Firm Demographics and the Great Recession∗
Gian Luca Clementi†
Berardino Palazzo‡
Peifan Wu§
NYU and NBER
Boston University
NYU
This version: March 20, 2017
Preliminary and Incomplete
Abstract
The last U.S. recession stands out not only for its depth, but also for the rather slow
recovery that followed it. What is less well known is that the number of productive
units also dropped substantially, while it barely budged in occasion of the 1981 recession, and kept increasing during the 1991 and 2001 recessions. To the extent that the
stock of establishments is a very slow–moving variable, a recession characterized by an
unusually large drop in establishments will necessarily be followed by a slow recovery.
In order to evaluate the quantitative significance of this mechanism, we build a general equilibrium business cycle model with heterogenous firms and endogenous entry
and exit, and calibrate it so that the implied unconditional average firm–level and
aggregate dynamics are consistent with the empirical evidence. Preliminary results
show that recessions characterized by a sizeable drop in entry rates stand out along
two dimensions: (1) The response of output in hump-shaped, meaning that output
can diverge from trend for years following a negative mean–reverting shock; (2) The
recovery is substantially slower.
Key words: Firm Heterogeneity, Entry, Exit
JEL Codes: Some JEL Codes
∗
We are grateful to Richard Rogerson, Stephen Terry, and Thomas Winberry for insightful comments
and suggestions. All errors are our responsibility only.
†
Department of Economics, Stern School of Business, New York University, and NBER. Email:
[email protected]. Web: http://people.stern.nyu.edu/gclement/
‡
Department of Finance, Questrom School of Business, Boston University. Email: [email protected].
Web: http://people.bu.edu/bpalazzo/
§
Department of Economics, Stern School of Business, New York University.
Email:
[email protected]. Web: http://peifanwu.weebly.com
1
Introduction
It is now widely recognized that the last economic recession stands out not only for
its depth, but also for the rather slow recovery the followed it. Scholars have tried to
rationalize this evidence in a variety of ways.
Our purpose is to evaluate the role that firm demographics may have played in generating both the slowdown and the subsequent slow recovery. To succinctly illustrate
our hypothesis, think of a model with heterogeneous firms, each of which produces an
homogenous good according to the production function y = skν l1−ν , where s is a productivity index, k is capital, l is labor, and ν > 0 is the capital share. As long as firms
are price–takers in both the product and labor markets, a little algebra allows to express
aggregate output as
h
iν
Y = COV s1/ν , k + E s1/ν E (k) N ν L(1−ν) ,
where N is the number of firms and L is aggregate employment. The covariance term is
an indicator of allocative efficiency. The remaining terms are self–explanatory.
1
When read through the lenses of such model, cyclical fluctuations will feature variation
in the average establishment size E(k) – the intensive margin – as well as the number of
establishments N , the extensive margin. If the number of firms and average firm size
evolve at different speeds, recoveries from recession may differ, depending on the extent
to which the preceding contraction operated via the extensive rather than the intensive
margin.
Data on the evolution of N in the United States is readily available, as the Census
Bureau publishes on the web its estimates of the number of active establishments – a
by-product of the Longitudinal Business Database. Unfortunately, the same cannot be
said of measures of E(k). However, we can proxy for such statistics by tracking L/N , also
available in the LBD.
Figure 1 shows the evolution of employment, the number of establishments, and average employment, around the last four recessions. All series are set equal to 1 in the year
classified as peak by the NBER Dating Committee.
The picture confirms that the 2008 recession featured a much larger drop of employment, as well as a slower recovery. What is less well known is that the number of
productive units also dropped – in a sort of dramatic fashion – while it barely budged in
1
The theory introduced in this paper does not differentiate between firms and establishments. For this
reason, we will label our production technologies in either way. In the data, we will always identify our
production units as establishments.
1
1981 Recession
1.10
1990 Recession
L
N
L/N
1.10
1.05
1.05
1.00
1.00
0.95
0.95
1980
1981
1982
1983
1984
1985
1990
1991
2001 Recession
1.10
1.05
1.05
1.00
1.00
0.95
0.95
2002
2003
2004
1993
1994
1995
2012
2013
2008 Recession
1.10
2001
1992
2005
2006
2008
2009
2010
2011
Figure 1: Dynamics of L, N , and L/N during recent recessions.
occasion of the 1981 recession, and kept increasing during the 1991 and 2001 recessions.
