Economics Report on Resource Reallocation, Productivity Dynamics, and Business Cycle

Description
In ecology, productivity or production refers to the rate of generation of biomass in an ecosystem. It is usually expressed in units of mass per unit surface (or volume) per unit time, for instance grams per square metre per day

ABSTRACT

Title of Dissertation / Thesis: RESOURCE REALLOCATION,
PRODUCTIVITY DYNAMICS,
AND BUSINESS CYCLES

Min Ouyang, Doctor of Philosophy, 2005

Dissertation / Thesis Directed By: Professor John Shea, Department of Economics
Professor John Rust, Department of Economics

This dissertation explores the interactions between resource reallocation,
productivity dynamics and business cycles. A theory that combines two driving forces for
resource reallocation, learning and creative destruction, is presented to reconcile several
empirical findings of gross job flows. The theory suggests a scarring effect, in addition to
the conventional Schumpeterian cleansing effect, of recessions on the allocation
efficiency of resources. I argue that while recessions kill off some of the least productive
businesses, they also impede the development of potentially good businesses -- the ones
that might have proven to be efficient in the future are cleared out and lose the
opportunity to realize their potential. Calibrations of the model using US manufacturing
job flows suggest that the scarring effect is likely to dominate the cleansing effect and
account for the observed pro-cyclical average productivity.

RESOURCE REALLOCATION, PRODUCTIVITY DYNAMICS,
AND BUSINESS CYCLES

by

Min Ouyang

Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park, in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
[2005]

Advisory Committee:

Professor John Shea, Chair
Professor John Rust, Co-chair
Professor John Haltiwanger
Professor John Horowitz
Professor Daniel Vincent

© Copyright by

[Min Ouyang]

[2005]

DEDICATION

To my family

Acknowledgments

I am grateful to John Shea, John Rust and John Haltiwanger, from whose instruction and
encouragement I have gained immeasurably. I owe special thanks to John Shea for his
particularly detailed comments that helped through this research, to John Rust for his C
codes of multi-dimensional interpolation, and to John Haltiwanger for providing data on
US manufacturing gross job flows. I am also indebted to the other members of my
proposal and dissertation committees, Daniel Vincent, Michael Pries, and John Horowitz,
for their advice and service. Comments from seminar participants at the department of
Economics of University of Maryland and the Midwest Macroeconomics Meetings 2004
are greatly appreciated.

Contents
1 Resource Reallocation and Business Cycles 1
2 Plant Life Cycle and Aggregate Employment Dynamics 8
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Evidence: Gross Job ?ows and plant age . . . . . . . . . . . . . . . . 10
2.3 A Model of Learning and Creative Destruction . . . . . . . . . . . . . 14
2.3.1 “All-Or-Nothing” Learning . . . . . . . . . . . . . . . . . . . . 16
2.3.2 Creative Destruction and Industry Equilibrium . . . . . . . . 19
2.4 Aggregate Employment Dynamics . . . . . . . . . . . . . . . . . . . . 23
2.4.1 Job Flows over the Plant Life Cycle . . . . . . . . . . . . . . . 23
2.4.2 Decomposition of Job ?ows . . . . . . . . . . . . . . . . . . . 25
2.4.3 The magnitude of job ?ows with no ?uctuations . . . . . . . . 30
2.4.4 Cyclical job ?ows with industry ?uctuations . . . . . . . . . . 33
2.5 Quantitative Implications . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.1 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5.2 Simulations of aggregate employment dynamics . . . . . . . . 36
2.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 The Scarring E?ect of Recessions 46
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.2 A Renovating Industry with Learning . . . . . . . . . . . . . . . . . . 50
3.2.1 Firms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2.2 “All-Or-Nothing” Learning . . . . . . . . . . . . . . . . . . . . 52
3.2.3 The Recursive Competitive Equilibrium . . . . . . . . . . . . 56
3.3 Cleansing and Scarring . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.1 The Steady State . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.3.2 Comparative Statics: Cleansing and Scarring . . . . . . . . . . 63
3.3.3 Sensitivity Analysis . . . . . . . . . . . . . . . . . . . . . . . . 69
3.4 Quantitative Implications with Stochastic Demand Fluctuations . . . 73
3.4.1 Computational Strategy . . . . . . . . . . . . . . . . . . . . . 74
3.4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.3 Response to a Negative Demand Shock and Simulations of U.S.
Manufacturing Job Flows . . . . . . . . . . . . . . . . . . . . 84
3.4.4 Simulation of U.S. Manufacturing Job Flows . . . . . . . . . . 89
3.5 Sensitivity Analysis of the Dominance of Scarring over Cleansing . . . 90
3.6 Cost of Business Cycles with Heterogeneous Firms . . . . . . . . . . . 93
3.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4 Appendix 99
1 Resource Reallocation and Business Cycles
Ever since the foundation of real business cycle theory in Kydland and Prescott
(1982), the empirical regularities seen in productivity dynamics over business cycles
have attracted a great amount of research attention. In recent years with longitudinal
micro business data bases becoming more available, our understanding of aggregate
productivity as well as its measurements have much improved.
1
We now know that
the representative ?rm paradigm does not hold in the real world. As a matter of
fact, economies across time and regions are characterized by a large and pervasive
restructuring process due to entry, exit, expansion and contraction of businesses.
2
This gives the economy great ?exibility and potentially allows economic resources to
be used where they will be most productive. Businesses that use outdated technolo-
gies, or produce products ?agging in popularity, experience employment decreases.
And the displaced workers can then be re-employed by entrants or businesses that
are expanding. Davis and Haltiwanger (1999) document that, in the U.S., roughly
thirty percent of productivity growth over a ten-year horizon is accounted for by more
productive entering businesses displacing less productive exiting ones.
A body of literature has arisen attempting to empirically synthesize the micro-
economic and macroeconomic patterns of reallocation.
3
Much of them have centered
on the creation and destruction of jobs, de?ned by Davis, Haltiwanger and Schuh
(1996) (hereafter DHS) as gross job ?ows. A key stylized fact in this literature is
that job reallocation exceeds that necessary to implement observed net job growth.
1
The most heavily examined one is the Longitudinal Research Data (LRD) provided by U.S.
Census of Bureau.
2
Davis and Haltiwanger (1999) report that in most western economies roughly 1 in 10 jobs is
created and 1 in 10 jobs is destroyed every year.
3
Due to data limitations, most of the evidence comes from the manufacturing sectors.
1
This implies that jobs are continually being reallocated across businesses within the
same industry. DHS document that this is true even when looking at very narrowly
de?ned industries (four-digit) within speci?c geographic regions.
4
Hence, the large
and pervasive job ?ows seem to re?ect businesses’ idiosyncratic characteristics and
the resulting heterogeneity in their individual labor demand.
This dissertation is an attempt at providing a theoretical framework with hetero-
geneous businesses that relates resource reallocation to productivity dynamics over
business cycles. I combine two driving forces for job ?ows — learning and creative de-
struction. There has been a long tradition in the profession of examining each force
separately. The idea of creative destruction traces back to Schumpeter (1942), and
has been formalized into a class of vintage models by Caballero and Hammour (1994
and 1996) and Campbell (1997).
5
Firm learning, originated by Jovanovic (1982), can
be seen in Ericson and Pakes (1995) and more recently in Moscarini (2003) and Pries
(2004).
Both theories on their own can match some of empirical evidence, but not all.
The vintage models of creative destruction assume that new technology can only be
adopted by constructing new businesses, so that technologically sophisticated busi-
nesses enter to displace older, out-moded ones. This is supported by the fact, as
documented by DHS, that entry and exit of businesses account for a large fraction of
job reallocation. However, while holding some appeal, this prediction runs counter to
4
Davis and Haltiwanger (1999) document that, employment shifts among the approximately
450 four-digit industries in the U.S. manufacturing sector account for a mere 13% of excess job
reallocation. Simultaneously cutting the U.S. manufacturing data by state and two-digit industry,
region, size class, age class and ownership type, between-sector shifts account for only 39 percent of
excess job reallocation. The same ?nding holds up in studies for other countries(e.g. Nocke 1994).
5
Another important paper that formalizes Schumpeterian idea is Aghion and Hawitt (1992).
They develop a theoretical model in which endogneous innovations drive creative destruction and
growth.
2
the prevalent ?ndings that failure rates decrease sharply with business age (Dunne,
Roberts, and Samuelson 1989), and that productivity rises with business age (Aw,
Chen and Roberts 1997, Jensen, McGuckin and Stiroh 2000). The learning models
formalize the idea that businesses learn over time about initial conditions relevant
to success and business survival. As learning diminishes with age, its contribution
to job ?ows among businesses in the same birth cohort decreases. While providing
an appealing interpretation of the strong and pervasive negative relationship between
employer age and the magnitude of gross job ?ows, the learning models fail to ex-
plain the large gross job ?ows among mature businesses. Moreover, neither learning
nor creative destruction alone can link business age with relative volatility of job de-
struction to creation, while these two have displayed a positive relationship in U.S.
Manufacturing.
6
In Chapter Two, I show that the empirical ?ndings above can be potentially
reconciled by a model that combines learning with creative destruction. I focus on
two salient facts of gross job ?ows: the ?rst is that young plants display greater
turnover rates than old plants; the second is that, although job destruction is more
volatile than job creation in general, this asymmetry is weaker for younger plants.
I then present a framework where two forces interact together to drive micro-level
job ?ows: creative destruction reallocates labor into technologically more advanced
production units; while learning leads labor to production units with good businesses.
With demand ?uctuations, learning generates relative symmetric responses of creation
and destruction, while the creative destruction force makes job destruction more
responsive. Since old businesses are surer about their true idiosyncratic productivity,
6
DHS document that in U.S. Manufacturing, creation and destruction are almost equally volatile
for young businesses while old businesses features more volatile destruction.
3
the learning force weakens with age. Hence, Chapter Two interprets the observed
cyclical pattern of job ?ows as the dominance of learning for young businesses and
the dominance of creative destruction for old ones. I use the model to assess job-
?ow magnitude over a business’s life cycle analytically and calibrate the model to
match the data quantitatively. Calibration results show that my model does well
in matching young businesses’ higher job-?ow magnitude as well as their relative
symmetric volatility of job creation and destruction. However, it cannot fully account
for the magnitude of job ?ows among mature businesses because of my assumption
of a simpli?ed all-or-nothing form of learning.
Chapter Three takes a further step to relate cyclical resource reallocation to cycli-
cal productivity dynamics. It explores how recessions a?ect the allocative e?ciency of
resources and hence the average productivity. The conventional cleansing view argues
that recessions promote more e?cient resource allocation by driving out less produc-
tive units and freeing up resources for better uses. Using Chapter Two’s framework of
learning and creative destruction, I posit that recessions create an additional “scar-
ring” e?ect by reducing the learning opportunities of “potentially good ?rms.” I show
that as a recession arrives and persists, the reduced pro?tability truncates the process
of learning, limits the realization of truly good ?rms, and thus pulls down average
labor productivity. Calibrating my model using data on job ?ows from U.S. manu-
facturing sector, I ?nd that the scarring e?ect is likely to dominate the conventional
cleansing e?ect, and can account for the observed procyclical average productivity.
To be consistent with the evidence provided by DHS, in Chapter Two I call the
basic production unit underlying gross job ?ows “a plant”, which refers to a physical
location where production takes place. DHS argue that the plant represents the ?nest
4
level of disaggregation available in the Longitudinal Research Data for calculating job
creation and destruction statistics. To ?t into the literature of industry organization,
in Chapter Three a production unit is called “a ?rm”. Chapter Three also features a
simpler version of the model by assuming each ?rm employs only one worker, so that
a job is created when a ?rm enters and a job is destroyed when a ?rm exits. Under
this set-up, learning does not drive job creation but still does promote productivity
growth. Since Chapter Three focuses on productivity dynamics, this simpli?cation
does not a?ect my main results.
Di?erent theories of job ?ows have existed in the literature. One branch of theories
emphasize the matching of employees and employers (see Mortensen and Pissarides
1994). In their analysis, job destruction is more volatile than job creation because,
although job destruction takes place instantaneously, job creation can not due to the
time-consuming matching process. Another branch focuses on nonconvex adjustment
costs (see Caballero 1992, Campbell and Fisher 2000). In their environment, the
cross-sectional distribution of production units, in terms of where they stand relative
to their adjustment thresholds, may yield asymmetries in the cyclical dynamics of job
creation and destruction. Foote (1997) gives another interesting story regarding the
relative volatility of job creation and destruction. He connects (S, s) idiosyncratic
productivity adjustments with trend employment growth and predicts that a growing
industry features a more responsive creation margin while a declining industry a more
responsive destruction margin. Nevertheless, none of these theories have incorporated
the observed strong and persistent relationship between business age and job-?ow
patterns. My theory in Chapter Two builds on this relationship. Campbell and
Fisher (2004) also links business age with job-?ow volatility by modeling substitution
5
between structured and unstructured jobs over a plant life cycle. While it is di?cult
to de?ne structured and unstructured jobs empirically, their work does not feature the
observed pro-cyclical entry rate and counter-cyclical exit rate. Chapter Two shows
that these patterns are present in the cyclical response of my model.
Chapter Three posits that, with a scarring e?ect pulling down productivity by
limiting businesses’ learning opportunities, the observed intense reallocation during
recessions may contribute to the procyclical behavior of productivity. Barlevy (2002)
proposes a di?erent story. He argues that during recessions, workers are more likely
to stuck in mediocre matches with reduced worker ?ows, so that fewer high quality
matches are created. Besides resource reallocation, the literature have provided other
explanations for the cyclical behavior of productivity including cyclical technological
shocks, increasing returns to scale, and factor utilization. Basu (1996) empirically
investigate their merits using a panel on US manufacturing inputs and outputs from
1953 to 1984, and highlights the relative importance of factor utilization. But with
industry-level data, he cannot assess the contribution of cross-business resource re-
allocation. Baily, Bartelsman, and Haltiwanger (2001) provide a more disaggregated
empirical exploration. They ?nd productivity to be more procyclical at the plant
level than at the industry level, and posit that short-run reallocation yields a coun-
tercyclical contribution to productivity. However, as I elaborate in Chapter Three,
the cleansing e?ect takes place immediately while the scarring e?ect takes place grad-
ually. Can reallocation in the longer run yield a procyclical contribution to produc-
tivity with the stronger dominance of the scarring e?ect? This remains an interesting
empirical question.
This dissertation tries to theoretically improve our understanding of the link be-
6
tween resource reallocation and productivity dynamics over business cycles. I hope it
will be a base for future research that looks more intensively into this direction. There
are many potential connections that have yet to be fully explored. For instance, Kyd-
land and Prescott (1982) argue that a representative-agent real business cycle model
with technological shocks can account for most of the observed aggregate ?uctuations.
However, later empirical work by Basu (1997) suggests that the technological residual
interacts very little with output and input sequences once we control for increasing
returns, cyclical utilization and resource reallocation. Can a heterogeneous-agent
model with resource reallocation reconcile these papers by showing that the cyclical
resource reallocation is a natural response of the economy to technological shocks?
I believe there are many bene?ts to be gained from answering this question. The
resulting ?ndings will undoubtedly allow economists to learn more about the sources
and consequences of business cycles.
7
2 Plant Life Cycle and Aggregate Employment Dy-
namics
2.1 Introduction
Research on aggregate employment dynamics has focused on two separate compo-
nents: the number of jobs created at expanding and newly born plants (job creation)
and the number of jobs lost at declining and dying plants (job destruction). A key
stylized fact in this literature is that patterns of job creation and destruction di?er
signi?cantly by plant age. In magnitude, job ?ow rates are larger for younger plants;
In cyclical responses, job destruction varies more over time than job creation for old
plants; but for young plants, their variations are much more symmetric.
This chapter proposes an explanation. I highlight the following relative advan-
tages and disadvantages of young and old plants in market competition. Intuitively,
old plants tend to be more productive since they have survived long; but they may
be using out-dated technologies or producing products ?agging in popularity. On
the contrary, young plants, although lacking market experience, are more likely to be
technologically updated. If these are true over the plant life cycle and plants’ employ-
ment positively depends on their productivity, then there are multiple margins for a
plant to create or destroy jobs as it ages. Cyclical aggregate employment dynamics
involve the interactions of these di?erence driving forces.
I embody this intuition in an industry model whose employment dynamics are
driven by two forces — learning and creative destruction. In my model, technology
grows exogenously over time. Only entrants have access to the most advanced tech-
nology. Plants enter the market with the leading technology, but di?er in idiosyncratic
8
productivity. A plant’s idiosyncratic productivity is not directly observable, but can
be learned over time. A plant increases its employment (creates jobs) when it learns
its true idiosyncratic productivity as high (a good plant); it exits (destroys jobs) when
learning its true idiosyncratic productivity as low (a bad plant). Meanwhile, as newly
born plants continually enter with more advanced technology, incumbents becomes
more and more technologically outdated. They tend to destroy jobs and eventually
leave the market at a certain age. This gives rise to a creative destruction process
that allows technologically more advanced entering plants to replace outdated ones.
The resulting employment dynamics match the observed magnitude of job ?ows
over the plant life cycle. Because learning diminishes with age, job creation and de-
struction decline with plant age; while large job ?ows still exist among mature plants
with outdated plants being replaced due to creative destruction. The model also
matches the observed cyclical pattern of job ?ows with plant age. The learning force
generates relative symmetric responses to business cycles on the creation and destruc-
tion sides, while the creative destruction force makes job destruction more responsive.
Since learning diminishes with plant age, the symmetric response of learning domi-
nates for young plants and the asymmetric response of creative destruction dominates
for old plants. Therefore, my model suggests that the variance ratio of job destruction
over job creation increases with plant age, as shown in the data.
Other work has also studied the sources and macroeconomic implications of the
relative variance of job creation and job destruction. Foote (1997) connects (S,s)
idiosyncratic productivity adjustments with trend employment growth and predicts a
tight relationship between trend growth and volatility of creation relative to destruc-
tion. In his analysis, a growing industry features a more responsive creation margin
9
while a declining industry a more responsive destruction margin. Although his model
succeeds in explaining the di?erences in relative gross-?ow volatility across sectors, it
cannot account for the high volatility of destruction within manufacturing or the high
volatility of creation within service sector. This paper di?ers from Foote’s work by
emphasizing the within-sector di?erences in relative gross-?ow volatility arising from
plant-level heterogeneity. Campbell and Fisher (2004) link plant age with relative
volatility of creation and destruction by modeling substitution between structured
and unstructured jobs over a plant life cycle. While it is di?cult to de?ne struc-
tured and unstructured jobs empirically, their work does not feature the observed
pro-cyclical entry rate and counter-cyclical exit rate. These patterns are present in
the cyclical response of my model.
The remainder of this chapter is organized as follows. In the next section, I
describe the di?erences in young and old plants’ employment dynamics. In Section
3, I present my model, with which I analytically analyze the job ?ows over the plant
life-cycle in Section 4. A calibrated version of the model is studied numerically in
Section 5. I conclude in section 6.
2.2 Evidence: Gross Job ?ows and plant age
This section describes the observations of employment dynamics over the producer’s
life cycle that motivate my theory of learning and creative destruction. Two salient
facts emerge from the analysis carried out by Davis and Haltiwanger (1999). The
?rst is that young plants display greater turnover rates than old plants. The second
is that, although job destruction is more volatile than job creation in general, this
asymmetry is weaker for younger plants.
10
A. Means
Plant type E(Cb) E(Cc) E(C) E(Dd) E(Dc) E(D)
all 0.42 4.77 5.20 0.64 4.89 5.53
young 1.52 6.00 7.52 1.24 5.33 6.56
old 0.12 4.42 4.54 0.47 4.77 5.24
B. Standard deviations
Plant type ?(Cb) ?(Cc) ?(C) ?(Dd) ?(Dc) ?(D)
all 0.26 0.78 0.89 0.23 1.50 1.66
young 1.06 1.23 1.80 0.66 1.67 2.07
old 0.07 0.78 0.78 0.22 1.50 1.60
C. Variance ratio of job destruction to creation
plant type ?(D)
2
/?(C)
2
?(Dc)
2
/?(Cc)
2
all 3.49 3.64
young 1.32 2.80
old 4.18 3.69
Table 1: Quarterly gross job ?ows from plant birth, plant death, and continuing
operating plants in the US manufacturing sector: 1973 II to 1988 IV. Cb denotes
job creation from plant birth, Dd job destruction from plant death, Cc and Dc job
creation and destruction from continuing operating plants. C and D represent gross
job creation and destruction. C=Cc+Cb, D=Dd+Dc. All numbers are in percentage
points.
My data source is DHS’s observations of job creation and destruction rates for
the US manufacturing sector. For a given population of plants, the job creation rate
in a period is de?ned as the total number of jobs added since the previous period at
plants that increased employment, divided by the average of total employment in the
current and previous periods. The job destruction rate is similarly de?ned in terms
of employment losses at shrinking plants. The di?erence between job creation and
destruction is the rate of job growth. As proposed by DHS, the sum of job creation
and destruction rates is used as a measure of job reallocation across plants.
For my comparison of young and old plants’ employment dynamics, I use DHS’s
quarterly job creation and destruction series for plants in three di?erent age categories.
11
72q1 73q1 74q1 75q1 76q1 77q1 78q1 79q1 80q1 81q1 82q1 83q1 84q1 85q1 86q1 87q1 88q1
0
5%
10%
15%
72q1 73q1 74q1 75q1 76q1 77q1 78q1 79q1 80q1 81q1 82q1 83q1 84q1 85q1 86q1 87q1 88q1
0
5%
10%
15%
A: Young Plants
B: Old Plants
Figure 1: Job ?ows at young and old plants, 1972:2 — 1988:4. Dashed lines represent
the job creation series; solid lines represent job destruction.
As recommended by DHS,
7
I aggregate the two categories that include the youngest
plants and refer to this combination as “young”. These plants are usually less than
10 years old and account for 22.5% of total employment on average over the sample
period. The remaining are referred to as “old”.
8
Table 1A reports the sample means of the overall job creation (C), overall de-
struction (D), job creation from plant birth (Cb), job destruction from plant death
(Dd), and job creation and destruction from continuing operating plants (Cc and
7
See DHS, p.225.
8
Because of the sample design, the threshold between young and old plants changes slightly over
time. The minimum age of old plants is between 9 years and 13 years. See DHS, p. 225, for details.
12
Dc) for young and old plants separately, as well as for the US manufacturing sector
as a whole.
9
Table 1B reports sample standard deviations. The sample covers the
statistics from the second quarter of 1972 to the fourth quarter of 1988.
As shown in Table 1A and 1B, young plants’ average job creation and destruction
are both higher than those for old plants. So are the standard deviations. Table 1C
reports the relative variability of job creation and destruction. For the US manufac-
turing sector as a whole, the variance ratio of job destruction to job creation equals
3.49, so job destruction ?uctuates much more than job creation. The variance ratio
is also considerably higher than one for old plants, 4.18. However, a more interesting
?nding, as noted by DHS, is that young plants’ job creation and destruction rates
have approximately equal variances.
Figure 1 reinforces the above impression of greater variability of job ?ows at young
plants. It illustrates that job creation and destruction rates at young plants are visibly
more volatile. Moreover, the time series variability of creation and destruction seems
more symmetric for young plants than for old plants.
Because the observed frequency of plant exit declines with age and entering plants
are young by de?nition, it is important to consider the possibility that young and old
plants’ di?erent variances only re?ect the concentration of entry and exit among young
plants. If I exclude the contributions of plant birth and death to the job creation and
destruction, the negative relationship between magnitude of job ?ows and plant age
is still evident. As shown in Table 1A, the average job creation and destruction from
9
Notice that average job creation from plant birth is positive even for old plants. It seems strange
at the ?rst sight, since by de?nition, an old plant cannot be newly born. DHS de?ne plant birth and
plant age di?erently. Plant age is caculated based on the ?rst time a plant has positive employment,
while plant birth is recorded as plants starting up. Most of the starting up plants age zero, but some
old plants’ employment can drop to zero temporarily and start to increase again. Also notice that
the contribution of “old plant birth” to old plants’ job creation is very small.
13
continuing young plants are higher than those of continuing old plants. So are the
standard deviations shown in Table 1B. Table 1C suggests that, excluding plant birth
and death, job destruction still varies more than job creation: the variance ratio of
job destruction to creation is 2.80 for continuing young plants, and 3.69 for continuing
old plants. Moreover, this asymmetry is weaker for continuing young plants, as it is
for young plants’ overall job ?ows that include the contributions of plant birth and
death.
Table 1 and Figure 1 reveal a sharp relationship between plant age and job re-
allocation rates.
10
Davis and Haltiwanger (1999) report that this relationship exists
in very narrowly de?ned (four-digit) manufacturing industries, even with detailed
controls for size and other producer characteristics. This highlights the connection
between a producer’s life-cycle and its employment dynamics, which is modeled in
the next section.
2.3 A Model of Learning and Creative Destruction
Consider an industry of plants that produce a single good for sale in a competitive
product market. Plants use a single factor of production, labor, that they hire in
a competitive labor market. I refer to an employee working one period as a job.
Each plant can be thought of as an “institutional adoption of technology” with the
following three characteristics:
1. vintage;
2. idiosyncratic productivity;
10
Similar patterns have also been found in data on job ?ows in France, Canada, Norway, Nether-
lands, Germany and U.K. See Davis and Haltiwanger (1999).
14
3. a group of workers (jobs).
There is an exogenous technological progress {A
t
}
?
0
with a positive growth rate
? so that:
11
A
t
= A
0
· (1 +?)
t
,
where A
0
is a constant. With a as plant age, apparently the vintage of a plant of age
a in period t is,
A
t?a
= A
0
· (1 +?)
t?a
,
Assuming discrete time, each period a continuum of plants enter embodied with the
latest technology. Incumbents do not have access to the latest technology.
12
so that
young plants have technological advantages over old plants.
At the time of entry, a plant is endowed with idiosyncratic productivity ?, so
that plants of the same vintage(age) cohort di?er in idiosyncratic productivity. ? can
represent the talent of the manager as in Lucas (1978), or alternatively, the location of
the store, the organizational structure of the production process, or its ?tness to the
embodied technology.
13
The key assumption regarding ? is that its value, although
?xed at the time of entry, is not directly observable.
11
See Caballero and Hammour (1994).
12
This serves as a short-coming of my theory, and of the whole branch of vintage models. Not
allowing re-tooling, a business’s technology level is ?xed and even good plants exit eventually. But
in the real world, some plants may stay long by keep updating their technology. DHS report that, in
US manufacturing, a large fraction of labor is concentrated on a small number of old plants. Their
?nding is not present in my model, in which old plants tend to hire less labor due to out-moded
technology. For a model that allows re-tooling, see Cooper, Haltiwanger, and Power (1999).
13
Since a ?rm is identical to a job under this set-up, ? can also be interpreted as “match quality.”
See Pries (2004).
15
Production takes place through a group of workers. n
t
represents the employment
level of a plant in period t. The period-t output of this plant is given by
A
t?a
· X
t
· (n
t
)
?
,
where ? is between zero and one, and
X
t
= ? +?
t
.
The shock ?
t
is an i.i.d. random draw from a ?xed distribution that masks the
in?uence of ? on output. I set the wage rate to 1 as a normalization, and let P
t
denote
the equilibrium output price in period t. Then the pro?t generated by a plant of age
a and idiosyncratic productivity ? in period t equals P
t
· A
t?a
· X
t
· (n
t
)
?
?n
t
. Both
output and pro?t are directly observable. Since the plant knows its vintage, it can
infer the value of X
t
. The plant uses its observations of X
t
to learn about ?.
2.3.1 “All-Or-Nothing” Learning
Plants are price takers and pro?t maximizers. They attempt to resolve the uncer-
tainty about ? to decide on an employment level and whether to continue or terminate
production. The random component ?
t
represents transitory factors that are inde-
pendent of the idiosyncratic productivity ?. By assuming that ?
t
has mean zero, I
have E
t
(x
t
) = E
t
(?) +E
t
(?
t
) = E
t
(?).
Given knowledge of the distribution of ?
t
, a sequence of observations of x
t
allows
the plant to learn about its ?. Although a continuum of potential values for ? is
more realistic, for simplicity it is assumed here that there are only two values: ?
g
for
16
a good plant and ?
b
for a bad plant. Furthermore, ?
t
is assumed to be distributed
uniformly on [??, ?]. Therefore, a good plant will have x
t
each period as a random
draw from a uniform distribution over [?
g
??, ?
g
+?], while the x
t
of a bad plant is
drawn from an uniform distribution over [?
b
??, ?
b
+w]. Finally, ?
g
, ?
b
and ? satisfy
0 < ?
b
?? < ?
g
?? < ?
b
+? < ?
g
+?.
Pries (2004) shows that the above assumptions give rise to an “all-or-nothing”
learning process. With an observation of x
t
within (?
b
+?, ?
g
+?], the plant learns with
certainty that it is a good plant; conversely, an observation of x
t
within [?
b
??, ?
g
??)
indicates that it is a bad plant. However, an x
t
within [?
g
??, ?
b
+?] does not reveal
anything, since the probabilities of falling in this range as a good plant and as a bad
plant are the same (both equal to
2?+?
b
??g
2?
).
This all-or-nothing learning simpli?es my model considerably. Since it is ?
e
instead
of ? that a?ects plants’ decisions, there are three types of plants corresponding to the
three values of ?
e
: plants with ?
e
= ?
g
, plants with ?
e
= ?
b
, and plants with ?
e
=
?
u
, the prior mean of ?. I de?ne “unsure plants” as those with ?
e
= ?
u
. I further
assume that the unconditional probability of ? = ?
g
is ?, and let p ?
?g??
b
2?
denote the
probability of the true idiosyncratic productivity being revealed every period. Hence
a plant’s life-cycle is incorporated into the model as follows. A ?ow of new plants
enter the market as unsure; thereafter, every period they stay unsure with probability
1?p, learn they are good with probability p·? and learn they are bad with probability
p· (1??). The evolution of ?
e
from the time of entry is a Markov process with values
17
0
0.5
0
d
e
n
s
i
t
i
e
s

