t,
:
t
. c
t,
c
it
)[:
t
i
_
where :
t
i
= ¦:
i,1,
:
i,2,
......:
i,t
¦ is the history of realizations of the signal process for …rm
i up until time t. The stochastic process of the signals that the …rm chooses is an
endogenous variable. Knowing how its signals a¤ect its information set and hence
its optimal input demand decisions, each …rm chooses the precision of the signals it
receives. The endogeneity of the signals’ noise is the main di¤erence between rational
inattention in this model and signal extraction.
8
In order to ensure the stationarity
of the attention allocation problem, I assume that the …rm in at period 0 receives an
in…nite sequence of past signals :
0
i
= ¦:
i,1
. .....:
i,2,
:
i,1,
:
i,0
¦. Formally the problem
of …rm i in period 0 is
max
fc
.I
g2S
1
_
1

t=0
~
,
t
(/

it,
|

it,
n
t,
:
t
. c
t,
c
it
)
_
(1.5)
subject to
1(¦n
t,
:
t
. c
t,
c
it
¦; ¦:
it
¦) _ i (1.6)
8
See Sims (2003) for a discussion on signal extraction models and rational inattention.
14
where 1(.) stands for the average ‡ow of information between the states the …rm is
trying to track and the signals it chooses to receive regarding those states, and i is
the maximum amount of information the …rms can process per period. Without any
further constraints on the structure of signals, the problem that …rms face in period
0 implies that …rms choose the joint distribution of signals and state variables, which
captures all the information signals contain about the state vector. This obviously
makes the solution quite di¢cult due to the curse of dimensionality. To avoid such
a problem I impose restrictions on the set of signals and take a quadratic approx-
imation of the objective function to allow for a much easier solution to the …rm’s
problem. I make the following assumptions on the set o. First, signals today do not
contain any information about future shocks. Second, the vector of signals that a
…rm receives can be partitioned into a subset of signals regarding only the aggregate
state (n
t
. :
t
. c
t
) and another subset of signals regarding the idiosyncratic state c
it
,
so that :
it
=
¹
it
. :
1
it
)
0
, where ¦:
¹
it
. n
t
. :
t
. c
t
¦. ¦:
1
it
. c
it
¦ are independent (this can be
true only if ¦n
t
. :
t
. c
t
¦. ¦c
it
¦ are independent, which is assumed to be the case). The
partition assumption implies that paying attention to the aggregate state and the
idiosyncratic state are two separate activities. Third, ¦:
¹
it
. :
1
it
. n
t
. :
t
. c
t,
c
it
¦ follows a
stationary Gaussian process. Gaussianity of the signals implies Gaussianity of the pos-
terior distribution, which can be shown to be optimal when the optimization problem
is quadratic (Sims, 2006). Given the tractability of a quadratic Gaussian (LQG) set-
ting, I take a log-quadratic approximation of the objective function. The question of
15
how good such an approximation is will be addressed in the calibration section of the
chapter. All the noise in the signals is assumed to be idiosyncratic, which is consistent
with the idea that errors in tracking the state of the economy come from constraints
in the ability to process information, not constraints in the availability of information
(Sims 2003, 2006).
9
The problem is set such that …rms are assumed to choose the nature of their
signals in period 0. This is not a restriction since it is optimal for the …rm to choose
its signal structure once and for all. Given the log-quadratic approximation of the
pro…t function, the objective function of the …rm will depend only on conditional
variances. In addition, given the stationary Gaussian environment that the …rms
operate in, conditional variances are independent of realizations and constant over
time. In period zero, the …rm correctly anticipates future conditional variances and
has no incentive to reallocate attention.
10
Perfect Information
Before solving the imperfect information problem, I summarize the solution to the
…rm’s problem under perfect information, which will be used in the attention alloca-
tion problem of each …rm.
9
The above mentioned assumptions also appear in Ma´ckowiak and Wiederholt (2009a,b)
and Paciello (2007).
10
See Ma´ckowiak and Wiederholt (2009a)
16
Proposition 1 Under perfect information, that is, when …rms perfectly observe
¦c
t
. c
it
. n
t
. :
t
¦ every period, the log-linearized decision rules for the …rm are
^
|
1
it
=
1
1 ÷c ÷o
[c
t
+c
it
÷(1 ÷c) ^ n
t
÷c^ :
t
] (1.7)
^
/
1
it
=
1
1 ÷c ÷o
[c
t
+c
it
÷o ^ n
t
÷(1 ÷o)^ :
t
] (1.8)
and aggregate labor and capital follow
^
1
t
=
1
1 ÷c ÷o
[c
t
÷(1 ÷c) ^ n
t
÷c^ :
t
] (1.9)
^
1
t
=
1
1 ÷c ÷o
[c
t
÷o ^ n
t
÷(1 ÷o)^ :
t
] (1.10)
Proof. See Appendix G.
It is important to emphasize that under perfect information, the aggregate
economy looks exactly like the representative agent RBC model with decreasing
returns to scale (DTRS) technology on …rms’ side, where the aggregates depend only
on aggregate technology shocks and idiosyncratic shocks disappear. Solving for the
full-information equilibrium is important in drawing out the main di¤erences rational
inattention introduces to aggregate behavior, which are that idiosyncratic volatility
matters for aggregate behavior and that aggregate volatility responds more than
one-for-one to a change in the volatility of aggregate TFP.
Rational Inattention
I start by taking a log-quadratic approximation of the pro…t function expressed
in terms of log deviations from steady state. Denoting ^ c
t,
c
it,
^
/
it,
^
|
it,
^ n
t
. ^ :
t
) =
17
c
oI
. c
o
.I
.

1c
^
I
.I
.

1c
^
|
.I
. nc
^ &I
. :c
^ vI
). where bars denote steady state values and carats
denote percentage deviations from steady state, the second order Taylor approxima-
tion of ^ : around (0,0,0,0,0,0) is given by
~ c
t,
c
it,
^
/
it,
^
|
it,
^ n
t
. ^ :
t
) · ^ 0. 0. 0. 0. 0. 0. 0) + ^ :
1
c
t
+ ^ :
2
c
it
+ ^ :
3
^
/
it
+ ^ :
4
^
|
it
+ ^ :
5
^ n
t
+ ^ :
6
^ :
t
+
^ ¬
11
2
c
2
t
+
^ ¬
22
2
c
2
it
+
^ ¬
33
2
^
/
2
it
+
^ ¬
44
2
^
|
2
it
+
^ ¬
55
2
^ n
t
+
^ ¬
66
2
^ :
t
+^ :
12
c
t
c
it
+ ^ :
13
c
t
^
/
it
+ ^ :
14
c
t
^
|
it
+ ^ :
15
c
t
^ n
t
+ ^ :
16
c
t
^ :
t
+^ :
23
c
it
^
/
it
+ ^ :
24
c
it
^
|
it
+ ^ :
25
c
it
^ n
t
+ ^ :
26
c
it
^ :
t
+^ :
34
^
/
it
^
|
it
+ ^ :
35
^
/
it
^ n
t
+ ^ :
36
^
/
it
^ :
t
+ ^ :
45
^
|
it
^ n
t
+ ^ :
46
^
|
it
^ :
t
+ ^ :
56
^ n
t
^ :
t
Using the approximated pro…t function, the optimal capital and labor inputs that
the …rm chooses are
^
|

it
= c
1
o
1[c
t
[:
t
i
] +c
1
1
1[c
it
[:
t
i
] +c
1
&
1[n
t
[:
t
i
] +c
1
v
1[:
t
[:
t
i
] (1.11)
^
/

it
= c
1
o
1[c
t
[:
t
i
] +c
1
1
1[c
it
[:
t
i
] +c
1
&
1[n
t
[:
t
i
] +c
1
v
1[:
t
[:
t
i
] (1.12)
where ¦/

it
. |

it
¦ stand for optimal capital and labor input under rational inattention.
11
For comparison the solution of …rm i in period t under full information is:
^
|
1
it
= c
1
o
c
t
+c
1
1
c
it
+c
1
&
n
t
+c
1
v
:
t
(1.13)
^
/
1
it
= c
1
o
c
t
+c
1
1
c
it
+c
1
&
n
t
+c
1
v
:
t
(1.14)
where ¦/
1
it
. |
1
it
¦ stand for the optimal choices of labor and capital under full-
information. As one can see from the equations above,
^
|

it
= 1
_
^
|
1
it
[:
t
i
_
and
^
/

it
=
11
Coe¢cients in the capital and labor input choices are as follows: c
1
o
= (
¬
34
¬
13
¬
33
÷ ¬
14
),
c
1
1
= (
¬
34
¬
23
¬
33
÷¬
24
), c
1
&
= (
¬
34
¬
35
¬
33
÷¬
45
), c
1
v
= (
¬
34
¬
36
¬
33
÷¬
46
), c
1
o
=
¬
34
¬
33
(
¬
34
¬
13
¬
33
÷¬
14
) ÷
¬
13
¬
33
,
c
1
1
=
¬
34
¬
33
(
¬
34
¬
23
¬
33
÷¬
24
)÷
¬
23
¬
33
, c
1
&
=
¬
34
¬
33
(
¬
34
¬
35
¬
33
÷¬
45
)÷
¬
35
¬
33
, and c
1
v
=
¬
34
¬
33
(
¬
34
¬
36
¬
33
÷¬
46
)÷
¬
36
¬
33
.
Equations (13) and (14) are identical to equations (7) and (8).
18
1
_
^
/
1
it
[:
t
i
_
. A …rm operating under imperfect information chooses inputs on the basis
of the perceived states (1[c
t
[:
t
i
]. 1[c
it
[:
t
i
]), whereas a …rm operating under full infor-
mation chooses inputs on the basis of the actual state (c
t
. c
it
). Anytime the input
choices di¤er from those prevalent under full information, there is a loss in pro…ts.
This loss can be measured by subtracting from ^ c
t
. c
it
.
^
/

it
.
^
|

it
. ^ n
t
. ^ :
t
) the equivalent
expression under full information ^ c
t
. c
it
.
^
/
1
it
.
^
|
1
it
. ^ n
t
. ^ :
t
), which simpli…es the atten-
tion allocation problem without a¤ecting the solution since the perfect information
pro…ts are independent of the signal choice.
The loss function is given by
1 = ^ c
t
. c
it
.
^
/

it
.
^
|

it
. ^ n
t
. ^ :
t
) ÷ ^ c
t
. c
it
.
^
/
1
it
.
^
|
1
it
. ^ n
t
. ^ :
t
)
which can be simpli…ed to
1 =
^ :
33
2
(
^
/

it
÷
^
/
1
it
)
2
+
^ :
44
2
(
^
|

it
÷
^
|
1
it
)
2
+ ^ :
34
(
^
/

it
÷
^
/
1
it
)(
^
|

it
÷
^
|
1
it
)
using (1.13), (1.14) and the fact that ^ :
3
= ^ :
4
= 0. Here ^ :
44
=

1 o
2
÷ n

1, ^ :
33
=

1 c
2
÷ :

1 and ^ :
34
=

1 co. The …rst term of the loss function measures the loss in
pro…ts due to the suboptimal capital choice, whereas the second term measures the
loss due to suboptimal labor decision. The last term in captures how the mistake in
one variable a¤ects the cost of a mistake in the other variable.
The attention allocation problem can now be stated as
min
fc
.I
g
1
_
1

t=0
,
t
_
^ :
33
2
(
^
/

it
÷
^
/
1
it
)
2
+
^ :
44
2
(
^
|

it
÷
^
|
1
it
)
2
+ ^ :
34
(
^
/

it
÷
^
/
1
it
)(
^
|

it
÷
^
|
1
it
)
_
_
(1.15)
19
subject to
^
|
1
it
=
1
1cc
(c
t
+c
it
÷(1 ÷c) ^ n
t
÷c^ :
t
) (1.16)
^
/
1
it
=
1
1cc
(c
t
+c
it
÷o
t
^ n
t
÷(1 ÷o)^ :
t
) (1.17)
^
|

it
= 1
_
^
|
1
it
[:
t
i
_
(1.18)
^
/

it
= 1
_
^
/
1
it
[:
t
i
_
(1.19)
1(¦n
t,
:
t
. c
t,
c
it
¦; ¦:
it
¦) _ i (1.20)
The result that the input choices under rational inattention are linear projections of
the optimal choices under perfect information is due to the objective function being
quadratic. Given the assumption that signals regarding idiosyncratic and aggregate
states are orthogonal, the information ‡ow can be expressed as the sum of information
‡ow that aggregate signals reveal for aggregate states, and the information ‡ow that
idiosyncratic signals reveal for idiosyncratic states. Formally,
1(¦n
t,
:
t
. c
t,
c
it
¦; ¦:
it
¦) = 1(¦n
t,
:
t
. c
t
¦; ¦:
¹
it
¦) +1(¦c
it
¦; ¦:
1
it
¦)
where :
¹
it
and :
1
it
represent the set of signals regarding the aggregate and idiosyncratic
states respectively. In this model there is only one idiosyncratic state whose true
realization …rms would like to track, namely the idiosyncratic component in …rm-level
TFP. On the other hand there are multiple aggregate states that …rms are interested
in tracking. In the multiple state case there is an additional constraint that needs to
be satis…ed

¹
_
¹jS
/
20
where
¹
is the prior variance-covariance matrix of the aggregate state vector and

¹jS
/ is the posterior variance-covariance of the same aggregate vector conditional
on the set of signals received. That is, the di¤erence between the prior and posterior
variance-covariance matrix must be positive semi-de…nite. This constraint is otherwise
called the non-subsidization constraint, which places a restriction on the precision of
signals. Without this constraint, the decision-maker can improve the precision of
one signal by erasing information (forgetting) about another variable (which can be
achieved without violating the constraint on information processing capacity, equation
(1.20). One can think of this condition as a type of irreversibility constraint on the
amount of information acquired about a particular state variable. Further details on
how information ‡ow is derived can be found in appendix E.
1.3.2 Households
The household sector is represented by a representative consumer which has access
to perfect information and a complete set of Arrow Debreu contingent securities. By
perfect information I mean that the household knows the whole history of the rele-
vant states including period t realizations. Households maximize expected discounted
utility given by
max 1
0
1

t=0
,
t
_
C
1¸
t
÷1
1 ÷¸
÷o
1
1+ç
t
1 +·
_
where C
t
is aggregate (average) consumption, 1
t
is the household’s supply of labor, ¸
is the coe¢cient of relative risk aversion, · is the inverse labor supply elasticity and
21
o captures the level of disutility of labor. Households make their decisions subject to
the following budget constraint
C
t
+1
t+1
= n
t
1
t
+ (1 + :
t
÷d)1
t
+
t
(1.21)
where n
t
and :
t
are the wage and rental rate respectively, d is the depreciation rate
of capital and
t
is the dividend yield from households’ ownership of …rms. Labor is
assumed to be homogeneous.
The transversality condition is
lim
T!1
1
0
[:
T
t=0
(1 +:
t+1
)
1
]1
T+1
= 0 (1.22)
Knowing the history of {n
t
. :
t
¦ including the period t realization, households
choose period t’s consumption, labor supply and next period’s capital holdings
{C
t
. 1
t+1,
1
t
¦. First order conditions obtained from the household’s problem are as
follows
C
¸
t
n
t
= o1
ç
t
(1.23)
C
¸
t
= ,1
t
[C
¸
t+1
(1 +:
t+1
÷d)] (1.24)
1.3.3 Equilibrium
The set of conditions to be satis…ed in equilibrium include …rst order conditions for
the household problem
C
¸
t
n
t
= o1
ç
t
(1.25)
C
¸
t
= ,1
t
[C
¸
t+1
(1 +:
t+1
÷d)] (1.26)
22
the resource constraint:
C
t
+1
t+1
÷(1 ÷d)1
t
= 1
t
(1.27)
labor and capital market equilibrium, where the prevalent wage and rental rate are
determined
1
c
(n
t
. :
t
. c
t
) =
_
i
1
o

t
i
)di (1.28)
1
c
(n
t
. :
t
. c
t
) =
_
i
1
o

t
i
)di (1.29)
market clearing condition
1
t
=
_
¸
it
di
and the aggregate and idiosyncratic components of …rm-level TFP, which are assumed
to follow independent AR(1) processes.
c
it
= j
1
c
it1
+n
it
(1.30)
c
t
= j
¹
c
t
+
t
(1.31)
_
c
it
di = 0 (1.32)
1.4 Special Case: No Capital and White Noise Disturbances
In order to illustrate the main mechanism in the model here I solve a special case of
the incomplete information model where disturbances follow a white noise process and
capital is …xed. The main di¤erences from the benchmark case are that households
cannot save, the production function is ¸
it
= c
oI
c
o
.I
|
c
it
. and the model is static. Such a
setting yields an analytic solution, which clari…es the main mechanism in the chapter.
23
1.4.1 Full Information
The equilibrium amount of aggregate hours employed in production, the wage rate
and the level of consumption in the economy under full information are
^
1
1
t
=
1 ÷¸
1 +· ÷o +o¸
c
t
(1.33)
^ n
1
t
=
· +¸
1 +· ÷o +o¸
c
t
(1.34)
^
C
1
t
=
1 +·
1 +· ÷o +o¸
c
t
(1.35)
The solution under full information shows, once again, that the aggregate variables
in the economy are determined only by the aggregate component of TFP and that
no characteristic of the idiosyncratic environment matters for aggregate dynamics. In
the following subsection I will show analytically how macroeconomic dynamics under
rational inattention do depend on the idiosyncratic environment and how this leads
to a volatility ampli…cation e¤ect.
1.4.2 Attention Allocation Problem
In this section I assume that the common and idiosyncratic components of …rm-level
TFP follow Gaussian white noise processes with respective variances o
o
and o
oi
.
Each …rm’s attention allocation problem becomes
min
fc
.I
g
1
_
1

t=0
,
t
:
33
2
(
^
|

it
÷
^
|
1
it
)
2
_
(1.36)
subject to
24
^
|
1
it
=
1
1c
(c
t
+c
it
÷ ^ n
t
) (1.37)
^
|

it
=
1
1c
(1 [c
t
[:
t
i
] +1 [c
it
[:
t
i
] ÷1 [ ^ n
t
[:
t
i
]) (1.38)
1 (¦ ^ n
t
. c
t,
c
it
¦ ; ¦:
it
¦) _ i (1.39)
There are three variables of interest to the …rms, namely the aggregate and idio-
syncratic component of TFP as well as the average wage in the economy. I start with
the guess that in equilibrium the wage rate satis…es n = ,c
t
and solve the attention
allocation problem as a function of such a guess. Instead of tracking three variables,
the …rms track only the aggregate and idiosyncratic component of TFP.
Given the quadratic nature of the objective function and the Gaussian white
noise process assumed for the states, one can prove that the optimal signals that
…rms choose take the form of "the true state + white noise".
12
Hence, we have
:
1it
= c
t
+n
it
(1.40)
:
2it
= c
it
+
it
(1.41)
where n
it
~ `(0. o
2
&
) and
it
~ `(0. o
2
.
).
12
Given that a
t
is assumed to follow a white noise process , n
t
is also white noise with a
variance of c
2
o
2
o
.
25
After receiving the signals regarding the two exogenous states, …rms form their
posteriors using Bayes’ Rule
1(c
t
[:
1it
) =
o
2
o
o
2
o
+o
2
&
:
1it
1(c
it
[:
2it
) =
o
2
o
.
o
2
o
.
+o
2
.
:
2it
These posteriors are substituted in the …rm’s objective function and the attention
allocation problem becomes
min
¬
2
a
¬
2
u
,
¬
2
a.
¬
2
s
_
_
_
_
1 ÷,
1 ÷o
_
2
o
2
o
_
_
_
1
o
2
a
o
2
u
+ 1
_
2
+
1
o
2
a
o
2
u
_
_
+
_
1
1 ÷o
_
2
o
2
o
.
_
_
_
_
1
o
2
a.
o
2
s
+ 1
_
_
2
+
1
o
2
a.
o
2
s
_
_
_
_
_
(1.42)
subject to
1
2
log
2
_
1 +
o
2
o
o
2
&
_
+
1
2
log
2
_
1 +
o
2
oi
o
2
.
_
_ i (1.43)
where each …rm minimizes its losses due to imperfect information by choosing the
signal-to-noise ratios ¦
o
2
a
o
2
u
.
o
2
a.
o
2
s
¦.
Optimal signal-to-noise ratios for each signal are
o
2
o
o
2
&
=
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
0 if (1 ÷,)
2 o
2
a
o
2
a.
_ 2
2i
(1 ÷,)
oa
o
a.
2
i
÷1 if (1 ÷,)
2 o
2
a
o
2
a.
¸ (2
2i
. 2
2i
)
2
2i
÷1 if (1 ÷,)
2 o
2
a
o
2
a.
_ 2
2i
o
2
oi
o
2
.
=
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
2
2i
÷1 if (1 ÷,)
2 o
2
a
o
2
a.
_ 2
2i
2
ì
(1,)
¬a
¬
a.
÷1 if (1 ÷,)
2 o
2
a
o
2
a.
¸ (2
2i
. 2
2i
)
0 if (1 ÷,)
2 o
2
a
o
2
a.
_ 2
2i
26
For each signal there are two possible corner solutions: one in which the …rm
chooses to allocate no attention (information ‡ow) at all and one where it chooses
to allocate all of the attention at its disposal. Zero information ‡ow allocated to a
signal implies that the signal-to-noise ratio of that signal is zero. That is, the …rm
chooses to receive an in…nitely noisy signal regarding that particular state. When a
particular signal receives all of the information ‡ow, its signal-to-noise ratio represents
the maximum precision that the signal can have given the limits on the ability to
process information.
The guess regarding the average wage rate in the economy implies a guess
regarding the average equilibrium labor employed in the economy via the general
equilibrium e¤ects from the household equilibrium conditions. Hence we have
1
t
=
, ÷¸
· +¸o
c
t
(1.44)
as the implied guess for aggregate labor.
Using the results above, I solve for the …xed point, in which the aggregate response
of labor to aggregate shocks equals the initial guess (1.44)
,