The number of establishments kept dropping until 2011, and as of 2013 was still more
than 2% below its 2008 level. By comparison, the ratio L/N had bottomed out in 2010
and by 2013 was already back to its 2008 value.
The evolution of the number of establishments is the result of entry and exit. In
Figure 2, we take a closer look at the cohorts of establishments born from 2007 through
2009. Let Njt denote the number of establishments born in year j that are still active
in year t. Further, let Ljt be their total employment in the same year. In the left panel,
we show the fraction of total employment accounted for by each cohort during the first
P
five years since birth. That is, we plot Ljt / tι=0 Lιt for j = 2007, 2008, 2009 and t = j
through t = j + 5.
In the central panel, we show the number of establishments divided by the total,
P
i.e. Njt / tι=0 Nιt . Finally, in the right panel, is the average employment divided by the
P
P
average size of incumbents, i.e. (Ljt /Njt )/( tι=0 Lιt / tι=0 Nιt ).
The cohorts born in 2007 through 2009 account for a fraction of employment lower
than the mean cohort. From the other two panels, it can be inferred that firms belonging
2
Employment Share
0.075
Extensive Margin
0.16
Intensive Margin
1.0
mean
2007
2008
2009
0.070
0.14
0.9
0.12
0.8
0.10
0.7
0.08
0.6
0.06
0.5
0.065
0.060
0.055
0.050
0.045
0.040
0.035
0
1
2
3
4
0.04
5
0
1
2
3
4
5
0.4
0
1
2
3
4
5
Figure 2: Extensive Margin Vs. Intensive Margin
to the same cohorts were not particulary small, when benchmarked against incumbents.
The lower employment share appears to be the direct result of a lower cohort size.
The central panel hints that the survival rate does not change much across cohorts,
which implies that the initial shock to the cohort size is permanent. In turn – see the left–
most panel – this translates into a permanent effect on the employment share accounted
for by the cohort.
The upshot of this preliminary analysis is that the years following 2007 were characterized by unusually low entry rates, which in turn led to a decline and very slow recovery
of the number of establishments. The main purpose of this paper is to try to quantify the
role played by these facts in shaping the dynamics of major macroeconomic aggregates in
the aftermath of the last recession. In other words, we are asking: To what extent are the
unusually low entry rates that characterized the recession responsible for the low recovery
that followed?
To this end, we build a general equilibrium model with heterogenous firms and endogenous entry and exit that displays genuine aggregate dynamics. We parameterize it
in such a way that key features of the model’s implied firm and aggregate dynamics are
3
consistent with unconditional micro and macro moments, respectively.
Business cycles are caused by two sources of exogenous disturbances: TFP shocks, i.e.
shocks to a common productivity component, and shocks to the cost that new productivity
units incur when entering the marketplace. Our choice to focus on TFP shocks is not
driven by the belief that changes in the pace of technological progress may have played a
decisive role in shaping the Great Recession.
Our goal is to highlight how a shock that directly affects the entry margin alone –
such as a temporary increase in the cost of entry – affects aggregate fluctuations that are
mostly driven by economy–wide disturbances. The decision to select TFP shocks rather
than demand shocks, say, is dictated by the need of keeping the model tractable.
When hit with a negative temporary TFP shock only, the model produces a much
more persistent decline in employment and output than an off-the-shelf model with a fixed
number of establishments. The drop in the common productivity factor leads to a decline
in entry, which in turn translates into a temporary but persistent drop in the number
of establishments. Such dynamics lead to a slower recovery of output, employment, and
consumption, with respect to a more standard model featuring a fixed measure of firms.
Entry end exit dynamics enhance persistence of all aggregate series.2
In the preliminary illustration of the empirical evidence that we have presented above,
the Great Recession stands out for the unusually large decline in the number of establishments. In our theoretical framework, the simplest way to mimic such feature is to allow
for a positive shock to the average cost of entry.
We compound the TFP shock with a temporary but persistent 10% increase to the
cost of entry – large enough to further lower the entry rate from 10.15% – resulting from
the TFP shock – to 9.35%. While the contemporaneous decline in output and employment
is unchanged, the shock to the cost of entry drive both series to decline for five years past
the occurrence of the shock, before converging back to the steady state. As a result, the
recession is deeper and the recovery slower.