o
f

t
y
p
e
s
age
unsure
good
bad
Figure 2: Dynamics of a Birth Cohort with Learning: the distance between the
concave curve and the bottom axis measures the density of plants with ?
e
= ?
g
; the
distance between the convex curve and the top axis measures the density of plants
with ?
e
= ?
b
; and the distance between the two curves measures the density of unsure
plants (plants with ?
e
= ?
u
).
(?
g
, ?
u
, ?
b
), an initial probability distribution
µ
0, 1, 0
¶
,and a transition matrix
_
_
_
_
_
_
1 0 0
p · ? , 1 ?p , p · (1 ??)
0 0 1
_
_
_
_
_
_
.
If plants were to live forever, eventually all uncertainty would be resolved because
the market would provide enough information to reveal each plant’s idiosyncratic pro-
ductivity. The limiting probability distribution as a goes to ?is
µ
?, 0, (1 ??)
¶
.
Because there is a continuum of plants, it is assumed that the law of large numbers
18
applies, so that both ? and p are not only the probabilities but also the fractions of
unsure plants with ? = ?
g
, and of plants who learn ? each period, respectively. Hence,
ignoring plant exit for now, the densities of the three groups of plants in a cohort of
age a as
µ
? · [1 ?(1 ?p)
a
] , (1 ?p)
a
, (1 ??) · [1 ?(1 ?p)
a
]
¶
,
which implies an evolution of the idiosyncratic-productivity plant distribution within
a birth cohort as shown in Figure 2, with the horizontal axis depicting the age of a
cohort over time. The densities of plants that are certain about their idiosyncratic
productivity, whether good or bad, grow as a cohort ages. Moreover, the two “learn-
ing curves” (depicting the evolution of densities of good plants and bad plants) are
concave. This feature is de?ned as the decreasing property of marginal learning in
Jovanovic (1982): the marginal learning e?ect decreases with plant age, which, in
my model, is re?ected by the fact that the marginal number of learners decreases
with cohort age. The convenient feature of all-or-nothing learning is that, on the one
hand, it implies that any single plant learns “suddenly”, which allows us to easily
keep track of the cross-section distribution of beliefs while, on the other hand, it still
implies “gradual learning” at the cohort level.
2.3.2 Creative Destruction and Industry Equilibrium
I nowturn to the supply and demand conditions in this model, and to the economics of
creative destruction. I model a perfectly competitive industry in partial equilibrium.
Plants of di?erent vintages and beliefs may coexist. The mass of plants with age a
and belief ?
e
in period t is denoted f
t
(?
e
, a).
The following sequence of events is assumed to occur within a period. First, entry
19
and exit occur by observing the aggregate state and perfectly predicting the current-
period price. Second, each surviving plant adjusts its employment and produces.
Third, the aggregate price is realized. Fourth, plants observe revenue and update
beliefs. Then, another period begins.
I assume costless employment adjustment each period so that a plant adjusts
its employment to solve a static pro?t maximization problem. With ?
e
as a plant’s
current belief of its idiosyncratic productivity and P
t
as the equilibrium price, I denote
?rm’s employment as n
t
(?
e
, a). That is,
n
t
(?
e
, a) = arg max
nt?0
P
t
· A
t?a
· X
t
· (n
t
)
?
?n
t
(2.1)
= [? · P
t
· A
0
· (1 +?)
t?a
· ?
e
]
1
1??
.
The corresponding expected value of the single-period pro?t maximized with respect
to n
t
is,
?
t
(?
e
, a; P, A) ? (?
?
1??
??
1
1??
) ·
£
P
t
· A
0
· (1 +?)
t?a
· ?
e
¤
1
1??
. (2.2)
Let W > 0 be the expected present value of the plant’s ?xed factor (its ”man-
agerial ability” or ”advantageous location”) if employed in a di?erent activity. The
value of W is the same for all plants in the industry regardless of their vintages and
idiosyncratic productivity. If a plant believes that the expected present discounted
value of staying is less than W, it chooses to exit.
Thus, the exit decision of a plant is forward-looking: plants have to form expecta-
tions about both current and future pro?ts. It is a dynamic problem with ?ve state
variables: 1) ?
e
, the plant’s belief about ?; 2) P, the expected price sequences under
20
possible paths of demand realizations; 3) A ? {A
t
}
?
0
, the technology sequence; 4)
time t, which determines where one is along the price sequence; 5) age a, which, com-
bined with time t, gives the vintage A
t?a
.
14
Let V
t
(?
e
, a; P, A) be the value of staying
in the market for t’th period for a plant with age a, when the plant’s belief is ?
e
, price
sequence is P and technology sequence is A. Then a plant has V that satis?es:
V
t
(?
e
, a; P, A) = ?(?
e
, a; P, A) +? · E
t
{max[W, V
t+1
(?
e
0
, a + 1; P, A)]|?
e
}
I assume that parameters are such that W > V
t
(?
b
, a; P, A) for any a, t: the present
discounted value of life-time pro?t as a bad idiosyncratic productivity at any age is
always lower than the outside option value. Therefore, bad plants always exit.
Proposition 2.1: V
t
(?
e
, a; P, A) is strictly decreasing in a, holding
?
e
constant, and strictly increasing in ?
e
, holding a constant; therefore,
there is a cut-o? age a
t
(?
e
; P, A) for each idiosyncratic productivity, such
that ?rms of ?
e
and age a ? a
t
(?
e
; P, A) exit before production takes place
in period t; furthermore, a
t
(?
g
; P, A) ? a
t
(?
u
; P, A).
See the appendix for proof. This follows from the fact that plants with smaller a
and higher ?
e
have higher expected value of staying. As V is strictly decreasing in a,
plants with belief ?
e
that are older than a
t
(?
e
; P, A) exit at the beginning of period
t; as the expected value of staying is strictly increasing in ?
e
, the exit age of good
plants is older than that of unsure plants.
The industry also features continual entry. To ?x the size of entry, I furthermore
14
P is a?ected by demand parameter D, as well as the distribution of heterogeneous plants. See
sub-section 3.2.3 for a more strict de?nition of the recursive competitive equilibrium.
21
assume that each entrant has to pay an entry cost c to enter the market:
c
t
= c(f
t
(?
e
, 0)), c (·) > 0, c
0
(·) ? 0.
I let the entry cost depend positively on the amount of entry to capture the idea
that, for the industry as a whole, fast entry is costly and adjustment may not take
place instantaneously. This can arise from a limited amount of land available to build
production sites or an upward-sloping supply curve for the industry’s speci?c capital.
The free entry condition equates a plant’s entry cost to its value of entry, and can be
written as
V
t
(?
u
, 0; P, A) = c (f
t
(?
u
, 0; P, A)) .
As more new plants enter, the entry cost is driven up until it reaches the value of
entry. At this point, entry stops.
Let Q
t
represent the period-t aggregate output level. An equilibrium in this in-
dustry is a path
©
P
t
, Q
t
, {f
t
(?
e
, a)}
?
e
=?
u
or ?
g
, a?0
ª
, which satis?es the following con-
ditions: 1) plants’ entry, exit and employment decisions are optimal; 2) the evolution
of {f
t
(?
e
, a)} is generated by the appropriate summing-up of plants’ entry, exit and
learning: 3) P
t
is such that
D
t
= Q
t
· P
t
, ?t (2.3)
, where D
t
is an fully observable exogenous demand parameter that captures aggregate
conditions. Industry cycles are driven by the ?uctuations of D
t
.
Proposition 2.2: With time-invariant demand level D
t
= D, the value
of P
t
· A
t
is also time-invariant; when demand ?uctuates, P
t
· A
t
?uctuates
with D
t
.
22
Proposition 2.2 suggests that P
t
· A
t
moves with the value of D
t
. Aggregate
?uctuations a?ect individual plant decisions through the ?uctuations of P
t
· A
t
.
2.4 Aggregate Employment Dynamics
This section uses the model to assess the impact on industry-level employment dynam-
ics of learning and creative destruction over the business cycle. Since industry-level
employment dynamics are computed by aggregating the individual decisions of plants,
I begin with the plant-level employment policy:
n
t
(?
e
, a) = [? · P
t
· A
0
· (1 +?)
t?a
· ?
e
]
1
1??
= [? · P
t
·
P
t
· A
t
(1 +?)
a
· ?
e
]
1
1??
Because n
t
(?
e
, a) depends positively on belief ?
e
, a plant increases its employment
(creates jobs) when it learns it is good, and exits (destroys jobs) once it learns it is
bad. Hence, the evolution of ?
e
captures the learning e?ect. With age a a?ecting
n
t
negatively, a plant tends to decrease its employment (destroys jobs) as it grows
older (a increases), and eventually exits (destroys jobs). Therefore, (1 +?)
a
captures
the creative destruction e?ect. Whether a plant creates or destroys jobs also depends
further on P
t
· A
t
, the product of equilibrium price and the industry-wide leading
technology. I call the impact of P
t
· A
t
the industry shock e?ect. These three e?ects
interact together to drive plant-level and thus aggregate employment dynamics.
2.4.1 Job Flows over the Plant Life Cycle
Aggregate dynamics in job creation and destruction re?ect the number of plants
choosing to adjust employment and the magnitude of their adjustment. In my model,
23