=
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
¸ if
_
o
2
a.
o
2
a
(1 ÷¸)2
i
ç+¸
1+çc+c¸
_
1 ÷
ç+¸c
ç+¸
_
o
2
a.
o
2
a
2
i
_
if
_
o
2
a.
o
2
a
<
(1c¸+¸c)2
ì
1c+(ç+¸c)(12
2ì
)
(ç+¸c)(12
2ì
)+¸(1c)
1c+(ç+¸c)(12
2ì
)
if
_
o
2
a.
o
2
a
<
(1c¸+¸c)2
ì
1c+(ç+¸c)(12
2ì
)
27
Using the assumptions on the signals and the derived information ‡ow constraint,
the interior solution to the attention allocation problem is as follows
13
^
1
t
=
^
1
1
t
_
1 ÷
1
1 ÷¸
_
o
2
oi
o
2
o
2
i
_
(1.45)
^
C
t
=
^
C
1
t
_
1 ÷
o
1 +·
_
o
2
oi
o
2
o
2
i
_
(1.46)
^ n
t
= ^ n
1
t
_
1 ÷
· +¸o
· +¸
_
o
2
oi
o
2
o
2
i
_
(1.47)
where ¦1
1
t
. C
1
t
. n
1
t
¦ are the full information solutions for labor, consumption and
wage rate respectively, as de…ned in equations (1.33), (1.34) and (1.35).
The solution under rational inattention, equations (1.45), (1.46) and (1.47), di¤ers
from the full information solution in two important ways. First, rational inattention
leads to dampened responses of all aggregate variables to a change in aggregate TFP.
Second, the responses of all aggregate variables to an innovation in aggregate TFP
are a function of aggregate and idiosyncratic TFP volatility. The latter is the key
result of this model. Endogeneizing the information set in a rational inattention sense
introduces a …rst-order e¤ect of aggregate and idiosyncratic shock volatilities. The
key parameter for this result is the relative volatility of idiosyncratic to aggregate
shocks, o
2
oi
,o
2
o
. As this ratio increases, idiosyncratic TFP is relatively more volatile
compared to aggregate TFP, which leads to a reallocation of attention (information
‡ow) towards the idiosyncratic state at the cost of less attention being allocated to
the aggregate state. The less information allocated to aggregate TFP, the stronger the
13
See appendix for details on these derivations.
28
dampening of the responses of macroeconomic aggregates to an aggregate TFP shock.
It is important to note that even though the model is solved using log-linearization
methods, endogeneizing the information set leads to a …rst-order e¤ect of aggregate
and idiosyncratic TFP volatilities on the impulse responses of endogenous variables.
In this way I can isolate the second-moment e¤ect on equilibriumoutcomes originating
only from the imperfect information part of the model. The result that the response
of macroeconomic variables to aggregate TFP is a function of relative volatility leads
to another result, which I will call the volatility ampli…cation e¤ect. A 1% change in
aggregate TFP volatility leads to more than a 1% change in the volatility of macroeco-
nomic aggregates. A standard RBC model solved using higher order approximations
to account for potential second-moment e¤ects has almost no volatility ampli…cation,
i.e. a 1% change in aggregate TFP volatility leads to an approximately 1% change in
macroeconomic volatility. Hence, the two main results that imperfect information in
the form of rational inattention delivers are a dampening in the response of all macro-
economic aggregates to an innovation in aggregate TFP, and an ampli…cation in the
response of macroeconomic volatility to a change in aggregate TFP volatility. The
…rst result is the usual result of imperfect information settings. Inability to see the
true state of the economy with no error leads to a smoother response and potentially
a delay, as shown below in the numerical solution for more generalized stochastic
processes. The ampli…cation in volatility occurs because a decline in the volatility of
the aggregate TFP shock has the direct e¤ect of lowering the volatility of the aggre-
29
gate outcome, as well as the indirect e¤ect of inducing agents to pay less attention to
aggregate shocks, leading to an additional moderating e¤ect.
In order to see this ampli…cation e¤ect analytically, I compute the elasticity of
each aggregate variable’s volatility with respect to the volatility in aggregate TFP:
c
·ov(A)
o
2
a
=
_
0·ov(A)
0o
2
s
__
o
2
s
·ov(A)
_
. The volatility elasticities for each aggregate variable
with respect to o
2
o
are
c
·ov(Y )
o
2
a
=
1
1 ÷
c
1+ç
_
o
2
a.
o
2
a
2
i
1 (1.48)
c
·ov(1)
o
2
a
=
1
1 ÷
1
1¸
_
o
2
a.
o
2
a
2
i
1 (1.49)
These elasticities are the main concern of this chapter. In the white noise case,
this ampli…cation e¤ect is determined by the relative volatility of the idiosyncratic
versus the common component of TFP, the information processing capacity, the risk
aversion coe¢cient, the degree of decreasing returns to scale and the elasticity of
labor supply. As the relative volatility increases, more attention is allocated to the
idiosyncratic state, and …rm-level actions respond less to aggregate states. This leads
to a higher volatility ampli…cation. As the capacity to process information increases,
the more the economy moves towards full-information since more capacity is available
to allocate to each state. Hence, the higher the information processing capacity, the
lower the volatility ampli…cation.
In order to explain the relationship between behavioral and technological parame-
ters a¤ecting volatility ampli…cation, I run the following thought experiment: suppose
the economy experiences a decrease in the volatility of the common component of
30
TFP. On the labor demand side of the economy, that is …rms, the fall in aggregate
volatility will lead to a reallocation of attention away from the aggregate states and
towards the idiosyncratic state. This in turn will lead …rms to respond less to aggre-
gate shocks. After aggregating all …rms’ responses, this leads to a lower volatility in
aggregate labor demand. On the supply side of the labor market, that is households,
a fall in the volatility of the common component of TFP will lead to a decline in labor
supply volatility. Given that the labor market must be in equilibrium, the change
in volatilities for labor demand and supply of labor must be the same. This implies
that wage volatility must change in equilibrium. This change in wage volatility intro-
duces general equilibrium e¤ects in the attention allocation problem. One can show
that for a risk aversion coe¢cient less than one (¸ < 1). the higher the CRRA, the
bigger the change in wage volatility required to restore labor market equilibrium for
any given change in common TFP volatility. In this experiment, the bigger the fall
in wage volatility, the bigger the fall in the volatility of the aggregate state that each
…rm wants to track. Hence, there is another round of attention reallocation in favor
of idiosyncratic variables and the same process repeats itself. To see how the labor
supply elasticity and returns to scale a¤ect aggregate output volatility, one can use
the following equations governing household labor supply and the resource constraint:
1
t
= c
t
+o1
t
= c
t
+
c
ç
(\
t
÷¸C
t
), so that a given change in wage volatility will lead
to higher changes in the volatility of output the closer production technology is to
31
constant returns to scale (higher o) and the higher the Frisch elasticity of labor supply
(lower ·). Thus, higher o and lower · increase the volatility ampli…cation e¤ect.
At the unique interior solution the optimal amount of information allocated to
the aggregate shock is
i
¹
=
1
2
log
2
_
1
1 ÷o +· +¸o
_
o
o
o
o
.
(1 ÷o)(1 ÷¸) + 2
i
(· +¸o)
__
+
1
2
i (1.50)
and the amount of information allocated to the idiosyncratic state is:
i
1
= i ÷i
¹
(1.51)
Equation (1.50) shows that the amount of attention allocated to each variable depends
on preference and technology parameters as well as the ratio of aggregate versus
idiosyncratic volatility. Below I consider an experiment, designed to mimic the Great
Moderation, in which preference and technology parameters do not change over time,
while changes in the volatility of each shock a¤ect the allocation of attention across
states.
1.5 Numerical Solution of the Benchmark Model
This section provides the numerical solution to the benchmark model with serially
correlated shocks presented in Section 3, which is a dynamic stochastic general equi-
librium model similar to the standard RBC model with the exception of rational
32
inattention on the part of …rms. I explore how accounting for an endogenous infor-
mation set a¤ects the transmission mechanism of aggregate technology shocks to the
economy.
1.5.1 Calibration
The period in the model is set to one quarter. Parameters that govern preferences
and production technology are calibrated such that they match long-run values of
postwar US aggregates. I follow standard calibration procedure as explained in Cooley
and Prescott (1995) and Prescott (1986). Using steady state equations, , is chosen
to match an annual real rate of return of 4%, which implies a value of 0.99 for ,. The
depreciation rate of 0.02 …xes the investment to capital ratio. Choosing a value of 1
for the coe¢cient of relative risk aversion reconciles the long-run observations for the
US economy of constant per-capita leisure and steadily increasing real wages (Cooley,
1995).
There has been an extensive empirical literature trying to estimate the curvature
of the pro…t function, which captures the decreasing returns to scale in the production
function. Important papers include Thomas (2002), Thomas and Khan (2007), Cooper
and Haltiwanger (2005), Fuentes, Gilchrist and Rysman (2006), and Hennessy and
Whited (2005). The estimated curvature ranges from 0.5 to 0.9. In the benchmark
model I follow Thomas and Khan (2007) and set the labor share to 0.64 and capital
share to 0.245.
33
The parameter · determining the inverse of the Frisch elasticity of labor is set
at 0.1 following Gali et al. (2005), who takes this value from micro estimates of the
elasticity of labor supply with respect to the real wage. The parameter controlling
the level of disutility of labor o is then chosen such that households spend 1/3 of
their time working.
14
Parameters governing the persistence and standard deviation of
the aggregate TFP shock are obtained using the quarterly series on TFP computed
by Fernald (2007). I …t equation (1.31) to the detrended data for both the pre and
post-1984 periods and obtain an autocorrelation coe¢cient of 0.98 for both periods
and standard deviations of 0.0092 for 1960-1983 and 0.006 for 1984-2005 respectively.
This implies a 34% decline in the volatility of the innovations in aggregate TFP and
a 15% decline in the volatility of TFP itself.
Idiosyncratic TFP process
I use the evidence on …rm-level data compiled by Davis, Haltiwanger, Jarmin and
Miranda (2006) to determine the parameters governing the process of …rm-level pro-
ductivity. There is only one moment in the model that can be exactly matched to the
data and that is the standard deviation of …rm-level employment growth rate. On the
other hand, assuming an AR(1) process for the idiosyncratic TFP process, there are
two parameters to be pinned down: the autocorrelation coe¢cient and the standard
deviation. Given that both parameters cannot be pinned down, I …x the persistence
14
This number comes from microeconomic evidence on time allocation studies, such as
Ghez and Becker (1975).
34
parameter to di¤erent values and compute the implied standard deviation for the
TFP process by matching the model’s implications to the data.
There is little consensus on the persistence of idiosyncratic TFP shocks. Ideally
this parameter should be estimated using …rm-level panel data accounting for both
common and idiosyncratic components to …rm-level TFP. There is little evidence on
…rm-level shocks but Foster, Haltiwanger and Syverson (2008) provide direct per-
sistence estimates of plant-level TFP shocks, which are around 0.80. Cooper and
Haltiwanger (2006), also using plant-level data estimate the persistence parameter
of the idiosyncratic shock to be around 0.89. In this chapter, in the absence of the
relevant …rm-level data required to compute the idiosyncratic TFP, I conduct an
indirect inference exercise. I match the model’s predictions for …rm-level employ-
ment dynamics with moments from …rm-level employment growth rate data provided
by Davis, Haltiwanger, Jarmin and Miranda (2006). The moments available from
these studies are 10-year window rolling standard deviations of …rm-level employ-
ment growth rates. The …rm-level data in these studies is annual, whereas my model
economy is quarterly. I aggregate the model to an annual frequency and obtain the
…rm-level growth rate in employment. Given the log-linearized version of the model
and the additive form of the …rst order conditions, I can exactly pin down the volatility
parameter of the idiosyncratic TFP process once I make an assumption on the per-
sistence of the idiosyncratic TFP. The indirect inference exercise is done using the
full-information version of the model. Inferring the parameters of the idiosyncratic
35
process assuming perfect information has two advantages. First, it saves computa-
tional time and second, equilibrium …rm-level responses to idiosyncratic shocks under
rational inattention match almost perfectly the behavior under perfect information,
since …rms under my benchmark calibration optimally allocate close to 95% of their
information ‡ow to tracking the idiosyncratic state.
The …rst order condition with respect to labor for …rm i in the full information
model is as follows
1
it
=
1
1 ÷c ÷o
[c
t
÷(1 ÷c)n
t
÷c:
t
+c
it
]
where c
t
is the aggregate TFP shock, whose parameters I take as given from Fernald
(2007), and c
it
is the idiosyncratic TFP.
Under full information, the equilibrium behavior of n
t
and :
t
is independent of the
idiosyncratic TFP. Assuming aggregate and idiosyncratic TFP are AR(1) processes,
their dynamics can be expressed as MA(·): c
t
= j
¹
c
t1
+
t
can be represented as
c
t
= c
¹
(1)
t
and c
it
= j
1
c
it1
+ n
it
can be represented as c
t
= c
1
(1)n
it
, where
lag polynomials c
1
(1) and c
¹
(1) are functions of their respective auto-correlation
coe¢cients. As a result, the model’s decision rules can also be expressed as MA
processes, which yields the following representation of the …rst order condition above
1
it
=
1
1 ÷c ÷o
_
c
¹
(1)
t
÷(1 ÷c)\(1)
t
÷c1(1)
t
+c
1
(1)n
it
¸
There are two unknown parameters in this decision rule, namely the persistence
and standard deviation of the idiosyncratic TFP process. Given that the only …rm-
level moment available to me is the standard deviation of …rm-level employment
36
growth rate, I experiment with di¤erent persistence parameters suggested from the
literature and then back out the implied standard deviation.
The …rm-level data are in the form of 10-year window rolling standard deviations
of …rm-level employment growth rates
o
it
=
_
1
10
5

c=4
(q
it+c
÷ q
i
)
2
_
1¸2
where q
it
is the …rm-level growth rate in employment and q
i
is its 10-year average. I
compute the model-equivalent measure and calculate the implied idiosyncratic TFP
volatility. For each sub-period (before and after 1984), I simulate the model 100
times with each simulation consisting of 300 periods. I then aggregate the model to
an annual frequency and compute a time-series of the rolling standard deviation for
the …rm-level employment growth rate. I average the 10-year window rolling standard
deviation for each sub-period and compute the implied idiosyncratic TFP. Table 1.1
reports the implied idiosyncratic standard deviation as well as the implied ratio of
idiosyncratic-to-aggregate volatility for di¤erent assumed persistence parameters for
the idiosyncratic shock.
The results show that in order to match the annual data on …rm-level volatility,
the implied standard deviation for innovations of idiosyncratic TFP prior to 1984
ranges between 0.15 and 0.17, which is 15-19 times higher than the standard devi-
ation for aggregate TFP for the pre-1984 period. The implied standard deviation
for the post-1984 era ranges between 0.13 to 0.16, which is 22-25 times than that of
aggregate TFP over this period. The ratio of idiosyncratic-to-aggregate TFP volatility
37
Table 1.1: Implied standard deviation for the Idiosyncratic TFP shock
pre 1984 post 1984 % change
Average standard deviation
( …rm-level employment growth rate data)
0.4996 0.4730 -9.46
Idiosyncratic TFP persistence j
1
= 0.95
Implied o
&
0.1746 0.1653 -9.46
Implied ratio
ou
os
19.036 27.510 44.51
Idiosyncratic TFP persistence j
1
= 0.5
Implied o
&
0.1537 0.1456 -9.47
Implied ratio
ou
os
16.763 24.226 44.52
Idiosyncratic TFP persistence j
1
= 0.3
Implied o
&
0.1435 0.1359 -9.47
Implied ratio
ou
os
15.645 22.610 44.52
38
has increased, despite a decline in both idiosyncratic and aggregate TFP volatility,
because the decline in aggregate TFP volatility has been substantially higher than
that of idiosyncratic TFP. This is the key stylized fact that will enable the cali-
brated model with rational inattention to generate a volatility ampli…cation e¤ect
when applied to the Great Moderation episode. For the benchmark model below, I
choose the persistence parameter for the idiosyncratic TFP process to be equal to
that of the aggregate TFP process, j
1
= 0.95. By setting the persistence parameter
equal across the two processes I can focus on the relative volatility ratio as the main
variable that determines the allocation of attention.
The structural parameters that this calibration exercise is most sensitive to are
the ones that govern the return to scale technology of the production function, o and
c. Table 1.2 shows the implied standard deviation, holding the persistence parameter
…xed at j
1
= 0.95. Stronger the decreasing returns to scale, larger is the implied
volatility for the idiosyncratic TFP process.
Calibrating the upper bound on information ‡ow i
The value of i , the maximum information processing capacity, has implications for
the per period loss of pro…ts for each …rm due to imperfect tracking of state variables
as well as for the marginal value of information. As Sims (2003, 2006) shows, the
Log-Quadratic-Gaussian setting is a good approximation when the marginal value of
information ‡ow is low and a bad approximation when the marginal value of informa-
39
Table 1.2: Implied standard deviation for the idiosyncratic TFP process - changing
returns to scale parameters
pre 1984 post 1984 % change
Average standard deviation
( …rm-level employment growth rate data)
0.4996 0.4730 -9.46
Returns to Scale o +c = 0.85 (o = 0.57. c = 0.28)
Implied o
&
0.2518 0.2384 -9.47
Implied ratio
ou
os
27.453 39.674 44.52
Returns to Scale o +c = 0.90 (o = 0.60. c = 0.30)
Implied o
&
0.1678 0.1589 -9.47
Implied ratio
ou
os
18.295 26.444 44.54
Returns to Scale o +c = 0.95 (o = 0.63. c = 0.32)
Implied o
&
0.0839 0.0795 -9.48
Implied ratio
ou
os
9.1474 13.23 44.63
Returns to Scale - Benchmark o +c = 0.896 (o = 0.64. c = 0.256)
Implied o
&
0.1746 0.1653 -9.46
Implied ratio
ou
os
19.036 27.510 44.51
40
tion ‡ow is high. Hence, i is chosen in such a way as to imply a low marginal value
of information. More speci…cally, as in Ma´ckowiak and Wiederholt (2009a,b), one can
…x the marginal value of information and let i be determined endogenously, or …x i
and let the marginal value of information be determined within the model. In both
cases the marginal value of information must be a reasonably low number. I pick the
latter strategy, because my goal is to evaluate the e¤ect of changes in the stochastic
processes of underlying shocks keeping …xed the information processing technology.
In the benchmark calibration, i = 4.7 bits, which implies a marginal value of informa-
tion of 0.04% of a …rm’s steady state output and an expected per-period loss in pro…ts
of 0.07% of a …rm’s steady state output. I think these are reasonably low numbers.
Table 1.3 summarizes the benchmark calibration.
1.5.2 Results
Figure 1.2 displays impulse responses of aggregate variables to a one standard devia-
tion positive shock to aggregate TFP under perfect information and rational inatten-
tion. All impulse responses presented in the chapter represent percentage deviations
from the nonstochastic steady state. For a given volatility of aggregate TFP, rational
inattention leads to a dampening and delay in the responses of output, labor, con-
sumption and investment to an innovation in aggregate TFP as compared to perfect
information. This is due to a combination of reasons. First, agents in the economy
are limited in their ability to process information, which implies imperfect tracking of
41
Table 1.3: Benchmark Parameters
Parameter Values Description
, 0.99 discount factor
¸ 1 coe¢cient of relative risk aversion
· 0.1 the inverse of labor supply elasticity
d 0.02 depreciation rate
c 0.256 capital’s share in output
o 0.64 labor’s share in output
o 2.95 the level of disutility of labor
i 4.7 upper bound on information ‡ow (bits)
j
¹
0.95 persistence parameter for aggregate TFP process
j
1
0.95 persistence parameter for idiosyncratic TFP process
o
.
(pre-1984) 0.0092 standard deviation of the innovation in aggregate TFP
o
.
(post-1984) 0.006 standard deviation of the innovation in aggregate TFP
o
&
(pre-1984) 0.1746 standard deviation of the innovation in idiosyncratic TFP
o
&
(post-1984) 0.1653 standard deviation of the innovation in idiosyncratic TFP
42
the true state vector in the economy. The degree of this imperfection depends on how
tight the information capacity constraint is. The tighter the constraint, the less precise
the signals and the more dampening and delay will be observed. Existing studies on
RBC models with rational inattention (e.g. Luo and Young 2009) have found signif-
icant departures from perfect information outcomes for a very low maximum bound
on information ‡ow (around .30 bits per time period, which is a quarter). In this
model, a low information ‡ow devoted to tracking the aggregate shock is an optimal
outcome, which is the second explanation for the …ndings in Figure 1. Agents in this
economy are endowed with 4.7 bits per period of information ‡ow, but they optimally
choose to allocate only 5% of this information ‡ow to aggregate conditions. Hence,
with most of the information ‡ow allocated to the idiosyncratic environment, agents
in the economy have a smooth and delayed response to an innovation in aggregate
TFP.
Because …rms optimally devote most of their attention to idiosyncratic outcomes,
their response to idiosyncratic shocks under rational inattention is almost identical
to that under perfect information, as shown in Figure 1.3. Labor and capital inputs
are a¤ected equally by the idiosyncratic shock. Hence, the impulse responses for both
labor and capital to an innovation in the idiosyncratic TFP shock will be the same.
Next I calculate the second moments implied by the benchmark model using the
pre-1984 estimated aggregate and idiosyncratic TFP volatilities. I simulate the model
200 times, with each simulation consisting of 300 periods. I apply the HP …lter to the
43
Figure 1.2: Impulse Response to an aggregate TFP shock
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
2 4 6 8 10 12 14 16 18
Rational Inattention
Perfect Information
-2
0
2
4
6
8
10
2 4 6 8 10 12 14 16 18
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2 4 6 8 10 12 14 16 18
.32
.36
.40
.44
.48
.52
.56
2 4 6 8 10 12 14 16 18
%
d
e
v
i
a
t
i
o
n
f
r
o
m
s
t
e
a
d
y
s
t
a
t
e
HOURS OUTPUT
CONSUMPTION INVESTMENT
44
Figure 1.3: Impulse response of …rm level input (labor and capital) choices to an
innovation in idiosyncratic TFP
-0.4
0.0
0.4
0.8
1.2
1.6
2.0
2 4 6 8 10 12 14 16 18 20
Rational Inattention
Full Information
45
Figure 1.4: Business Cycle Statistics - Perfect Information vs Rational Inattention
Cross Correlation of Output with :
(Full Information)
Variable SD (%) x(-4) x(-3) x(-2) x(-1) x x(+1) x(+2) x(+3) x(+4)
C 0.62 0.50 0.59 0.69 0.78 0.85 0.54 0.28 0.07 -0.09
I 11.61 0.12 0.28 0.48 0.72 0.99 0.78 0.59 0.41 0.27
L 1.72 0.10 0.26 0.46 0.71 0.99 0.78 0.60 0.43 0.29
Y 2.39 0.21 0.36 0.55 0.76 1.00 0.76 0.55 0.36 0.20
Cross Correlation of Output with :
(Rational Inattention)
Variable SD (%) x(-4) x(-3) x(-2) x(-1) x x(+1) x(+2) x(+3) x(+4)
C 0.6 0.59 0.68 0.75 0.82 0.87 0.66 0.45 0.25 0.06
I 7.84 0.26 0.44 0.63 0.82 0.98 0.87 0.74 0.60 0.46
L 0.8 0.48 0.66 0.82 0.92 0.93 0.80 0.66 0.52 0.37
Y 1.75 0.37 0.53 0.70 0.86 1.00 0.86 0.70 0.53 0.37
simulated data and compute the moments presented in Figure 1.4. Major di¤erences
between the perfect information and rational inattention models are observed in the
volatility of aggregate variables. Note that given the simplifying assumption that the
household sector in the economy has full information, there is little di¤erence in the
volatility of consumption. However, investment, hours and output are markedly less
volatile under rational inattention as compared to the perfect information RBCmodel.
This is expected given the low information ‡ow agents in the economy allocate to the
aggregate environment and the consequent dampening. Another e¤ect of rational
inattention in an otherwise standard RBC setting is that the delay in the response of
aggregate variables leads to stronger autocorrelations and cross-correlations.
46
Comparing Two Di¤erent TFP Volatility Regimes: Great Moderation as
a Case Study
Figure 1.5 plots the impulse responses of aggregate variables to an innovation in aggre-
gate TFP under di¤erent TFP-volatility regimes and di¤erent information structures.
The "high volatility" impulse responses correspond to an economy with aggregate
TFP calibrated to the US data prior to 1984. The "low volatility" impulse responses
correspond to an economy with TFP calibrated to the post-1984 period. Following the
evidence of Fernald (2009), I assume that TFP innovations are 34% less volatile post
1984. As the economy moves from high to low aggregate TFP volatility, the impulse
responses of output and hours experience a bigger change under rational inattention
as compared to full information. As the economy is hit by less volatile aggregate TFP
shocks, …rms optimally choose to reallocate their attention towards tracking idiosyn-
cratic TFP, and therefore respond less to innovations in aggregate TFP. This is the
mechanism that leads to the volatility ampli…cation e¤ect.
The magnitude of this ampli…cation e¤ect, which is the main result of this chapter,
is summarized in Table 1.4. I simulate the models 200 times, with each simulation con-
sisting of 300 periods. I then HP …lter the simulated data and compute the volatility
of output, hours, consumption and investment. For the model under rational inatten-
tion, a 34% decline in the standard deviation of the innovation to aggregate TFP leads
to a 46% decline in the volatility of aggregate output, a 72% decline in the volatility
of hours, a 33% decline in the volatility of consumption and a 50% decline in the
47
Figure 1.5: Impulse Responses to an aggregate TFP shock across di¤erent TFP
volatility regime and information structures
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
2 4 6 8 10 12 14 16
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2 4 6 8 10 12 14 16
High aggregate TFP volatility
Low aggregate TFP volatility
-.1
.0
.1
.2
.3
.4
.5
2 4 6 8 10 12 14 16
0.2
0.4
0.6
0.8
1.0
1.2
2 4 6 8 10 12 14 16
%
d
e
v
i
a
t
i
o
n
s
f
r
o
m
s
t
e
a
d
y
s
t
a
t
e
OUTPUT, Perfect Infomation OUTPUT, Rational Inattention
HOURS, Perfect Information HOURS, Rational Inattention
48
volatility of investment. Under perfect information, when aggregated, the model col-
lapses to a standard RBC model with decreasing returns to scale. In that case a 34%
decline in aggregate TFP volatility leads to only 34% decline in the volatility of all
macroeconomic variables. Hence, the model under rational inattention di¤ers from
the full information model in two ways. First, it ampli…es changes in the volatility of
aggregate TFP. Second, the response to changes in the volatility of TFP is di¤erent
across aggregate variables. It is stronger for hours and weaker for consumption. The
lack of volatility ampli…cation for consumption is because for simplicity households
are assumed to have in…nite information processing capacity, i.e. perfect information
about the state of the economy. The reason why volatility of hours responds more
than that of output under rational inattention but not under perfect information can
be explained as follows. Under perfect information both labor and output depend
on the true state of technology (aggregate TFP). Under rational inattention hours
depend on the perceived state of technology (1[c
t
[:
t
] ), whereas output is determined
by the true state of technology as well as hours employed in production according to
the production function. Changes in the volatility of aggregate TFP lead to bigger
changes in the volatility of the perceived state, as the latter is a function of atten-
tion allocation. Because output is a function of these two states (c
t
and 1[c
t
[:
t
]), in
percentage terms its volatility will change by more than the change in TFP volatility
and by less than the change in hours volatility. See Appendix D for the proof.
49
Table 1.4: Great Moderation: Data versus RBC and Rational Inattention (RI)
( % standard deviations)
Series Output Hours Consumption Investment
Data (1961 - 2006) 1.55 1.78 0.78 4.56
Data (1961 - 1983) 1.90 2.01 0.92 5.41
Data (1983 - 2006) 0.94 1.44 0.56 3.15
Data (late/early) 0.49 0.72 0.61 0.58
Rational Inattention (pre 1984) 1.75 0.80 0.60 7.84
Rational Inattention (post 1984) 0.95 0.33 0.40 3.92
RI (late/early) 0.54 0.28 0.67 0.50
RBC (pre 1984) 2.39 1.72 0.62 11.61
RBC (post 1984) 1.58 1.14 0.41 7.65
RBC (late/early) 0.66 0.66 0.66 0.66
o
.
(j:c1984) = 0.0092. o
.
(jo:t1984) = 0.006.
os(jcct1984)
os(jvc1984)
= 0.66
50
1.6 Shutting Down the Idiosyncratic Channel: Rational Inat-
tention versus Attention Allocation
In this section I explore the extent to which allowing for idiosyncratic volatility mat-
ters for aggregate dynamics. There are two dimensions of rational inattention that are
important for this chapter. First, …rms have imperfect information about the state
vector due to their limited ability to process information. Second, the presence of the
idiosyncratic shocks forces the …rms to allocate attention to tracking the idiosyncratic
state, at the cost of less information being allocated to the aggregate environment.
Changes in the volatility of idiosyncratic and/or aggregate shocks do not a¤ect the
total precision of …rms’ signals, but do a¤ect the way precision is allocated across sig-
nals. The direction in which the relative volatility of the shocks changes determines
the direction of attention reallocation. In the case where there is no idiosyncratic
volatility to compete for attention, all information processing capacity will be allo-
cated to improving the precision of signals regarding the aggregate state. In this case
a change in the volatility of aggregate shocks does not change the amount of informa-
tion ‡ow that goes to tracking the true state of the economy. In such an environment
there is no volatility ampli…cation e¤ect.
1.6.1 Rational Inattention Problem for the Firm
To illustrate the importance of idiosyncratic volatility to my results, I examine an
alternative model in which …rms face only aggregate shocks, but are still subject to
51
imperfect information in the form of a capacity constraint on per period information
‡ow. My setting is the standard RBC model with an information processing constraint
placed on the side of the representative …rm.
min 1
_
1