The remainder of the paper is organized as follows. In section 2 we introduce the
model. In section 3, we describe the solution method and the calibration. In section 4,
we consider the effects of hitting our economy with a 1% negative TFP shock. In Section
5, we conduct our key exercise, compounding the TFP shock with a 10% temporary but
persistent increase in the average cost of entry. Section 6 concludes.
2
This is consistent with the partial equilibrium analysis in Clementi and Palazzo (2016).
4
2
Model
Time is discrete and the horizon is infinite. At each time t, a positive mass of pricetaking firms produce an homogeneous good by means of the production function yt =
θ
zt st ktν lt1−ν , where ν ∈ (0, 1), θ ∈ (0, 1), kt denotes physical capital in place, and lt is
labor. Total factor productivity is the product of a common (zt ) and an idiosyncratic
(st )component, distributed according to autoregressive processes with log–normal and
orthogonal innovations, i.e.
log zt+1 = ρz log zt + σz ǫz
(1)
log st+1 = ρs log st + σs ǫs
(2)
where ǫz ∼ N (0, 1) and ǫs ∼ N (0, 1).
Adjusting capital entails both fixed and variable adjustment costs. When installed
capital is kt , a gross investment xt requires firms to sustain a cost
g (xt , kt ) = 1
xt
∈[−a,a]
/
kt
c0 kt + c1
xt
kt
2
kt ,
(3)
where a ∈ (0, 1) and c1 > 0 are constants common to all firms, and c0 is the firm–specific
realization of a i.i.d. random variable whose time–invariant distribution G is common
across firms. Notice that the fixed cost applies only when the gross investment rate in
absolute value is greater than a.
As is customary in the business cycle literature, firms choose capital one period in
advance and hire labor on the spot market, after having observed their productivity realization. Once production is completed, firms find out the overhead cost for the following
period cf > 0. We assume that cf is distributed according to the common and time–
invariant distribution Gf . Firms may decide not to pay cf and exit the market instead.
2.1
Incumbents’ Decision Problem
Firms’ decisions depend on the idiosyncratic state variables (kt , st ) and on prices, which
in equilibrium will be functions of the aggregate state vector s, to be defined below. Given
the wage w (s), the indirect profit function is
π (s, k; s) = max zs kν l1−ν
l
θ
− w (s) l.
(4)
Letting Ve and Vx denote the value of continuing in operation and exiting, respectively,
5
firm value at the beginning of the period is
Z
n
o
max Ve (s, k; s) − cf , Vx (s, k; s) dGf (cf ) .
V (s, k; s) = π (s, k; s) +
(5)
cf
2.1.1
Value of continuing in operation
Contingent on a draw for c0 , the program (6)–(7) yields the value of choosing the unconstrained optimal investment rate:
Vu (s, k, c0 ; s) = max −x − c0 k − c1
x
s.t. k′ = k(1 − δ) + x.
x 2
k
k + β E m s′ , s V s′ , k′ ; s′ |s; s
(6)
(7)
Instead, the program (8)–(9) yields the payoff of choosing the efficient level of capital
among the values that dispense the firm from paying c0 .
Vc (s, k; s) =
max
x∈[−ak,ak]
−x − c1
s.t. k′ = k(1 − δ) + x.
x 2
k
k + β E m s′ , s V s′ , k′ ; s′ |s; s
(8)
(9)
It follows that the expected value of staying in operation at least one more period is
Z
max {V u (s, k, c0 ; s) , V c (s, k; s)} dG0 (c0 )
(10)
Ve (s, k; s) =
c0
2.1.2
Value of exiting
Upon exit, a firm gets to sell the undepreciated portion of it capital stock and incurs the
adjustment cost of reducing the capital to zero. We assume that the scrap value of capital
is a simple monotone function of aggregate productivity. That is,
Z
γ
Vx (s, k; s) = z k(1 − δ) − k c0 dG0 (c0 ) − c1 k(1 − δ)2 .