age

0
Exit Margin
of Unsure Plants
Exit Margin
of Good Plants
Learning Margin
?? Exit of Bad Plantss
Entry Margin

unsure plants
good plants
maximum age of
unsure plants
maximum age of
good plants
Figure 3: Dynamics of a Birth Cohort with both Learning and Creative Destruction.
the distance between the lower curve (extended as the horizontal line) and the bottom
axis measures the density of good ?rms; the distance between the two curves measures
the density of unsure ?rms.
the response of plants varies systematically on both of these dimensions over the life-
cycle. Proposition 2.1 suggests that, because of creative destruction, the evolution of
the idiosyncratic-productivity distribution within a birth cohort shown in Figure 1
will be truncated by exit ages of unsure and good plants, as shown in Figure 3:
Figure 3 displays the life-cycle dynamics of a representative cohort with the hori-
zontal axis depicting the cohort age across time. All plants enter as unsure. As the
cohort ages and learns, bad plants are thrown out and good plants are realized. After
a certain age, all unsure plants exit because their vintage is too old to survive with
?
e
= ?
u
. However, plants with ?
e
= ?
g
stay. Subsequently, the cohort contains only
good plants and its size remains constant because learning has stopped. Eventually,
24
the vintage of the cohort will be too old even for good plants to survive.
Figure 3 also implies a job creation and destruction schedule over the plant life-
cycle. First, because all newly born plants begin with zero employment, they begin
their lives by job creation. As they age, the learning e?ect drives job creation among
plants that discover they are good, and drives job destruction among plants that
discover they are bad. Meanwhile, the creative destruction e?ect drives aging plants
that do not learn to destroy jobs. Upon a certain age, all plants end their lives by
job destruction.
As I have elaborated in Section 2.3, the concave learning curves suggest that the
marginal number of learners decreases as a cohort ages. Hence, as plants grow older,
the learning e?ect weakens. Fewer and fewer plants create or destroy jobs because of
learning. Once all unsure plants have left, learning stops completely. On the contrary,
the creative destruction e?ect strengthens with plant age. According to Proposition
2.1, older plants are more likely to exit (destroy jobs). At a certain age, all unsure
plants destroy jobs by exit; as the remaining good plants grow older, eventually they
destroy jobs too.
2.4.2 Decomposition of Job ?ows
To show the above intuition mathematically, I assume a variable x
t
such that
x
t
=
P
t
P
t?1
.
Let C(a, x
t
) denote the job creation rate of a cohort aged a in period t. I decompose
C(a, x
t
) into the sum of job creation from entry, denoted C
entry
(a, x
t
), job creation
from learning, denoted C
learn
(a, x
t
), and job creation from price increases, denoted
25
C
price
t
(a, x
t
):
C(a, x
t
) = C
entry
(a, x
t
) +C
learn
(a, x
t
) +C
price
(a, x
t
).
Apparently, both C
learn
(a, x
t
) and C
price
(a , x
t
) are zero for an entering cohort (a = 0),
so that
C(0, x
t
) = C
entry
(0, x
t
) =
(?P
t
A
t
?
u
)
1
1??
1
2
h
0 + (?P
t
A
t
?
u
)
1
1??
i = 2, ?x
t
.
For an incumbent cohort (a > 0), C
entry
(a, x
t
) = 0; its job creation from other
components are
C
learn
(a, x
t
) =
½
[x
t
?
g
]
1
1??
??
1
1??
u
¾
p?f
t?1
(?
u
, a ?1)
1
2
x
1
1??
t
·
f
t
(?
u
, a)?
1
1??
u
+f
t
(?
g
, a)?
1
1??
g
¸
+
1
2
·
f
t?1
(?
u
, a ?1)?
1
1??
u
+f
t?1
(?
g
, a ?1)?
1
1??
g
¸
, (2.4)
and
C
price
(a, x
t
) =
µ
max
½
0, x
1
1??
t
?1
¾¶
_
¸
_
(1 ?p) f
t?1
(?
u
, a ?1)?
1
1??
u
+
f
t?1
(?
g
, a ?1)?
1
1??
g
_
¸
_
1
2
x
1
1??
t
·
f
t
(?
u
, a)?
1
1??
u
+f
t
(?
g
, a)?
1
1??
g
¸
+
1
2
·
f
t?1
(?
u
, a ?1)?
1
1??
u
+f
t?1
(?
g
, a ?1)?
1
1??
g
¸
. (2.5)
Here I have assumed that x
t
?
g
> ?
u
.
15
C
learn
(a, x
t
) depends on the number
15
This assumption, which remains valid in my calibration exercise in the next sub-section, has
its support from U.S. manufacturing job-?ow facts. As documented in DHS, quarterly job creation
among continuing operating plants, modeled here as the sum of POS
learn
t
(a) and POS
price
t
(a),
stayed strictly positive from 1972 to 1988. Notice that, if x
t
?
g
? ?
u
instead so that POS
learn
t
(a)
26
of plants who learn in period t they are good, represented by p?f
t?1
(?
u
, a ? 1),
and how many jobs each of them creates, shown as (x
t
?
g
)
1
1??
? ?
1
1??
u
. C
price
t
(a, x
t
)
captures possible job creation by plants not on the learning margin, including old
good plants and unsure plants that have not learned, the number of which are denoted
f
t?1
(?
g
, a?1) and (1 ?p) f
t?1
(?
u
, a?1). The term C
price
(a, x
t
) is driven by industry
shocks; if x
t
> 1, the term will be zero.
Similarly, the job destruction rate for an incumbent cohort aged a (a > 0) in
period t , denoted D(a, x
t
), can be decomposed as
D(a, x
t
) = D
learn
(a, x
t
) +D
exit
(a, x
t
) +D
price
(a, x
t
),
where D
learn
(a, x
t
) denotes job destruction from learning, D
exit
(a, x
t
) denotes that
from the exit of unsure and good plants, and D
price
(a, x
t
) denotes that from decreases
in price.
D
learn
(a, x
t
) =
?
1
1??
u
p(1 ??)f
t?1
(?
u
, a ?1)
1
2
x
1
1??
t
·
f
t
(?
u
, a)?
1
1??
u
+f
t
(?
g
, a)?
1
1??
g
¸
+
1
2
·
f
t?1
(?
u
, a ?1)?
1
1??
u
+f
t?1
(?
g
, a ?1)?
1
1??
g
¸
(2.6)
drops to zero, POS
price
t
(a) would also be zero since ?
g
> ?
u
, together with x
t
?
g
? ?
u
, suggests
x
t
? 1. In that case, there would be no job creation from continuing operating plants and my model
would not be able to match DHS’s documented facts of gross job ?ows.
27
D
price
(a, x
t
) =
µ
max
½
0, 1 ?x
1
1??
t
¾¶
_
¸
_
(1 ?p) f
t?1
(?
u
, a ?1)?
1
1??
u
+
f
t?1
(?
g
, a ?1)?
1
1??
g
_
¸
_
1
2
x
1
1??
t
·
f
t
(?
u
, a)?
1
1??
u
+f
t
(?
g
, a)?
1
1??
g
¸
+
1
2
·
f
t?1
(?
u
, a ?1)?
1
1??
u
+f
t?1
(?
g
, a ?1)?
1
1??
g
¸
, (2.7)
D
learn
(a, x
t
) captures job destruction on the learning margin, with p(1??)f
t?1
(?
u
, a?
1) representing the number of plants who learn they are bad. It equals zero for cohorts
without unsure plants in period t ?1 (f
t?1
(?
u
, a?1) = 0) since learning among these
plants has stopped. D
price
(a, x
t
) represents possible job destruction by plants not on
the learning margin. This job destruction is driven by industry shocks and occurs as
long as x
t
< 1. The plants a?ected include old good plants and unsure plants that
have not learned.
The magnitude of D
exit
(a, x
t
), job destruction from the exit of unsure or good
plants, is more complicated due to shifts of the exit margins. Let a
t
(?
u
) represent
the period-t exit age of unsure plants and a
t
(?
g
) that of good plants. When a
t
(?
u
) >
a
t?1
(?
u
), unsure exit margin extends to an older age and no unsure plants are exiting.
Similarly, no good plants are exiting when a
t
(?
g
) > a
t?1
(?
g
). If both margins extend
to older ages, then no plants are exiting and D
exit
(a, x
t
) must be zero for any cohorts.
On the contrary, with a
t
(?
u
) ? a
t?1
(?
u
), the unsure exit margin stays at the
same age or shifts to a younger age, so that one or more cohorts of unsure plants are
exiting. It can be shown that
D
exit
(a, x
t
) =
?
1
1??
u
f
t?1
(?
u
, a ?1)
1
2
x
1
1??
t
f
t
(?
g
, a)?
1
1??
g
+
1
2
·
f
t?1
(?
u
, a ?1)?
1
1??
u
+f
t?1
(?
g
, a ?1)?
1
1??
g
¸
if a < a
t?1
(?
u
) + 1 and a ? a
t
(?
u
) .
28
Similarly, with a
t
(?
g
) ? a
t?1
(?
g
), the good exit margin stays at the same age or shifts
to a younger age, so that one or more cohorts of good plants are exiting. For these
cohorts, job destruction reaches its maximum value two, shown as follows:
16
D
exit
(a, x
t
) =
?
1
1??
g
f
t?1
(?
g
, a ?1)
1
2
f
t?1
(?
g
, a ?1)?
1
1??
g
= 2
if a < a
t?1
(?
g
) + 1 and a ? a
t
(?
g
) .
I conclude this sub-section by relating above decomposition of job ?ows to the
underlying driving forces. I argued earlier that plant-level employment is a?ected
by the learning e?ect, the creative destruction e?ect, and the industry shock e?ect.
Apparently, C
learn
and D
learn
come from the learning e?ect. The creative destruc-
tion e?ect drives C
entry
and D
exit
: youngest plants enter, and the oldest plants exit.
However, the industry shock e?ect and the creative destruction e?ect together drive
C
price
and D
price
.
To see why, recall that the industry shock e?ect is de?ned as the impact of P
t
A
t
.
With constant P
t
A
t
, there is no industry shock e?ect, but D
price
would still be positive
because
x
t
=
P
t
P
t?1
=
P
t
A
t
P
t?1
A
t?1
(1 +?)
=
1
(1 +?)
< 1, when P
t
A
t
= P
t?1
A
t?1
.
This is due to the creative destruction e?ect: a plant becomes more technologically
outdated by
1
(1+?)
as it ages for another period. To see how creative destruction a?ects
C
price
, let me assume that P
t
A
t
increases so that the industry shock e?ect is present.
16
Here I implicitly assume a
t?1
(?
u
) < a
t
(?
g
) ? 1, so that any cohorts with exiting good plants
included no unsure plants in period t ?1. This will the case in my calibration exercises.
29
In that case, C
price
may not be positive because x
t
may not be greater than one: the
impact of P
t
A
t
’ s increase on x
t
needs to overcome the impact of
1
(1+?)
. Hence, the
creative destruction e?ect a?ects C
price
by dampening the industry shock e?ect.
2.4.3 The magnitude of job ?ows with no ?uctuations
This sub-section establishes analytically a negative relationship between the average
magnitude of job ?ows and plant age in a version of my model with no ?uctuations.
Variations in D
t
serve as the source of economic ?uctuations in my model. Proposition
2.2 establishes that time-invariant D
t
implies a time-invariant P
t
A
t
. With time-
invariant P
t
A
t
, the expected value of staying across a and ?
e
would also be time-
invariant, which implies time-invariant size of entry and time-invariant exit ages.
Moreover, with P
t
A
t
= P
t?1
A
t?1
, x
t
equals a less-than-one value
1
(1+?)
. Hence,
C
price
(a,
1
(1+?)
) is zero so that job creation includes only C
entry
and C
learn
, while
D
price
(a,
1
(1+?)
) stays positive. Let C
?
(a) denote the job creation rate of a cohort aged
a with no ?uctuations, D
?
(a) the job destruction rate, and a
u
?
and a
g
?
the exit ages
of unsure and good plants. I have:
C
?
(a) = C
entry
(a,
1
(1 +?)
) = 2, if a = 0
C
?
(a) = C
learn
(a,
1
(1 +?)
)
=
½
h
?
g
(1+?)
i 1
1??
??
1
1??
u
¾
p?
1
2
h
1
(1+?)
i 1
1??
·
(1 ?p)?
1
1??
u
+
?(1?(1?p)
a
)
(1?p)
a?1
?
1
1??
g
¸
+
1
2
·
?
1
1??
u
+
?(1?(1?p)
a?1
)
(1?p)
a?1
?
1
1??
g
¸
, if 0 < a < a
u
?
C
?
(a) = 0, otherwise
30
Proposition 2.3: without demand ?uctuations, the job creation rate
weakly decreases in cohort age.
Job creation strictly decreases in age for cohorts younger than a
u
?
, because
?(1?(1?p)
a
)
(1?p)
a?1
increases in a. According to the all-or-nothing learning described in Sub-section 2.2,
?(1 ?(1 ?p)
a
) is the fraction of good plants in a cohort aged a, and (1 ?p)
a?1
the
fraction of unsure plants. The ratio of good plants to unsure plants increases in a
because of learning: for older cohorts, more plants have learned. Job creation drops
to and stays at zero for the group of plants older than a
u
?
, since these plants have
already learned that they are good.
I also have:
D
?
(a) = D
learn
(a,
1
(1 +?)
) +D
price
(a,
1
(1 +?)
)
=
?
1
1??
u
p(1 ??)+
µ
1 ?
h
1
(1+?)
i 1
1??
¶
_
¸
_
(1 ?p) ?
1
1??
u
+
?(1?(1?p)
a?1
)
(1?p)
a?1
?
1
1??
g
_
¸
_
1
2
h
1
(1+?)
i 1
1??
·
(1 ?p)?
1
1??
u
+
?(1?(1?p)
a
)
(1?p)
a?1
?
1
1??
g
¸
+
1
2
·
?
1
1??
u
+
?(1?(1?p)
a?1
)
(1?p)
a?1
?
1
1??
g
¸
, if 0 < a < a
u
?
31
D
?
(a) = D
price
(a,
1
(1 +?)
) +D
exit
(a,
1
(1 +?)
)
=
µ
1 ?
h
1
(1+?)
i 1
1??
¶·
?(1?(1?p)
a?1
)
(1?p)
a?1
?
1
1??
g
¸
+?
1
1??
u
1
2
x
1
1??
t
·
?(1?(1?p)
a
)
(1?p)
a?1
?
1
1??
g
¸
+
1
2
·
?
1
1??
u
+
?(1?(1?p)
a?1
)
(1?p)
a?1
?
1
1??
g
¸
, if a = a
u
?
D
?
(a) = D
price
(a,
1
(1 +?)
) =
1 ?
h
1
(1+?)
i 1
1??
2
µ
1 +
h
1
(1+?)
i 1
1??
¶, if a
u
?
< a < a
g
?
D
?
(a) = D
exit
(a,
1
(1 +?)
) = 2, if a = a
g
?
Proposition 2.4: For small enough ?, job destruction weakly decreases
in cohort age for a 6= a
u
?
and a 6= a
g
?
.
For cohorts younger than a ? a
u
?
, D
?
(a) decreases in a because learning implies
that the ratio of good to unsure plants (
?(1?(1?p)
a
)
(1?p)
a?1
) increases in a. For cohorts with
a
u
?
< a < a
g
?
, although learning has stopped, plants gradually decrease employment
(destroy jobs) due to technological obsolescence or creative destruction. Their job
destruction rate is as shown in 2.7. Notice that for small enough ?, the value of
1?
[
1
(1+?)
]
1
1??
2