t=0
,
t
(
^ :
33
2
(
^
/

t
÷
^
/
1
t
)
2
+
^ :
44
2
(
^
|

t
÷
^
|
1
t
)
2
+ ^ :
34
(
^
/

t
÷
^
/
1
t
)(
^
|

t
÷
^
|
1
t
))
_
(1.52)
subject to
^
|
1
t
=
1
1 ÷c ÷o
(c
t
÷(1 ÷c) ^ n
t
÷c^ :
t
) (1.53)
^
/
1
t
=
1
1 ÷c ÷o
(c
t
÷o
t
^ n
t
÷(1 ÷o)^ :
t
) (1.54)
^
|

t
= 1
_
^
|
1
t
[:
t
i
_
(1.55)
^
/

t
= 1
_
^
/
1
t
[:
t
i
_
(1.56)
1(¦n
t,
:
t
. c¦; ¦:
it
¦) _ i (1.57)
If we remove the most important shock (idiosyncratic shock) and hold i constant,
…rms will have enough information ‡ow to track the aggregate shock almost perfectly
and the results under rational inattention and perfect information will be indistin-
guishable. There will be no delay or dampening in the responses of hours, output and
investment to an innovation in aggregate TFP, and there will be no volatility ampli-
…cation. This is only due to the fact that …rms have an abundance of information
processing ability on their hands.
To make the exercise interesting, suppose instead that agents are endowed with
much less information processing capacity than in the benchmark model. In partic-
ular, suppose i equals 0.23 bits, which is the amount of information ‡ow per period
52
allocated to aggregate shocks in the benchmark model. In this case rational inat-
tention will lead to dampened and delayed responses in aggregate outcomes to the
aggregate technology shock, but there will be no volatility ampli…cation. This is due
to the fact that changes in underlying shock volatility do not lead to changes in the
information ‡ow allocated to that shock (since it is the only shock). To make this
point clear, I set i = 0.23 in the imperfect information model with only aggregate
shocks and compare its volatility ampli…cation e¤ects (if any) with the benchmark
and the RBC models. Table 1.5 shows that even when the model under Rational Inat-
tention with only aggregate shocks is calibrated to yield less volatility than the RBC
model, it still maintains a linear relationship between the volatility of the aggregate
shock and the volatility of aggregate outcomes. That is, a 34% decline in the volatility
of the aggregate technology shock leads to 34% decline in the volatility of aggregate
variables just as in the standard perfect information RBC model.
1.7 Can Changes in the Volatility of the Idiosyncratic Envi-
ronment Cause Changes in the Macroeconomic Environ-
ment ?
In this section I ask whether changes in the idiosyncratic shock process alone can
generate changes in the dynamics of macroeconomic aggregates. In the following
numerical exercise I examine how an economy under rational inattention responds
53
Table 1.5: Rational inattention (RI) without the attention allocation problem
( percent standard deviations )
Series Output Hours Consumption Investment
Data (1961 - 2006) 1.55 1.78 0.78 4.56
Data (1961 - 1983) 1.90 2.01 0.92 5.41
Data (1983 - 2006) 0.94 1.44 0.56 3.15
Data (late/early) 0.49 0.72 0.61 0.58
Rational Inattention (pre 1984) 1.75 0.80 0.60 7.84
Rational Inattention (post 1984) 1.15 0.53 0.39 5.13
RI (late/early) 0.66 0.66 0.66 0.66
RBC (pre 1984) 2.39 1.72 0.62 11.61
RBC (post 1984) 1.58 1.14 0.41 7.65
RBC (late/early) 0.66 0.66 0.66 0.66
o
.
(j:c ÷1984) = 0.0092. o
.
(jo:t ÷1984) = 0.006.
os(jcct1984)
os(jvc1984)
= 0.66
54
to an increase in the volatility of idiosyncratic shocks. The "low volatility" impulse
responses correspond to an economy with idiosyncratic TFP calibrated to US data
prior to 1984. The "high volatility" impulse responses correspond to an economy
with idiosyncratic TFP being hypothetically 25% more volatile. Everything else is
kept unchanged.
Figure 1.6 plots the impulse responses of output and hours to an innovation in
aggregate TFP when the economy moves from a low-volatility to a high-volatility
idiosyncratic environment under rational inattention and perfect information. Under
perfect information, the response of variables to an innovation in aggregate TFP is the
same under high or low idiosyncratic volatility. That is, under perfect information, the
nature of the idiosyncratic environment plays no role for aggregate dynamics. On the
other hand, under rational inattention, the volatility of the idiosyncratic environment
matters for the aggregate dynamics. The more volatile the idiosyncratic shock, the
more dampened the response of aggregate variables to an innovation in aggregate
TFP, as shown in the second row in Figure 1.6.
Table 1.6 shows the magnitude of the decline in aggregate volatility due to an
hypothetical 25% increase in the standard deviation of the innovations in the idiosyn-
cratic TFP. The perfect information case as expected is not a¤ected by changes in the
idiosyncratic environment. However, the rational inattention case o¤ers a role for the
idiosyncratic environment in aggregate dynamics. Changes in idiosyncratic volatility
change the allocation of attention, which a¤ects the equilibrium behavior of agents
55
Figure 1.6: Impulse response of output and hours to an innovation in aggregate TFP
across di¤erent idiosyncratic volatility regimes
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
2 4 6 8 10 12 14 16
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
2 4 6 8 10 12 14 16
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
2 4 6 8 10 12 14 16
High idiosyncratic volaitlity
Low idiosyncratic volatility
-.1
.0
.1
.2
.3
.4
.5
2 4 6 8 10 12 14 16
%
s
t
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
f
r
o
m
s
t
e
a
d
y
s
t
a
t
e
OUTPUT, Perfect Information
HOURS, Perfect Information
OUTPUT, Rational Inattention
HOURS, Rational Inattention
56
Table 1.6: 25% increase in idiosyncratic TFP volatility and no change in aggregate
TFP volatility
( percent standard deviations )
Series Output Hours Consumption Investment
11
|c&
1.75 0.80 0.60 7.84
11
IijI
1.56 0.51 0.6 6.69
RI
IijI
,RI
|c&
0.89 0.64 1.00 0.85
11C
|c&
2.39 1.72 0.62 11.61
11C
IijI
2.39 1.72 0.62 11.61
RBC
IijI
,RBC
|c&
1.00 1.00 1.00 1.00
o
&
(/iq/) = 0.2242. o
&
(|on) = 0.1746. o
.
(/iq/) = o
.
(|on)
in the economy. In other words, the transmission mechanism of aggregate shocks in
the economy is a function in part of the stochastic properties governing idiosyncratic
shocks. Keeping all other benchmark parameters unchanged, an increase of 25 % in
the standard deviation of idiosyncratic shocks leads to a 11% decline in volatility of
aggregate output and a 36% decline in that of aggregate hours.
Reconciling a contemporaneous increase in idiosyncratic volatility and a decrease
in macroeconomic volatility is of particular importance when looking at another estab-
lished fact during the Great Moderation episode, which is the increased household-
level consumption and income volatility (Gottschalk and Mo¢tt (2002),Comin,
Groshen, and Rabin (2006), Hyslop (2001)). Increased household level volatility in
57
the mid 1980s in the face of a decline in macroeconomic activity during the same
period has stirred considerable research. Abras (2010) documents the rise in earnings
instability associated with a moderation in the aggregate as well as …rm level activity.
Augmenting the benchmark model with rational inattention in the side of the
consumers as well as …rms, could potentially reconcile the contemporaneous increase
in household level volatility and the decline in macroeconomic volatility. I will pursue
this extension of my model in my future research.
1.8 Sensitivity Analysis
In this section I examine the model’s implication for di¤erent structural parameters
such as the labor supply elasticity, the upper bound on information processing con-
straint, the assumed persistence parameter for the idiosyncratic TFP process, as well
as di¤erent household preference speci…cation.
1.8.1 Labor Supply Elasticity
Given the preferences used in the benchmark model, labor supply elasticity is de…ned
as
1
ç
. I compare the magnitude of the volatility ampli…cation for di¤erent labor elas-
ticity values. It must be noted that changes in · should be associated with changes
in o in order to maintain the same steady state value of time spent working that
we observe in the data (1
cc
= 1,3). All other parameters are kept unchanged.
15
As
15
Not changing the other parameters does not have an e¤ect on the steady state. The
only steady state value that labor supply elasticity a¤ects is hours (1).
58
the labor supply elasticity falls ( as · increases) the volatility ampli…cation e¤ect
does not change for hours but it falls for output and investment. More speci…cally,
a 34% decline in aggregate TFP innovations leads to a decline in the volatility of
aggregate output which varies in magnitude from 43% in the case of in…nitely elastic
labor supply to 38% in the case of unit elastic labor supply. The decline in aggregate
investment volatility ranges from 46% to 40%. The decline in the volatility of hours
remains roughly unchanged as labor supply elasticity changes.
16
The intuition of why labor supply elasticity is important for the volatility ampli-
…cation e¤ect, can be found by looking at the equilibrium conditions in the labor
market and how information is being processed. Equation (1.58) and (1.59) show the
aggregate labor supply and labor demand equations, which I repeat here for conve-
nience
·
^
1
t
+¸
^
C
t
= ^ n
t
(1.58)
^
1
1
t
= 1
_
1
1cc
(c
t
÷(1 ÷c) ^ n
t
÷c^ :
t
)[:
t
¸
(1.59)
Notice that given the assumption that households have full information, the labor
supply decision does not depend on information processing constraints. Labor demand
on the other hand, depends on the history and the set of signals that all …rms in the
economy receive. In equilibrium, labor demand and labor supply must equal each
other, which implies that all the ‡uctuations in labor demand must be matched by
16
Experiments show that volatility of hours is sensitive to changes in the labor supply
elasticity alone. But in this exercise we must change the parameter that governs the disutility
of labor as well in order to maintain the same steady state. As one can see from Table 1.7
lower labor supply need to be associated with higher labor supply disutility.
59
‡uctuations in labor supply and vice versa. This adjustment is done via the wage
rate in the economy, as can be seen from equation (1.58).
17
For a lower labor supply
elasticity (high ·), higher ‡uctuations in the wage rate would be required to reach the
labor market equilibrium. Higher the volatility in wage rates, stronger the incentive
of …rms to pay more attention to the aggregate environment. And in fact, in all the
numerical examples, lower the labor supply elasticity (higher ·), higher is the amount
of information processing capacity allocated to the aggregate environment. Hence, the
reason why lower labor supply elasticity is associated with lower volatility ampli…-
cation e¤ect, which numerically is shown on Table 1.7, is that …rms allocate more
attention to the aggregate state. And …rms allocate more attention to the aggregate
state because wage rate volatility is bigger.
While changing the structural parameters the marginal value of information might
change as well, questioning, in this case, how reasonable the assumed value of i is.
In all my experiments in this sensitivity analysis, the marginal value of information
remains roughly the same. Hence, comparing two models with di¤erent structural
parameters while maintaining i unchanged is a valid exercise.
1.8.2 Upper Bound on Information Processing Capacity i
As described in the benchmark calibration section of the model, the upper bound on
the capacity to process information is chosen such that the loss in pro…ts due to the
17
The adjustment mechanism in the general equilibrium is more complicated than this,
but focusing on the wage-channel captures the importance of labor supply elasticity.
60
Table 1.7: Robustness check - changing Labor Supply Elasticity
(% standard deviation)
Series Output Hours Consumption Investment
· = 0. o = 2.61. i = 5
Pre-1984 2.01 1.22 0.62 9.29
Post-1984 1.14 0.53 0.41 5.06
Late/Early 0.57 0.43 0.66 0.54
· = 0.1. o = 2.95. i = 5
Pre-1984 1.89 1.01 0.59 8.66
Post-1984 1.07 0.41 0.39 4.67
Late/Early 0.57 0.41 0.66 0.54
· = 0.3. o = 3.77. i = 5
Pre-1984 1.70 0.70 0.56 7.70
Post-1984 1.02 0.31 0.38 4.41
Late/Early 0.60 0.44 0.68 0.57
· = 1. o = 8.85. i = 5
Pre-1984 1.50 0.35 0.52 6.65
Post-1984 0.93 0.15 0.36 3.98
Late/Early 0.62 0.43 0.69 0.60
o
.
(j:c ÷1984) = 0.92. o
.
(jo:t ÷1984) = 0.6.
os(jcct1984)
os(jvc1984)
= 0.66
61
lack of full information on the side of the …rms is small enough not to induce them
to invest in additional information processing capacity.
18
In Table 1.8 I report the
model’s results for di¤erent values of i. As expected the higher the maximum capacity
to process information, the lower the volatility ampli…cation e¤ect due to rational
inattention. The reason is that higher i leads to higher capacity being allocated to
the aggregate TFP as well as idiosyncratic TFP. This means that …rms will be able
to observe the aggregate state more accurately and the aggregate dynamics approach
those under full information.
19
1.8.3 Persistence of the Idiosyncratic TFP Process
The assumed persistence parameter for the idiosyncratic TFP process is one of the
important parameters that a¤ect the allocation of attention by …rms. As discussed by
Ma´ckowiak and Wiederholt (2009a), changes in the persistence of an AR(1) process
(keeping variance constant) have ambiguous e¤ects on the amount of attention allo-
cated to that variable. On one hand, a lower persistence, everything else equal, makes
a process more di¢cult to track and hence it leads to more attention being allo-
cated to it. On the other hand, a lower persistence may also increase or decrease the
18
The idea is that what matters for …rms pro…ts is the idiosyncratic variables and not the
aggregate ones. Hence this leads …rms to allocate almost all of the information processing
capacity to processing information about idiosyncratic variables. In this sense the mistakes
…rms make regarding the aggregate state have very little impact on their own pro…ts but
substantial impact on the aggregate dynamics.
19
Please refer to Table 1.4 for comparison with the full information case. Within the same
TFP-volatility regime, the higher i is, the closer the volatility of aggregate variables is to
those under full information.
62
Table 1.8: Robustness check - changing the upper bound of Information Processing
Capacity
(% standard deviation)
Series Output Hours Consumption Investment
i = 4.7
Pre-1984 1.75 0.80 0.60 7.84
Post-1984 0.95 0.22 0.40 3.92
Late/Early 0.54 0.28 0.67 0.50
i = 4.9
Pre-1984 1.85 0.96 0.60 8.47
Post-1984 1.05 0.38 0.39 4.57
Late/Early 0.57 0.40 0.65 0.54
i = 5.1
Pre-1984 1.93 1.06 0.60 8.90
Post-1984 1.11 0.48 0.39 4.95
Late/Early 0.58 0.45 0.65 0.56
i = 5.3
Pre-1984 1.95 1.11 0.60 9.07
Post-1984 1.16 0.56 0.39 5.27
Late/Early 0.59 0.50 0.65 0.58
i = 6.15
Pre-1984 2.15 1.38 0.60 10.21
Post-1984 1.34 0.80 0.39 6.26
Late/Early 0.62 0.58 0.65 0.61
o
.
(j:c ÷1984) = 0.92. o
.
(jo:t ÷1984) = 0.6.
os(jcct1984)
os(jvc1984)
= 0.66
63
marginal value of information, which leads to an increase or decrease in the attention
allocation to that variable. In this model, lowering the persistence of the idiosyncratic
shock while holding everything else constant leads to less attention allocated to the
idiosyncratic shock.
In the calibration of the idiosyncratic TFP process, Table 1.1, di¤erent assumed
persistence parameters lead to di¤erent implied volatilities as well. More speci…cally,
lower persistence parameters are associated with lower implied volatilities.
20
Hence, in
order to evaluate the e¤ect of a lower persistence parameter, I have to use the implied
volatility associating with it. Table 1.1 shows that the calibration exercise for j
1
= 0.5
yields an implied standard deviation of 0.1537 and 0.1456 for pre-1984 and post-1984
periods respectively. Table 1.9 shows the results on aggregate volatility and the ampli-
…cation e¤ect of rational inattention for the benchmark calibration (j
1
= 0.95) and
for a less persistent idiosyncratic TFP process (j
1
= 0.5). Results show that within a
given volatility regime for the aggregate TFP (pre or post -1984), a lower persistence
for the idiosyncratic TFP process (which is also associated with a lower volatility
as well) leads to more attention being allocated to the aggregate state and less to
the idiosyncratic one. This implies a better tracking of the aggregate environment
and hence a greater volatility for each aggregate variable. Given the higher level of
attention allocated to the aggregate conditions, the volatility ampli…cation e¤ect that
rational inattention produces is lower in this case. A 34% reduction in the volatility of
20
I note that because I am matching the volatility of the growth rate of employment as
oppose to is level, the relationship between the persistence parameter and the volatility of
the idiosyncratic TFP process is a positive one.
64
Table 1.9: Robustness check - Persistence of the idiosyncratic TFP process
(% standard deviation)
Series Output Hours Consumption Investment
j
1
= 0.95. o
&
(j:c ÷84) = 0.1746. o
&
(jo:t ÷84) = 0.1653
Pre-1984 1.75 0.80 0.60 7.84
Post-1984 0.95 0.22 0.40 3.92
Late/Early 0.54 0.28 0.67 0.50
j
1
= 0.5. o
&
(j:c ÷84) = 0.1537. o
&
(jo:t ÷84) = 0.1456
Pre-1984 1.85 0.95 0.60 8.45
Post-1984 1.05 0.38 0.39 4.55
Late/Early 0.57 0.40 0.65 0.54
aggregate TFP innovations, leads to a 43% reduction in aggregate output volatility
as opposed to the 46% reduction in the benchmark calibration. It is important to
mention that direct estimates of plant-level and aggregate TFP shock persistence
parameter (Cooper and Haltiwanger, 2006) point in the direction of higher idiosyn-
cratic persistence as compared to the aggregate, which in this model works in favor
of higher volatility ampli…cation.
1.8.4 Di¤erent Household Preferences
Here I explore the implications that the form of household preferences has for the
volatility ampli…cation e¤ect. I compare the results for the benchmark separable
65
preferences versus the preferences assumed in Greenwood-Hercowitz-Ho¤man (GHH,
1988).
21
The speci…cation of preferences determines the dynamics on the labor supply
side of the economy and hence a¤ects the feedback mechanism between imperfect
information on the side of the …rms and the household sector. The GHH preference
function is as follows:
l(C
t
. 1
t
) =
(C
t
÷o1
u
t
)
1¸
÷1
1 ÷¸
. · 0. i 1
Whereas the preferences in the benchmark model are:
l(C
t
. 1
t
) =
C
1¸
t
÷1
1 ÷¸
÷o
1
1+ç
t
1 +·
The main di¤erence between these two types of preferences is the equilibrium labor
supply. Under GHH preferences, labor supply is independent of consumption, due to
the absence of wealth e¤ects. Both preference speci…cations lead to a volatility ampli-
…cation e¤ect, but of di¤erent magnitude. In the numerical experiments, I calibrate
the two di¤erent models such that they yield the same steady state equilibrium. I
…nd that the ampli…cation is smaller in magnitude for GHH preferences. The absence
of wealth e¤ects leads to less reallocation of attention in response to a change in
the volatility of aggregate shocks. The intuition is the following: when the economy
faces a decline in the volatility of aggregate shocks, this will lead …rms in all cases to
reallocate attention away from the aggregate environment, which will be re‡ected in
the weights they put on various shocks in their demand for inputs. Such changes in
21
I consider Cobb-Douglas preferences as well. Results show that ampli…cation is similar
for separable and Cobb-Douglas preferences.
66
the input demand by …rms will have to be matched by changes in the input supply
of households. Under GHH preferences labor supply responds di¤erently to changes
in labor demand than under the benchmark preference speci…cation. In particular,
the change in labor supply is accomplished only through a change in the wage rate
rather than consumption. For preference speci…cations with wealth e¤ects and hence
a negative covariance between consumption and labor supply, a larger change in the
wage rate will be required to match a given change in the demand for labor by …rms.
This leads to bigger volatility ampli…cation for preference speci…cations which allow
for wealth e¤ects.
Table 1.10 reports the parameters used in the numerical solution for each pref-
erence speci…cation. Labor supply elasticity for GHH preferences is
1
i1
. i 1 and
for our benchmark preferences is
1
ç
. · 0. In this section’s exercise I set i such
that it produces the same labor supply elasticity as in the benchmark model, that is,
i = · + 1 while adjusting the parameters governing disutility of labor o such that
both models yield the same steady state results.
Table 1.11 shows the numerical results. First, within the same subperiod, GHH
preferences lead to higher aggregate volatility. Second, as mentioned above rational
inattention produces less of a volatility ampli…cation e¤ect in the case of GHH prefer-
ences. Third, there is a smaller asymmetry across aggregate variables in terms of the
reduction in volatility as a response to a less volatile aggregate TFP process. This is
a desirable feature since the benchmark model predicts strong counter factual results
67
Table 1.10: GHH and Benchmark Preferences - Parameters
Parameter Values Description
Common Parameters
, 0.99 discount factor
¸ 1 coe¢cient of relative risk aversion
d 0.02 depreciation rate
c 0.256 capital’s share in output
o 0.64 labor’s share in output
i 5 upper bound on information ‡ow (bits)
j
¹
0.95 persistence parameter for aggregate TFP process
j
1
0.95 persistence parameter for idiosyncratic TFP process
o
.
(pre-1984) 0.0092 standard deviation of the innovation in aggregate TFP
o
.
(post-1984) 0.006 standard deviation of the innovation in aggregate TFP
o
&
(pre-1984) 0.1746 standard deviation of the innovation in idiosyncratic TFP
o
&
(post-1984) 0.1653 standard deviation of the innovation in idiosyncratic TFP
GHH Preferences
i 1.3 labor supply elasticity:
1
i1
o 1.76 the level of disutility of labor
Benchmark Preferences
· 0.3 labor supply elasticity:
1
ç
o 3.77 the level of disutility of labor
68
in terms of the disproportionate response of hours as compared to other aggregate
variables. As shown in appendix D, the reaction of hours in terms of volatility reduc-
tion as compared to output will always be greater but the extent of this di¤erence
depends on the type of preferences being considered.
1.9 Endogenous Information Processing Capacity (i)
In this subsection I explore the implications of rational inattention when the …rms in
addition to deciding how to allocate information, also decide how much information
processing capacity they want to acquire. I assume that …rms face a cost function
C(i) when acquiring additional i.
The attention allocation problem of the …rms as described by equations (1.42)
and (1.43), can be restated in terms of information ‡ow allocated to aggregate versus
idiosyncratic variables as opposed to signal-to-noise ratios. Let i
¹
and i
1
, denote the
amount of information allocated to the aggregate and idiosyncratic shock respectively.
Any given pair ¦i
¹
. i
1
¦ is associated with the following signal-to-noise ratios:
o
2
a
o
2
u
=
2
i
/
÷1 and
o
2
a.
o
2
s
= 2
i
1
÷1. This comes from the information ‡ow constraint, equation
(1.43)
1
2
log
2
_
1 +
o
2
o
o
2
&
_
. ¸¸ .
i
/
+
1
2
log
2
_
1 +
o
2
oi
o
2
.
_
. ¸¸ .
i
1
_ i
It implies that choosing ¦i
¹
. i
1
¦ is the same as choosing the signal-to-noise ratios.
The objective loss function that …rms face due to imperfect information (1.42) can
69
T
a
b
l
e
1
.
1
1
:
G
H
H
v
s
B
e
n
c
h
m
a
r
k
P
r
e
f
e
r
e
n
c
e
s
-
R
a
t
i
o
n
a
l
i
n
a
t
t
e
n
t
i
o
n
(
R
I
)
v
e
r
s
u
s
s
t
a
n
d
a
r
d
R
B
C
m
o
d
e
l
(
p
e
r
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e
n
t
s
t
a
n
d
a
r
d
d
e
v
i
a
t
i
o
n
s
)
G
H
H
P
r
e
f
e
r
e
n
c
e
s
B
e
n
c
h
m
a
r
k
P
r
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f
e
r
e
n
c
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S
e
r
i
e
s
O
u
t
p
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t
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o
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r
s
C
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s
u
m
p
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t
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I
(
p
r
e
1
9
8
4
)
2
.
2
4
1
.
5
7
1
.
4
2
6
.
7
0
1
.
7
0
0
.
7
0
0
.
5
6
7
.
7
0
R
I
(
p
o
s
t
1
9
8
4
)
1
.
3
9
0
.
9
3
0
.
8
7
4
.
2
9
1
.
0
2
0
.
3
1
0
.
3
8
4
.
4
1
R
I
(
l
a
t
e
/
e
a
r
l
y
)
0
.
6
2
0
.
5
9
0
.
6
1
0
.
6
4
0
.
6
0
0
.
4
4
0
.
6
8
0
.
5
7
R
B
C
(
p
r
e
1
9
8
4
)
2
.
5
5
1
.
9
6
1
.
6
7
6
.
9
0
2
.
1
1
1
.
2
7
0
.
5
6
1
0
.
1
4
R
B
C
(
p
o
s
t
1
9
8
4
)
1
.
6
6
1
.
2
8
1
.
0
9
4
.
5
0
1
.
3
8
0
.
8
3
0
.
3
6
6
.
6
3
R
B
C
(
l
a
t
e
/
e
a
r
l
y
)
0
.
6
5
0
.
6
5
0
.
6
5
0
.
6
5
0
.
6
5
0
.
6
5
0
.
6
5
0
.
6
5
o
.
(
j
:
c
÷
1
9
8
4
)
=
0
.
9
2
.
o
.
(
j
o
:
t
÷
1
9
8
4
)
=
0
.
6
.
o
s
(
j
c
c
t