2.2
(11)
Entrant Firms
At each time t, there is a constant measure of potential entrants M > 0. Each candidate draws a signal of its idiosyncratic productivity q and a cost of entry ce > 0. The
distributions – denoted as Gq and Ge , respectively, are time invariant and common to all
firms.
Conditional on q, the value of entering is
′
′
′ ′ ′
−k
+
β
E
m
s
,
s
V
s
,
k
;
s
Ve (q; s) = max
|q,
s
,
′
k
6
(12)
Firms will enter if and only if Ve (q; s) ≥ ce .
2.3
Households
The economy is populated by a unit measure of ex-ante identical households, who own
the firms and receive utility from consumption and disutility from labor according to the
utility function
u(ct , nt , ht ) =
n1+α
t
ct − ht − χ 1+α
1−σ
1−σ
,
where σ > 0, α > 0, χ > 0 and ht denotes the habit stock. Following Campbell and
Cochrane (1999), we characterize the evolution of habit by defining the surplus consumption ratio as
bt ≡
ct − ht
ct
and assuming
log bt+1 = (1 − ρb ) log b̄ + ρb log bt +
1
ct+1
.
− 1 log
ct
b̄
(13)
Notice that this implies that current habit depends on all past values of consumption. As
in Campbell and Cochrane (1999), consumption habit is external: households do not take
into account the impact of consumption choices on future values of the habit stock.
We assume complete markets. This implies that in equilibrium households will share
the same intertemporal marginal rate of substitution, which is also the valid stochastic
discount factor for the firm. This also implies that the firm’s investment decisions are
consistent with household optimization, so that the household’s optimization problem
can be written as
W (s) = max
1+α 1−σ
cb(s) − χ n1+α
1−σ
s.t. c ≤ Π (s) + w (s) n
c,n
+ βE W ′ s′ |s
(14)
(15)
where Π(s) denotes total profits generated by firms.
2.4
Equilibrium
The aggregate state vector is s = (z, b−1 , c−1 , µ), where z is the common productivity
component, b−1 is previous period’s surplus consumption ratio, c−1 is the previous period’s
consumption ,and µ is the distribution of firms over idiosyncratic shock and capital.
A recursive competitive equilibrium consists of: Value functions V (s, k; s), Vx (s, k; s),
7
Ve (q; s); policy functions π (s, k; s), l (s, k; s), ka (s, k; s), kn (s, k; s), ke (q; s), ĉf (s, k; s),
ĉ0 (s, k; s), ĉe (q; s); aggregate prices and quantities w (s), m (s′ , s), c (s), n (s); distributions E (s, k; s), µ (s, k; s), such that
1. (Incumbent Firm Optimization) Given w (s) and m (s′ , s), incumbent firm holding
capital k and facing idiosyncratic productivity shock s solve firm’s maximization
problem (10). l (s, k; s), ka (s, k; s), kn (s, k; s), ĉf (s, k; s), ĉ0 (s, k; s) are associated
policy functions, V (s, k; s) is the associated value function and π (s, k; s) stands for
firm’s current period profit. All costs encountered are deadweight loss.
2. (Entrant Firm Optimization) Given w (s), entrant firms solve (12) with solution
ĉe (q; s) and ke (q; s).
3. (Entrant Firm Distribution) For all Borel sets S × K ∈ R+ × R+ ,
Z Z
1ke (q;s)∈K dQ (q) d Pr s′ |q
E (S × K; s) = M
S
K
4. (Household Optimization) Given w (s), Π (s) and c (s) and n (s) solve the utility
maximization problem ((14)-(15)).