1+
[
1
(1+?)
]
1
1??

is close to zero, so that D
?
(a)|
a?a
u
? > D
?
(a)|
a
u
?
0 so that
A
t
= A
0
· (1 +?)
t
,
50
where A
0
is a constant. When a ?rm that enters the industry, it embodies the leading
technology, which becomes its vintage and will a?ect its production afterward. I
assume that, only entrants have access to the updated technology, incumbents cannot
retool. Since technology grows exogenously, young ?rms are always technologically
more advanced than old ?rms. With a as the ?rm age, the vintage of a ?rm of age a
in period t is A
t?a
. Apparently:
A
t?a
= A
0
· (1 +?)
t?a
.
At the time of entry, a ?rm is endowed with idiosyncratic productivity ?. Hence,
?rms of the same vintage di?er in idiosyncratic productivity. ? can represent the
talent of the manager as in Lucas (1978), or alternatively, the location of the store,
the organizational structure of the production process, or its ?tness to the embodied
technology.
28
The key assumption regarding ? is that its value, although ?xed at
the time of entry, is not directly observable. We can think of some real-world cases
that re?ect this assumption. For example, when a ?rm adopts new technology or
introduces a new product, it needs to make many decisions, such as picking a manager
to take charge of the production or choosing a location to sell the product. Although
all ?rms try to make the best decisions possible, the outcome of their choices is
uncertain and will be tested via market performance. Furthermore, their investments
are irreversible; once a manager has signed the contract and a store is built, it becomes
costly to make a new choice. Hence, the value of ?, as the consequence of a ?rm’s
random decisions, is unobservable and remains constant afterward.
28
Since a ?rm is identical to a job under this set-up, ? can also be interpreted as “match quality.”
See Pries (2004).
51
A ?rm of age a and idiosyncratic productivity ? produces output in period t,
according to
q
t
(a, ?) = A
t?a
· x
t
= A
0
· (1 +?)
t?a
· x
t
, (3.1)
where
x
t
= ? +?
t
.
The shock ?
t
is an i.i.d. random draw from a ?xed distribution that masks the
in?uence of ? on output. I set the operating cost of a ?rm (including wages) to 1
by normalization, and let P
t
denote the output price in period t. Then the pro?t
generated by a ?rm of age a and idiosyncratic productivity ? in period t is
?
t
(a, ?) = P
t
· A
0
· (1 +?)
t?a
· (? +?
t
) ?1. (3.2)
Both q
t
(a, ?) and ?
t
(a, ?) are directly observable. Since the ?rm knows its vintage,
it can infer the value of x
t
. The ?rm uses its observations of x
t
to learn about ?.
3.2.2 “All-Or-Nothing” Learning
Firms are price takers and pro?t maximizers. They attempt to resolve the uncertainty
about ? to decide whether to continue or terminate the production. The random
component ?
t
represents transitory factors that are independent of the idiosyncratic
productivity ?. Assuming that ?
t
has mean zero, we have
E
t
(x
t
) = E
t
(?) +E
t
(?
t
) = E
t
(?).
Given knowledge of the distribution of ?
t
, a sequence of observations of x
t
allows
the ?rm to learn about its ?. Although a continuum of potential values for ? is
52
more realistic, for simplicity it is assumed here that there are only two values: ?
g
for a good ?rm and ?
b
for a bad ?rm. Furthermore, ?
t
is assumed to be distributed
uniformly on [??, ?]. Therefore, a good ?rm will have x
t
each period as a random
draw from a uniform distribution over [?
g
??, ?
g
+?], while the x
t
of a bad ?rm is
drawn from an uniform distribution over [?
b
??, ?
b
+w]. Finally, ?
g
, ?
b
and ? satisfy
0 < ?
b
?? < ?
g
?? < ?
b
+? < ?
g
+?.
Pries (2004) shows that the above assumptions give rise to an “all-or-nothing”
learning process. With an observation of x
t
within (?
b
+?, ?
g
+?], the ?rm learns with
certainty that it is a good idiosyncratic productivity; conversely, an observation of x
t
within [?
b
??, ?
g
??) indicates that it is a bad idiosyncratic productivity. However,
an x
t
within [?
g
??, ?
b
+?] does not reveal anything, since the probabilities of falling
in this range as a good ?rm and as a bad ?rm are the same (both equal to
2?+?
b
??
g
2?
).
This all-or-nothing learning simpli?es my model considerably. I let ?
e
represent
the expected ?. Since it is ?
e
instead of ? that a?ects ?rms’ decisions, there are three
idiosyncratic productivity of ?rms corresponding to the three values of ?
e
: ?rms with
?
e
= ?
g
, ?rms with ?
e
= ?
b
, and ?rms with ?
e
= ?
u
, the prior mean of ?. I de-
?ne “unsure ?rms” as those with ?
e
= ?
u
. I further assume that the unconditional
probability of ? = ?
g
is ?, and let p ?
?g??
b
2?
denote the probability of true idiosyn-
cratic productivity being revealed every period. Firms enter the market as unsure;
thereafter, every period they stay unsure with probability 1 ?p, learn they are good
with probability p · ? and learn they are bad with probability p · (1 ??). Thus, the
evolution of ?
e
from the time of entry is a Markov process with values (?
g
, ?
u
, ?
b
), an
53
0
0.5
0
d
e
n
s
i
t
i
e
s

o
f

t
y
p
e
s
age
unsure
good
bad
Figure 8: Dynamics of a Birth Cohort: the distance between the concave curve and
the bottom axis measures the density of ?rms with ?
e
= ?
g
; the distance between the
convex curve and the top axis measures the ?rms with ?
e
= ?
b
; the distance between
the two curves measures the density of unsure ?rms (?rms with ?
e
= ?
u
).
initial probability distribution:
µ
0, 1, 0
¶
,
and a transition matrix
_
_
_
_
_
_
1 0 0
p · ? , 1 ?p , p · (1 ??)
0 0 1
_
_
_
_
_
_
.
54
If ?rms were to live forever, eventually all uncertainty would be resolved because
the market would provide enough information to reveal each ?rm’s idiosyncratic pro-
ductivity. The limiting probability distribution as a goes to ? is
µ
?, 0, (1 ??)
¶
.
Because there is a continuum of ?rms, it is assumed that the law of large numbers
applies, so that both ? and p are not only the probabilities but also the fractions of
unsure ?rms with ? = ?
g
, and of ?rms who learn ? each period, respectively. Hence,
ignoring ?rm exit for now, I have the densities of three groups of ?rms in a cohort of
age a as
µ
? · [1 ?(1 ?p)
a
] , (1 ?p)
a
, (1 ??) · [1 ?(1 ?p)
a
]
¶
,
which implies an evolution of the idiosyncratic-productivity ?rm distribution within
a birth cohort as shown in Figure 8, with the horizontal axis depicting the age of a
cohort across time. The densities of ?rms that are certain about their idiosyncratic
productivity, whether good or bad, growas a cohort ages. Moreover, the two “learning
curves” (depicting the evolution of densities of good ?rms and bad ?rms) are concave.
This feature is de?ned as the decreasing property of marginal learning in Jovanovic
(1982): the marginal learning e?ect decreases with ?rm age, which in my model is
re?ected by the fact that the marginal number of learners decreases with cohort age.
The convenient feature of all-or-nothing learning is that, on the one hand, it implies
that any single ?rm learns “suddenly”, which allows us to easily keep track of the
cross-section distribution of beliefs, while on the other hand, it still implies “gradual
learning” at the cohort level.
55
However, there is more that Figure 8 can tell. If we let the horizontal axis de-
pict the cross-sectional distribution of ?rm ages at any instant, then Figure 4 can be
interpreted as the ?rm distribution across ages and idiosyncratic productivity of an
industry that features constant entry but no exit. In this industry, cohorts contin-
uously enter in the same size and experience the same dynamics afterward, so that
at any one time, di?erent life-stages of di?erent birth cohorts overlap, giving rise to
the distribution in Figure 4. Under this interpretation, Figure 4 indicates that at any
instant older cohorts contain fewer unsure ?rms, because they have lived longer and
learned more.
3.2.3 The Recursive Competitive Equilibrium
The following sequence of events is assumed to occur within a period. First, entry
and exit occur after ?rms observe the aggregate state. Second, each surviving ?rm
pays a ?xed operating cost to produce. Third, the aggregate price is realized. Fourth,
?rms observe revenue and update beliefs. Then, another period begins.
With the above setup, this subsection considers a recursive competitive equilibrium
de?nition which includes as a key component the law of motion of the aggregate state
of the industry. The aggregate state is (F, D). F denotes the distribution (measure)
of ?rms across vintages and idiosyncratic productivity. The part of F that measures
the number of ?rms with belief ?
e
and age a is denoted f (?
e
, a). D is an exogenous
demand parameter; it captures aggregate conditions and is fully observable. The law
of motion for D is exogenous, described by D’s transition matrix. The law of motion
for F is denoted H so that F
0
= H(F, D). The sequence of events implies that H
captures the in?uence of entry, exit and learning.
56
Three assumptions characterize the equilibrium: ?rm rationality, free entry and
competitive pricing.
Firm Rationality: ?rms are assumed to have rational expectations; their decisions
are forward-looking. In period t, a ?rm with age a and belief ?
e
expects its pro?t in
period s ? t to equal
A
t?a
· E(P
s
|F
t
, D
t
) · ?
e
?1.
E
t
(P
s
|F
t
, D
t
) implies that ?rms need to observe (F, D) to predict the sequence of
prices from today onward. Therefore, the relevant state variables for a ?rm are its
vintage, its belief about its true idiosyncratic productivity, and the aggregate state
(F, D). I let V (?
e
, a; F, D) be the expected value, for a ?rm with belief ?
e
and age
a, of staying in operation for one more period and optimizing afterward, when the
aggregate state is (F, D). Then V satis?es:
V (?
e
, a; F, D) = E[? (?
e
, a) |F, D] +?E[max (0, V (?
e0
, a + 1; F
0
, D
0
)) |F, D] (3.3)
subject to
F
0
= H (F, D)
and the exogenous laws of motion for D and ?
e
( driven by all-or-nothing learning).
Since ?rms enter as unsure, ?rm rationality implies that entry occurs if and only
if V (?
u
, 0; F, D) > 0. Meanwhile, a ?rm with belief ?
e
and age a exits if and only if
V (?
e
, a; F, D) < 0.
Free entry: new ?rms are free to enter at any instant, each bearing an entry cost
c. The entry cost can be interpreted as the cost of establishing a particular location
or the cost of ?nding a manager. Assuming f (?
u
, 0; F, D) represents the size of the
57
entering cohort when the aggregate state is (F, D), and letting c represent the entry
cost, I have
c = C (f (?
u
, 0; F, D)) , c > 0 and C
0
? 0. (3.4)
I let the entry cost depend positively on the entry size to capture the idea that,
for the industry as a whole, fast entry is costly and adjustment may not take place
instantaneously. This can arise from a limited amount of land available to build
production sites or an upward-sloping supply curve for the industry’s capital stock.
29
The free entry condition equates a ?rm’s entry cost to its value of entry, and can be
written as
V (?
u
, 0; F, D) = C (f (?
u
, 0; F, D)) . (3.5)
As more new ?rms enter, the entry cost is driven up until it reaches the value of entry.
At this point, entry stops.
Competitive Pricing: the output price is competitive; the price level is given by
P (F, D) =
D
Q(F, D)
(3.6)
Q represents aggregate output; it equals the the sum of production over heterogeneous
?rms. Given (3.1), the sequence of events implies that:
30
Q(F, D) = Q(F
0
) = A
X
a
X
?
e
(1 +?)
?a
· ?
e
· f
0
(?
e
, a) , (3.7)
where A represents the industry leading technology when the aggregate state is (F, D).
29
See Subsection 3.3.1 for further discussion.
30
Q is the sum of realized output rather than expected output, since the contribution to aggregate
output by each ?rm depends on its true type ? rather than ?
e
. However, with a continuum of ?rms,
the law of large numbers implies that the random noises and the expectation errors cancel out in
each cohort, so that the sum of realized output equals the sum of expected output.
58
f
0
(?
e
, a) measures the number of operating ?rms with ?
e
and a after entry and exit.
f
0
(?
e
, a) belongs to F
0
, the updated ?rm distribution. Since F
0
= H(F, D), Q is a
function of (F, D).
(3.6) implies that high output drives down the price. (3.7) implies that Q depends
not only on the number of ?rms in operation, but also on their distribution. More
?rms yield higher output and drive down the price; the more the distribution is skewed
toward younger vintages and better idiosyncratic productivity, the higher the output
and the lower the price.
With the above three conditions, I have the following:
De?nition: A recursive competitive equilibrium is a law of motion H, a
value function V , and a pricing function P such that (i) V solves the
?rm’s problem; (ii) P satis?es (3.6) and (3.7); and (iii) H is generated
by the decision rules suggested by V and the appropriate summing-up of
entry, exit and learning.
An additional assumption is made to simplify the model:
Assumption: Given values for other parameters, the value of ?
b
is so low
that V (?
b
, a; F, D) is negative for any (F, D) and a.
This assumption implies that bad ?rms always exit, so that at any one time, there
are only two idiosyncratic productivity of ?rms in operation — unsure and good.
The following proposition characterizes the value function V and the correspond-
ing exit ages of heterogeneous ?rms.
Proposition 3.1: V (?
e
, a; F, D) is strictly decreasing in a, holding ?
e
constant,
and strictly increasing in ?
e
, holding a constant; therefore, there is a cut-
59
o? age a (?
e
; F, D) for each idiosyncratic productivity, such that ?rms of
idiosyncratic productivity ?
e
and age a ? a (?
e
; F, D) exit before produc-
tion takes place; furthermore, a (?
g
; F, D) ? a (?
u
; F, D).
The proof for Proposition 3.1 presented in the appendix is not restricted to all-
or-nothing learning. Hence, Proposition 3.1 holds for any learning process. It follows
from the fact that ?rms with smaller a and higher ?
e
have a higher expected value
of staying. As V is strictly decreasing in a, ?rms with belief ?
e
that are older than
a (?
e
; F, D) exit; as the expected value of staying is strictly increasing in ?
e
, a good
?rm stays longer than an unsure ?rm.
3.3 Cleansing and Scarring
The ?rm distribution F enters the model as a state variable, which makes it di?cult
to characterize the dynamics generated by demand ?uctuations. However, similar
studies ?nd that the e?ects of temporary changes in aggregate conditions are qualita-
tively similar to the e?ects of permanent changes.
31
Therefore, I begin in this section
with comparative static exercises on the steady-state equilibrium. The comparative
static exercises capture the essence of industry dynamics as well as how demand can
a?ect the labor allocation, and thus provide a more rigorous intuition for the scarring
and cleansing e?ects described in the introduction. In the next section, I will turn to
a numerical analysis of the model’s response to stochastic demand ?uctuations and
con?rm that the results from the comparative static exercises carry over.
31
See Mortensen and Pissarides (1994), Caballero and Hammour (1994 and 1996), and Barlevy
(2003).
60
3.3.1 The Steady State
I de?ne a steady state as a recursive competitive equilibrium with time-invariant
aggregate states.
32
It satis?es two additional conditions, (i) D is and is perceived
as time-invariant: D
0
= D. (ii) F is time-invariant: F
0
= H (F, D). Since H is
generated by entry, exit and learning, a steady state must feature time-invariant
entry and exit for F = H (F, D) to hold. Thus, a steady state equilibrium can be
summarized by {f(0), a
g
,a
u
}, with f (0) as the entry size, a
g
as the maximum age
for good ?rms, and a
u
as the maximum age for unsure ?rms. The next proposition
establishes the existence of a unique steady-state equilibrium. The proof is presented
in the appendix.
Proposition 3.2: With D constant over time, there exists a unique time-invariant
{f(0), a
g
, a
u
} that satis?es the conditions of ?rm rationality, free entry and competi-
tive pricing.
The steady-state labor distribution and job ?ows are illustrated in Figure 9. Like
Figure 8, there are two ways to interpret Figure 9. First, it displays the steady-state
life-cycle dynamics of a representative cohort with the horizontal axis depicting the
cohort age across time. Firms enter in size f (0) as unsure. As the cohort ages and
learns, bad ?rms are thrown out so that the cohort size declines; good ?rms are
realized, so that the density of good ?rms increases. After age a
u
, all unsure ?rms
exit because their vintage is too old to survive with ?
e
= ?
u
. However, ?rms with
?
e
= ?
g
stay. Afterwards, the cohort contains only good ?rms and the number of
good ?rms remains constant because learning has stopped. Good ?rms live until a
g
.
32
The term “steady state” follows Caballero and Hammour (1994). Despite its name, the steady-
state price decreases while the steady-state average labor productivity increases over time due to
technological progress.
61