1
9
8
4
)
o
s
(
j
v
c

1
9
8
4
)
=
0
.
6
6
70
be rewritten as
min
i
/
,i
1
[ :
33
[
2
_
_
1 ÷,
1 ÷o
_
2
o
2
o
2
2i
/
+
_
1
1 ÷o
_
2
o
2
o
.
2
2i
1
_
subject to
i = i
¹
+i
1
Below I explore the implications of having a linear and a convex cost structure.
1.9.1 Linear Costs in Acquiring Information Processing
Capacity
In addition to deciding how to allocate a given information processing capacity, …rms
also decide how much of this capacity to acquire. The …rm’s problem then becomes
min
i
/
,i
[ :
33
[
2
_
_
1 ÷,
1 ÷o
_
2
o
2
o
2
2i
/
+
_
1
1 ÷o
_
2
o
2
o
.
2
2(ii
/
)
_
÷C(i) (1.60)
where I have substituted the constraint i
1
= i÷i
¹
. Let’s consider a linear cost struc-
ture C(i) = ci. where c is the marginal cost of acquiring an additional information
processing capacity. The …rst order conditions for this problem are
2
2i
/
= (1 ÷,)
_
o
o
o
o
.
_
2
i
(1.61)
and
ln(2) [ :
33
[
_
1
1 ÷o
_
2
o
2
o
.
2
2(i
/
i)
= c (1.62)
The …rst equation captures the attention allocation decision for a given capacity i.
and the second equation balances the marginal bene…t and cost of acquiring additional
71
information processing capacity. Looking for an interior solution, equations (1.61) and
(1.62) will determine the optimal allocation of attention as well as the optimal amount
of information processing capacity. The solution to this system of equations is
2
i
=
_
ln(2) [ :
33
[
c
(1 ÷,)
(1 ÷o)
2
_
o
o
o
o
.
(1.63)
and
2
2i
/
=
_
1 ÷,
1 ÷o
_
2
_
ln(2) [ :
33
[
c
_
o
2
o
(1.64)
There are two important outcomes when i is endogenized assuming a linear
cost structure. First, as equation (1.63) shows, the amount of optimal information
processing capacity is an increasing function of the volatility of each shock.
22
This
implies that, as each shock becomes more volatile (keeping the volatility of the other
shock constant), it is optimal to increase the capacity to process information. Second,
and most important, the optimal amount of information allocated to the aggregate
shock is no longer a function of the ratio of idiosyncratic versus aggregate shock
volatility. The optimal amount of attention now depends on own-shock volatility not
on the relative volatility of shocks. This is important since it is in stark di¤erence
with the result obtained when i was held …xed. The reason for such a result is the
linear cost structure in obtaining new information processing capacity. In order to see
why this is the case, let’s focus on equations (1.61) and (1.63). When the volatility of
the aggregate shock increases, there are two e¤ects on the optimal level of attention
22
At this point of the problem I haven’t solved the …xed point problem yet (equilibrium
,), but as it will be shown later, , < 1, which ensures a positive coe¢cient in equation
(1.63).
72
allocated to the aggregate shock (i
¹
) : …rst, …rms would want to substitute capacity
away from the idiosyncratic shock (equation (1.61)), and second, …rms would also
like to increase their total information processing capacity (equation (1.63)). I call the
…rst the capacity substitution e¤ect and the second, the capacity acquisition e¤ect. In
this case as I have shown before, an increase (decrease) in the volatility of the aggre-
gate shock (keeping the idiosyncratic shock volatility constant), will lead to a higher
(lower) level of information ‡ow being allocated to the aggregate shock. The di¤erence
from the …xed-i case is that all this increase comes from a higher overall capacity
being acquired by the …rm not due to a substitution of attention across states. Going
back to equations (1.61) and (1.63), I explore the e¤ect that a change in the volatility
of the idiosyncratic shock has on the allocation of attention to the aggregate shock.
By looking at equation (1.61) as the volatility of the idiosyncratic shock increases
(o
o
.
), the attention allocated to the aggregate shock (i
¹
) will tend to decrease since
…rms would want to substitute information from the aggregate to idiosyncratic shock.
This captures the familiar substitution e¤ect I explored the previous section, where
i was held …xed. In this case however, endogeneizing i leads to an additional e¤ect.
When the volatility of the idiosyncratic shock increases, the overall capacity i will
also tend to increase, as can be seen from equation (1.63). In the case of a linear cost
structure these two opposing e¤ects cancel each other, leaving i
¹
unchanged.
23
As o
o
.
increases, all the new acquired capacity is fully devoted to an increase in attention to
23
Technically speaking, under a di¤erent cost structure, we would have a term C
0
(i)
instead of c in equations (1.63) and (1.61).
73
idiosyncratic shocks and no change in the attention allocated to the aggregate shock.
Hence, under a linear cost structure, changes in the idiosyncratic environment have
no e¤ect on the macroeconomic environment.
The equilibrium to this model is the solution to the …xed point problem between
the initial guess (1.44) and the actual aggregate labor’s law of motion.
24
1 =
1 ÷,
1 ÷o
_
1 ÷2
2i
/
_
The analytical solution to this problem, somewhat tedious, can be found in Appendix
A. For expositional purposes I run a simple numerical example to show the e¤ects of
endogeneizing information processing capacity on the main focuses of this chapter,
which is the elasticity of the aggregate shock volatility and aggregate outcome
volatility.
Changes in the volatility of the aggregate shock are ampli…ed even in the case of
endogenous i. That is, even though the model achieves a dichotomy between the idio-
syncratic and the aggregate environment in the face of changes in the idiosyncratic
volatility, it can still provide an ampli…cation of the volatility of aggregate shocks.
When the volatility of the aggregate TFP shock changes, the volatility of aggregate
variables will still change by more. Figure 1.7 shows that for both aggregate vari-
ables, the elasticity of aggregate volatility with respect to aggregate shock volatility
is greater than one. and that as the volatility of the aggregate shock gets larger, this
24
This expression comes from aggregating individual labor-input responses of all …rms
under rational inattention.
74
Figure 1.7: Elasticity of aggregate volatility with respect to aggregate shock
volatility. Linear cost in acquiring new information processing capacity.
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
.009 .010 .011 .012 .013 .014 .015
Aggregate shock volatility
(keeping idiosyncratic shock volatility constant)
Labor
Output
e
l
a
s
t
i
c
i
t
y
75
elasticity falls. The decline in elasticity is due to the non-linear nature of the informa-
tion ‡ow constraint.
25
However, as previously shown, there will be no change in the
volatility of aggregate variables if the volatility in the idiosyncratic shock changes.
1.9.2 Convex Costs in Acquiring Information Processing
Capacity
In this section I experiment with a convex cost structure in the acquisition of infor-
mation processing capacity and consider the implications that an endogenously deter-
mined i has on the equilibrium. The problem the …rm faces is represented by (1.60)
min
i
/
,i
[ :
33
[
2
_
_
1 ÷,
1 ÷o
_
2
o
2
o
2
2i
/
+
_
1
1 ÷o
_
2
o
2
o
.
2
2(ii
/
)
_
÷C(i)
where now C
0
(.) 0. C
00
(.) 0. For expositional purposes I choose C(i) = 2
i
. simply
because it allows for neater closed form expressions. The …rst order conditions for this
version of the problem are
26
2
2i
/
= (1 ÷,)
_
o
o
o
o
.
_
2
i
and
[ :
33
[
_
1
1 ÷o
_
2
o
2
o
.
2
2(i
/
i)
= 2
i
(1.65)
Solving this system of equations leads to
2
i
=
_
[ :
33
[
1 ÷,
(1 ÷o)
2
_1
2
o
1
2
o
o
1
2
o
.
(1.66)
25
See equation (1.43)
26
The …rst order condition wrt to i
¹
is the same and i repeat the equation here for
completeness.
76
and
2
2i
/
=
_
(1 ÷,)
3
(1 ÷o)
2
[ :
33
[
_1
2
_
o
3
o
o
o
.
_1
2
(1.67)
Allowing for non-linear costs in acquiring information processing capacity, recon-
nects the macro and the microeconomic environment, similar to Section 1.4, where
i was held …xed. Changes in the idiosyncratic environment can a¤ect the aggregate
volatility. In order to see why this is the case let’s focus on equations (1.61) and (1.66).
The attention allocated to the aggregate shock (i
¹
) is increasing in the volatility of
this shock since both the capacity substitution e¤ect and the capacity acquisition e¤ect
work in the same direction. As shown before such a result is true for a linear cost
function as well. The di¤erence between the two capacity acquisition technologies
lies in the way that optimal allocation of attention to one shock reacts to changes
in the volatility of the other shock. In this case, as the volatility of the idiosyncratic
shock increases, there are two opposing e¤ects on the optimal amount of attention
allocated to the aggregate shock. As equation (1.61) shows, for a …xed i there will
be a tendency to decrease the attention allocated to the aggregate environment (i
¹
).
On the other hand, as equation (1.66) shows, an increase in the noise of any shock
would make it optimal to increase the overall capacity (i). Hence, on one hand, there
is the capacity substitution e¤ect that lowers the attention allocated to the aggregate
shock and on the other hand, there is the capacity acquisition e¤ect that increases
this same attention. In the case of linear costs, these two e¤ects cancel each other out.
In the case of convex costs, on the other hand, the substitution e¤ect is greater than
77
the acquisition e¤ect and hence as equation (1.67) shows, an increase in the volatility
of the idiosyncratic shock lead to a reduction in the attention allocated to the aggre-
gate shock. Hence, in this setting we are able once more to connect the idiosyncratic
environment to the aggregate one.
Solving for the …xed point of this problem follows the same procedure as before.
The two main results that were obtained in the …xed-i case still hold in the case of
convex costs. That is, there will be an ampli…cation in the volatility of aggregate TFP
shock as well as an impact of changes in the idiosyncratic noise onto the volatility of
aggregate outcomes.
1.10 Conclusion
In a standard RBC model there is an almost linear relationship between the volatility
of aggregate TFP shocks and the volatility of aggregate variables such as output,
employment and investment. This chapter shows that endogenizing the information
set in an otherwise standard RBC model breaks this linear relationship. Following
the literature on rational inattention, agents in this economy are assumed to be con-
strained in their ability to process information and face the decision of how to allocate
this limited information ‡ow across many state variables of interest. The trade-o¤ they
face in terms of allocating limited attention across aggregate and idiosyncratic states
is the key aspect of the model that leads to a non-linear relationship between the
volatility of aggregate TFP and macroeconomic variables. The observed 34% decline
78
in TFP volatility fromthe pre-1984 to the post-1984 period can generate a 46%decline
in output volatility when agents rationally reallocate attention away from aggregate
shocks and towards idiosyncratic shocks.
Hence, rational inattention with attention allocation implies that equi-proportional
changes in the volatility of aggregate shocks are not necessary to generate a given
magnitude of change in the volatility of macroeconomic variables. One of the key
variables that determines the extent of this non-linear relationship between TFP
volatility and output volatility is the relative volatility of aggregate versus idio-
syncratic shocks. This variable determines how much attention is allocated to each
state variable, with more information ‡ow being directed towards the nosier variable.
Hence, a relatively more noisy idiosyncratic environment would lead to more atten-
tion being allocated towards idiosyncratic states at the cost of less information being
allocated to aggregate shocks. The contribution of this chapter is to bring forth the
importance of endogenous information sets as well as the interaction between the
aggregate and idiosyncratic environment in determining macroeconomic volatility.
There are several extension of this model that I intend to work in the future.
First, this model can be extended to allow for rational inattention on the side of
consumers as well as …rms. This would be particularly interesting since this model
could reconcile two established facts regarding the 1984-2006 period, that of increasing
household level earnings volatility and declining macroeconomic volatility (Gottschalk
and Mo¢tt (2002), Comin, Groshen, and Rabin (2006), Hyslop (2001)). As shown in
79
Section 1.7 of the chapter, the attention allocation mechanism can lead to a contem-
poraneous increase in idiosyncratic volatility and a decline in aggregate volatility.
A second extension of this model would be to allow for monetary shocks as another
aggregate shock in the economy. The reason for this is to address the observed decline
in in‡ation volatility that the U.S. has experienced during 1984-2006. This would be
complementary to the Ma´ckowiak and Wiederholt (2009b) DSGE model of rational
inattention where they allow for technology and monetary policy shocks.
Third, this model can be extended to allow for a time variation in the volatility
of the structural innovations. This would have implications for the time-variation in
the share of information allocated across shocks.
80
Chapter 2
Welfare Cost of Anticipated In‡ation in a Heterogeneous
Agent Model
2.1 Introduction
This chapter examines the redistributive e¤ects of monetary policy using a dynamic
general equilibrium model with heterogenous agents. I study the long-run e¤ects of
in‡ation on output, consumption and welfare, as well as the distribution of wealth
in the economy. Unlike in representative agent models, heterogeneity can potentially
allow for bene…cial e¤ects of in‡ation. Increases in the growth rate of money supply
can reduce wealth dispersion, increasing output and welfare.
This chapter builds on the two-sector search-theoretic model of Lagos and Wright
(2005), which provides micro-foundations for money. One sector is characterized by
decentralized trade where trading partners are matched randomly. This sector incor-
porates search and information frictions, which make money the essential medium
of exchange. In addition to decentralized trade, agents have access to another sector
where Walrasian markets operate and agents produce and trade goods, and adjust
81
their money balances. Money is the only asset in the economy. It is a medium of
exchange as well as the only store of value. For tractability purposes, preferences are
assumed to be quasi-linear, eliminating any wealth e¤ects in the demand for money
and making the distribution of money holdings at the end of the centralized market
session degenerate. This eliminates the extreme degree of market incompleteness. The
sector where centralized trade occurs basically insures against all trading shocks that
agents face in the decentralized market. All agents choose the same level of money
holdings to carry into the next period. The Lagos and Wright (2005) model provides
a tractable way of evaluating the welfare cost of in‡ation in an environment where the
role of money is an endogenous outcome of search and information frictions. Due to
its simplifying assumptions, which lead to a degenerate distribution of money hold-
ings, it cannot be used to analyze the redistributive aspect of in‡ationary policies
or their impact on the real economy and welfare. In order to study these aspects
of in‡ation I augment the Lagos and Wright model in two ways. First, I introduce
heterogeneity in discount factors and second, I allow the presence of productive cap-
ital in the economy.
1
I evaluate each element systematically, …rst by solving a model
where money is the only asset in the economy but where agents di¤er in their dis-
count factors, and second by solving a model where I allow for productive capital in
the economy. This type of ex-ante heterogeneity in a Lagos and Wright framework
1
One could introduce other forms of heterogeneity, such as, heterogeneity in preferences,
productivity, etc.
82
provides a non-degenerate distribution of money while keeping the model tractable.
2
Agents have either high or low discount factors. The result is a two point distribution
of money holdings and in‡ation tax has redistributive e¤ects. The following results
emerge from the …rst model. More patient agents hold more money than impatient
ones. As long as money is being injected in the economy using lump-sum transfers,
an increase in the growth rate of money supply has two e¤ects: A direct e¤ect of
redistributing wealth from the rich to the poor, since the poor will have less than
average money holdings, and an indirect e¤ect of reducing real money balances for
both agents. The sensitivity of the agent’s money demand to in‡ation will be di¤erent
for each type. In this model, the richer agents avoid the in‡ation tax faster than the
poor agents. The net e¤ect is that the direct redistributive e¤ect of in‡ation in favor
of the less wealthy is dominated by their weaker ability to evade the in‡ation tax.
Next, I measure the welfare cost of in‡ation for each type of agent. The in‡ation tax
seems to be a¤ecting the less wealthy more than the wealthy agents, making in‡ation
in this way a regressive tax. Erosa and Ventura (2002), using a di¤erent monetary
model, reach the same conclusion.
3
In the second model, I allow for the agents to accumulate human capital by allo-
cating a fraction of their time to skill acquisition activities. Human capital can be used
2
The quasilinearity assumption in the LW framework eliminates all the heterogeneity in
money holdings that would emerge from trading shocks that agents face in the decentralized
sector, but not any type of ex-ante heterogeneity such as heterogeneity in preferences,
discount factors or other structural parameters.
3
Erosa and Ventura (2002) build a model where agents hold money because buying
goods with credit is costly. They show how in‡ation is a regressive consumption tax because
wealthy agents have access to …nancial markets which allow them to avoid the in‡ation tax.
83
in productive activities in the decentralized and centralized sectors of the economy.
This model allows me to examine the e¤ect of in‡ation in a richer environment. The
presence of another capital in an heterogenous agents environment can provide an
additional channel for redistributive e¤ects of changes in the growth rate of money
supply. This channel is the economy-wide price of e¤ective labor, in this case the
wage rate. Firms in the centralized market produce an homogenous product using
aggregate e¤ective labor. The wage rate in the economy depends on the returns to
scale technology. It can be constant, in the case of constant returns to scale or depend
on average e¤ective labor, in the case of a decreasing returns to scale production tech-
nology. In the latter case, the wage rate provides a redistributive channel for in‡ation
other than the lump-sum transfer injections of money supply by the central bank.
The results of this model show that for constant returns to scale technology, increases
in the money growth rate lead to a reduction in aggregate production, consumption
and human capital accumulation, as well as hours of work and time spent in skill
acquisition. It also leads to an increase in the dispersion of wealth and skill-level
(human capital). The welfare analysis shows that, as in the …rst model, the poorer
agents (the less patient) su¤er more from the in‡ation tax than the richer agents do.
In the case of decreasing return to scale technology of production in the centralized
market, the wage will depend on economy-wide average e¤ective labor. In this case
the model predicts a reversal of the previous results. Aggregate consumption, human
capital and time spent in skill acquisition increase with a higher money growth rate,
84
whereas aggregate hours of work decline. Dispersion in wealth and human capital
falls. In this version of the model, patient agents hold less money and accumulate less
human capital in the steady state as compared to the impatient ones. Welfare cost
analysis shows that richer agents su¤er most from in‡ation tax and poorer agents can
actually bene…t from in‡ation. In this scenario the Friedman rule is not the optimal
policy.
Papers by Berentsen et al.(2005), Molico (2006), Bhattacharya et al. (2005),
Berentsen and Strub (2009) have built on the Lagos and Wright (2005) framework
to examine the redistributive e¤ects of in‡ation. Bhattacharya et al. (2005) examine
the redistributive e¤ects of in‡ation in a very similar framework to the …rst model of
this chapter, where money was the only asset in the economy. Berentsen and Strub
(2009) in a similar model to Bhattacharya et al. (2005) study alternative institutional
arrangements for the determination of monetary policy in a search-theoretic setting
where agents di¤er in terms of preferences. The main di¤erence between the …rst
model in this chapter and Bhattacharya et al. (2005), is that the later assumes a
di¤erent type of heterogeneity. Agents in their model are di¤erent in their consump-
tion preferences as opposed to having di¤erent discount factors. The resulting e¤ect
of in‡ation on welfare is also di¤erent. Bhattacharya et al. (2005)’s results show that
relatively richer agents su¤er more from in‡ation as opposed to the less wealthy. I
obtain the opposite result. In this model the ability of the wealthier agents to evade
the in‡ation tax dominates the transfer from the rich to the poor that lump-sum
85
injections of money provide. Molico (2006) studies the e¤ects of money growth in a
heterogenous agents model. The author also uses a search-theoretic model of money,
but not of the Lagos and Wright (2005) type. Molico (2006) results show that for low
in‡ation rates, lump-sum transfers of money compress the distribution of wealth and
improve welfare. The opposite is true for higher in‡ation rates. Their heterogeneity is
an endogenous one. Unlike in my …rst model, agents in Molico (2006) hold di¤erent
amounts of money because of the history of trading shocks they face.
4
Typically, search-theoretic models of money consider environments where money
is the only asset in the economy. It is a medium of exchange as well as the only
store of value. The …rst attempt to introduce another asset in a Lagos and Wright
framework, namely physical capital in the centralized market (CM), was by Aruoba
and Wright (2003). Physical capital was not introduced into decentralized market
(DM) production since claims on physical capital would compete with money as a
medium of exchange and potentially dominate it in a rate of return sense.
5
With
physical capital being used for production only in the CM, the model dichotomizes.
That is, one can solve for the DM production path and CM production and capital
accumulation separately, with in‡ation having no e¤ect on the latter. Money is super-
neutral in terms of capital accumulation and CM production. Aruoba, Waller and
Wright (2008) extend the Aruoba and Wright (2003) model by allowing physical
4
Molico (2006) does not make the quasilinearity assumption that Lagos and Wright as
well as this model does. Hence, agents depending on their trading shock history will end up
with di¤erent amounts of money in each period.
5
Lagos and Rochetau (2008) address the co-existence of money and capital as media-of-
exchange.
86
capital to be used for productive purposes in the DM. This breaks the aforementioned
dichotomy, so changes in the money growth rate have an e¤ect in production and
capital accumulation in the CM as well. The second model in this chapter di¤ers
from Aruoba, Waller and Wright (2008), in that it augments the search-theoretic
model of Lagos and Wright (2005) with the decision to accumulate another type of
capital, human capital, which is used for productive purposes in both markets. I also
allow for heterogeneity in discount factors. In this setting, I can explore the e¤ects of
in‡ation on aggregate output, human capital accumulation and welfare as well as the
distribution of wealth.
To my knowledge, Molico and Zhang (2005) is the only paper using search models
of money that allows for a portfolio allocation decision in a heterogenous agents
model by allowing agent to accumulate both money and capital (storable goods in
their model).
6
Their results show that a moderate rate of monetary expansion can
lead to an increase in steady-state aggregate output, aggregate consumption, capital
accumulation, and welfare. Also, the average fraction of time spent working might
decrease. My model is di¤erent from Molico and Zhang (2005) in that I consider an
ex-ante heterogeneity in discount factors and human capital as opposed to tangible
forms of capital. Their results are similar to the ones I obtain in the second model in
the case of decreasing returns to scale technology of production.
6
Even though their model has a two-sector economy, the sectors play di¤erent roles from
the sectors in the Lagos and Wright (2005) model.
87
This chapter is organized as follows. In section 2.2 I solve a Lagos and Wright (LW)
model which allows for heterogeneity in discount factors. In Section 2.4 I augment
the previous model to allow for human capital accumulation.
2.2 A Search Economy: Money is the Only Asset in the
Economy
There is a [0,1] continuum of in…nitely lived agents operating in a Lagos and Wright
type of economy. Time is discrete. Each period consists of two subperiods. It is also
assumed that there are two types of goods, a special good produced and traded in
the …rst subperiod and a general good traded in the second. In the …rst subperiod,
which I will call the Decentralized Market (DM), agents trade in pairwise meetings.
Agents receive a trading shock at the entrance of the DM. An agent can be in one of
the following states: she can consume but not produce or produce but not consume.
I assume that there is no double coincidence of wants (without loss of generality), so
that agents refrain from bartering and I can focus on single coincidence meetings.
Such search frictions generate endogenously the existence of an additional object
called money, which enables trade in the DM and which cannot be consumed or
produced by any agent. Money here is an intrinsically useless, non-perishable object
used as a medium of exchange. In the second subperiod, the Centralized (Walrasian)
Market, agents produce and consume a general good. They can transform one unit
of labor into one unit of the general good. Similar to the Lagos and Wright model, I
88
assume that preferences in the CM are quasilinear. Such an assumption implies that
independently of the trading shock during the DM, agents exit the centralized market
(CM) with the same level of money holdings. Achieving a degenerate distribution of
money holdings at the end of each period increases the tractability of the model
but it comes at the cost of ignoring the di¤erential impact that in‡ation has in the
economy. In this model, I allow agents to be di¤erent in their discount factors, which
allows me to examine the redistributive e¤ects of in‡ation while keeping the model
tractable. I assume there is a monetary authority, namely the central bank, which
injects money in the CM via lump-sum transfers denoted by t. Money supply evolves
via `
t
= (1 + .)`
t1
and t = .`
t1
. I start by examining the problem of an agent
in the CM. The agent chooses consumption (A) of the general good, hours of work
(H) and next period’s amount of money holdings
0
), which maximizes
\
c
) = max
A,1,n
0
¦l(A) ÷H +,(c)\
c