5. (Market Clearing) Labor market clears
Z
l (s, k; s) dµ (s, k; s) = n (s)
Good market clears
Z
Z Z
x
kdE (s, k; s)
(π (s, k; s) + Gf (ĉf ) V ) dµ (s, k; s) = Π (s) +
s k
Z
Z
a
a
n
n
max (k − g (k , k) , k − g (k , k)) dG0 (c0 ) − (1 − δ) k dµ (s, k; s)
+ (1 − Gf (ĉf ))
c0
6. (Law of Motions) The surplus consumption ratio follows (13), the stochastic discount
factor m (s′ , s) follows (??), aggregate shock follows (1), and for all Borel sets S×K ∈
R+ × R+ ,
µ′ (S × K; s)
Z
Z
′
′
= 1s ∈S Pr s |s Gf (ĉf ) (1ka ∈K G0 (ĉ0 ) + 1kn ∈K (1 − G0 (ĉ0 ))) dµ (s, k; s) + E (S × K; s)
s
k
8
3
Approximation and Calibration
3.1
Approximation Method
We start by computing the solution to the stationary version of the model, i.e. one where
σz = 0, by means of a rather standard non-linear projection method. We approximate
the value function with cubic splines. We guess a value for the wage rate, solve for the
parameters of the value function, and use the implied policy functions to compute the
stationary distribution. If the implied equilibrium wage rate is close enough to our initial
guess, we have found our appoximate solution. If not, we update our guess and keep going
until we have found a fixed point. See Appendix A.1 for details.
We then proceed to compute a linear approximation of the policy functions and the
law of motion of the distribution, along the lines of Reiter (2009). Conceptually, this
method is similar to the linear perturbation method employed in solving representative
DSGE models as we are perturbing with respect to the variance of the aggregate TFP
shock. We list out all the necessary conditions for equilibrium – first order conditions,
value functions, and market clear conditions – as well as the law of motion of the distribution. Differentiating the system and evaluating at the steady state leads to a quadratic
matrix equation that can be solved by Blanchard–Kahn. A detailed account of the implementation is provided in Appendix A.2.
3.2
Calibration
As is common in the business cycle literature, we pre–set certain parameter values and
pick others in order to match stylized facts. The preset parameters are listed in Table 1.
One period is assumed to be one year. We choose β = 0.987 to generate an annual
risk-free interest rate of 1.3% and δ = 0.1 to match the post–war investment–to–capital
ratio. The parameters of the production function are from Clementi and Palazzo (2016).
The inverse Frisch elasticity is 0.5, a value well within the range considered in that macro
literature (Chetty et al. (2011)) and the relative risk aversion coefficient is 1, indicating log
utility. The values of the investment rate that do not entail a fixed cost are [−0.002, 0.002].
The signal drawn before entry q, as well as the operating cost cf are Pareto–distributed,
while the fixed capital adjustment cost c0 and the entry cost are uniform.
We then proceed to select other parameters, so that the model is consistent with certain
stylized facts. We choose some parameters so that the stationary model reproduces key
empirical features of firm dynamics and then proceed to set others so that the full model
generates sensible dynamics of aggregate quantities.
9
Parameter
α
σ
ν
θ
β
δ
Description
Inverse Frisch Elasticity
Utility Curvature
Relative Capital Share
Span of Control
Time Discount Factor
Depreciation Rate
Value
0.5
1.0
0.3
0.8
0.987
0.1
Table 1: Fixed Parameter Values
3.2.1
Consistency with Micro Data
Consider the following decomposition:
L
N
P i,t = P i,t ×
j Lj,t
j Nj,t
Li,t
Ni,t
P
L
P j j,t
N
j,t
j
,
where Li,t is total employment at establishments of age i at time t and Ni,t is the total
number of establishments of i at time t. The term
PNi,t ,
j Nj,t
which we will refer to as
extensive
margin, is the fraction of age i–establishments at time t. The second term,
P
Li,t P j Lj,t
Ni,t / j Nj,t ,
is the average size of age i at time t, expressed as fraction of the average size
of all incumbents – the intensive margin.
Figure 3 shows the average evolution of both margins over the period 1978–2014,
according to the summary statistics of the LBD made available to the general public by
the Business Dynamic Statistics (BDS). The gray band is the min-max band.
0.14
1.0
0.9
0.12
0.8
Rate
Rate
0.10
0.7
0.08
0.6
0.06
0.04
0.5
0
1
2
3
Year since birth
4
5
0.4
0
1
2
3
Year since birth
4
5
Figure 3: Left: Extensive Margin. Right: Intensive Margin.
Given the nature of our exercise, we repute it necessary that our model reproduces
these features of the data. To this end, we set the top 7 parameters listed in Table 2
to minimize the distance between the 10 moments graphed in Figure 3 and their model–
10
generated counterparts.