age

0
Exit Margin
of Unsure Firms
Exit Margin
of Good Firms
Learning Margin
?? Exit of Bad Firms
Entry Margin
f(0)

unsure firms
good firms
maximum age of
unsure firms
maximum age of
good firms
Figure 9: The Steady-state Labor Distribution and Job Flows: the distance between
the lower curve (extended as the horizontal line) and the bottom axis measures the
density of good ?rms; the distance between the two curves measures the density of
unsure ?rms.
62
The vintage after a
g
is too old even for good ?rms to survive.
Second, Figure 9 also displays the ?rm distribution across ages and idiosyncratic
productivity at any one time, with the horizontal axis depicting the cohort age cross
section. At the steady state, ?rms of di?erent ages coexist. Since older cohorts have
lived longer and learned more, their size is lower and their density of good ?rms is
higher. Cohorts older than a
u
are of the same size and contain only good ?rms. No
cohort is older than a
g
.
Despite its time-invariant structure, the industry experiences continuous entry
and exit. With entry, jobs are created; with exit, jobs are destroyed. From a pure
accounting point of view, there are three margins for job ?ows: the entry margin, the
exit margins of good ?rms and unsure ?rms, and the learning margin. Two forces —
learning and creative destruction — interact together to drive job ?ows. At the entry
margin, creative destruction drives in new vintages. At the exit margins, it drives out
old vintages. At the learning margin, bad ?rms are selected out. Because of creative
destruction, average labor productivity grows at the technological pace ?. Because
of learning, the productivity distribution among older cohorts is more skewed toward
good ?rms. For cohorts older than a
u
, labor is employed only at good ?rms.
3.3.2 Comparative Statics: Cleansing and Scarring
The previous subsection has shown that for a given demand level, there exists a
steady-state equilibrium summarized by {f(0), a
g
, a
u
}. In this subsection, I establish
that across steady states corresponding to di?erent demand levels, the model delivers
the conventional cleansing e?ect promoted in the previous literature, as well as an
additional scarring e?ect. The two e?ects are formalized in Propositions 3.3 and 3.4.
63
Proposition 3.3: In a steady-state equilibrium, the exit age for ?rms with
a given belief is weakly increasing in the demand level and the job de-
struction rate is weakly decreasing in the demand level.
A detailed proof is included in the appendix. To understand Proposition 3.3,
compare two steady states with di?erent demand levels, D
h
> D
l
. For any time t,
(3.6) suggests that the steady state with D
l
features either a lower price, or a lower
output, or both. Now assume initially that the lower demand is fully re?ected as
a lower output and the prices of the two steady states are identical. Then ?rms’
pro?tability in the two steady states would also be identical: V
l
(?
e
, a) = V
h
(?
e
, a) for
any ?
e
and a. Free entry and the exit conditions suggest that identical value functions
lead to identical entry size and exit ages, and thus an identical ?rm distribution.
With ?rm-level output of a given age and idiosyncratic productivity independent
of demand, identical cross-sectional distributions imply identical aggregate output,
which contradicts our assumption. Therefore, we can conclude that the low-demand
steady state must feature a lower price compared to the high-demand steady state, so
that V
l
(?
e
, a) < V
h
(?
e
, a) for any ?
e
and a. Since V (?
e
, a) strictly decreases in a, the
cut-o? age that solves the V (?
e
, a) = 0 must be lower for lower demand. Intuitively,
lower demand tends to drive down the price so that some ?rms that are viable in a
high-demand steady state are not viable when demand is low.
Moreover, the following equation is derived by combining the exit conditions for
unsure and good ?rms:
µ
?
u
?
g
+
p??
1 +? ??
¶
(1 +?)
a
g
?a
u
= 1 +
p??
1 ??
?
p???
(1 ??) (1 +? ??)
?
a
g
?a
u
(3.8)
64
I prove in the appendix that (3.8) gives an unique solution for a
g
?a
u
as long as
?
g
> ?
u
. Since D does not enter (3.8), a
g
?a
u
is independent of demand:
d(a
g
?a
u
)
dD
= 0.
(3.8) suggests that the demand level does not a?ect the gap between the exit ages of
good and unsure ?rms.
The steady-state job destruction rate, denoted jd
ss
, equals the following:
33
jd
ss
=
1
a
u
· ? + [
1??
p
+ (a
g
?a
u
) · ?] · [1 ?(1 ?p)
a
u
+1
]
. (3.9)
Since (a
g
?a
u
) is independent of D, demand a?ects jd
ss
only through its impact on
a
u
:
d(jd
ss
)
d(D)
=
d(jd
ss
)
d(a
u
)
·
d(a
u
)
d(D)
. I prove in the appendix that
d(jd
ss
)
d(a
u
)
? 0, which, together
with
d(au)
d(D)
? 0, implies
d(jd
ss
)
d(D)
? 0. Put intuitively, a high-demand steady state allows
both unsure ?rms and good ?rms to live longer, so that fewer jobs are destroyed at
the exit margins.
To summarize, Proposition 3.3 argues that the steady state with lower demand
features younger exit ages and a higher job destruction rate. In other words, it
suggests that more ?rms are cleared out in an environment that is more di?cult for
survival.
If the above story suggested by comparative statics carries over when D ?uctuates
stochastically over time, then my model delivers a conventional “cleansing” e?ect, in
which average ?rm age falls during recessions so that recessions direct resources to
33
According to Davis and Haltiwanger (1992), the job destruction rate at time t is de?ned as:
2 ? Jobs destroyed in period t
[(number of jobs at the beginning of period t) + (number of jobs at the beginning of period t + 1)]
.
With constant total number of jobs, the steady-state job destruction rate equals the ratio of jobs
destroyed at the learning and exit margins over the total number of jobs. The expression of jd
ss
applies not only to a steady state, but also to any industry equilibrium that features time-invariant
entry and exit. See Subsection 4.2 for further discussions on jd
ss
.
65
younger, more productive vintages. However, once learning is allowed, we also need
to take into account the allocation of labor across idiosyncratic productivity. With
only two true idiosyncratic productivity, good and bad, the idiosyncratic productivity
distribution of labor can be summarized by the fraction of labor at good ?rms. A
higher fraction suggests a more e?cient cross-idiosyncratic productivity allocation of
labor. The next proposition establishes how the level of demand a?ects this ratio in
a steady state.
Proposition 3.4: In a steady state equilibrium, the fraction of labor at
good ?rms is weakly increasing in the demand level.
It can be shown that the steady-state fraction of labor at good ?rms, denoted l
ss
g
,
equals:
l
ss
g
= 1 ?
(1 ??)
p?a
u
1?(1?p)
au
+ (1 ??) +p?(a
g
?a
u
)
.
Again, since (a
g
?a
u
) is independent of D, demand a?ects l
ss
g
only through its impact
on a
u
:
d(l
ss
g
)
d(D)
=
d(l
ss
g
)
d(au)
·
d(au)
d(D)
. I prove
d(l
ss
g
)
d(au)
? 0 in the appendix, which, together with
d(a
u
)
d(D)
? 0, implies
d
(
l
ss
g )
d(D)
? 0.
My analysis suggests that the impact of demand on the fraction of labor at good
?rms comes from its impact on the exit age of unsure ?rms. To understand this result
intuitively, consider Figure 10.
Figure 10 displays the steady-state industry structures corresponding to two de-
mand levels.
34
The cleansing e?ect formalized in Proposition 3.3 is shown as the
34
The entry sizes of the two steady states, although di?erent, are normalized as 1. Since the
steady state features time-invariant entry and all cohorts are the same size, entry size matters only
as a scale.
66