0
)¦ (2.1)
subject to
A = c÷:
0
+t) +nH (2.2)
where c is the units of consumption good per unit of money (inverse of price level),
n is the wage rate
7
, \
c
) is the value function of a type c entering the CM with
money : and \
c
) is the value function of type c agent entering the DM with
7
We can think of an environment where there are …rms in the CM that employ only
labor using linear production technology, which implies a constant wage rate. We set n = 1
for now.
89
money :. After substituting the budget constraint into the value function I get
\
c
) = max
A,n
0
¦l(A) ÷[A ÷c÷:
0
+t)] +,(c)\
c

0
)¦ (2.3)
The …rst order conditions for the CM problem are
l
0
(A) = 1 (2.4)
c = ,(c)\
0
c

0
) (2.5)
The envelope condition is
\
0
c
) = c (2.6)
Every agent consumes the same A. and the decision for the next period’s money
holdings is independent of this period’s money holdings, but it does depend on the
agents type c. Hence, agents of the same type will exit the CM with the same money
holdings. 1
c
) = G(c). This means that the distribution of money holdings is
degenerate conditional on types.
In the DM agents come together in pairwise meetings. An agent can be in one
of three possible situations. She can receive a consumption shock with probability o
and hence be a buyer, she can receive a production shock with the same probability
o and be a seller, or with probability 1 ÷ 2o she can be neither a consumer nor a
producer. The only possible trades are goods for money, since I assumed no double
coincidence of wants (barter). Letting \
c
) denote the value function of a type c
individual entering the DM with :. I have
\
c
) = o
_
[÷c(¡( ~ )+\
c
+d( ~ )]d1( ~ +o[n(¡))+\
c
÷d))+(1÷2o)\
c
)
90
where ¡( ~ is the quantity produced by a seller, which depends only on money bal-
ances of the buyer, and d( ~ is the payment received by the seller. I am assuming,
as will be veri…ed below, that the quantity produced in the DM (¡) and the amount
that will have to be paid in exchange for the product (d) depend only on buyer’s and
not on seller’s money balances. The …rst term captures the value of being a seller in
the DM. Since the quantity produced by a seller (¡( ~ : )) will depend on the buyer’s
money balances, I integrate over the type distribution of buyers in the economy. The
second and last term capture the value of being a buyer and not trading in the DM
respectively. The marginal value of carrying money balances in the DM is
\
0
c
) = o
_
\
0
c
+d( ~ )d1( ~ +o[n
0
(¡)) +\
0
c
÷d)) + (1 ÷2o)\
0
c
)
(2.7)
Using the fact that \
0
c
) = c. equation (2.7) becomes
\
0
c
) = (1 ÷o)c +o[n
0
(¡))¡
0
) +c(1 ÷d
0
)) (2.8)
Following LW, I assume that terms of trade are determined by generalized Nash
Bargaining, where o is the buyer’s bargaining power. This problem is as follows
max
q,on
[n(¡) +\
c
÷d) ÷\
c
)]
0
[÷c(¡) +\
c
( ~ :+d) ÷\
c
( ~ ]
10
Surplus from trading for the buyer is [n(¡) +\
c
÷d) ÷\
c
)] and surplus from
trading for the seller is [÷c(¡) +\
c
( ~ :+d) ÷\
c
( ~ ]. Making use of the fact that for
each type c. \
c
+d) ÷\
c
) = dc I can rewrite the Nash Bargaining problem
91
as follows:
max
q,on
[n(¡) +dc]
0
[÷c(¡) +dc)]
10
As it is the case in LW, in equilibrium d = : must hold. The quantity of goods being
produced and consumed in the DM, ¡ = ¡) is the solution of
:c =
on
0
(¡)c(¡) + (1 ÷o)c
q
(¡)n(¡)
on
0
(¡) + (1 ÷o)c
q
(¡)
= q(¡) (2.9)
q
q
0. ¡ = ¡). ¡
0
) =
c
q
q
. d
0
) = 1 (2.10)
After substituting equation (2.9) into equation (2.8), I get:
\
0
c
) = c[1 ÷o +o
n
0
(¡)
q
q
(¡)
] (2.11)
Substituting equation (2.11), into equation (2.5) I can derive the equilibriumcondition
for this economy:
c
t
= ,(c)c
t+1
[1 ÷o +o
n
0
(¡
t+1
)
q
q
(¡
t+1
)
] (2.12)
At steady state, real money balances, :c. are constant. The law of motion for money
balances, :
t+1
= (1 + .):
t
, implies that the law of motion for prices follows c
t
=
(1 +.)c
t+1
. As a result, the steady state equation for this economy becomes:
1 +. = ,(c)[1 ÷o +o
n
0
(¡
c
)
q
q
(¡
c
)
] (2.13)
Proposition 2 At the steady state, relatively more patient agents choose to hold more
money and consume more of the DM goods.
92
Proof. Implicitly from the steady state equation we have ¡ = ¡(c) and one can
show that J¡,Jc 0.
J¡,Jc = ÷
1 +.
,(c)
2
o[
&qqjq&qjqq
j
2
q
]
0
Given that n
qq
< 0. q
q
0. q
qq
0 we have n
qq
q
q
÷ n
q
q
qq
< 0 . From the …rst order
condition of the Nash Bargaining problem equation (2.9) we have:
J:,Jc = ÷
÷q
q
0q
0c
c
0
There are two types of heterogeneity at the beginning of the CM, the trading shock
in the DM and our imposed heterogeneity in discount factors. The money holdings
of the agent entering the CM will depend on whether she was a buyer, a seller or no
trade occurred. Looking at the budget constraint (2.2), there will be a variation in the
hours of work as well. Using the fact that in steady state t = .` and :
0
c
= (1+.):
c
.
each type can …nd oneself in any of the three situations:
H
c
=
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
A ÷c(0 ÷(1 +.):
c
+.`) if previously a buyer in the DM, w/p o
A ÷c
c
+:
~ c
÷(1 +.):
c
+.`) if previously a seller in the DM, w/p oG(c)
A ÷c
c
÷(1 +.):
c
+.`) if previously no trade, w/p 1 ÷2o
(2.14)
where ` =
_
:
c
dG(c) is the average money balances in the whole economy. Given
the quasilinear preferences, the level of next period’s money balances is the same for
93
agents with the same discount factor. Hence, CM labor e¤ort absorbs the trading
shock in the DM. In order to carry out the welfare analysis, I denote welfare for a
type c agent as the sum of the expected steady state utility in the DM and in the
CM
(1 ÷,(c))\
c
= o[n(¡
c
) ÷c(¡
c
)] +l(A) ÷

H
c
(2.15)
where

H
c
is the expected hours of work for type c computed using (2.14).

H
c
= A ÷c(o +.)(` ÷:
c
) (2.16)
The welfare function then becomes
(1 ÷,(c))\
c
= o[n(¡
c
) ÷c(¡
c
)] +l(A) ÷A +c(o +.)(` ÷:
c
) (2.17)
I am interested in how higher rates of money growth rate a¤ect welfare, hence
J(1 ÷,(c))\
c
J.
= o
_
n
0
(¡
c
)
J¡
c
J.
÷c
0
(¡
c
)
J¡
c
J.
_
÷
J

H
c
J.
(2.18)
The …rst term denotes the e¤ect of a higher money growth rate on expected utility in
the DM, whereas the second term denotes the impact on expected CM utility. Since
the DM expected utility depends only on individual variables, it can only capture the
rate of return e¤ect of higher in‡ation. The second term re‡ects how expected utility
in the CM is a¤ected by a higher money growth rate. It is this term that captures
interesting redistributive e¤ects as we will show below. The change in the expected
hours of work for each agent as a response to the higher money growth rate is:
J

H
c
J.
= ÷
_
c(` ÷:
c
) + (o +.)
_
Jc`
J.
÷
Jc:
c
J.
__
(2.19)
94
The right hand side of equation (2.19) captures the two di¤erent redistributive aspects
of in‡ation. The …rst term denotes the static redistributive e¤ect. That is, assuming
prices are …xed (agents have not yet adjusted to in‡ation), one’s position in the
distribution of money holdings determines whether one bene…ts or not from a higher
money growth rate. Agents holding less than average money holdings will have to work
less in the CM. The second term, denotes the dynamic redistributive e¤ect. This term
captures the di¤erences in the responsiveness of money demand to in‡ation. That is,
the degree to which di¤erent agents evade the in‡ation tax. If agents holding less than
average money balances, have a stronger response to changes in the in‡ation rate, then
both these redistributive e¤ects work in the same direction, and an increase in the
money growth rate would lead to a redistribution of wealth from the rich to the poor.
If the opposite is true, that is, if agents holding more than average money balances,
have a stronger response to in‡ation, then, the net result of redistribution will depend
on which term dominates the other. Given that
0çA
0:
< 0.
0çno
0:
< 0. I can express this
term as : (o +.)([
0çno
0:
[ ÷[
0çA
0:
[) and equation (2.19) becomes
J

H
c
J.
= ÷
_
c(` ÷:
c
) + (o +.)
_
[
Jc:
c
J.
[ ÷[
Jc`
J.
[
__
(2.20)
Below, I provide an analytical example.
95
Example Suppose n(¡) = ln ¡. c(¡) = ¡. o = 1 == q(¡) = c(¡) = ¡. Then, the
equilibrium quantities of the DM good and money holdings for each type are:
¡ =
o,(c)
1 +. ÷,(c)(1 ÷o)
(2.21)
: =
1
c
[
o,(c)
1 +. ÷,(c)(1 ÷o)
] (2.22)
where, ¡(c
1
) ¡(c
1
). c
1
) c
1
).Consider an economy consisting of only two
types of agents, the patient (,
1
) and the impatient (,
1
).Both types have equal mass.
The aggregate amount of money in the economy is then de…ned as ` =
1
2
:
1
+
1
2
:
1
.
After inserting equations (2.21), (2.22) into equation (2.16) I get:

H
1
= A

÷
1
2
(o +.)
_
,
1
o
1 +. ÷,
1
(1 ÷o)
÷
,
1
o
1 +. ÷,
1
(1 ÷o)
_
One can show that :
J

H
1
J.
0.
J

H
1
J.
< 0
In this example, changes in the money growth rate generate a transfer from the
poor to the rich. Bhattacharya et al. (2005) examine the redistributive e¤ects of
in‡ation in a similar environment, where the agents are assumed to be heterogeneous
in their consumption preferences. The authors obtain the opposite results in terms
of the direction of the transfer. It is important to note at this point that the type
of heterogeneity a¤ects the direction of redistribution generated by in‡ation in the
economy. In the next section I provide a numerical solution to a version of the model
that does not allow for an analytical solution. The same results hold. That is, in a
96
search economy, where agents are heterogenous in their discount factors, changes in
the money growth rate generate a transfer from the poor to the rich.
The intuition for the above result can be found in the following tax/transfer argu-
ment. The real transfer each agent receives is
11 = c.` == J11,J. = c` +
Jc`
J.
The real in‡ation tax each agent incurs is
11 = c.:
c
== J11,J. = c:
c
+
Jc:
c
J.
The di¤erence between the two expressions is the net e¤ect of a higher money growth
rate .. The …rst term in each expression shows that before agents adjust to the new
higher prices they bene…t from a higher rate of monetary expansion as long as their
money holdings are below average. The second term in each of the above expressions
shows how fast the real transfer is falling and by how fast agents are able to evade
in‡ation tax. The scenario in this chapter is such that for the poor agents the rate at
which the real transfer is falling is bigger than the rate at which they are evading the
in‡ation tax.
It is important to note that the aggregate hours of work in the CM, H. remain
constant as the money growth rate changes. That is, the redistributive e¤ects of
in‡ation that allow patient agents to work less in the CM are exactly o¤set by the
increase in working hours for impatient agents. In this respect, monetary policy is
neutral in the CM. Nevertheless, redistributive e¤ects have a signi…cantly di¤erent
97
impact on the welfare cost of in‡ation for each group of agents, as I will show in the
next section.
2.3 Numerical Solution of the First Model
Here I provide some simulation results for parameter values that do not allow for ana-
lytical solutions that are easy to read. My parametrization (Table 2.1)and functional
form choice follows that of Lagos and Wright (2005).
l(r) = 1log(r)
1` : n(¡) =
(¡ +/)
1j
÷/
1j
1 ÷j
c(¡) = ¡
In Figure 2.1, I decompose the impact of in‡ation on welfare among di¤erent
subperiods. In the last row, we see how in‡ation a¤ects welfare at the DM at the
steady state. Expected utility is falling for both types in the DM. This basically
re‡ects the in‡ation tax argument of monetary expansion on DM activity. The erosion
of the purchasing power of money leads to lower money demand and hence less trade
in the DM. Expected utility in the CM for di¤erent types of agents on the other hand,
moves in opposite directions.
I compute the welfare cost of moving from 0% to 10% in‡ation in Table 2.2 for
di¤erent parametrizations of the model. The low type agents, the impatient and the
98
Figure 2.1: Welfare implications of expansionary monetary policy
-23.8
-23.6
-23.4
-23.2
-23.0
-22.8
-22.6
-22.4
.0 .1 .2 .3 .4 .5 .6 .7
-13.6
-13.4
-13.2
-13.0
-12.8
-12.6
.0 .1 .2 .3 .4 .5 .6 .7
0
1
2
3
4
5
6
7
.0 .1 .2 .3 .4 .5 .6 .7
0
1
2
3
4
.0 .1 .2 .3 .4 .5 .6 .7
CM Welfare (high beta)
CM Welfare (low beta)
DM Welfare (high beta) DM Welfare (low beta)
money growth rate
99
Table 2.1: Benchmark Parameter Values
Parameter Value Description
j 0.30 coe¢cient of risk aversion
o 0.50 buyer’s bargaining power
o 0.50 probability of a bilateral meeting
1 1.91 constant
b 0.001 constant
.
1
0.5 share of ,
1
types
,
1
0.94 discount factor for low-types
,
1
0.9615 discount factor for high-types
poor, su¤er more from expansionary monetary policies relative to high types. The
di¤erence between the welfare cost of in‡ation between the two types increases as the
degree of heterogeneity, the distance between the discount factors, increases. In an
economy populated by agents with the same (high type) discount factor, the welfare
cost of in‡ation will always be positive. If an economy is populated by both types
of agents, then the welfare cost of in‡ation for the high type decreases and can also
become negative. Redistributive e¤ects of in‡ation from low to high types can be
strong enough for in‡ation to be bene…cial for a group of individuals. I obtain welfare
bene…ts for high types when the di¤erence in discount factors or the share of impatient
agents is big enough (last three columns in Table 2.2).
100
T
a
b
l
e
2
.
2
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(
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8
101
2.4 An Augmented Search economy: Money and Human
Capital
In this section, I augment the Lagos and Wright model in yet another dimension.
Agents in the economy can decide to accumulate another type of capital, human
capital, which is used for productive purposes in both markets. Agents continue to
be heterogeneous in discount factors. In this setting, I can explore the e¤ects of
in‡ation on aggregate output, human capital accumulation and welfare as well as the
distribution of wealth
2.4.1 Model
Let \
c
. /) and \
c
. /) be the value functions for an agent in the CM and DM
respectively, holding : dollars and / units of human capital. Human capital in this
model is accumulated via time invested in education or in any knowledge acquiring
activity. During the CM, agents rent their e¤ective labor to …rms which employ only
labor. They choose what fraction of time to spend on accumulating human capital n
or on market activity : as well as choosing future money balances and how much of
the general good to produce. The CM problem for an agent becomes
\
c
. /) = max
a,&,a,n
0
,I
0
¦l(r) +¹(1 ÷n ÷ +,(c)\
c

0
. /
0
)¦ (2.23)
102
subject to
r = c÷:
0
+t) +n/: +: (2.24)
/
0
= (1 ÷o
I
)/ +,(n. /) (2.25)
`
t
(/
0
÷(1 ÷o
I
)/) _ 0 (2.26)
where n is the wage per unit of e¤ective labor (/. : is the fraction of time spent in
market activities, n is the fraction of time invested in education, 1 ÷n ÷: is leisure.
,(n. /) captures the production function for human capital, which depends on time
invested in education and on the level of current human capital. Firms operate under
a one-input (e¤ective labor) technology
1 =
__
:
c
/
c
dG(c)
_
¸
The parameter ¸ captures the returns to scale of the production technology. In the
case of decreasing returns, pro…ts generated by …rms are distributed to consumers as
:. A constant returns to scale production technology in the CM, implies a constant
wage rate (n = 1
_
). whereas a decreasing returns to scale technology implies that the
wage rate will depend on the economy-wide average e¤ective labor. In this chapter, I
will experiment with decreasing returns and constant returns to scale technology. This
will turn out to be an important distinction in terms of the qualitative results of this
chapter, since decreasing returns to scale technology introduces general equilibrium
e¤ects. After substituting : from the budget constraint (2.24) and n from (2.25) into
103
(2.23) I get
8
\
c
. /) = max
a,n
0
,&
¦l(r)+¹÷¹n÷¹[
r ÷c÷:
0
) ÷:
n/
]+,(c)\
c

0
. (1÷o
I
)/+,(n. /)¦
(2.27)
First order conditions for the CM problem are
r : l(r) =
¹
n/
(2.28)
:
0
:
¹c
n/
= ,(c)\
cn