The upper bound of the support of the fixed capital adjustment cost c0 is set to
match the fraction of firms conducting positive investment ratio as reported by Zwick and
Mahon (2014). Finally, the parameter χ is set so that the household spends 1/3 of her
time working. The model’s generated moments and their empirical counterparts are listed
in Table 3. The extent to which the model is consistent with the evolution of intensive
and extensive margins can also be garnered by glancing at Figure 4.
Parameter
ρs
σs
Description
Persistence of Idiosyncratic Shock
Std. Dev. of Idiosyncratic Shock Innovation
Value
0.70
0.0687
cf
ce
bp
M
Curvature Parameter of Gf
Upper Bound of Ge
Curvature Parameter of Gq
Measure of Potential Entrants
72.0
1.0
29.28
0.122
c1
c0
Quadratic Coefficient of Adjustment Cost
Upper Bound of G0
0.1856
0.0023
χ
Disutility of Labor
3.55
Table 2: Fitted Parameter Values – Micro
Extensive Margin
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
Data
0.1124
0.0876
0.0743
0.0646
0.0571
0.0509
Model
0.1063
0.0763
0.0596
0.0496
0.0430
0.0384
Positive Investment Rate
76.3%
76.4%
Intensive Margin
Year 0
Year 1
Year 2
Year 3
Year 4
Year 5
Data
0.5191
0.6342
0.6866
0.7327
0.7722
0.8095
Model
0.5255
0.6129
0.7104
0.8066
0.8939
0.9682
Table 3: Targeted Moments in Steady State
Next, we turn to evaluate predictions of the model that we did not target in our
calibration exercise. Figure 5 illustrates that, consistent with the data, survival is increasing in age, while growth is decreasing. Figure 6 Illustrates the evolution of a cohorts
of entrants. Consistent with the empirical evidence gathered by Cabral and Mata (2003),
the entrants’ distribution displays a think right tail. However, skewness decreases as the
cohort ages.
11
Extensive Margin
0.12
Percentage
0.1
0.08
0.06
0.04
0.02
0
1
2
3
4
5
4
5
Intensive Margin
1
Percentage
0.9
0.8
0.7
0.6
0.5
0
1
2
3
Figure 4: Solid: Model. Dashed: Data.
14
30
12
25
10
20
Percent
Percent
8
6
15
4
10
2
0
5
-2
0
5
10
15
20
25
30
0
5
10
15
20
25
30
Figure 5: Exit Rate and Conditional Growth over Age
3.2.2
Aggregate Dynamics
We are left with five parameters to calibrate, which do not affect the steady state, but
obviously contribute to shaping aggregate dynamics. The parameters and their values
are listed in Figure 4. They were chosen so that the model generates data–conforming
12
0.3
Age 1
Age 2
Age 5
Age 10
0.25
Measure
0.2
0.15
0.1
0.05
0
0
0.05
0.1
0.15
0.2
0.25
0.3
Capital
Figure 6: Evolution of Cohort
values for the standard deviation of detrended output as well as its autocorrelation, the
correlation of detrended output and risk–free rate, and the correlation of exit rate and
output. Simulated and empirical models are listed in Table 5. In the same table, we list
a number of over–identifying restrictions.
Parameter
ρz
σz
γ
ρb
b̄
Description
Persistence of Aggregate Productivity
Conditional Std. Dev. of Innovation
Cyclicality of scrap value of capital
Persistence of Surplus Consumption Ratio
Steady State of Surplus Consumption Ratio
Value
0.85
0.014
0.64
0.65
0.71
Table 4: Fitted Parameter Values – Aggregate Dynamics
4
The effect of economy–wide shocks
In this section we describe the response of our model to a 1% negative shock to total factor
productivity and we compare it to the response implied by a version of our framework
that does not feature either entry or exit. In that version, as is common in the macro
literature with firm heterogeneity, the measure of firms is fixed.
Refer to Figure 7. Solid lines describe the dynamics of aggregate output, consumption,
13
Targeted Moment
Data
Model
σ (Y )
ρ (Y, Y−1 )
Corr (r, Y )
Corr (%exit, Y )
2.05%
0.56
−0.21
2 · 10−5
2.04%
0.55
−0.21
0.03
Entry and Exit Dynamics
σ (%entry)
σ (%exit)
Corr (%entry, Y )
0.74%
0.67%
0.34
0.87%
0.16%
0.78
Business Cycle Dynamics
σ(C)/σ(Y )
σ(I)/σ(Y )
ρ (C, C−1 )
ρ (I, I−1 )
0.65
4.00
0.67
0.46
0.84
2.66
0.53
0.45
Not–targeted Moment
Table 5: Moments – Aggregate Dynamics
investment, and employment, in the case of the standard framework featuring a fixed
measure of firms. The dashed lines depict the same quantities, in our benchmark model
with entry and exit.