age

f( ,a)
0
Cleansing Effect
Cleansing Effect
Scarring Effect
Good firms
Unsure firms
Figure 10: Cleansing and Scarring
67
leftward shift of the two exit margins. The shifted margins clear out old ?rms that
could be either good or unsure. However, the leftward shift of the unsure exit margin
also reduces the number of older good ?rms. The latter e?ect, shown as the shaded
area in Figure 10, is the scarring e?ect of recessions.
The scarring e?ect stems from learning. New entrants begin unsure of their idio-
syncratic productivity, although a proportion ? are truly good. Over time, more and
more bad ?rms leave while good ?rms stay. Since learning takes time, the number
of “potentially good ?rms” that realize their true idiosyncratic productivity depends
on how many learning chances they have. If ?rms could live forever, eventually all
the potentially good ?rms would get to realize their true idiosyncratic productivity.
But a ?nite life span of unsure ?rms implies that if potentially good ?rms do not
learn before age a
u
, they exit and thus forever lose the chance to learn. Therefore,
a
u
represents not only the exit age of unsure ?rms, but also the number of learning
opportunities. A low a
u
allows potentially good ?rms fewer chances to realize their
true idiosyncratic productivity, so that the number of old good ?rms in operation
after age a
u
is also reduced.
Hence, the industry su?ers from uncertainty; it tries to select out bad ?rms but
the group of ?rms it clears at age a
u
includes some ?rms that are truly good. The
number of clearing mistakes the industry makes at a
u
depends on the size of the
unsure exit margin, which in turn depends on the value of a
u
.
35
When a drop in
demand reduces the value of a
u
, this reduces the number of learning opportunities,
allows fewer good ?rms to become old and thus shifts the labor distribution toward
bad ?rms.
35
The all-or-nothing learning suggests that the number of truly good ?rms cleared out at a
u
equals
f (0) (1 ?p)
a
u
?.
68
To summarize from Propositions 3.3 and 3.4, a low-demand steady state features
a better average vintage, yet a less e?cient cross-idiosyncratic productivity distrib-
ution of labor. If the comparative static results carry over when demand ?uctuates
stochastically, then recessions will have both a conventional cleansing e?ect, shifting
resources to better vintages, and a scarring e?ect, shifting resources to bad idio-
syncratic productivity. The two e?ects are directly related to each other: it is the
cleansing e?ect that signi?cantly reduces learning opportunities and hence prevents
more ?rms from realizing their potential.
When we move beyond steady states to allow for cyclical ?uctuations, the intu-
ition behind “cleansing and scarring” still carries over. Consider Figure 6. Both exit
margins shift as soon as demand drops so that the cleansing e?ect takes place imme-
diately.
36
However, the scarring e?ect takes place gradually. When a recession ?rst
arrives, the group of ?rms already in the shaded area in Figure 6 will not leave despite
the shift in exit margins, since they know their true idiosyncratic productivity to be
good. They leave gradually as the recession persists. At this point, the scarring e?ect
starts to take place: the reduced a
u
allows fewer good ?rms to survive past a
u
. The
shaded area would eventually be left blank, and the “scar” left by recessions would
surface.
3.3.3 Sensitivity Analysis
Two modi?cations are examined in this subsection to check the robustness of my
results from the comparative static exercises: ?rst, I allow the entry cost to be inde-
36
My numerical exercises imply that when demand falls, these margins initially shift more than
suggested by the comparative static exercises. The margins shift back partially as the recession
persists. A detailed discussion of this phenomenon is contained in Section 4.
69
pendent of entry size; second, I allow the process of learning to be more complicated
than “all-or-nothing”.
Entry Cost Independent of Entry Size The previous subsection has argued that
the shift of the exit margins creates both a cleansing e?ect and a scarring e?ect. Now,
focus on the entry side. How does demand a?ect entry, and how would alternative
assumptions on entry a?ect my results?
To address these questions, recall that the free entry condition requires V (?
u
, 0) =
C (f (?
u
, 0)), and C is assumed to depend positively on entry size. Since low demand
reduces the value of entry by driving down pro?tability, C
0
(f (?
u
, 0)) > 0 implies less
entry (smaller f (?
u
, 0)) for the low-demand steady state. Hence, an industry in my
model has two margins along which it can accommodate low demand. It can either
reduce entry, or increase exit by shifting the exit margins. The issue is which of
these two margins will respond when demand falls, and by how much. If the drop in
demand level can be fully incorporated as a decrease in entry size, the exit margins
might not respond.
The extreme case that the entry margin exclusively accommodates demand ?uc-
tuations is de?ned as the “full-insulation” case in Caballero and Hammour (1994).
They argue that creation (entry) “insulates” destruction (exit), and the extent of the
insulation e?ect depends on the cost of fast entry, that is, C
0
(f (?
u
, 0)). The full-
insulation case occurs when C
0
(f (?
u
, 0)) = 0. The intuition is as follows. If entry
cost is independent of entry size, then fast entry is costless and the adjustment on
the entry margin becomes instantaneous. When demand falls, entry will adjust to
such a level that aggregate output falls by the same proportion, which keeps price at
the same level. Then the value of staying remain una?ected, and the exit margins
70
do not respond. Hence, with entry cost independent of entry size, there is neither a
cleansing e?ect nor a scarring e?ect.
Two remarks can be made. First, in reality, an industry may not be able to create
all the necessary production units instantaneously. Goolsbee (1998) shows empirically
that higher investment demand drives up both the equipment prices and the wage
of workers producing the capital goods. His ?ndings suggest that as more ?rms
enter and increase the demand for capital, it becomes increasingly costly to purchase
capital. As another intuitive example, when more new stores are built, land prices
and rentals usually rise. Therefore, C
0
(f (?
u
, 0)) > 0 seems more reasonable. Second,
data does not support the assumption that C
0
(f (?
u
, 0)) = 0. In the full-insulation
case, job creation fully accommodates demand ?uctuations and job destruction does
not respond. This contradicts the large and robust evidence that job destruction is
more responsive than job creation to the business cycle.
37
More Complicated Learning As I have argued in subsection 2.2, the all-or-
nothing learning with a uniform distribution of random noise simpli?es the analysis
considerably. But how restrictive is it? Would the scarring e?ect carry over with a
more complicated process of learning?
In general, we can de?ne the scarring e?ect as a drop in the fraction of labor at
good ?rms. To look at the scarring e?ect from a di?erent angle, suppose we divide
?rms into two groups, young and old.
38
With l
o
g
denoting the fraction of labor at
good ?rms among the old, l
y
g
as the fraction among the young, f
y
as the density of
young ?rms and f
o
as the density of old ?rms, the fraction of labor at good ?rms for
37
See footnote 6.
38
The cut-o? age to de?ne “young” and “old” is arbitrarily chosen. Changing this cut-o? age does
not a?ect the analysis that follows.
71
the industry as a whole, l
g
, can be written as:
l
g
=
f
y
l
y
g
+f
o
l
o
g
f
y
+f
o
=
l
y
g
+l
o
g
f
o
f
y
1 +
f
o
f
y
.
The ?rst order derivative of l
g
with respect to
f
o
f
y
equals:
d (l
g
)
d
³
f
o
f
y
´ =
l
o
g
?l
y
g
1 +
f
o
f
y
.
which is greater than or equal to zero as long as l
o
g
?l
y
g
? 0, which should hold for any
learning process, since old ?rms have experienced more learning. Hence, the scarring
e?ect of recessions should occur under any idiosyncratic productivity of learning as
long as recessions reduce the ratio of old to young ?rms (
f
o
f
y
), which by de?nition
will be true in any model in which recessions cleanse the economy of older vintages.
Intuitively, the scarring e?ect suggests that recessions shift resources toward younger
?rms, so that there cannot be as much learning taking place as in booms.
Now suppose we assume a more complicated learning process with normally dis-
tributed random noise, so that the signals received by good ?rms are normally dis-
tributed around ?
g
and the signals received by bad ?rms are normally distributed
around ?
b
. In that case, a ?rm can never know for certain that it is good or bad, and
posterior beliefs are distributed continuously between ?
b
and ?
g
. The expected value
of staying would still depend positively on ?
e
and negatively on age. Thus, given the
aggregate state, there would be a cut-o? age for each belief, a (?
e
; F, D), such that
?rms with belief ?
e
do not live beyond a (?
e
; F, D).
With a recession, the value of staying across all ages and idiosyncratic productivity
falls, so that for each belief ?
e
, the cut-o? age a(?
e
; F, D) becomes younger. Hence,
72
the ?rm distribution tilts toward younger ages and
f
o
f
y
falls. Since
d(lg)
d(
f
o
f
y )
? 0, a fall
in
f
o
f
y
drives down the ratio of good ?rms and creates the scarring e?ect. Although
this analysis is preliminary,
39
we can still argue that recessions would allow for less
?rm learning, so the scarring e?ect would carry over even with a more complicated
process of learning.
3.4 Quantitative Implications with Stochastic Demand Fluc-
tuations
I establish in Section 3 that across steady states, variations in demand induce compet-
ing cleansing and scarring e?ects on productivity. In this section, I address whether
the two e?ects carry over when demand ?uctuates stochastically, and which e?ect
dominates quantitatively.
This section turns to numerical techniques to analyze a stochastic version of my
model in which the demand level follows a two-state Markov process with values
[D
h
, D
l
] and transition probability µ. Throughout this section, ?rms expect the cur-
rent demand level to persist for the next period with probability µ, and to change
with probability 1 ?µ.
I ?rst describe my computational strategy, which follows Krusell and Smith (1998)
by shrinking the state space into a limited set of variables and showing that these
variables’ laws of motion can approximate the equilibrium behavior of ?rms in the
simulated time series. Later in this section, I con?rm that the basic insights from
the comparative static exercises carry over with probabilistic business cycles. Then I
39
For instance, the analysis cannot address the relative sizes of the cleansing e?ect on young ?rms
versus old ?rms. Whether cleansing a?ects primarily young or old ?rms depends on the speci?cs of
the learning process.
73
examine whether the scarring e?ect is likely to be empirically relevant. Speci?cally, I
calibrate my model so that its equilibrium job destruction rate mimics the observed
pattern in U.S. manufacturing. As I have argued, recessions clear out old ?rms,
including some good ?rms that have not yet learned their idiosyncratic productivity.
Therefore, the model allows us to use the job destruction rate to make inferences on
the size of the cleansing and scarring e?ects.
3.4.1 Computational Strategy
The de?nition of the recursive competitive equilibrium in Section 2 implies that indi-
vidual decision rules can be generated from the value functions V ; by summing up
the corresponding individual decision rules, we can get the laws of motion H, then
trace out the evolution of industry structure. Therefore, the key computational task
is to map F, the ?rm distribution across ages and idiosyncratic productivity, given
demand level D, into a set of value functions V (?
e
, a; F, D). Unfortunately, the en-
dogenous state variable F is a high-dimensional object. The numerical solution of
dynamic programming problems becomes increasingly di?cult as the size of the state
space increases. To make the state space tractable, I de?ne a variable X such that
40
X (F) =
X
a
X
?
e
(1 +?)
?a
· ?
e
· f (?
e
, a) . (3.10)
Combining (3.9) with (3.6) and (3.7), I get
P (F, D) · A =
D
X (F
0
)
.
40
X can be interpreted as detrended output.
74
A is the leading technology; F
0
is the updated ?rm distribution after entry and
exit; X
0
corresponds to F
0
; P (F, D) is the equilibrium price in a period with initial
aggregate state (F, D). Since F
0
= H(F, D), the above equation can be re-written as
P (F, D) · A =
D
X (H (F, D))
Given these de?nitions, the single-period pro?tability of a ?rm of idiosyncratic pro-
ductivity ?
e
and age a, given aggregate state (F, D), equals
? (a, ?; F, D) =
D
X (H (F, D))
· (1 +?)
?a
· (? +?) ?1. (3.11)
Thus, the aggregate state (F, D) and its law of motion help ?rms to predict future
pro?tability by suggesting sequences of X’s from today onward under di?erent paths
of demand realizations. The question then is: what is the ?rm’s critical level of
knowledge of F that allows it to predict the sequence of X
0
s over time? Although
?rms would ideally have full information about F, this is not computationally feasible.
Therefore I need to ?nd an information set ? that delivers a good approximation of
?rms’ equilibriumbehavior, yet is small enough to reduce the computational di?culty.
I look for an ? through the following procedure. In step 1, I choose a candidate
?. In step 2, I postulate perceived laws of motion for all members of ?, denoted H
?
,
such that ?
0
= H
?
(?, D). In step 3, given H
?
, I calculate ?rms’ value functions on a
grid of points in the state space of ? applying value function iteration, and obtain the
corresponding industry-level decision rules — entry sizes and exit ages across aggregate
states. In step 4, given such decision rules and an initial ?rm distribution,
41
I simulate
41
I start with a uniform ?rm distribution across types and ages. My numerical exercises suggest
that the dynamic system of my model is stable and that the initial ?rm distribution does not a?ect
75
? {X}
H
?
H
x
(X, D
h
): log X
0
= 1.2631 + 0.8536 log X
H
x
(X, D
l
) : log X
0
= 2.4261 + 0.7172 log X
R
2
for D
h
: 0.9876
for D
l
: 0.9421
standard forecast error
for D
h
: 0.0000036073%
for D
l
: 0.000030068%
maximum forecast error
for D
h
: 0.000049895%
for D
l
: 0.00074675%
Den Haan &Marcet test sta-
tistic (?
2
7
)
0.8007
Table 5: The Estimated Laws of Motion and Measures of Fit
the behavior of a continuum of ?rms along a random path of demand realizations,
and derive the implied aggregate behavior – a time series of ?. In step 5, I use the
stationary region of the simulated series to estimate the implied laws of motion and
compare them with the perceived H
?
; if di?erent, I update H
?
, return to step 3 and
continue until convergence. In step 6, once H
?
converges, I evaluate the ?t of H
?
in terms of tracking the aggregate behavior. If the ?t is satisfactory, I stop; if not,
I return to step 1, make ?rms more knowledgeable by expanding ?, and repeat the
procedure.
I start with ? = {X} – ?rms observe X instead of F. I further assume that
?rms perceive the sequence of future coming X
0
s as depending on nothing more than
the current observed X and the state of demand. The perceived law of motion for X
is denoted H
x
so that X
0
= H
x
(X, D). I then apply the procedure described above
and simulate the behavior of a continuum of ?rms over 5000 periods. The results
are presented in Table 5 As shown in Table 5, the estimated H
x
is log-linear. The
?t of H
x
is quite good, as suggested by the high R
2
, the low standard forecast error,
the result.
76
Figure 11: Expected Value of Staying: aggregate state variables are D and log X (the
log of detrended output), ?rm-level state variables are ?rm age and belief (good or
unsure); the parameter choices underlying these ?gures are summarized in Table 2
and discussed in Subsection 4.2.
77
8 9
80
85
90
95
100
105
110
115
120
entry size
logX
8 9
40
45
50
55
60
65
70
75
80
85
90
exit ages
logX
of good firms in booms
of good firms in recessions
of unsure firms in booms
of unsure firms in recessions
in booms
in recessions
Figure 12: Industry-level Policy Functions: Entry Size and Exit Ages. Aggregate
states are D (booms or recessions) and log X (the log of detrended output).
78
and the low maximum forecast error. The good ?t when ? = {X} implies that ?rms
perceiving these simple laws of motion make only small mistakes in forecasting future
prices. To explore the extent to which the forecast error can be explained by variables
other than X, I implement the Den Haan and Marcet (1994) test using instruments
[1, X, µ
a
, ?
a
, ?
a
, ?
a
, r
u
], where µ
a
, ?
a
, ?
a
, ?
a
,r
u
are the mean, standard deviation,
skewness, and kurtosis of the age distribution of ?rms, and the fraction of unsure
?rms, respectively.
42
The test statistic is 0.8007, well below the critical value at the
1% level. This suggests that given the estimated laws of motion, I do not ?nd much
additional forecasting power contained in other variables. Nevertheless, I expand ?
further to include ?
a
, the standard deviation of the age distribution of ?rms. The
results when ? = {X, ?
a
} are presented in the appendix. The measures of ?t do not
change much.
43
Furthermore, the impact of changes in ?
a
on the approximated value
function is very small (less than 0.5%). This con?rms that the inclusion of information
other than X improves the forecast accuracy by only a very small amount.
Figure 11 displays the value of staying for heterogeneous ?rms as a function of
a, ?
e
, D and X (log X). Figure 12 displays the corresponding optimal exit ages and
entry sizes. The properties of value functions and exit ages stated in Proposition
3.2 are satis?ed in both ?gures: given the aggregate state, the value of staying is
increasing in the perceived idiosyncratic productivity ?
e
and decreasing in ?rm age;
42
Den Haan and Marcet (1994) o?er a statistic for computing the accuracy of a simulation. It has
an asymptotic ?
2
distribution under the null that the simulation is accurate. The statistic for my
industry is given by TB
0
T
A
?1
T
B
T
, where B
T
=
1
T
X
u
t+1
?h(G
t
), A
T
=
1
T
X
u
2
t+1
?h(G
t
) h(G
t
)
0
,
u
t+1
is the expectation error for X
t+1
(or log X
t+1
), and h(G
t
) is some function of variables dated
t. I choose h(G
t
) = [1, X, µ
a
, ?
a
, ?
a
, ?
a
, r
u
], which gives my test statistic 7 degrees of freedom.
43
Actually the ?t during recessions becomes worse to some extent. Young (2002) adds an additional
moment to the original Krusell & Smith approach, and also gets a worse measure of ?t for the bad
state (recessions). He attributes this result to numerical error.
79
parameters (pre-chosen) value
productivity of bad ?rms: ?
b
1
productivity of good ?rms: ?
g
3.5
quarterly technological pace: ? 0.007
quarterly discount factor: ? 0.99
parameters (calibrated) value
high demand: D
h
2899
low demand: D
l
2464
prior probability of being a good ?rm: ? 0.14
quarterly pace of learning: p 0.08
persistence rate of demand: µ 0.58
entry cost function 0.405 + 0.52 ? f(0, ?
u
)
Table 6: Base-line Parameterization of the Model
and good ?rms exit at an older age than unsure ?rms.
To conclude, Table 5 and Figures 11 and 12 suggest that my solution using X
to approximate the aggregate state closely replicates optimal ?rm behavior at the
equilibrium.
44
Therefore, I use the solution based on ? = {X} to generate all the
series in the subsequent analysis.
3.4.2 Calibration
Table 6 presents the assigned parameter values. Some of the parameter values are pre-
chosen. The most signi?cant in this group are the relative productivity of good and
bad ?rms. I follow Davis and Haltiwanger (1999), who assume a ratio of high-to-low
productivity of 2.4 for total factor productivity and 3.5 for labor productivity based
on the between-plant productivity di?erentials reported by Bartelsman and Doms
44
These results were robust when I experimented with di?erent parameterizations of the model.
Although they suggest that my approximation is good, one could say that these are self-ful?lling
equilibria: because everyone perceives a simple law of motion, they behave correspondingly so that