0
. /
0
) (2.29)
n : ¹ = ,(c)\
cI

0
. /
0
),
&
(n. /) (2.30)
There is an important distinction between this model and the model employed by
Aruoba, Waller and Wright (2008), where …rms in the CM use physical capital instead
of human capital. In this model, preference are quasilinear in hours of work as well,
but production is carried out using e¤ective hours of work. As will be clear below, this
leads to di¤erent results in terms of trade determination and equilibrium. I note that
even though …rst order conditions for r. :
0
. /
0
depend on the current level of human
capital (/), the joint distribution of . /) is still degenerate conditional on types. I
assume that the initial distribution of knowledge (human capital) is degenerate across
agents and there is no human capital depreciation when moving from the DM into
the CM.
9
8
I am ignoring the illiquidity constraint for the moment since it turns out that in the
steady state in does not bind.
9
In this model, the only type of heterogeneity besides the discount factor di¤erences
is due to idiosyncratic trading shocks in the DM. The only variable it a¤ects is the level
of money holdings agents carry into the CM (buyers carry less or no cash, seller carry
more cash and so on). Hence, conditional on types (patient versus impatient), when agents
104
Envelope conditions:
\
cn
. /) =
¹c
n/
(2.31)
\
cI
. /) =
¹:
/
+
¹
,
&
(n. /)
[1 ÷o +,
I
(n. /)] (2.32)
There are two important points to notice from the envelope conditions which lead
to di¤erent qualitative results: the …rst is that the marginal value of holding cash in
the CM depends in the level of human capital, and the second is that the marginal
value of human capital in the CM depends on the level of money balances. As we
will see later on, the …rst will lead the DM terms of trade to depend on seller’s and
buyer’s levels of human capital even though the buyer does not use human capital in
the DM. This will also have di¤erent implications for the hold-up problem compared
to Aruoba, Waller and Wright (2008).
Decentralized market
In the DM, buyers consume ¡ amount of special goods and derive utility n(¡). A
seller incurs a utility cost c(¡.
~
/) from producing ¡ using labor and human capital.
10
As
before, I consider single coincidence meeting, where the probability to meet a trading
partner is o. Let \
c
. /) denote the value function of a type c agent entering the
DM with : money holdings and / units of human capital. As will be shown from the
enter the CM they are identical in the level of human capital since we assume there is no
depreciation when moving from DM into the CM.
10
Here again as in the CM, production requires e¤ective hours of work. Consider a produc-
tion function ¡ =
~
/)
¸
and disutility of labor in the DM measured by ·) =
a
1+)
1+ç
,where
c is the Frisch labor supply elasticity. Then the utility cost of producing ¡ for a seller with
~
/ level of human capital is c(¡,
~
/) =
_
q
1+)
¸
~
I
1+)
_
1
1+ç
, c 0, ¸ 6 1.
105
Nash Bargaining stage, ¡ = ¡. /.
~
/) and d = :.
\
c
. /) = o
_
[÷c(¡( ~ :.
~
/. /). /) +\
c
+d( ~ . /)]dH( ~ :.
~
/)
+o
_
[n(¡. /.
~
/)) +\
c
÷d). /)d1(
~
/) + (1 ÷2o)\
c
)
(2.33)
where each of the right hand side terms represents the expected value of being a seller,
a buyer or a non-trader respectively. I now, turn to the terms of trade determination
in the DM.
Terms of Trade Determination: Generalized Nash Bargaining
Terms of trade are determined by maximizing gains from trade:
max
q,on
[n(¡) +\
c
÷d. /) ÷\
c
. /)]
0
[÷c(¡.
~
/) +\
c
( ~ :+d.
~
/) ÷\
c
( ~ :.
~
/)]
10
where . /) are money holdings and human capital of the buyer and ( ~ :.
~
/) are
money holdings and human capital of the seller, and o is the bargaining power of the
buyer. Given the linearity in money holdings : of \
c
. we have
\
c
+d. /) ÷\
c
. /) = d
¹c
n/
Hence the problem can be written as
max
q,on
[n(¡) ÷d
¹c
n/
]
0
[÷c(¡.
~
/) +d
¹c
n
~
/
]
10
106
The solution to the Nash Bargaining problem is as follows:
¡. ~ :. /.
~
/) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
¡. /.
~
/) if : < :

(/.
~
/)
¡

(/.
~
/) if : _ :

(/.
~
/)
d. ~ :. /.
~
/) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
: if : < :

(/.
~
/)
:

if : _ :

(/.
~
/)
where ¡

de…nes the optimal quantity produced during DM and can be found by
solving
n
0
(¡

)/ =
~
/c
q
(¡

.
~
/) (2.34)
and :

de…nes the optimal amount of money required for purchasing ¡

:

=
n
¹c
_
o
~
/c(¡

.
~
/) + (1 ÷o)n(¡

)/
_
(2.35)
For cases when : < :

. ¡. /.
~
/) is the ¡ that solves
nç¹
&
= q(¡. /.
~
/) with
q(¡. /.
~
/) =
on
0
(¡)c(¡.
~
/) + (1 ÷o)c
q
(¡.
~
/)n(¡)
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
Implicitly, ¡ = ¡. /.
~
/). and J¡. /.
~
/),: = c¹,(nq
q
) 0. J¡. /.
~
/),J/ =
÷q
I
,q
q
_0. J¡. /.
~
/),J
~
/ = ÷q
~
I
,q
q
_0. Each partial derivative is signed as follows
q
q
=
c
q
n
q
(
0&q
~
I
+ (1 ÷o)
cq
I
) +o(1 ÷o)(c
qq
n
q
÷c
q
n
qq
)(n ÷c)
_
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
_
2
0
q
I
=
(1 ÷o)c
q
(on
q
c + (1 ÷o)c
q
n)
/
2
_
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
_
2
_ 0
107
q
~
I
=
on
q
c
I
_
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
_
+o(1 ÷o)n
q
c
qI
_
&
~
I
÷
c
I
_
_
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
_
2
+
0&q
~
I
2
(on
q
c + (1 ÷o)c
q
n)
_
0&
0
(q)
~
I
+
(10)cq(q,
~
I)
I
_
2
_ 0
It is important at this point to discuss whether in equilibrium, the amount of
money held by each type of buyer satis…es : < :

or not. In the Lagos and Wright
model as well as in Aruoba, Waller and Wright (2008), one can show that in equi-
librium : < :

and hence d) = :. This implies that agents hold less than the
optimal amount of money, and all their money balances are used to purchase the DM
good. In these papers this result is due to the lack of heterogeneity among sellers.
In my model this is not the case. As one can see from equations (2.34) and (2.35),
the optimal amount of money to purchase ¡

will depend on seller’s type. When
deciding how much money to bring into the DM, buyers have to take into account
the possible type of their future trading partner (seller’s type). Meetings with less
productive sellers (the ones having low human capital) will require more money bal-
ances and meetings with more productive sellers will require less money. Buyers know
their own type but not the type of their trading partner. It might be very well the
case that for some meetings : < :

and for some others : :

. This would intro-
duce additional heterogeneity in the model and endanger its tractability. Whether
one condition holds versus the other will depend on the structural parameters of the
model. In appendix H, I solve a simple model with seller heterogeneity, and derive
the condition under which : < :

is satis…ed. Unfortunately, I cannot derive such a
108
condition for the full model, due to its complexity. Hence, I solve the model for cases
where : < min ¦:

1
. :

1
¦.
The marginal value of carrying money balances in the DM:
\
cn
. /) = o
_
\
cn
+d( ~ . /)d1( ~ +o
_
[n
0
(¡. /.
~
/))¡
n
. /.
~
/)
+\
cn
÷d). /)]d1(
~
/) + (1 ÷2o)\
cn
. /)
(2.36)
Substituting \
0
c
) =
¹ç
&I
. and the Nash Bargaining outcomes ¡
n
. /.
~
/) =
ç¹
&jq
. d
0
) = 1 above we have:
\
cn
. /) = (1 ÷o)
¹c
n/
+o
_
[n
0
(¡)c¹,(nq
q
(¡. /.
~
/)d1(
~
/) (2.37)
Marginal value of human capital:
\
cI
. /) = o
_
[÷c
q
(¡( ~ :.
~
/. /). /)¡
I
( ~ :.
~
/. /) ÷c
I
(¡( ~ :.
~
/. /). /) +\
cI
+d( ~ . /)]d1( ~
+o
_
[n
0
(¡. /.
~
/))¡
I
. /.
~
/) +\
cI
÷d). /)]d1(
~
/) + (1 ÷2o)\
cI
. /)
(2.38)
Using Envelope Conditions, partial derivatives from Nash Bargaining and some
algebra, \
cI
. /) becomes:
\
cI
. /) = o
_
[c
q
(¡. /)
j
I
(q,
~
I,I)
jq
) ÷c
I
(¡. /)]d1(
~
/) ÷o
_
n
0
(¡)
j
I
(q,I,
~
I)
jq(q,I,
~
I)
d1(
~
/)
+
¹
&I
2
[r ÷: ÷(. +o)(c` ÷c] +
¹
)u(&,I)
(1 ÷o +,
I
(n. /))
(2.39)
109
Equilibrium Conditions
Substituting (2.37), (2.39), into (2.29) and (2.30) as well using ¹c:,n = q(¡. /.
~
/)
we get the equilibrium conditions for a type c agent:
q(¡. /.
~
/)
:/
= ,(c)
q(¡
0
. /
0
.
~
/
0
)
:
0
_
1 ÷o
/
0
+o
_
n
0
(¡
0
)
q
q
(¡
0
. /
0
.
~
/
0
)
d1(
~
/)
_
(2.40)
¹
,
&
(n. /)
= ,(c)
_
¸
¸
¸
¸
¸
¸
_
o
_
[c
q
(¡
0
. /
0
)
j
I
(q
0
,
~
I
0
,I
0
)
jq(q
0
,I
0
,
~
I
0
)
) ÷c
I
(¡
0
. /
0
)]d1(
~
/) ÷o
_
n
0
(¡
0
)
j
I
(q
0
,I
0
,
~
I
0
)
jq(q
0
,I
0
,
~
I
0
)
d1(
~
/)
+
¹
&I
2
[r
0
÷:
0
÷(. +o)(c
0
`
0
÷c
0
:
0
)] +
¹
)u(&
0
,I
0
)
(1 ÷o +,
I
(n
0
. /
0
))
_
¸
¸
¸
¸
¸
¸
_
(2.41)
l
0
(r) =
¹
n/
Given that the dynamics are computationally demanding, I choose to focus on the
steady state analysis even though I amaware of potentially important dynamics of this
model. At the steady state, I make use of the following: `
0
= (1+.)`. :
0
= (1+.):.
c = (1 +.)c
0
. n = ¸[
_
:
c
/
c
dG(c)]
¸1
and pro…ts, : = (1 ÷¸)[
_
:
c
/
c
dG(c)]
¸
. Note
that the wage depends on aggregate e¤ective labor supply in the economy. Hence, the
two aggregate variables that enter each type’s decision problem are `.
_
:
c
/
c
dG(c).
i.e. aggregate money supply and aggregate e¤ective labor supply.
Steady state equations:
1 +. = ,(c)
_
1 ÷o +o
_
n
0
(¡
c
)/
q
q
(¡
c
. /
c
. /
~ c
)
dG(~ c)
_
(2.42)
110
¹
,
&
(n
c
. /
c
)
= ,(c)
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
o
_
[c
q
(¡
c
. /
c
)
j
I
(qo,I
~ o
,Io)
jq(qo,I
~ o,
Io)
) ÷c
I
(¡
c
. /
c
)]dG(~ c) ÷o
_
n
0
(¡
c
)
j
I
(qo,Io,I
~ o
)
jq(qo,Io,I
~ o
)
dG(~ c)
+
¹
¸
2
4
_
aoIooG(c)
3
5
¸1
I
2
[r ÷(. +o)(c` ÷c:
c
) ÷(1 ÷¸)
__
:
c
/
c
dG(c)
_
¸
+
¹
)u(&o,Io)
(1 ÷o +,
I
(n
c
. /
c
))
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
(2.43)
l
0
(r
c
) =
¹
n/
c
(2.44)
:
c
=
r
c
÷.(c` ÷c:
c
) ÷:
n/
c
(2.45)
From the Nash Bargaining solution we also have:
:
c
c¹
n
= q(¡. /
c
. /
~ c
)
As shown in the Nash Bargaining stage, the fact that the terms of trade now
depend on seller’s and buyer’s human capital, results in novel insights regarding
incentives to accumulate human capital. Equations (2.46) and (2.47) refer to the
net surplus of a buyer and a seller entering the DM respectively.
`o
1
= n(¡) +\
c
÷d. /) ÷\
c
. /) = n(¡) ÷d
¹c
n/
(2.46)
`o
S
= ÷c(¡.
~
/) +\
c
( ~ :+d.
~
/) ÷\
c
( ~ :.
~
/) = ÷c(¡.
~
/) +d
¹c
n
~
/
(2.47)
111
As one can see from equations (2.46), (2.47) the marginal value of holding money
in the CM is inversely related to the level of human capital. That is, bringing an
additional unit of money into the CM has a higher value for the less skilled agents.
Bringing more human capital into the DM as a seller has two e¤ects: it lowers the
production cost, which tends to increase the net surplus, and it also reduces the
marginal value of carrying cash into the CM, which lowers the net surplus. Bringing
more human capital into DM as a buyer lowers the marginal cost of carrying no
cash into the CM, so it increases the net surplus of the buyer. For cash constrained
buyers, this implies that the higher the level of human capital of the buyer, the lower
the amount of production ¡ in the DM (J¡. /.
~
/),J/ < 0). When the buyer has
full bargaining power o = 1. to capture all the bene…ts of the surplus s/he creates
by bringing : to the DM, J¡. /.
~
/),J/ = 0. The quantity traded in the DM is
insensitive to the buyer’s level of human capital. Giving full bargaining power to the
buyer implies that the incentives to invest in human capital will be determined by
the returns in the CM as well as its e¤ect on the seller’s production in the DM. From
a seller’s perspective, a higher
~
/ in the DM has two opposing e¤ects: it lowers the
cost of production when trade occurs, but any amount of : received in exchange is
associated with a lower marginal value in the CM. The net e¤ect on the quantity
traded ¡ is that more skilled sellers lead to higher ¡. J¡. /.
~
/),J
~
/ 0.
112
2.5 Numerical Solution of the Second Model
In this subsection, I show simulations to check the long run e¤ect of in‡ation on the
real economy, the distribution of wealth, and welfare. Given that the wage is set at
the marginal product of aggregate e¤ective labor supplied in the economy, there are
potential externalities taking e¤ect through wages, so I experiment with the returns to
scale technology in the CM
11
. When I refer to comparisons between an heterogenous
agents model and a model with a degenerate distribution the baseline experiment
underlined is as follows: I …rst compute the steady state results under the assumption
that all agents have the same discount factor, which leads to a degenerate distribution
of assets. I then introduce a group of agents with a lower discount factor and track
down how the behavior of the old group changed.
Parametrization
CM : 1
c
= 21(1) = 21
¸
. 1 =
_
:
c
/
c
dG(c)
,(n. /) = n/
j
. j ¸ (0. 1]
l(r) = 1log(r)
DM : n(¡) =
(q+b)
1¡
b
1¡
1j
c(¡.
~
/) =
_
q
1+)
¸
~
I
1+)
_
1
1+ç
11
Experimenting with returns to scale in the DM is of little relevance, since production
takes place in single-agent …rms (simple individual production) and there no channel in the
production technology connecting di¤erent types of agents.
113
Table 2.3: Benchmark Parameter Values - Human Capital Augmented Model
Parameter Value Description
j 1 coe¢cient of risk aversion
o 0.745 buyer’s bargaining power
o 0.26 probability of bilateral meetings
1 1.30 constant
/ 0.0001 constant
· 1 labor elasticity
¸ 0.85 returns to scale
.
1
0.5 share of low-type (impatients
2 0.1985 constant
j 0.5 returns to human capital accumulation
o 0.04 human capital depreciation
¹ 4 disutility of labor in CM
,
1
0.94 discount factor for low-type
,
1
0.9615 discount factor for high-type
The cost function c(.) comes from the production technology ¡ =
~
/)
¸
and the
disutility of labor in the DM measured by ·) =
a
1+)
1+ç
.where · is the Frisch labor
supply elasticity
12
. Then the utility cost of producing ¡ for a seller with
~
/ level of
human capital is c(¡.
~
/) =
_
q
1+)
¸
~
I
1+)
_
1
1+ç
. · 0. ¸ 6 1.Table 2.3 summarizes the
benchmark parametrization used in the model.
12
Here, as assumed in the CM, production is carried out using e¤ective units of labor.
114
Time period in the model is one year. I use the values from Aruoba, Waller and
Wright (2008) for the overlapping parameters and functional forms. Human capital
is increased via a concave function, following (Ortigueira, 2000).
Constant returns to scale and homogenous agents
I …rst show what the long run e¤ects of a higher money growth rate are on hours of
work, time devoted to human capital accumulation and consumption, when there is
no ex-ante heterogeneity. Agents can fully smooth their idiosyncratic trade shocks by
adjusting their non-leisure time in the CM, which leads to a degenerate distribution of
assets. As we can see from Figure 2.2, a higher rate of money growth leads to less time
spent in education, a lower steady state level of human capital, lower consumption,
and a constant amount of hours devoted to market activity. For the working hours
to remain constant it must be the case, as one can see from equation (2.45), that
CM consumption and human capital are decreasing by exactly the same amount. In
order to understand the intuition behind the human capital response to in‡ation in
the steady state, one should make clear where the returns to accumulating human
capital come from. There are bene…ts to bringing human capital as a seller into the
DM since it lowers production cost, so anything that taxes DM activity (in this case
in‡ation) will lower the returns to human capital in the DM. Returns to holding
human capital in the CM come in the form of higher labor income. Hence, in the face
of a higher money growth rate, there are two opposing incentives to accumulation of
115
human capital. Here, accumulation of human capital comes in the form of a greater
share of time devoted to skill acquisition. From the law of motion for human capital,
one can derive the steady state level of time devoted to education as follows:
n
c
= o/
1j
c
Hence, agents with a higher level of human capital devote more time to education, but
do so at a decreasing rate. Marginal utility of CM consumption is inversely related
to the level of human capital (equation (2.44)).This means that as human capital
decreases, marginal utility of consumption increases, which indicates a falling level
of consumption. The main result of the model, when I abstract from heterogeneity,
is that monetary policy has real e¤ects on aggregate consumption, human capital
accumulation, and time devoted to skill acquisition, and has no e¤ect on aggregate
hours of work. It must be noted that the e¤ect of monetary policy is very small for
CM variables and has a much bigger e¤ect on DM consumption.
Constant returns to scale and heterogeneous agents
I now introduce a new group of agents with a lower discount factor ("the impatients").
As previously stated, more patient agents accumulate more money and more human
capital. The introduction of heterogeneity in this environment does not change any
of the qualitative results in the degenerate case, except for the hours of work. As one
can see from Figure 2.3The high types, which happen to be the rich group, tend to
work more in the face of higher in‡ation. It is important to note that this does not
116
Figure 2.2: The long-run e¤ects of expansionary monetary policy - Costant Returns
to Scale technology
.30
.31
.32
.33
.34
.35
.00 .05 .10 .15 .20 .25 .30 .35 .40
.2172
.2174
.2176
.2178
.2180
.2182
.2184
.2186
.00 .05 .10 .15 .20 .25 .30 .35 .40
29.5
29.6
29.7
29.8
29.9
.00 .05 .10 .15 .20 .25 .30 .35 .40
1.900
1.905
1.910
1.915
1.920
1.925
.00 .05 .10 .15 .20 .25 .30 .35 .40
.2
.3
.4
.5
.6
.7
.00 .05 .10 .15 .20 .25 .30 .35 .40
0.6
0.7
0.8
0.9
1.0
1.1
.00 .05 .10 .15 .20 .25 .30 .35 .40
Hours
Ti me spent i n educati on (ski ll acqui si tion).
Human capital CM consumption
Real balances DM consumption
money growth rate
117
mean that the rich group is experiencing an increase in welfare cost relative to the
poor, as will be shown in the welfare analysis section. CM variables are little a¤ected
by changes in money growth rate, as before. DM trade, on the other hand is more
in‡uenced by these changes. The quantity of goods being produced in the DM is
determined by the type of buyer and seller that meet in the pairwise meetings. The
largest amount of DM trade occurs between a high-type buyer and high-type seller.
Given the redistributive e¤ects of in‡ation, we are also interested in how in‡ation
impacts wealth and human capital distribution in the long run. Figure 2.4 shows that
dispersion of money holdings and human capital increases with in‡ation. This implies
that, in steady state, real balances and human capital decline at a faster pace for low
types as money growth rate increases.
Figure 2.5 shows the impact of changes in the money growth rate on aggregate
variables in the economy. Monetary policy has real e¤ects. Even though these e¤ects
are quite small in terms of quantities, they can be signi…cant in welfare terms, as will
be shown in the next section.
Decreasing Returns to Scale Technology
Besides the lump sum transfer of money in the amount t = .` that agents get in
the CM, returns to scale technology is the only other channel through which another
aggregate (total supply of labor) can interact with individual variables, and poten-
tially lead to qualitatively di¤erent results. As mentioned, before given that wage is
118
Figure 2.3: Type-speci…c long-run e¤ects of expansionary monetary policy -
Constant returns to scale
1.998
1.999
2.000
2.001
2.002
2.003
2.004
2.005
2.006
2.007
.860
.865
.870
.875
.880
.885
.890
.895
.900
.905
.00 .05 .10 .15 .20 .25 .30 .35 .40
.324
.325
.326
.327
.328
.329
.330
.331
.312
.314
.316
.318
.320
.322
.324
.326
.00 .05 .10 .15 .20 .25 .30 .35 .40
.2226
.2227
.2228
.2229
.2230
.2231
.2232
.1468
.1472
.1476
.1480
.1484
.1488
.1492
.00 .05 .10 .15 .20 .25 .30 .35 .40
.2
.3
.4
.5
.6
.7
.08
.12
.16
.20
.24
.28
.00 .05 .10 .15 .20 .25 .30 .35 .40
High Type (Left Axis)
Low Type (Righ Axis)
30.94
30.96
30.98
31.00
31.02
31.04
31.06
31.08
31.10
31.12
31.14
13.48
13.52
13.56
13.60
13.64
13.68
13.72
13.76
13.80
13.84
13.88
.00 .05 .10 .15 .20 .25 .30 .35 .40
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
.00 .05 .10 .15 .20 .25 .30 .35 .40
Q_LL
Q_LH
Q_HL
Q_HH
CM consumption
Hours
Time spent in education Human capital
Real money balances
money growth rate
DM trade
money growth rate
119
Figure 2.4: Distribution of wealth and human capital - Constant returns to scale
3.20
3.25
3.30
3.35
3.40
3.45
.00 .05 .10 .15 .20 .25 .30 .35 .40
3.48
3.52
3.56
3.60
3.64
.00 .05 .10 .15 .20 .25 .30 .35 .40
Inequality in real money holdings
(m_high/m_low)
Inequality in human capital
(h_high/h_low)
money growth rate
set at the marginal product of aggregate e¤ective labor supplied in the economy, there
are potential externalities taking e¤ect through wages. I note that there are di¤erent
ways as well to allow for externalities in this model. For example, human capital
production function may depend on the economy wide level of education. To save
space, I do not present the degenerate case results for decreasing returns to scale, but
they are qualitatively the same as Figure 2.2. Figure 2.6 shows the e¤ect of mone-
tary expansion on hours of work, education, and human capital for each type of agent
under DRTS technology. Introducing DRTS changes steady-state results in a substan-
tial way. First, there is a change in roles. The impatient agents in this case are the
120
Figure 2.5: Long run e¤ect of in‡ation on aggregate variables - Constant returns to
scale
3.66
3.67
3.68
3.69
3.70
3.71
3.72
3.73
.00 .05 .10 .15 .20 .25 .30 .35 .40
56.8
57.0
57.2
57.4
57.6
57.8
.00 .05 .10 .15 .20 .25 .30 .35 .40
.1930
.1935
.1940
.1945
.1950
.1955
.00 .05 .10 .15 .20 .25 .30 .35 .40
.316
.318
.320
.322
.324
.326
.00 .05 .10 .15 .20 .25 .30 .35 .40
money growth rate
CM Consumption Human Capital
Time spent in education Hours of work
121
ones who hold more money, accumulate more human capital, work and study more.
DRTS makes the wage sensitive to the heterogeneity. Under DRTS, the introduction
of a di¤erent group of agents, which will ultimately supply di¤erent levels of e¤ective
labor in the CM, will a¤ect the wage rate, and hence the CM margin of human capital
accumulation. It is the case that as the money growth rate increases, DRTS makes the
return to accumulating human capital in the CM dominate the negative impact on
the DM return for patient (poor) agents. Second, due to the additional redistributive
e¤ects provided, now, by a non-constant wage rate, which depends on the economy’s
aggregate e¤ective labor, CM consumption is decreasing for the low type. Unlike in
the CRTS case, steady state human capital and time spent in skill acquisition are
increasing in the rate of money growth.
13
Figure 2.7 shows that the relatively richer agents (impatients) react to changes
in the money growth rate by reducing real balances faster than the poor agents do.
The steady state level of human capital, on the other hand, is increasing at a slower
pace for the richer agents. This implies that in the long run di¤erences in wealth and
human capital are diminishing.
Figure 2.8 displays the long run e¤ect of changes in money growth rate for aggre-
gate variables. Aggregate human capital, CM consumption, and time spent in skill
acquisition increase with the money growth rate, whereas aggregate hours of work
decrease.
13
It must be noted that such overturning results do not hold for all ¸ < 1. The degree of
decreasing returns to scale must be low enough to yield this sections results. In my numerical
examples, ¸ _ 0.85.
122
Figure 2.6: Type speci…c long-run e¤ects of monetary expansion. Decreasing
Returns to Scale CM production technology
.098
.099
.100
.101
.102
.103
.104
.105
.106
.107
.6574
.6575
.6576
.6577
.6578
.6579
.6580
.6581
.6582
.6583
.00 .05 .10 .15 .20 .25 .30 .35 .40
.108
.112
.116
.120
.124
.128
.132
.136
.140
.296
.297
.298
.299
.300
.301
.302
.303
.304
.00 .05 .10 .15 .20 .25 .30 .35 .40
.0568
.0572
.0576
.0580
.0584
.0588
.0592
.146710
.146711
.146712
.146713
.146714
.146715
.146716
.00 .05 .10 .15 .20 .25 .30 .35 .40
.00
.04
.08
.12
.16
.20
.00 .05 .10 .15 .20 .25 .30 .35 .40
High Type ( Left Axis)
Low Type (Right Axis)
2.02
2.04
2.06
2.08
2.10
2.12
2.14
2.16
13.4522
13.4524
13.4526
13.4528
13.4530
13.4532
13.4534
13.4536
.00 .05 .10 .15 .20 .25 .30 .35 .40
.05
.10
.15
.20
.25
.30
.35
.40
.00 .05 .10 .15 .20 .25 .30 .35 .40
Q_LL
Q_LH
Q_HL
Q_HH
money growth rate
CM consumption Hours
Human Capital
Real money balances DM Trade
Time invested in Education
123
Figure 2.7: Dispersion in wealth and human capital - Decreasing Returns to Scale
CM production technology
5.96
6.00
6.04
6.08
6.12
6.16
.00 .05 .10 .15 .20 .25 .30 .35 .40
6.2
6.3
6.4
6.5
6.6
6.7
.00 .05 .10 .15 .20 .25 .30 .35 .40
Inequality on money holdings
(m_low/m_high)
Inequality in human capital
(h_low/h_high)
money growth rate
Discussion on Decreasing Returns to Scale
In this section I address the results obtained under the DRTS calibration, with
particular focus on the conditions under which, the patient agents in the steady
state accumulate less human capital. The …rst order condition that determines the
equilibrium amount of human capital (hours spent in education) is given by equation
(2.30), which I repeat here for convenience.
¹ = ,(c)\
cI