A striking finding is that the responses of all quantities are more persistent in the
scenario with entry and exit. The reason can be appreciated by considering the decomposition of aggregate output introduced in Section 1, which we restate here, adjusted for
the presence of decreasing returns to scale and aggregate shocks:
i1−(1−ν)θ
h
νθ
1
νθ
1
N 1−(1−ν)θ L(1−ν)θ
Y = z COV(s 1−(1−ν)θ , k 1−(1−ν)θ ) + E(s 1−(1−ν)θ )E(k 1−(1−ν)θ )
(16)
Figure 8 illustrates a first key difference: In our benchmark model, the measure of
active firms drops during the recession and reverts back to its steady–state rather slowly.
In a sense, the model with entry and exit features two types of capital, both of which are
slow–moving and contribute to generate a response to aggregate shocks whose persistence
is greater than the persistence of the exogenous disturbance: Beyond physical capital,
there is the number of establishments.
i1−(1−ν)θ
h
νθ
1
νθ
1
,
In Figure 9, we plot the the term COV(s 1−(1−ν)θ , k 1−(1−ν)θ ) + E(s 1−(1−ν)θ )E(k 1−(1−ν)θ )
1
νθ
labeled residual, as well as the two addenda between square brackets, i.e. COV(s 1−(1−ν)θ , k 1−(1−ν)θ )
14
Consumption
0
Investment
1
0
-0.5
-1
-2
-1
-3
-4
-1.5
-5
0
10
20
30
40
0
Employment
0
10
20
30
40
30
40
Output
0
-0.2
-0.5
-0.4
-0.6
-1
-0.8
-1.5
-1
-1.2
-2
0
10
20
30
40
0
10
20
Figure 7: Recovery from the Recession – I
1
νθ
and E(s 1−(1−ν)θ )E(k 1−(1−ν)θ ), respectively.
In either model, the time–variation of the covariance is rather limited, signalling that
the dynamics of misallocation are relatively unimportant in our framework. In the case
without entry or exit, the distribution of idiosyncratic productivity is time–invariant by
1
assumption, therefore the element E(s 1−(1−ν)θ ) is constant. It follows that the dynamics
of the residual is driven uniquely by the average capital stock.
With entry and exit, the residual rises after the negative shock hits, then falls below
steady–state, and finally reverts back, although at a lower speed than in the scenario with
1
a fixed number of firms. This feature clearly depends from the response of E(s 1−(1−ν)θ ).
When aggregate productivity is below mean, only firms with higher idiosyncratic pro1
ductivity find it optimal to enter. As a result, the moment E(s 1−(1−ν)θ ) rises above its
unconditional mean, and then slowly reverts back to steady state. This explains why the
residual is actually above steady state for a short period after the shock hits, and also why
it displays so much more persistence. What is really important for our purposes is that
1
E(s 1−(1−ν)θ ) converges from above: The distribution of idiosyncratic productivity worsens
as the exogenous component of aggregate productivity increases.
15
0.2
0
0
-0.2
-0.2
-0.4
-0.4
Percent Deviation
Percent Deviation
0.2
-0.6
-0.8
-1
-1.2
-0.6
-0.8
-1
-1.2
-1.4
-1.4
Output
Labor
Number of Firms
Shock
-1.6
Output
Labor
Number of Firms
Shock
-1.6
-1.8
-1.8
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
Figure 8: Decomposition I. Left: With entry and exit. Right: Without.
1.2
1
1
0.8
0.8
0.6
0.6
Percent Deviation
Percent Deviation
1.2
0.4
0.2
0
-0.2
0.4
0.2
0
-0.2
-0.4
-0.4
Residual
Expectation
Covariance
-0.6
-0.8
Residual
Expectation
Covariance
-0.6
-0.8
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
Figure 9: Decomposition II Left: With entry and exit. Right: Without.