the aggregate states turn out as predicted. However, it has been di?cult to prove theoretically the
existence of such self-ful?lling equilibria in my model.
80
(1997). Since labor is the only input in my model, I normalize productivity of bad
?rms as 1 and set productivity of good ?rms as 3.5. I allow a period to represent one
quarter and set the quarterly discount factor ? = 0.99. Next, I need to choose ?, the
quarterly pace of technological progress. In a model with only creative destruction,
Caballero and Hammour (1994) choose the quarterly technological growth rate as
0.007 by attributing all output growth of US manufacturing from 1972 (II) to 1983
(IV) to technical progress. To make comparison with their results convenient in the
coming subsections, I also choose ? = 0.007. Caballero and Hammour (1994) assume
a linear entry cost function c
0
+ c
1
f(0, ?
u
) with f(0, ?
u
) denoting the size of entry,
which is also applied in my calibration exercises.
The remaining undetermined parameters are: p, the pace of learning; ?, the
probability of being a good ?rm; D
h
and D
l
, the demand levels; µ, the probability
with which demand persists; and c
0
and c
1
, the entry cost parameters. The values of
these parameters are chosen so that the job destruction series in the calibrated model
matches properties of the historical series from the U.S. manufacturing sector. Their
values are calibrated in the following manner.
First, I match the long-run behavior of job destruction. My numerical simulations
suggest that the dynamic system eventually settles down with constant entry and exit
along any sample path where the demand level is unchanging. The industry structures
at these stable points are similar to those at the steady states, which allows me to use
steady state conditions for approximation.
45
I let a
g
and a
u
represent the maximum
ages of good ?rms and unsure ?rms at the high-demand steady state and a
g
0
and
45
However, a stable point is di?erent from a steady state. In a steady state, ?rms perceive
demand as constant, while in a stable point, ?rms perceive demand to persist with probability µ,
and to change with probability 1 ?µ.
81
a
u
0
represent the exit ages at the low-demand steady state. The steady-state job
destruction rate, denoted jd
ss
, is given by (3.9).
Second, I match the peak in job destruction that occurs at the onset of a recession.
My model suggests that the jump in the job destruction rate at the beginning of a
recession comes from the shift of exit margins to younger ages. I assume that when
demand drops, the exit margins shift from a
g
and a
u
to a
g
0
and a
u
0
immediately, so
that the job destruction rate at the beginning of a recession, denoted as jd
max
, is
approximately:
46
jd
max
=
2 ·
?
h
1 ?(1 ?p)
a
u
+1
i
(a
g
?a
g
0
) +
h
1
p
+? ?1 ?
1
p
(1 ?p)
a
u
?a
u
0
i
(1 ?p)
a
u
0
+1
+
(1 ??)
?(a
u
+a
u
0
) +
(1??)
p
h
2 ?(1 ?p)
a
u
+1
?(1 ?p)
a
u
0
+1
i
+
?
h
1 ?(1 ?p)
au+1
i
(a
g
?a
u
) +?
h
1 ?(1 ?p)
au
0
+1
i
(a
g
0
?a
u
0
)
(3.12)
Third, I match the trough in job destruction that occurs at the onset of a boom.
My model suggests that when demand goes up, the exit margins extend to older ages,
so that for several subsequent periods job destruction comes only from the learning
margin, implying a trough in the job destruction rate. The job destruction rate at
this moment, denoted as jd
min
, is approximately:
jd
min
=
(1 ??)
h
1 ?(1 ?p)
au
0
+1
i
a
u
0
· ? + [
1??
p
+ (a
g
0
?a
u
0
) · ?] · [1 ?(1 ?p)
au
0
+1
]
(3.13)
46
As I have noted earlier, the calibration exercises suggest that when a negative aggregate demand
shock strikes, the exit margins shift more than a
g
0
and a
u
0
. The bigger shift implies a bigger jump
in job destruction, This is why I require neg
max
to lie below 11.60%. I experiment with di?erent
demand levels to ?nd those that generate the closest ?t.
82
Descriptive Statistics Mean Min. Max. Std.
Value 5.6% 2.96% 11.60% 1.66%
Table 7: Descriptive Statistics of Quarterly Job Destruction in U.S. Manufacturing
(1972:2-1993:4), constructed by Davis and Haltiwanger.
NowI turn to data for conditions on jd
ss
, jd
max
, and jd
min
. Table 7 lists descriptive
statistics for the job destruction series of the U.S. manufacturing sector from 1972:2
to 1993:4 compiled by Davis and Haltiwanger. This data places three restrictions on
the values of p, ?, a
g
, a
u
, a
g
0
and a
u
0
. First, the implied jd
ss
with either (a
g
, a
u
) or
(a
g
0
, a
u
0
) must be around 5.6%.
47
Second, the implied jd
max
must not exceed 11.6%.
Third, the implied jd
min
must be above 3%. Additionally, (a
g
, a
u
) and (a
g
0
, a
u
0
) must
satisfy (3.8), the gap between the exit ages of good and unsure ?rms suggested by
the steady state. There are six equations in total to pin down the values of these six
parameters. Using a search algorithm, I ?nd that these conditions are satis?ed for
the following combination of parameter values: p = 0.06, ? = 0.18, a
g
= 78, a
u
= 62,
a
g
0
= 73, a
u
0
= 57. By applying these a
g
, a
u
, a
g
0
and a
u
0
to the steady state industry
structure, I ?nd D
h
= 2899 and D
l
= 2464.
The value of µ is calibrated to match the observed standard deviation of the job
destruction rate. In my model, the job destruction rate jumps above its mean when
demand drops and falls below when demand rises. Thus, the frequency of demand
switches between D
h
and D
l
determines the frequency with which the job destruction
rate ?uctuates between 11.6% and 3%, which in turn a?ects the standard deviation
of the simulated job destruction series. My calibration exercises suggest µ = 0.58.
Finally, the entry cost parameters are adjusted to match the observed mean job
creation rate of 5.19%.
47
The job destruction rate implied by (a
g
0
, a
u
0
) is slightly higher since a
g
0
< a
g
and a
u
0
< a
u
.
83
3.4.3 Response to a Negative Demand Shock and Simulations of U.S.
Manufacturing Job Flows
With all of the parameter values assigned, I approximate ?rms’ value functions apply-
ing the computational strategy described in subsection 4.1. With the approximated
value functions, the corresponding decision rules and an initial ?rm distribution, I
can investigate the dynamics of my model’s key variables along any particular path
of demand realizations, and study the model’s quantitative implications.
Scarring and Cleansing over the Cycle To assess the e?ect of a negative de-
mand shock, I start with a random ?rm distribution and simulate my model with
demand level equal to D
h
for the ?rst 200 quarters. Regardless of the initial ?rm
distribution, I ?nd that the exit age of good ?rms settles down to 76, the exit age of
unsure ?rms settles down to 62, the job destruction rate converges to 5.38%, and the
fraction of good ?rms converges to 49.8%. This suggests that my model is globally
stable. Once the key variables converge, I simulate the e?ects of a negative demand
shock that persists for the next 87 quarters.
The dynamics of the job destruction rate and the job creation rate are illustrated in
Panel 1 of Figure 13, with the quarter labeled 0 denoting the onset of a recession. The
job destruction rate goes up from 5.38% to 10.84% on impact. Thus, the immediate
e?ect of a negative demand shock is to clear out some ?rms that would have stayed
in had demand remained high. After 70 quarters, the job destruction rate converges
to 5.63%, still above its original value. Hence, the conventional cleansing e?ect of
demand on job destruction that I establish analytically in steady state carries over
with probabilistic cycles.
84
Unlike the job destruction rate, the job creation rate drops from 4.69% to 4.32%
when a recession strikes, rises gradually and converges later. This matches the ?nding
of Davis and Haltiwanger (1992) that the job creation rate falls during recessions and
co-moves negatively with the job destruction rate over the cycle.
48
The analysis of the steady state also suggests that recessions will bring a scarring
e?ect by shifting labor resources toward bad ?rms. As shown in Panel 2 of Figure
13, the fraction of labor at good ?rms drops from 49.8% to 48.07% when the negative
demand shock strikes and converges to 47.87% after 70 quarters. This implies that
the negative demand shock shifts the cross-idiosyncratic productivity ?rmdistribution
toward bad ?rms. Hence, the scarring e?ect suggested by the steady-state analysis
also carries over with probabilistic business cycles.
Two remarks are in order regarding the response of the fraction of labor at good
?rms to a negative demand shock. First, the initial drop in l
g
at the onset of a
recession contradicts my argument in Section 3.2 that the scarring e?ect takes time
to work. My calibration exercises suggest that this feature is robust and can be
understood as follows. Recessions shift both exit margins to younger ages. While the
shift of the exit margin for unsure ?rms clears out both bad ?rms and good ?rms,
the shift of the exit margin for good ?rms clears out only good ?rms, so that in total
more good ?rms are cleared out than bad ?rms initially and l
g
drops at the onset
of a recession. Since l
g
eventually converges to a value below the initial drop, and
the initial drop in l
g
also stems from learning, this result does not hurt my argument
that in a model with learning, recessions create a scarring e?ect by shifting resources
toward bad ?rms.
48
Davis and Haltiwanger (1999) report a correlation coe?cient of ?0.17 of job destruction and
job creation for the U.S. Manufacturing from 1947:1-1993:4.
85
0 20 40 60 80 100
0.04
0.05
0.06
0.07
0.08
0.09
0.1
0.11
Panel 1: Job Reallocation Rates
0 20 40 60 80 100
0.475
0.48
0.485
0.49
0.495
0.5
Panel 2: Fraction of Good Firms
0 20 40 60 80 100
0.99
0.995
1
1.005
1.01
1.015
Panel 3: Average Labor Productivity
0 20 40 60 80 100
?20
?15
?10
?5
0
5
x 10
?3
Panel 4: Scar
Job Destruction Rate
Job Creation Rate
Vin
Prod
Figure 13: Response to a Negative Demand Shock: vin is the detrended average
labor productivity driven only by the cleansing e?ect, prod is the detrended average
labor productivity driven by both the cleansing e?ect and the scarring e?ect. Scar =
prod?vin. The horizontal axis denotes quarters, with the quarter labeled 0 denoting
the onset of a recession.
86
Second, the response of l
g
shown in Panel 2 is hump-shaped: it drops initially,
increases gradually, then declines again. This feature is mainly due to the response of
the exit margins over the cycle. When a recession ?rst strikes, the exit margins over-
shift to the left, and shift back gradually as the recession persists. As the exit margin
for unsure ?rms shifts back, more good ?rms are allowed to reach their potential;
meanwhile, as the exit margin for good ?rms shifts back, no old good ?rms exit for
several quarters. Hence, l
g
increases after the initial drop. The exit margins reach
their stable points after about 20 quarters. From then on, l
g
starts to fall, with old
good ?rms gradually being cleared out but not enough new good ?rms being realized.
Another part of this hump-shaped response comes from the entry margin. Because
they have had no time to learn, newly entered cohorts have the least e?cient cross-
idiosyncratic productivity ?rm distribution in the industry, so that entry tends to
drive down l
g
. When entry falls in a recession, the negative impact of entry on l
g
is
also reduced, which contributes to part of the increase in l
g
after the initial drop.
To summarize, despite some transitory dynamics, Panel 1 and Panel 2 of Figure
13 suggest that both the conventional cleansing e?ect established in Proposition 3.2,
and the scarring e?ect established in Proposition 3.3, carry over with probabilistic
business cycles.
Implications for Productivity Next, I turn to the quantitative implications of
the model for the cyclical behavior of average labor productivity. With one worker
per ?rm setup and ?rm-level productivity given by
A·?
(1+?)
a , average labor productivity
is a?ected by A, the level of the leading technology, and the ?rm distribution across
a and ?. While technological progress drives A, and thus average labor productivity,
to grow at a trend rate ? (the technological pace), demand shocks add ?uctuations
87
around this trend by a?ecting the labor distribution across a and ?.
To analyze the ?uctuations of average labor productivity over the cycle, I de?ne
de-trended average labor productivity as the average of
?
(1+?)
a over heterogeneous ?rms.
In evaluating this measure, recall that there are two competing e?ects. On the one
hand, the cleansing e?ect drives down the average a by lowering the cut-o? ages for
each idiosyncratic productivity, causing average labor productivity to rise. On the
other hand, the scarring e?ect drives down the average ? by shifting resources away
from good ?rms, causing average labor productivity to fall. To separate the two
e?ects, I generate two indexes for average labor productivity. The ?rst index is the
average of
?
(1+?)
a across all ?rms in operation, de?ned as the following:
prod =
P
f
³
?
e
(1+?)
a
´
· f (?
e
, a)
P
f
f (?
e
, a)
.
This measure is a?ected by both cleansing and scarring e?ects. The other index is
the average of
1
(1+?)
a across all existing ?rms, de?ned as:
vin =
P
f
³
1
(1+?)
a
´
· f (?
e
, a)
P
f
f (?
e
, a)
.
This measure is a?ected only by the cleansing e?ect. To compare the relative mag-
nitude of these two e?ects, their initial levels are both normalized as 1. Since only
the cleansing e?ect drives the dynamics of vin but both cleansing and scarring e?ects
drive the dynamics of prod, the gap between vin and prod re?ects the magnitude of
the scarring e?ect. A scarring index measures this gap. It is de?ned as:
88
scar = prod ?vin.
Panel 3 in Figure 13 traces the evolution of vin and prod in response to a negative
demand shock. As the negative demand shock strikes, the cleansing e?ect alone
raises the average labor productivity to 1.013 while the scarring e?ect brings the
average labor productivity down to 0.9974. After 70 quarters, prod converges to 0.9947
while vin converges to 1.0126. The dynamics of the scarring index in response to a
negative demand shock is plotted in Panel 4 of Figure 13. The scarring index remains
negative following a negative demand shock and eventually converges to ?0.0179.
This matches the predictions of my model that the scarring e?ect plays against the
conventional cleansing e?ect during recessions by shifting resources away from good
?rms, driving down the average labor productivity.
3.4.4 Simulation of U.S. Manufacturing Job Flows
To gauge whether the scarring e?ect is likely to be relevant at business cycle frequen-
cies, I simulate my model’s response to random demand realizations generated by
the model’s Markov chain. I perform 1000 simulations of 87 quarters each. Results
are presented in Table 8. The reported statistics are means (standard deviations)
based on 1000 simulated samples. Sample statistics for U.S. Manufacturing data for
the 87 quarters from 1972 (II) to1993(IV) are included for comparison. In the table,
jd and jc represent the job destruction and job creation rate; prod and q represent
de-trended average labor productivity and de-trended output.
Table 8 suggests that my calibrated model can replicate the observed patterns of
job ?ows; moreover, the positive correlation coe?cient of 0.1675 between prod and q
implies that my model generates procyclical average labor productivity for the U.S.
89
simulation statistics data
jd
mean
5.29%(0.0100%) 5.6%
jd
std
1.65%(0.3100%) 1.66%
jc
mean
4.72%(0.0581%) 5.19%
jc
std
0.72%(0.0595%) 0.95%
corr(prod, q) 0.1675(0.7504) 0.5537
?
Table 8: Means (std errors) of 1000 Simulated 87-quarter Samples: jd is the job de-
struction rate, jc is the job creation rate, prod is detrended average labor productivity,
q is detrended aggregate output. Data comes from the U.S. Manufacturing job ?ow
series for 1972:2-1993:4, compiled by Davis and Haltiwanger. *Detrended average
labor productivity is calculated as output per production worker, with output mea-
sured by industrial production index. The quarterly series of industrial production
index of U.S. manufacturing sector for 1972:2-1993:4 comes from the Federal Reserve
and the series of total production workers comes from the Bureau of Labor Statistics.
manufacturing sector in the relevant period. Put di?erently, under my benchmark
calibration the scarring e?ect on cyclical productivity dominates the cleansing e?ect.
3.5 Sensitivity Analysis of the Dominance of Scarring over
Cleansing
In the baseline parameterization of subsection 4.2, I followed Caballero and Ham-
mour (1994) in setting the quarterly technological pace ? equal to 0.007. The value
was estimated by attributing all output growth of the U.S. manufacturing sector to
technological progress, which may exaggerate the technological pace in the relevant
period. An alternative estimate of ?, has been provided by Basu, Fernald and Shapiro
(2001), who estimate TFP growth for di?erent industries in the U.S. from1965 to 1996
after controlling for employment growth, factor utilization, capital adjustment costs,
quality of inputs and deviations from constant returns and perfect competition. They
estimate a quarterly technological pace of 0.0037 for durable manufacturing, a pace
90
Calibration Results ? = 0.003 ? = 0.007
calibrated p 0.0830 0.0800
calibrated ? 0.1200 0.1420
Response to a Negative
Demand Shock
vin (when a recession
strikes)
1.0052 1.0130
vin (70 quarters after a re-
cession strikes)
1.0029 1.0126
prod (when a recession
strikes)
0.9866 0.9974
prod (70 quarters after a re-
cession strikes)
0.9820 0.9947
scar (when a recession
strikes)
?0.0186 ?0.0156
scar (70 quarters after a re-
cession strikes)
?0.0209 ?0.0179
Table 9: Sensitivity Analysis to a Slower Technological Pace (I): prod is detrended
average labor productivity, driven by both the cleansing and the scarring e?ects, vin
is the component of detrended average labor productivity driven only by the cleansing
e?ect, scar = prod - vin. Other parameter values are as shown in Table 2.
of 0.0027 for non-durable manufacturing and an even slower pace for other sectors.
How would a slow pace of technological progress a?ect the magnitudes of the
scarring and cleansing e?ects? To address this question, I re-calibrate my model
assuming ? = 0.003, matching the same moments of job creation and destruction as
before, and simulate responses to a negative demand shock. The results are presented
in Table 9 together with results from the baseline parameterization.
The calibration results in Table 9 suggest that the model with ? = 0.003 needs a
faster learning pace (p = 0.083 compared to 0.08) and a smaller prior probability of
?rms’ being good (? = 0.120 compared to 0.142) to match the observed moments of
91
simulation sta-
tistics with
? = 0.003
simulation sta-
tistics with
? = 0.007
data
jd
mean
5.73%(0.0799%) 5.29%(0.0100%) 5.6%
jd
std
1.42%(0.2800%) 1.65%(0.3100%) 1.66%
jc
mean
5.14%(0.0565%) 4.72%(0.0581%) 5.19%
jc
std
0.34%(0.0059%) 0.37%(0.0535%) 0.95%
corr(prod, q) 0.4819(0.5212) 0.1675(0.7504) 0.5537
Table 10: Sensitivity to A Slower Technological Pace (II): Means (std errors) of 1000
Simulated 87-quarter Samples. De?nitions, measures and data sources are the same
as Table 4.
job ?ows.
49
The simulated responses suggest that slower technological progress mag-
ni?es the scarring e?ect, weakens the cleansing e?ect, and magni?es the procyclical
behavior of productivity.
This result can be explained as follows. First, slower technological progress implies
that the force of creative destruction is weak. A lower ? weakens the technical dis-
advantage of old ?rms and allows both good ?rms and unsure ?rms to live longer, so
that less job destruction occurs at the exit margins. A lower ? also implies a smaller
cleansing e?ect on average labor productivity. A recession clears out marginal ?rms
by shifting the exit margins toward younger ages. The size of the shift is pinned
down in my calibration exercises by matching jd
max
? 11.6%. Given the shift of exit
margins, a slower technological pace shrinks the productivity di?erence between the
vintages that have been killed and the ones that have survived, so that the impact of
49
Consider (9), the expression of jd
ss
, for intuition. My calibration exercises look for parameter