0
. /
0
),
&
(n. /)
This equation implies that agents with a higher discount factor (the patient) require
a lower future marginal value of human capital (lower \
cI

0
. /
0
)). The concavity of
124
Figure 2.8: The long run e¤ect of monetary policy on aggregate variables -
Decreasing returns to scale
7.72
7.74
7.76
7.78
7.80
7.82
.00 .05 .10 .15 .20 .25 .30 .35 .40
.3785
.3790
.3795
.3800
.3805
.3810
.3815
.00 .05 .10 .15 .20 .25 .30 .35 .40
.204
.206
.208
.210
.212
.214
.216
.218
.00 .05 .10 .15 .20 .25 .30 .35 .40
.1016
.1018
.1020
.1022
.1024
.1026
.1028
.00 .05 .10 .15 .20 .25 .30 .35 .40
Human capital CM consumption
Hours of work Time spent in education
money growth rate
125
the value function, implies a tendency for the patient to accumulate more human cap-
ital. Another additional factor however, is the nature of the human capital production
function ,(n. /). Heckman (1975) and Heckman, Lochner and Taber (1998) estimates
show that human capital production function displays strong increasing returns to
scale. As it is the case in this paper as well, the human capital production function
exhibits increasing returns to scale.
14
Equation (2.30) reconciles the human capital
production function and the discount factor. For an increasing returns to scale human
capital production function, the more patient agents have an incentive to invest less
time in education and accumulate less human capital today since the return to edu-
cation will be increasing in the level of human capital.
15
Hence, for a high discount
factor there are two opposing tendencies in the accumulation of human capital and
time invested in education. In order to see how the returns to scale in human capital
production interact with the returns to scale in the CM good production, I rewrite
14
It must be noted that the results are sensitive to the degree of IRTS for )(n, /
_
).
15
Consider a general human capital production function widely used in the literature
)(n, /) = (n/)
j
. For DRTS, j < 1,2, for CRTS j = 1,2 and for IRTS j 1,2. This
functional form implies that at the steady state )
&
(n, /) = jc
µ1
µ
/
2
1
µ
.
This implies that
)
&I
(n, )) = jc
µ1
µ
(2 ÷
1
j
)/
1
1
µ
_
¸
¸
¸
_
¸
¸
¸
_
0, if IRTS, j 1,2
< 0, if DRTS, j < 1,2
= 0, if CRTS, j = 1,2
126
the steady state equation (2.43) associated with equation (2.30).
¹ = ,(c)
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
o
_
[c
q
(¡
c
. /
c
)
j
I
(qo,I
~ o
,Io)
jq(qo,I
~ o,
Io)
) ÷c
I
(¡
c
. /
c
)]dG(~ c) ÷o
_
n
0
(¡
c
)
j
I
(qo,Io,I
~ o
)
jq(qo,Io,I
~ o
)
dG(~ c)
+
¹
&I
2
[r ÷(. +o)(c` ÷c:
c
) ÷:]
+
¹
)u(&o,Io)
(1 ÷o +,
I
(n
c
. /
c
))
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
,
&
(.)
(2.48)
The …rst line on the right hand side re‡ects the returns to human capital in terms of
DM production, taking into consideration the e¤ect it has on the terms of trade. The
second and the third line re‡ect the expected returns to human capital when entering
the CM as a buyer, a seller or a non-trader.
16
Ignoring for a moment the DM aspect
of human capital accumulation, a higher discount factor is associated on one hand,
by a higher human capital stock (look at the …rst term in the second line), and on the
other hand, with a lower human capital stock if there are increasing returns to human
capital production function, and higher (constant) human capital stock in the case
of decreasing (constant) returns to scale in the human capital production technology.
In this chapter, I use an increasing returns to scale technology. This implies that for
the patient agent there are two opposing incentives in terms of accumulating human
capital.
Under CRTS technology in the production of the CM good, n = 1 and : = 0.
the tendency to accumulate higher levels of human capital for the patient agent
16
Expected returns, depending on the various trading partner matches.
127
dominates the increasing returns to scale factor in human capital production. On the
other hand, decreasing returns to scale technology for CM production lead to n =
¸
__
:
c
/
c
dG(c)
_
¸1
and : = (1 ÷¸)
__
:
c
/
c
dG(c)
_
¸
, which implies that another
factor, namely the wage rate can alter the incentives to accumulate human capital.
Equation (2.48) becomes
¹ = ,(c)
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
o
_
[c
q
(¡
c
. /
c
)
j
I
(qo,I
~ o
,Io)
jq(qo,I
~ o,
Io)
) ÷c
I
(¡
c
. /
c
)]dG(~ c) ÷o
_
n
0
(¡
c
)
j
I
(qo,Io,I
~ o
)
jq(qo,Io,I
~ o
)
dG(~ c)
+
¹
¸
__
:
c
/
c
dG(c)
_
¸1
. ¸¸ .
u
I
2
[r ÷(. +o)(c` ÷c:
c
) ÷(1 ÷¸)
__
:
c
/
c
dG(c)
_
¸
. ¸¸ .
¬
+
¹
)u(&o,Io)
(1 ÷o +,
I
(n
c
. /
c
))
_
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
¸
_
,
&
(.)
A patient agent (high discount factor), through the wage-channel has the incentive to
reduce the amount of human capital stock, since higher human capital depresses the
wage rate. In this case for strong enough decreasing returns in the CM production
technology and increasing returns technology in human capital production, can lead
to patient agents to accumulate less human capital, as shown in the numerical results
in this section. I note that such a result is sensitive to the returns to scale parameters
in both production technologies and hence it is not a general result
2.6 Welfare Analysis
As mentioned previously, looking at the behavior of hours of work in the above graphs
can be misleading in terms of welfare analysis. I measure welfare as expected utility
128
in the steady state (2.49), which consists of expected DM utility and expected CM
utility. Similar to the previous section, in order to assess the welfare e¤ects of in‡ation,
I look at the steady state expected utility of an agent type c entering the DM with
. /)
(1 ÷,(c))\
c
= o1[n(¡
c
) ÷c(¡
c
. /)] +l(r
c
) ÷1
c
+n
c
) (2.49)
The …rst term denotes the expected utility from the DM. Note that, here, I take the
expectation not only with respect to the trading status, but also with respect to all
four possible meetings (type c
i
buyer meets type c
)
seller, (i. ,) ¸ (1. H)). I compute
the expected working hours as I did in the previous section:
1
c
+n
c
) = :
c
+n
c
= n
c
+
r
c
÷c(o +.)(` ÷:
c
) ÷:
n/
c
(2.50)
We are interested in how higher rates of money growth rate a¤ect welfare :
J(1 ÷,(c))\
c
J.
= o
J1[n(¡
c
) ÷c(¡
c
. /
c
)]
J.
+l
0
(r
c
)
Jr
c
J.
÷
J( :
c
+n
c
)
J.
(2.51)
J( :
c
+ n
c
)
J.
=
Jn
c
J.
+
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
_
÷c(` ÷:
c
) ÷(o +.)(
0çA
0:
÷
0çno
0:
) +
0ao
0:
÷
0¬
0:
¸
n/
c
÷[r
c
÷c(o +.)(` ÷:
c
) ÷:][
0&
0:
/
c
+n
0Io
0:
]
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
n
2
/
2
c
(2.52)
129
Substituting n = ¸
__
:
c
/
c
dG(c)
¸
¸1
above
0( ao+ &o)
0:
=
0&o
0:
+
[ç(Ano)(o+(
OçL
O:

Oçro
O:
)]
¸[
R
aoIooG(c)]
¸1
Io
+
Oio
O:

Or
O:
¸[
R
[aoIooG(c)
¸1
]Io
÷
[aoç(o+(Ano)¬][
Ou
O:
Io+¸[
R
aoIooG(c)]
¸1
OIo
O:
]
¸
h
[
R
[aoIooG(c)]
¸1
i
2
I
2
o
(2.53)
What is important to notice is that, now, there is another channel, namely the human
capital channel, through which a higher money growth rate can redistribute among
types. Given that it would be very tedious to show analytically how each welfare term
is a¤ected as we change the rate of money growth, and how it di¤ers across agents
I choose to rely on the numerical results. I proceed by examining each of the three
terms that a¤ect welfare: expected utility from the DM , utility from CM consumption,
and expected utility from leisure in the CM.
In Figure 2.9, I decompose CM expected utility into utility from leisure and utility
from consumption. The poor agents (patients) bene…t from the introduction of the
impatients because, as compared to the degenerate distribution case, this leads to an
increase in CM utility and welfare from in‡ation. DM welfare, on the other hand, is
decreasing for both types.
For completeness, I do the same exercise with CRTS technology. Figure 2.10 shows
that the presence of a di¤erent group of agents in the economy leads to di¤erent
welfare costs of in‡ation. The relatively poor agents bear a higher cost of in‡ation as
compared to the richer agents.
17
17
The quati…cation of such a statement can be found in Table 4.
130
F
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2
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:
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g
r
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r
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131
F
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u
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2
.
1
0
:
W
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f
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c
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a
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132
In Tables 2.4 and 2.5, I present the welfare cost of moving from 0% to 10% in‡ation
rate for various parameter values. For CRTS technology (Table 2.5 )in CM production
I observe the following: for all parameter values, poor agents su¤er more fromin‡ation,
indicating a regressive in‡ation tax. The lower the weight of the poor agents, and the
bigger the di¤erence in discount factors, the greater is the dispersion in welfare cost
between di¤erent agents. Allowing for DRTS technology (Table 2.4 ), on the other
hand, o¤ers another channel of redistributive e¤ects. In this case, the relatively poor
agents (high types in this case) bene…t from in‡ation, and the rich agents bear the
cost of higher money growth rates. The welfare cost for rich agents is systematically
smaller than in an economy with CRTS technology (constant wages). Overall, allowing
for another productive asset, whose return depends on economy-wide prices, allows
for yet another channel of redistribution through which in‡ationary policies operate.
When that is the case, considering representative agent models overstates the welfare
cost of in‡ation as well as hiding the fact that a section of the economy can actually
bene…t from it.
2.7 Conclusion
This chapter studies the long-run redistributive e¤ects of monetary policy in a micro-
founded model of money. It builds on the search-theoretic model of Lagos and Wright
(2005) in two important dimensions. First, I introduce heterogeneity while keeping
the distribution of money holdings tractable. This version of the model allows us to
133
T
a
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2
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4
:
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134
T
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a
t
i
e
n
t
s
"
)
2
.
5
5
3
.
1
4
2
.
7
2
C
a
s
e
3
-
C
a
s
e
1
w
i
t
h
j
=
0
.
5
2
.
5
8
2
.
8
8
2
.
6
3
C
a
s
e
4
-
C
a
s
e
1
w
i
t
h
.
1
=
0
.
3
2
.
7
5
6
.
7
1
2
.
7
2
C
a
s
e
4
-
C
a
s
e
1
w
i
t
h
o
=
1
4
.
0
5
7
.
0
7
4
.
5
8
C
a
s
e
5
-
C
a
s
e
1
w
i
t
h
o
=
0
.
3
2
.
3
9
2
.
8
0
2
.
4
1
C
a
s
e
6
-
C
a
s
e
1
w
i
t
h
o
=
0
.
5
2
.
4
6
3
.
1
4
2
.
5
0
C
a
s
e
7
-
C
a
s
e
1
w
i
t
h
,
1
=
0
.
9
2
2
.
6
5
4
.
2
1
2
.
6
3
135
examine the redistributive e¤ects of changes in the money growth rate when money is
injected via lump-sum transfers. Heterogeneity in discount factors results in a regres-
sive in‡ation tax. Wealthy agents are less a¤ected by the in‡ation tax than the less
wealthy.
Second, I introduce human capital as a productive asset, which can be used in both
DM and CM markets. This capital provides a link between the in‡ation tax in the
DM and CM activity, breaking in this way the super-neutrality of money in the CM.
I examine the e¤ect of an increase in the money growth rate on output, welfare and
the distribution of wealth and human capital accumulation. I discover two channels
of redistributive e¤ects of in‡ation. One is the usual e¤ect generated by lump-sum
transfers of money injected into the economy. The other e¤ect is through the wage
rate, which under decreasing returns to scale technology depends on economy-wide
e¤ective labor. My numerical results show that in‡ationary monetary policy can lead
to a long-run increase in output, consumption, and time spent in skill acquisition
activities and a decrease in the time spent working as well as a lower dispersion in
the distribution of wealth and human capital.
136
Appendix A
Endogeneizing Information Processing Capacity (i)
The …xed point solution for the linear cost case to the problem under endogeneous i
is the one that solves the following system of equations:
2
2i
/
= (1 ÷,)
_
o
o
o
o
.
_
2
i
[ :
33
[
_
1
1 ÷o
_
2
o
2
o
.
2
2(i
/
i)
= 2
i
and
, ÷¸
· +¸o
c
t
=
1 ÷,
1 ÷o
_
1 ÷2
2i
/
_
The , which veri…es the inital guess is the solution to the following equation
1
1
,
2
+1
2
, +1
3
= 0
where 1
1
= ÷
_
1 ÷o +
c
j¬
33
jln(2)o
2
a
(· +¸o)
_
. 1
2
= 1÷o+¸÷¸o+2
c
j¬
33
jln(2)o
2
a
(· +¸o) .
1
3
= ¸(1 ÷o) +
c
j¬
33
jln(2)o
2
a
(· +¸o) (1 ÷(1 ÷o)
2
)
137
Appendix B
Data
Data on macroeconomic aggregates are taken from Federal Reserve Economic Data
(FRED) dataset and Bureau of Labor Statistics (BLS) The data series include sea-
sonally adjusted, quarterly, billions of chained 2000$, real gross national product, real
personal consumption expenditures of durable, non-durable goods and services, real
private …xed investment, hours and employment.
138
Appendix C
Non-stochastic steady state
In the deterministic steady state there are no technology shocks : c
it
= c
t
= 0. Given
that technology is the only source of heterogeneity in the model, in this case all …rms
are exactly the same. From the household …rst order conditions I have:

C
¸
n = o

1 (C.1)
1 = ,(1 +: ÷d) (C.2)
For the representative …rm (due to lack of heterogeneity in the deterministic steady
state) I have:
n = o

1
c

1
c1
(C.3)
: = c

1
c1

1
c
(C.4)
From the aggregate resource constraint and the production function I have:

C =

1 +d

1 (C.5)

1 =

1
c

1
c
(C.6)
There are 6 equations and 6 unknowns, so I can solve for
_

1 .

C.

1.

1. n. :
_
.
139
Appendix D
Why volatility ampli…cation is stronger for aggregate hours
of work than aggregate output
Suppose 1
t
= q(.
t
. 1
t
). where .
t
= c
oI
and q(.) is any production function. After
log-linearizing output around .
t
= 1. 1
t
=

1 I have:
^
1
t
=
q
:
(1.

1)

1
c
t
+
q
1
(1.

1)

1
^
1
t
Under rational inattention
^
1
t
= ,(
ou
os
)c
t
. Assume for simplicity that c
t
=
t
. Then
I have:
^
1
t
=
_
j1,

1)

Y
+
j
1
(1,

1)

Y
,(
ou
os
)
_

t
. The volatilities of labor and output are
\ c1
t
) = ,(
o
&
o
.
)
2
o
2
.
and
\ c
^
1
t
) =
_
q
:
(1.

1)

1
+
q
1
(1.

1)

1
,(
o
&
o
.
)
_
2
o
2
.
The elasticities of \ c1
t
) and \ c
^
1
t
) with respect to o
2
.
are :
c
·ov(1)
o
2
s
= 1 +
2,
o
2
s
(.)o
2
.
,(.)
and
c
·ov(Y )
o
2
s
= 1 +
2,
o
2
s
(.)o
2
.
j1,

1)
j
1
(1,

1)
+,(.)
140
Given that
j1,

1)
j
1
(1,

1)
is always positive,
c
·ov(Y )
o
2
s
< c
·ov(1)
o
2
s
141
Appendix E
Derivation of the information ‡ow constraint
In this subsection I will derive the information rate for one and two-dimensional
discrete parameter Gaussian processes using frequency-domain methods.
E.1 Information rate of discrete parameter one-dimensional
Gaussian processes
Let A = ¦r(t)¦. 1 = ¦¸(t)¦ be one-dimensional, real-valued, discrete parameter
, wide-sense stationary and stationarily correlated processes. The information rate
between these two processes can be written as follows
1
A,Y
= ÷
1
4:
_
¬
¬
log(1 ÷[:
AY
(.)[
2
)d.
where
[:
AY
(.)[
2
= ¦
j]
^Y
(.)j
2
]
^^
(.)]
Y Y
(.)
,)
^Y
(.)6=0
0,)
^Y
(.)=0
where, ,
AA
(.) and ,
Y Y
(.) are spectral densities of process A and 1 respectively, and
,
AY
(.) is the cross-spectral density. [:
AY
(.)[
2
is also called the coherence between
142
the processes at frequency .. which is the frequency-domain analog of the correlation
coe¢cient.
As an example of this, assume that A and 1 can be expressed as in…nite-
order moving average: A =

1
|=0
d
|

t|
= 1(1)
t
and 1 =

1
|=0
:
1
|

t|
+

1
|=0
:
1
|
j
1
t|
(1)
t
= `
1
(1)
t
+ `
1
(1)j
1
t
. where 1(1). `
1
(1). `
1
(1) are in…nite
lag polynomials and ¦
t
¦. ¦j
1
t
¦ are Gaussian mutually independent white noise
processes with o
2
.
and unit variance respectively and independent of each other.
Spectral density functions for A
1
and 1
1
are:
,
AA
(.) =
o
2
.
2:
1(c
i.
)1(c
i.
)
,
Y Y
(.) =
o
2
.
2:
`
1
(c
i.
)`
1
(c
i.
) +
1
2:
`
1
(c
i.)
`
1
(c
i.
)
and the cross-spectral density is
,
AY
(.) =
o
2
.
2:
1(c
i.
)`
1
(c
i.
)
where 1(c
i.
) = d
c
+d
1
c
i.
+d
2
c
2i.
+...d
T
c
Ti.
+... 1(c
i.
) = d
c
+d
1
c
i.
+d
2
c
2i.
+
...d
T
c
Ti.
+... `
1
(c
i.
) = :
1
c
+:
1
1
c
i.
+:
1
2
c
2i.
+...:
1
T
c
Ti.
+... `
1
(c
i.
) = :
1
c
+
:
1
1
c
i.
+:
1
2
c
2i.
+...:
1
T
c
Ti.
+.. and `
1
(c
i.
) = :
1
c
+:
1
1
c
i.
+:
1
2
c
2i.
+...:
1
T
c
Ti.
+...
`
1
(c
i.
) = :
1
c
+:
1
1
c
i.
+:
1
2
c
2i.
+...:
1
T
c
Ti.
+... Using the spectral and cross-spectral
densities, the information rate between these two one-dimensional processes becomes:
1
A,Y
= ÷
1
4:
_
¬
¬
log(
1
1 +
o
2
s
A
1
(c
..
)A
1
(c
..
)
.
1
(c
..
).
1
(c
..
)
)d.
where
o
2
s
A
1
(c
..
)A
1
(c
..
)
.
1
(c
..
).
1
(c
..
)
is also de…ned as the signal-to-noise ratio. Hence, one can
express the information rate between two moving average Gaussian processes in terms
143
of their moving average coe¢cients. This information ‡ow constraint will be used in
the dynamic version of the model with labor only as the input choice to be made by
the …rms.
E.2 Information rate of discrete parameter multi-dimensional
Gaussian processes
Derivations in this section follow the book "Information and information stability of
random variables and processes" by M. S. Pinsker (1964).
The multidimensional case of the problem applies to the benchmark model in the
paper, where the …rms’ optimal input choices are those of capital and labor.
Let A = ¦r
1
(t). r
2
(t). ...r
a
(t)¦. 1 = ¦¸
1
(t). ¸
2
(t). ...¸
n
(t)¦ be n and m-dimensional,
real-valued, discrete parameter , wide-sense stationary and stationarily correlated
processes respectively. The information rate between these two processes can be
written as follows:
1
A,Y
= ÷
1
4:
_
¬
¬
log
det ¹
~
A
~
Y
(.)
det ¹
~
A
(.) det ¹
~
Y
(.)
d.
where det ¹
A
(.) = det [[,
a
.
a
¡
(.)[[
i,)=1,.....,a
. det ¹
Y
(.) = det [[,
j
.
j
¡
(.)[[
i,)=1,.....,n
.
det ¹
AY
(.) = det [[,
a
.
j
¡
(.)[[
i,)=1,.....,a+n
and det ¹
~
A
(.) is a non-vanishing principal
minor of highest order ’:
0
of the determinant det ¹
A
(.). det ¹
~
Y
(.) is a non-vanishing
principal minor of highest order ’:
0
of the determinant det ¹
Y
(.). and det ¹
~
A
~
Y
(.)
is the principal minor of order
0
: + :
0
of the determinant det ¹
AY
(.) which contains
144
det ¹
~
A
(.) and det ¹
~
Y
(.). ,
12
(.) refers to the cross-spectrum between variable ’1’
and ’2’.
The model in this chapter requires the computation of the information rate
between two-dimensional Gaussian processes. The information ‡ow relevant in the
model is the information ‡ow between the full information pro…t maximizing decisions
of capital and labor, and the actual decisions under limited information. In turn,
this can be interpreted as the information rate between the variable the …rms are
trying to track (the pro…t maximizing decisions) and the signals they get regarding
the pro…t maximizing decisions, which are the actual decisions.
We have 1(¦|
1
it
¦. ¦/
1
it
¦; ¦|

it
¦. ¦/

it
¦) = 1(¦|
1¹
t
¦. ¦/
1¹
t
¦; ¦|
¹
t
¦. ¦/
¹
t
¦)+1(¦|
11
it
¦. ¦/
11
it
¦; ¦|
1
it
¦. ¦/
1
it
¦).
where subscript 1 stands for full information optimal decisions and subscript + stands
for actual decisions for capital and labor, and where ¹ stands for aggregate compo-
nents while 1 stands for the idiosyncratic components. The equality above comes from
the fact that common and idiosyncratic components of the …rm-level productivity
shock are independent from each other. Hence, I can separate the aggregate from the
idiosyncratic component in each decision rule
1
. In order to compute the information
‡ow, I use the moving average representation of decision rules for capital and labor
derived under full and incomplete information. The following derivation involves the
information ‡ow pertaining to the aggregate component of the decision rules.
1
This same procedure is followed in Ma´ckowiak and Wiederholt (2009a)
145
|
1¹
t
= 1(1)
t
. /
1¹
t
= 1(1)
t
. |
¹
it
= `
1
(1)
t
+ `
1
(1)j
1
it
. /
¹
it
= `
1
(1)
t
+
`
1
(1)j
1
it
. where
t
~ \`(0. o
2
.
). j
1
it
and j
1
it
~ \`(0. 1). where ¦
t
¦. ¦j
1
it
¦ and
¦
t
¦. ¦j
1
it
¦ are pairwise independent from each other but {j
1
it
¦. ¦j
1
it
¦ do not need to
be independent. This setting applies to an environment where there is a single agent
(the …rm’s decision maker) that chooses the optimal pair of labor and capital inputs.
The objective of the …rm is to track the full information pro…t-maximizing levels of
labor and capital using an optimal set of signals. Since there is only one decision
maker within the …rm that jointly chooses labor and capital inputs, it is reasonable
to assume that information processing will lead to optimal signals being correlated.
This chapter allows for this possibility, which expands the set of choice variables for
the …rm when they solve their attention allocation problem. Firms now will choose
not only the extent of the noise in each signal but also their correlation across signals.
After calculating the spectral and cross-spectral densities as well as using the
de…nition for information ‡ow for multi-dimensional Gaussian processes I obtain:
1(¦|
1¹
t
¦. ¦/
1¹
t
¦; ¦|
¹
t
¦. ¦/
¹
t
¦) =
÷
1
4¬
_
¬
¬
log
1
1+
¬
2
s
L
1
(c
..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)
+
¬
2
s
L
1..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)
d.
where ¸ = 1(j
1
it
j
1
it
).
By looking at the pro…t-maximizing decision rules for each …rm , the idiosyncratic
component for both labor and capital input decisions is the same, namely the idio-
146
syncratic TFP component. In this case the …rm chooses to receive only one signal
whose noise will be a choice variable.
|
11
it
= /
11
t
= ¹
2
(1)n
t
. |
1
it
= /
1
it
= o(1)n
it
+1(1)·
it
. where n
it
~ \`(0. o
2
&
). ·
it
~ \`(0. 1).
1(¦|
11
it
¦. ¦/
11
it
¦; ¦|
1
it
¦. ¦/
1
it
¦ = ÷
1
4:
_
¬
¬
log
1
1 +
o
2
u
S(c
..
)S(c
..
)
T(c
..
)T(c
..
)
d.
147
Appendix F
Algorithm
The algorithm used here to solve the model is similar to Paciello (2008).
Step 1:
Under both types of information structures, I solve the model by log-linearizing
around the deterministic steady-state. It is well-known that under full-information
log-linearization, eliminates second-moment e¤ects. However, under incomplete infor-
mation with information processing constraints, there are …rst-order e¤ects of the
volatility of underlying shocks, even though the model is log-linearized.
Full Information
148
Under full-information the following equations must hold in equilibrium:
·
^
1
t
+¸
^
C
t
= ^ n
t
^
C
t
= 1(
^
C
t+1
÷
^ v
I+1
¸
)
^
1
t
=

C

Y
^
C
t
+

1

Y
(
^
1
t+1
÷(1 ÷d)
^
1
t
)
^
|
1
it
=
1
1cc
(c
t
+c
it
÷(1 ÷c) ^ n
t
÷c^ :
t
)
^
/
1
it
=
1
1cc
(c
t
+c
it
÷o ^ n
t
÷(1 ÷o)^ :
t
)
c
it
= j
1
c
it1
+n
it
. n
it
~ \`(0. o
2
&
)
c
t
= j
¹
c
t
+
t
.
t
~ \`(0. o
2
.
)
The …rst two equations come from household problem, the third one the resource
constraint, the third is from the resource constraint, the fourth and the …fth equations
are optimal labor and capital decisions taken by …rms under full-information, and
the last two equations are the assumed processes for the common and idiosyncratic
components of …rm-level TFP. Given the assumption of decreasing returns to scale
one can determine optimal hours of work and capital, unlike the case of constant
returns to scale, where only the capital-to-labor ratio can be pinned down. Part of
step 1 involves making a guess for the deviation of capital and labor decisions under
rational inattention from the pro…t-maximizing decisions (under full information)
1
.
The guess takes the following form: qnc::
1
= |

it
÷|
1
it
and qnc::
1
= /

it
÷/
1
it
1
This step is similar to formulating a guess regarding the actual labor and capital deci-
sions under rational inattention.
149
Using the guess I compute the implied dynamics for the model for the aggre-
gate variables. The set of equations that must hold in equilibrium for the aggregate
dynamics under rational inattention are the following:
·
^
1
t
+¸
^
C
t
= ^ n
t
^
C
t
= 1(
^
C
t+1
÷
^ v
I+1
¸
)
^
1
t
=

C

Y
^
C
t
+

1

Y
(
^
1
t+1
÷(1 ÷d)
^
1
t
)
¸
t
= c
t
+o|
t
+c/
t
c
t
= j
¹
c
t
+
t
.
t
~ \`(0. o
2
.
)
Obtaining the average wage and rental rate I can compute the pro…t-maximizing
decision rules for capital and labor, which are used in solving the attention allocation
problem:
|
1¹
t
=
1
1 ÷c ÷o
(c
t
÷(1 ÷c)n
t
÷c:
t
)
and
/
1¹
t
=
1
1 ÷c ÷o
(c
t
÷on
t
÷(1 ÷o):
t
)
One can express all variables as moving averages. For instance, c
t
= ¹
1
(1)
t
. n
t
=
\(1)
t
. :
t
= 1(1)
t
. Substituting these moving average representations into |
1¹
t
and /
1¹
t
I obtain: |
1¹
t
= 1(1)
t
. /
1¹
t
= 1(1)
t
. where 1(1) =
1
1cc
(¹
1
(1) ÷
(1 ÷ c)\(1) ÷ c1(1))
t
and 1(1) =
1
1cc
(c
t
÷ o\(1) ÷ (1 ÷ o)1(1))
t
. The
idiosyncratic part of the pro…t-maximizing decision rules is simply |
11
it
= /
11
it
=
150
1
1cc
c
it
=
1
1cc
¹
2
(1)n
it
. where ¹
2
(1)n
it
is a moving average representation of the
idiosyncratic component of the …rm-level TFP shock.
Step 2. Having obtained the pro…t -maximizing decision rules for capital and
labor I can now solve the attention allocation problem that …rms face. Each …rm
minimizes the losses it incurs due to incomplete information, subject to an information
processing constraint.
1o:: =
1
2
1[:
33
(/
it
÷/
1
it
)
2
+ 2:
34
(/
it
÷/
1
it
)(|
it
÷|
1
it
) +:
44
(|
it
÷|
1
it
)
2
] =
1
2
1[:
33
(/
¹
it
÷/
1¹
it
)
2
+:
33
(/
1
it
÷/
11
it
)
2
+:
44
(|
¹
it
÷|
1¹
it
)
2
+:
44
(|
1
it
÷|
11
it
)
2
+2:
34
(/
¹
it
÷/
1¹
it
)(|
¹
it
÷|
1¹
it
) + 2:
34
(/
1
it
÷/
11
it
)(|
1
it
÷|
11
it
)] =
1
2
1[:
33
(/
¹
it
÷/
1¹
it
)
2
+:
44
(|
¹
it
÷|
1¹
it
)
2
+ 2:
34
(/
¹
it
÷/
1¹
it
)(|
¹
it
÷|
1¹
it
)]
+
1
2
1[:
33
(/
1
it
÷/
11
it
)
2
+:
44
(|
1
it
÷|
11
it
)
2
+ 2:
34
(/
1
it
÷/
11
it
)(|
1
it
÷|
11
it
)]
where
|
1¹
t
= 1(1)
t
/
1¹
t
= 1(1)
t
|
¹
it
= `
1
(1)
t
+`
1
(1)j
1
it
/
¹
it
= `
1
(1)
t
+`
1
(1)j
1
it
(F.1)
151
where
t
~ \`(0. o
2
.
). j
1
it
and j
1
it
~ \`(0. 1).where ¦
t
¦. ¦j
1
it
¦ and¦
t
¦. ¦j
1
it
¦
are pairwise independent and 1(j
1
it
j
1
it
) = ¸.
|
11
it
= /
11
t
= ¹
2
(1)n
t
|
1
it
= /
1
it
= o(1)n
it
+1(1)·
it
(F.2)
where n
it
~ \`(0. o
2
&
).and ·
it
~ \`(0. 1). Lag polynomials 1(1) and 1(1)
come from step 1 given the initial guess whereas the moving average coe¢cients on
the actual decisions are what the …rms choose.
Information ‡ow can also be expressed as the sum of information ‡ow between
idiosyncratic variables and information ‡ow between aggregate variables.
1(¦|
1
it
¦. ¦/
1
it
¦; ¦|

it
¦. ¦/

it
¦) = 1(¦|
1¹
t
¦. ¦/
1¹
t
¦; ¦|
¹
t
¦. ¦/
¹
t
¦) +1(¦|
11
it
¦. ¦/
11
it
¦; ¦|
1
it
¦. ¦/
1
it
¦)
= ÷
1
4¬
_
¬
¬
log
1
1+
¬
2
s
L
1
(c
..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)
+
¬
2
s
L
1..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)
d.
÷
1
4¬
_
¬
¬
log
1
1+
¬
2
u
S(c
..
)S(c
..
)
T(c
..
)T(c
..
)
d.
152
The attention allocation problem becomes:
max
fn
1
,n
1
,a
1
,a
1
,c,tg
1
2
(
1
1cc
)
2
¦o
2
.
:
33

T
|=0

1
|
÷c
|
)
2
+:
33

T
|=0

1
|
)
2
+
o
2
.
:
44

T
|=0

1
|
÷d
|
)
2
+:
44

T
|=0

1
|
)
2
+ 2:
34
o
2
.

T
|=0

1
|
÷c
|
)
1
|
÷d
|
)+
2:
34
¸

T
|=0
:
1
|
:
1
|
o
2
&

44
+:
33
+ 2:
34
)

T
|=0

|
÷c
2|
)
2
+

44
+:
33
+ 2:
34
)

T
|=0
(t
|
)
2
¦
subject to
÷
1
4¬
_
¬
¬
log
1
1+
¬
2
s
L
1
(c
..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)
+
¬
2
s
L
1..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)
d.
÷
1
4¬
_
¬
¬
log
1
1+
¬
2
u
S(c
..
)S(c
..
)
T(c
..
)T(c
..
)
d. _ i
where ¦:
1
. :
1
. :
1
. :
1
. :. t¦ are the lag polynomial coe¢cients in equations (F.1)
and (F.2).
As previously derived, the information ‡ow is a function of moving average coef-
…cients, which also appear in the loss function. As an example, consider the choice of
:
1
. :
1
:
(
1
1cc
)
2
o
2
.
:
44

1
|
÷d
|
) +:
34
o
2
.

1
|
÷c
|
) =
÷
A
4¬
¬
_
¬
0
0
B
@log
1
1+
¬
2
s
L
1
(c
..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)
+
¬
2
s
L
1..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)
1
C
A
0n
1
I
153
and
(
1
1cc
)
2

44
+ 2:
34
¸:
1
|
):
1
|
=
÷
A
4¬
_
¬
¬
0
0
B
@log
1
1+
¬
2
s
L
1
(c
..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)
+
¬
2
s
L
1..
)L
1
(c
..
)
(1_
2
)^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)

¬
2
s
_
1_
2
L
1
(c
..
)L
1
(c
..
)
^
1
(c
..
)^
1
(c
..
)
1
C
A
0a
1
I
where ` is the shadow price of information. The complete solution of the attention
allocation stage consists of 6T+1 equations and 6T+1 unknowns, which are solved
numerically. Once this stage is solved I obtain¦|

it
¦¦/

it
¦. which are the actual decisions
under rational inattention. As a next step I compute the di¤erence between these
decision rules and pro…t maximizing decision rules. If |

it
÷|
1
it
,= qnc::
1
and /

it
÷/
1
it
,=
qnc::
1
I update the guess by the following rule:
qnc::
1
ac&
= cqnc::
1
+ (1 ÷c)(|

it
÷|
1
it
)
and
qnc::
1
ac&
= cqnc::
1
+ (1 ÷c)(/

it
÷/
1
it
)
154
Appendix G
Perfect Information Case
In this section I compute the equilibrium dynamics of the full-information version
of the model in which …rms know the entire history of state variables, including
their period t realization. Under full information the model collapses to a standard
RBC model with DRTS technology in the production function. Hence, the perfect
information solution is not only important in comparing the two di¤erent information
structures but also because it nests a well known benchmark, that of a standard RBC
model.
The household part of the economy is the same as in the benchmark model. Given
that there are no adjustment costs to the …rm of changing the number of workers or
capital, their problem is static.
The …rm’s problem is:
max
|
.I,
I
.I
_
c
oI
c
o
.I
/
c
it
|
c
it
÷n
t
|
it
÷:
t
/
it
_
(G.1)
The implied …rst order conditions are:
n
t
= oc
oI
c
o
.I
/
c
it
|
c1
it
(G.2)
:
t
= cc
oI
c
o
.I
/
c1
it
|
c
it
(G.3)
155
Which implies :
n
t
:
t
=
_
o
c
_
/
it
|
it
(G.4)
All …rms have the same capital-to-labor ratio. The DRTS assumption allows me to
pin down …rm-speci…c levels of labor and capital demand:
|
it
=
_
cc
a
I
c
a
.I (
ou
I
¿r
I
)
o
&I
_ 1
1o¿
(G.5)
/
it
= |
it
(
&I
vI
c
c
) (G.6)
The market clearing conditions are 1
t
=
_
/
it
di, 1
t
=
_
|
it
di, 1
t
=
_
¸
it
di.
_
c
it
di =
0.The resource constraint is:
C
t
+1
t+1
÷(1 ÷d)1
t
= 1
t
(G.7)
Log-linearized version of the Perfect Information Model
Given that the imperfect information model will be solved in a Linear Quadratic
Gaussian framework, I need the log-linearized FOC of the perfect information case
to make a consistent comparison as well to build a quadratic loss function. The log-
linearization is done around the non-stochastic steady state.
156
The log-linearized set of …rst order conditions for the household and …rms are:
·
^
1
t
+¸
^
C
t
= ^ n
t
(G.8)
^
C
t
= 1
_
^
C
t+1
÷
^ v
I+1
¸
_
(G.9)
^ n
t
= c
t
+c
it
+c
^
/
1
it
+ (o ÷1)
^
|
1
it
(G.10)
^ :
t
= c
t
+c
it
+ (c ÷1)
^
/
1
it
+o
^
|
1
it
(G.11)
^
/
1
it
÷
^
|
1
it
= ^ n
t
÷ ^ :
t
(G.12)
1
^ jit
= c
t
+c
it
+c
^
/
1
it
+o
^
|
1
it
(G.13)
^
|
1
it
=
1
1cc
(c
t
+c
it
÷(1 ÷c) ^ n
t
÷c^ :
t
) (G.14)
^
/
1
it
=
1
1cc
(c
t
+c
it
÷o
^
\
t
÷(1 ÷o)^ :
t
) (G.15)
^
1
t
=

C

Y
^
C
t
+

1

Y
_
^
1
t+1
÷(1 ÷d)
^
1
t
_
(G.16)
Aggregate Equilibrium Conditions under Perfect Information
By aggregating the …rm-speci…c …rst order conditions I obtain 6 equations, three
of which are equations (G.8), (??) and (??), and 6 unknowns ¦1
t
. 1
t+1
. C
t
. 1
t
. n
t
. :
t
¦:
^
1
t
= c
t
+c
^
1
t
+o
^
1
t
(G.17)
^
1
t
=
1
1cc
(c
t
÷(1 ÷c) ^ n
t
÷c^ :
t
) (G.18)
^
1
t
=
1
1cc
(c
t
÷o
^
\
t
÷(1 ÷o)^ :
t
) (G.19)
157
Appendix H
Nash Bargaining Solution and Seller Heterogeneity
In this section, I solve a simple model with seller heterogeneity and derive the condi-
tion under which : :

holds in equilibrium. I follow closely the methodology used
in Lagos and Wright (2005).
Suppose agents di¤er in their productivity in the DM. That is, when an agent
enters the DM as a seller, she can produce according to c
c
(¡) = c¡. The productivity
parameter c can take two values, ¦c
1
. c
1
¦. The rest of the model is similar to Lagos
and Wright.
In the CM, agents solve the following problem
\) = l(A) ÷H +,\
0
)
s.t
A = H +c÷:
0
)
158
First order conditions and the Envelope condition are as follows
l
0
(A) = 1
c = ,\
0

0
)
\
0
) = c
In the DM, with probability o an agent can be a seller or a buyer, or doesn’t trade
at all. If the agent gets to be a buyer, she can be a a type c
1
(low productivity) or
a type c
1
(high productivity) producer The value function of an agent with money
holdings : entering the DM is
\
c
) = o [÷c¡( ~ :. c) +\+d( ~ :. c)] + (H.1)
o
_
,
_
n(¡. c
1
) +\÷d)
¸
+ (1 ÷,)
_
n(¡. c
1
) +\÷d)
¸_
+ (1 ÷2o)\)
where , denotes the share of type H agents. Equation (H.1) implies that the terms
of trade in the DM depend only on the buyers money balances and the sellers
productivity. Below, I solve the Nash Bargaining stage, where terms of trade are
determined.
max
q,on
[n(¡) ÷cd]
0
[÷c¡ +cd]
10
159
The solution to this problem is
¡. c) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
¡. c) if : < :

(c)
¡

(c) if : _ :

(c)
d. c) =
_
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
_
: if : < :

(c)
:

if : _ :

(c)
For cases when : :

I have
n
0
(¡

) = c (H.2)
c:

= (1 ÷o)n(¡

) +oc¡

(H.3)
For cases when : < :

.we have
c: =
(1 ÷o)n(¡)c +con
0
(¡)¡
on
0
(¡) + (1 ÷o)c
(H.4)
d = :
where ¡. c) solves equation (H.4). At this point, similar to section 2.4.1, I need to
prove that in equilibrium : < :

for both types and hence d = :. Focusing on this
equilibrium, helps the tractability of the model since, otherwise there would be an
additional source of heterogeneity. I proceed by showing the conditions under which
: < :

holds for both types. Following Lagos and Wright, one can showthat the value
function (H.1) can be shown to be \
t
) = ·
t
) +c
t
:+ max¦÷c
t
:
0
+,\
t+1

0
)¦
160
where
·
c,t
) = o
_
,
_
n(¡. c
1
) ÷c
t
d
t
)
¸
+ (1 ÷,)
_
n(¡. c
1
) ÷c
t
d
t
)
¸_
+o
_
c
t
d( ~ ÷
q
c
_
+l(A) ÷A
By repeated substitution, one can show that
\
c,t

t
) = ·
c,t

t
) +c
t
:
t
+
1

)=t
,
)t
max
n
¡+1
_
÷c
)
:
)+1
+,
_
·
)+1

)+1
) +c
)+1
:
)+1
¸_
(H.5)
One needs to check under which conditions an equilibrium exists. In order to do this,
I look at the slope of objective function (H.5) as :
t+1
÷:

t+1
from below. In a model
with sellers heterogeneity, there are two optimal quantities and money holdings, as
shown in equations (H.2) and (2.35). Hence, I need check the equilibrium condition
for each type. When a buyer comes across a high-productivity seller (c
1
), the optimal
amount of money held is less than if the buyer were to meet a low-productivity seller
(c
1
).
For a H-type (low productivity), the slope of (H.5) as :
t+1
÷ :

t+1
from below,
is
lim
n
I+1
!n

1,I+1
J\
1,t

t
)
J:
t+1
= ÷c
t
+,c
t+1
+,oc
t+1

where
= ,
c
2
1
o(1 ÷o)n
qq
(¡

1
)(c
1
¡

1
÷n) +c
2
1
÷1
Except for c
t
= ,c
t+1
or o = 1. < 0. hence, in equilibrium, :
t+1
< :

1,t+1
.
161
For a L-type (high productivity), the slope of (H.5) as :
t+1
÷:

1,t+1
from below,
is
lim
n
I+1
!n

1,I+1
J\
1,t

t
)
J:
t+1
= ÷c
t
+,c
t+1
+,oc
t+1

where
= ,
c
1
_
oc
1
+ (1 ÷o)c
1
¸
c
1
(1 ÷o)on
qq
(¡

1
) (c
1
¡

1
÷n(¡

1
)) +c
1
c
1
[oc
1
+ (1 ÷o)c
1
]
+(1 ÷,)
c
2
1
o(1 ÷o)o(1 ÷o)n
qq
(¡

1
)(c
1
¡

1
÷n) +c
2
1
÷1
If < 0 then one can safely say that : < :

holds for both types of trade, between
a buyer and high or low-type seller. Hence, in order to justify solving the model with
sellers heterogeneity, under the situation when : < :

. one has to restrict model’s
parameters such that < 0. We need
,
c
1[0c
1
+(10)c
1
]
c
1
(10)0&qq(q

1
)(c
1
q

1
&(q

1
))+c
1
c
1
[0c
1
+(10)c
1
]
+(1 ÷,)
c
2
1
0(10)0(10)&qq(q

1
)(c
1
q

1
&)+c
2
1
< 1
162
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