values that satisfy three moment conditions on job ?ows, one of which is that jd
ss
? 5.6%. Propo-
sition 3 establishes that jd
ss
decreases with the exit ages (a
g
and a
u
). It can be further shown that
it increases in p but decreases in ?. A slower technological pace weakens the technical disadvantage
of old ?rms and extends their life span so that both a
g
and a
u
tend to increase. Hence, the job
destruction rate would decrease if p and ? remain the same. A faster learning pace and a lower prior
probability of being good are thus needed to match the observed mean job destruction. Thus, the
paramerization of my model with ? = 0.003 suggests that more job destruction comes from learning
rather than creative destruction.
92
the cleansing e?ect on average labor productivity declines.
Second, when I assume a lower ?, I must also assume a higher p and a lower ? to
match the moments of job destruction. This re-calibration implies a larger role for
learning in job destruction: ?rms not only learn faster, but are more likely to learn
that they are bad. This also gives a larger scarring e?ect on average labor produc-
tivity: a faster learning pace implies a higher opportunity cost of not allowing unsure
?rms to survive; a smaller prior probability of being good suggests that learning has
a greater marginal impact on cross-idiosyncratic productivity e?ciency.
Table 10 reports the simulation statistics of 1000 simulated 87-quarter samples
when ? = 0.003. Results when ? = 0.007 and sample statistics from data are included
for comparison. My model with ? = 0.003 generates a correlation coe?cient of 0.4819
between detrended average labor productivity and detrended output. Productivity is
strongly procyclical, almost as much as in the data.
3.6 Cost of Business Cycles with Heterogeneous Firms
This sub-section explores possible welfare cost of business cycles with the scarring
e?ect’s presence. Suppose my modeled industry’s output is consumed by a represen-
tative consumer, whose utility depends positively on the level of consumption. Then
lower output implies lower consumption and consequently lower welfare. Since the
scarring e?ect drives down average labor productivity during recessions, it can lead
to lower equilibrium output, and hence imply a welfare cost.
To proceed, I compare the output series of two industries, a cyclical industry
whose demand follows a Markov chain and a steady-state industry with time-invariant
demand. I let the steady-state industry’s demand equal the cyclical industry’s average
93
0 20 40 60 80 100 120 140 160 180 200
5400
5500
5600
5700
5800
5900
6000
steady?state industry
cyclical industry
Figure 14: Time Series of Detrended Output of a Cyclical industry and a Steady-
state Industry. Dashed line represents the average of cyclical output. This ?gure is
generated using the base-line calibration, with the steady-state industry’s demand
level assumed equal to the cyclical industry’s average demand level.
94
0 20 40 60 80 100 120 140 160 180 200
Figure 15: Time Series of Output with Trend of a Cyclical Industry and a Steady-
state Industry. This ?gure is generated with the base-line calibration, assuming the
steady-state industry’s demand equal to the cyclical industry’s average demand.
95
demand. The results are shown in Figure 14 and Figure 15. Both ?gures are generated
with baseline calibrations. The cyclical industry’s demand switches between 2899 and
2464 with probability 0.42 (1?0.58). This implies an average demand of 2657, which
is applied as the steady-state industry’s demand.
Figure 14 presents the time series of the two industries’ de-trended output. The
cyclical industry’s de-trended output ?uctuates around a mean of 5515.8, below the
steady-state industry’s de-trended output of 5910.6. Figure 15 shows the two indus-
tries’ output series with technological progress added. Both industries’ outputs grow.
But only the cyclical industry’s output ?uctuates around the growth trend. Moreover,
the cyclical output series stay strictly below the steady-state output series.
Figure 14 and Figure 15 suggest that, in my model’s framework, more output
would be produced if business cycles could be eliminated. Hence, business cycles
may possibly bring a welfare cost of a representative consumer who consumes the
industry’s output.
The discussion of welfare cost of business cycles traces back to Lucas (1987), who
put forth an argument that the welfare gains from reducing the volatility of aggregate
consumption is negligible. Subsequent work that revisited Lucas calculation continued
to ?nd only small bene?ts from reducing the consumption volatility, reinforcing the
perception that business cycles do not matter. However, with a heterogeneous-?rm
setup, my model argues from the supply side that the business cycles reduces the
average output by a?ecting production e?ciency. In Lucas’ argument, eliminating
business cycles only eliminates the consumption volatility, but does not a?ect the
mean of consumption. My model suggests that, eliminating business cycles may
raise up the mean by providing more output.
50
Although my analysis here is very
50
Since consumption grows over time, this mean refers to the mean of detrended consumption.
96
preliminary,
51
it does point out an interesting research direction.
3.7 Conclusion
How do recessions a?ect resource allocation? My theory suggests learning has impor-
tant consequences for this question. I posit that in addition to the cleansing e?ect
proposed by previous authors, recessions create a scarring e?ect by interrupting the
learning process. Recessions kill o? potentially good ?rms, shift resources toward
bad ?rms and exacerbate the allocative ine?ciency in an industry. The empirical
relevance of the scarring e?ect is examined in Section 4. Using data on U.S. man-
ufacturing job ?ows, I ?nd that the scarring e?ect dominates the cleansing e?ect in
the U.S. manufacturing sector from 1972 to 1993, and can account for the observed
degree of procyclical productivity.
The scarring e?ect stems from learning. Recessions bring a scarring e?ect by
limiting the learning scope. Figure 3 of the paper provides intuition. Recessions force
?rms to exit at earlier ages. The shortened ?rm life allows less learning time, so that
fewer truly good ?rms get to realize their potential and the shaded area in Figure 3
would disappear. The decrease in the fraction of labor at good ?rms implies a less
e?cient allocation of labor during recessions.
My theory highlights a ?rm’ age as an indicator for its number of learning op-
portunities. The existing empirical literature documents that ?rm age has important
A higher mean of detrended cosumption (output) is shown in Figure 15: although both growing
over time, the steady-state series with a higher detrended mean stays strictly above the other series.
Another interesting exploration of the welfare cost of business cycles is Barlevy (2003), who posits
that eliminating cycles may give rise to a higher growth rate of consumption.
51
A more careful exploration of this question should study cyclical labor supply and cyclical
equilibrium labor input. Seperating the cleansing e?ect from the scarring e?ect is also important.
97
explanatory power for micro-level job ?ow patterns.
52
My model predicts that the
mean and the dispersion of ?rm age both decline during recessions, while the pro-
ductivity dispersion within an age cohort goes up on average. These are testable
hypotheses with detailed data on the age distribution of ?rms over the cycle.
The empirical relevance of the scarring e?ect remains to be explored in a wider
framework. My calibration exercises have focused on the U.S. manufacturing sec-
tor, where job destruction is more responsive to business cycles than job creation.
However, Foote (1997) documents that in services, ?re, transportation and commu-
nications, retail trade, and wholesale trade, job creation is more volatile than job
destruction. Would relatively more responsive job creation hurt the dominance of the
scarring e?ect? It could, since recessions leave “scars” by killing o? potentially good
?rms on the destruction side. It may not, because a larger decline in job creation
also introduces fewer potentially good ?rms on the creation side. Whether “scarring”
dominates “cleansing” in sectors other than manufacturing remains an interesting
question.
52
See Caves (1998) for an extensive review of recent ?ndings on ?rm turnover and industrial
dynamics.
98
4 Appendix
Proof of Proposition 2.1 and 3.1(three steps):
Step1: to prove that
V (?
e
,a;F,D)
?a
< 0:
Proof. Compare two ?rms with same belief ?
e
, but di?erent ages a
1
> a
2
. To prove
V (?
e
,a;F,D)
?a
< 0, I need to show that
V (?
e
, a
1
; F, D) < V (?
e
, a
2
; F, D) .
Suppose that the aggregate state is (F, D) at the beginning of period t
0
. I assume
there are n di?erent possible paths of demand realizations from t
0
onward, each with
probability p
i
, where i = 1, ..., n. I also assume that under the i’th path of demand
realizations, the ?rm with a
1
expects itself to exit at the end of period t
i
1
? t
0
and
the ?rm with a
2
expects itself to exit at the end of period t
i
2
? t
0
, then:
V (?
e
, a
1
; F, D) =
n
X
i=1
t
i
1
X
t=t
0
©
?
t?t
0
E
£
?
i
t
(?
e
, a
1
+t ?t
0
) |F, D
¤ª
· p
i
,
and
V (?
e
, a
2
; F, D) =
n
X
i=1
t
i
2
X
t=t
0
©
?
t?t
0
E
£
?
i
t
(?
e
, a
2
+t ?t
0
) |F, D
¤ª
· p
i
,
where ?
i
t
(?
e
, a
1
+t ?t
0
) is the expected pro?t (of a ?rm with current age a
1
and current belief ?
e
) at period t ? t
0
under demand path i. Firms have rational
expectations and expect a price sequence {P
i
t
(F, D)}
t?t
0
conditional on the realization
99
of path i. Since price is competitive and ?rms are price takers, I must have:
V (?
e
, a
1
; F, D) =
n
X
i=1
t
i
1
X
t=t
0
©
?
t?t
0
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤ª
· p
i
and
V (?
e
, a
2
; F, D) =
n
X
i=1
t
i
2
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
· p
i
.
There are three possibilities for any i.
Possibility 1, if t
i
1
= t
i
2
= t
i
:
since A(t
0
?a
1
) < A(t
0
?a
2
),
(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1 < A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
holds for any t. Hence,
t
i
X
t=t
0
©
?
t?t
0
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤ª
<
t
i
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
Possibility 2, if t
i
1
< t
i
2
:
100
then it must be true that,
t
i
2
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
=
t
i
1
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
+
t
i
2
X
t=t
i
1
+1
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
,
and hence,
t
i
1
X
t=t
0
©
?
t?t
0
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤ª
<
t
i
2
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
,
Possibility 3, if t
i
1
> t
i
2
:
when it comes to period t
i
2
under path i, the ?rm aged a
1
+t
i
2
?t
0
chooses to stay
and the ?rm aged a
2
+t
i
2
?t
0
decides to leave. Based on the exit condition, it must
be true that,
V
¡
?
e
, a
1
+t
i
2
?t
0
; F
0
, D
0
¢
> 0andV
¡
?
e
, a
2
+t
i
2
?t
0
; F
0
, D
0
¢
< 0.
The ?rm aged a
1
+t
i
2
?t
0
chooses to stay to capture the potential pro?t
t
i
1
X
t=t
i
2
+1
n
?
t?t
i
2
·
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤
o
101
and he expects those future pro?ts can cover any possible cost if demand path does
not goes as expected. Since
t
i
1
X
t=t
i
2
+1
n
?
t?t
i
2
·
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤
o
<
t
i
1
X
t=t
i
2
+1
n
?
t?t
i
2
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤
o
,
the ?rm aged a
2
+ t
i
2
? t
0
should have expected even higher potential pro?ts in the
future which is worth waiting for. Hence, it must not choose to leave at period t
i
2
.
Therefore, t
i
1
> t
i
2
cannot be true.
1), 2) and 3) help me conclude that:
t
i
1
X
t=t
0
©
?
t?t
0
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤ª
<
t
i
2
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
holds for any i. Then it must be true that,
n
X
i=1
t
i
1
X
t=t
0
©
?
t?t
0
£
A(t
0
?a
1
) ?
e
P
i
t
(F, D) ?1
¤ª
p
i
<
n
X
i=1
t
i
2
X
t=t
0
©
?
t?t
0
·
£
A(t
0
?a
2
) ?
e
P
i
t
(F, D) ?1
¤ª
p
i
or
V (?
e
, a
1
; F, D) < V (?
e
, a
2
; F, D) .
102
Step 2: to prove
V (?
e
,a;F,D)
??
e
> 0.
Proof. It is similar to the proof of
V (?
e
,a;F,D)
?a
> 0.
Step 3: to prove the existence of cut-o? age a (?
e
; F, D) and a (?
e0
; F, D) ?
a (?
e
; F, D), for ?
e0
> ?
e
.
Proof. The existence of a (?
e
; F, D) is straightforward. Holding ?
e
constant, V (?
e
, a; F, D)
is monotonically decreasing in a, then there must be a (?
e
; F, D) such that
V (?
e
, a (?
e
; F, D) ; F, D) > 0
but
V (?
e
, a (?
e
; F, D) + 1; F, D) ? 0.
And since
V (?
e
,a;F,D)
??
e
> 0, I have:
V
³
?
e
0
, a (?
e
; F, D) ; F, D
´
> V (?
e
, a (?
e
; F, D) ; F, D) = 0 holds for any ?
e0
> ?
e
.
Therefore, it must be true that a (?
e0
; F, D) ? a (?
e
; F, D).
PROOF OF PROPOSITION 3.2 (three steps):
Proof. Step 1: to show that a steady state features time-invariant P
t
A
t
, such that
P
t
A
t
= PA, ? t, where P
t
represents the equilibrium price and A
t
represents the
leading technology in period t.
The condition of competitive pricing tells that:
D
t
= P
t
· Q
t
.
103
Q
t
is the aggregate output over heterogeneous ?rms.
Q
t
=
X
a
X
?
e
A
t
?
e
f
t
(?
e
, a) (1 +?)
?a
.
so that:
D
t
= P
t
A
t
·
X
a
X
?
e
?
e
f
t
(?
e
, a) (1 +?)
?a
. (1)
By de?nition, a steady state features constant level of demand, D
t
= D (? t). and
time-invariant ?rm distribution. Let f (?
e
, a) denote the number of ?rms with (?
e
, a)
and a
g
, a
u
denote the maximum ages for good ?rms and unsure ?rms in operation,
respectively. The above equation can be rewritten as:
D = P
t
A
t
·
½
a
u
P
a=0
£
?
u
f (?
u
, a) (1 +?)
?a
¤
+
a
g
P
a=1
£
?
g
f (?
g
, a) (1 +?)
?a
¤
¾
so that
P
t
A
t
=
D
½
a
u
P
a=0
[?
u
f (?
u
, a) (1 +?)
?a
] +
a
g
P
a=1
[?
g
f (?
g
, a) (1 +?)
?a
]
¾.
Hence, P
t
A
t
must be time-invariant. I let P
t
A
t
= PA.
Step 2: solve for a
g
?a
u
by ?rms’ exit conditions.
At a steady state, the aggregate state {D, F} is perceived to be time-invariant.
Thus, good ?rms know they will live until a
g
, and unsure ?rms know they will live
until a
u
. The time-invariant decision rules at the steady state imply time-invariant
value functions. Let V (?
e
, a) represent the steady-state expected value of staying of
a ?rm with belief ?
e
and age a.
104
Since a
g
denote the maximum age of good ?rms in operation, and V (?
g
, a) de-
creases in a monotonically, the condition of ?rm rationality suggests it must be true
for a
g
that:
V (?
g
, a
g
) = 0
?
g
PA(1 +?)
?a
g
?1 = 0
so that
PA =
(1 +?)
a
g
?
g
. (2)
Similarly, exit condition for unsure ?rms suggest:
V (?
u
, a
u
) = 0
?
u
PA(1 +?)
?a
u
?1 +?p?V (?
g
, a
u
+ 1) = 0
?
u
PA(1 +?)
?a
u
?1 +?p?
a
g
X
a=au+1
?
a?a
u
?1
£
?
g
PA(1 +?)
?a
?1
¤
= 0
With (15) plugged in, I have (8):
µ
?
u
?
g
+
p??
1 +? ??
¶
(1 +?)
a
g
?a
u
= 1 +
p??
1 ??
?
p???
(1 ??) (1 +? ??)
?
a
g
?a
u
(8)
which can be re-written as:
F (a
g
?a
u
) = G(a
g
?a
u
)
Proposition 1 suggests that a
g
?a
u
? 0. To establish the existence of a
g
?a
u
? 0
that satis?es the above equation, I need to show that F and G cross each other at a
105
positive value of a
g
?a
u
.
G
0
= ?
p???
(1 ??) (1 +? ??)
?
a
g
?a
u
ln? > 0, but
G
00
= ?
p???
(1 ??) (1 +? ??)
?
a
g
?a
u
(ln ?)
2
< 0
moreover,
F (0) =
?
u
?
g
+
p??
1 +? ??
, and
G(0) = 1 +
p??
1 +? ??
.
and:
F (0) < G(0)
because
?u
?g
< 1 by de?nition (?
u
= ??
g
+ (1 ??) ?
b
and ?
g
> ?
b
). F (0) < G(0)
suggests that the curve of F starts at a
g
? a
u
= 0 below the curve of G. F
0
> 0
and G
0
> 0 imply that both of F and G increase monotonically in a
g
?a
u
. F
00
> 0
suggests that F is convex but G
00
< 0 suggests that G is concave. Hence, F and G
must cross once at a positive value of a
g
?a
u
, as shown in the following ?gure:
a a
g u
? 0
( ) F a a
g u
?
( ) G a a
g u
?
( ) F 0
( ) G 0
a a
g u
? 0
( ) F a a
g u
?
( ) G a a
g u
?
( ) F 0
( ) G 0
106
Therefore, (8) alone determines a unique value for a
g
?a
u
.
Step 3, solve for f (0) and a
g
by combining the free entry condition and the com-
petitive pricing condition:
V (?
u
, 0) = C (f (0))
where f (0) represents the size of the entering cohort. With time-invariant life-cycle
dynamics for each cohort shown in Figure 2, I have:
V (?
u
, 0) =
a
u
X
a=1
?
a
·
PA?
u
(1 +?)
a
?1
¸
?(?
u
, a) +
ag
X
a=1
?
a
·
PA?
g
(1 +?)
a
?1
¸
?(?
g
, a)
where ?(?
u
, a) denotes the probability of staying in operation at age a as an unsure
?rm, and ? (?
g
, a) denotes the probability of staying in operation at age a as a good
?rm. All-or-nothing learning suggests that:
?(?
u
, a) = (1 ?p)
a
for 0 ? a ? a
u
,
?(?
g
, a) = ?[1 ?(1 ?p)
a
] for 0 ? a ? a
u
,
?(?
g
, a) = ?
h
1 ?(1 ?p)
au+1
i
for a
u
+ 1 ? a ? a
g
Plugging ?(?
u
, a), ?(?
g
, a) and PA =
(1+?)
a
g
?g
into V (?
u
, 0), I have:
(1 +?)
a
g
?
g
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
_
au
P
a=1
?
a
_
¸
_
(1 ?p)
a
³
?
u
(1+?)
a ?1
´
+
?(1 ?(1 ?p)
a
)
³
?
g
(1+?)
a ?1
´
_
¸
_
+
?
³
1 ?(1 ?p)
a
u
+1
´ ag
P
a=a
u
+1
?
a
³
?g
(1+?)
a ?1
´
+
?
u
?1
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
_
= C (f (0)) (3)
107
Plugging PA =
(1+?)
a
g
?g
back into (14) and applying the steady state industry
structure suggested by all-or-nothing learning and exit ages, I have:
f(0) ·
(1 +?)
ag
?
g
_
¸
¸
_
(?
u
???
g
)
au
P
a=1
³
1?p
1+?
´
a
+??
g
ag
P
a=1
³
1
1+?
´
a
+
??
g
(1 ?p)
a
u
+1
a
g
P
a=a
u
+1
³
1
1+?
´
a
_
¸
¸
_
= D (4)
a
g
? a
u
has been given by (8). The left-hand sides of (16) and (17) are both
monotonically increasing in a
g
; The left-hand side and the right-hand side of (16) are
both monotonically increasing in f (0). Hence, with a
u
replaced by a
g
? (a
g
?a
u
),
(16) and (17) jointly determine a
g
and f (0).
Therefore, for any D, there exists a steady state that can be captured by {f (0) , a
g
, a
u
}.
PROOF OF PROPOSITION 3.3:
Proof. To prove that
d(ag)
dD
? 0 and
d(a
u
)
dD
? 0 at the steady state, combining (16 )
with (17) and replacing a
u
by a
g
?(a
g
?a
u
) gives the following:
(1 +?)
a
g
?
g
_
¸
¸
_
(?
u
???
g
)
a
u
P
a=1
³
1?p
1+?
´
a
+??
g
a
g
P
a=1
³
1
1+?
´
a
+
??
g
(1 ?p)
au+1
ag
P
a=a
u
+1
³
1
1+?
´
a
_
¸
¸
_
·
c
?1
_
_
_
_
_
_
_
_
_
_
(1 +?)
ag
?
g
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
_
a
u
P
a=1
?
a
_
¸
_
(1 ?p)
a
³
?u
(1+?)
a ?1
´
+
?(1 ?(1 ?p)
a
)
³
?
g
(1+?)
a ?1
´
_
¸
_
+
?
³
1 ?(1 ?p)
a
u
+1
´ a
g
P
a=a
u
+1
?
a
³
?
g
(1+?)
a ?1
´
+
?
u
?1
_
¸
¸
¸
¸
¸
¸
¸
¸
_
¸
¸
¸
¸
¸
¸
¸
¸
_
_
_
_
_
_
_
_
_
_
_
= D
108
The left-hand is monotonically increasing in a
g
. Hence,
d(ag)
dD
? 0. With a
g
? a
u
independent of D as suggested by (8),
d(a
u
)
dD
=
d(a
g
?(a
g
?a
u
))
dD
? 0.
PROOF OF PROPOSITION 3.4:
Proof. Since r
g
= 1 ?
(1??)
p?au
1?(1?p)
a
u
+(1??)+p?(a
g
?a
u
)
and a
g
?a
u
is independent of D,
d (r
g
)
d (D)
=
d (r
g
)
d (a
u
)
·
d (a
u
)
d (D)
Proposition 2 has established that
d(a
u
)
d(D)
? 0. Therefore,
d(rg)
d(D)
? 0 if and only if
d(r
g
)
d(au)
? 0.
With
au
1?(1?p)
au
= x,
d(rg)
d(a
u
)
=
d(rg)
d(x)
·
d(x)
d(a
u
)
. Since
d(rg)
d(x)
> 0,
d(rg)
d(a
u
)
? 0 if and only if
d(x)
d(a
u
)
? 0.
Hence, I need to prove that
d(x)
d(a
u
)
? 0.
1 ?(1 ?p)
au
is plotted in the following graph as a function of a
u
. Since
d
³
1 ?(1 ?p)
a
u
´
d (a
u
)
= ?(1 ?p)
a
u
· ln(1 ?p) > 0
but
d
2
³
1 ?(1 ?p)
a
u
´
d (a
u
)
2
= ?(1 ?p)
au
· (ln(1 ?p))
2
< 0,
the curve is concave.
a
u
1 1 ? ? ( ) p
au
?
a
u
1 1 ? ? ( ) p
au
a
u
1 1 ? ? ( ) p
au
?
a
u
1 1 ? ? ( ) p
au
109
Clearly, it indicates that x =
a
u
1?(1?p)
au
= cot (?) .The concavity of the curve
suggests that as a
u
increases, the angle of ? shrinks and cot (?) increases. Therefore,
x increases in a
u
.
Results from two-moment Krusell-Smith approach:
110
? {X, ?
a
}
H
?
booms ( log X):
log X
0
= 0.1261 + 0.9653 log X + 0.3246?
a
recessions( log X):
?
0
a
= 0.0079 + 0.0076 log X + 0.8988?
a
booms (?
a
):
log X
0
= ?0.1485 + 0.9291 log X + 1.0317?
a
recessions(?
a
):
?
0
a
= 0.0789 + 0.0166 log X + 0.6924?
a
R
2
booms ( log X): 0.9940
recessions( log X): 0.9287
booms (?
a
): 0.9571
recessions(?
a
): 0.5812
standard forecast
error
booms ( log X): 0.0000069741%
recessions( log X): 0.000068307%
booms (?
a
): 0.00012513%
recessions(?
a
):0.00097406%
maximum forecast
error
booms ( log X): 0.000087730%
recessions( log X):0.0016626%
booms (?
a
):0.0014396%
recessions(?
a
):0.028074%
Den Haan &
Marcet test statis-
tic (?
2
7
)
0.9216
111
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