5
The effect of shocks to the entry margin
In this section, we consider the dynamics of our economy when the TFP shock introduced
above is supplemented by a 10% positive shock to the mean cost of entry. This is our
attempt to generate a recession with an unusually large decline in the number of establishments, where the benchmark is clearly the case where the only shock is economy–wide.
Figure 10 reproduces the dynamics of the entry rate. The shock to the cost of entry
has the effect of roughly doubling the contemporaneous decline in the entry rate. In
figure 11, we reproduce the response of consumption, investment, employment, and output
to the scenario analyzed above. Notably, the contemporaneous decline in employment and
outpout is the same across the two scenario. With the shock to the cost of entry, however,
output and employment keep dropping with respect to the steady state for five periods
after the shock.
16
0.108
0.106
0.104
Percent
0.102
0.1
0.098
0.096
0.094
0.092
0
10
20
30
40
50
60
70
80
90
100
Figure 10: Dynamics of Entry Rate. Dashed: With shock to cost of entry.
Consumption
0
Investment
0
-1
-0.5
-2
-1
-3
-1.5
-4
-2
-5
0
20
40
60
80
100
0
Employment
0
20
40
60
80
100
80
100
Output
0
-0.2
-0.5
-0.4
-0.6
-1
-0.8
-1.5
-1
-1.2
-2
0
20
40
60
80
100
0
20
40
60
Figure 11: Recovery from the Recession. Dashed: With shock to cost of entry.
The mechanism driving this result can be evinced from Figures 12 and 13, which
illustrate the decomposition (16) in the scenarios with and without shock to the cost of
17
entry, respectively.
In absence of a shock to the cost of entry, the measure of firms reaches its minimum
at a lever 0.6% smaller than in steady state. With the shock to entry cost, however, the
mass of firms gets to be as low as 1% smaller than steady state.
Now refer to Figure 13. An higher cost of entry, via selection, leads to slightly higher
entrants’ mean size and productivity. This effect goes in the direction of reducing the
impact of the shock on aggregate quantities. Given our parameter values, however, the
extensive margin documented above dominates.
0.2
0
0
-0.2
-0.2
-0.4
-0.4
Percent Deviation
Percent Deviation
0.2
-0.6
-0.8
-1
-1.2
-0.6
-0.8
-1
-1.2
-1.4
-1.4
Output
Labor
Number of Firms
Shock
-1.6
Output
Labor
Number of Firms
Shock
-1.6
-1.8
-1.8
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
90
100
1.2
1.2
1
1
0.8
0.8
0.6
0.6
Percent Deviation
Percent Deviation
Figure 12: Decomposition I. Left: With shock to cost. Right: Without.
0.4
0.2
0
-0.2
0.4
0.2
0
-0.2
-0.4
-0.4
Residual
Expectation
Covariance
-0.6
-0.8
Residual
Expectation
Covariance
-0.6
-0.8
0
10
20
30
40
50
60
70
80
90
100
0
10
20
30
40
50
60
70
80
Figure 13: Decomposition II Left: With shock to cost. Right: Without.
6
Conclusion
TBA
18
90
100
References
Cabral, L. and J. Mata (2003): “On the evolution of the firm size distribution: Facts
and theory,” The American Economic Review, 93, 1075–1090.
Campbell, J. Y. and J. H. Cochrane (1999): “By force of habit: A consumptionbased explanation of aggregate stock market behavior,” Journal of political Economy,
107, 205–251.
Chetty, R., A. Guren, D. Manoli, and A. Weber (2011): “Are micro and macro
labor supply elasticities consistent? A review of evidence on the intensive and extensive
margins,” The American Economic Review, 101, 471–475.
Clementi, G. L. and B. Palazzo (2016): “Entry, exit, firm dynamics, and aggregate
fluctuations,” American Economic Journal: Macroeconomics, 8, 1–41.
Klein, P. (2000): “Using the generalized Schur form to solve a multivariate linear rational
expectations model,” Journal of Economic Dynamics and Control, 24, 1405–1423.
Kopecky, K. A. and R. M. Suen (2010): “Finite state Markov-chain approximations
to highly persistent processes,” Review of Economic Dynamics, 13, 701–714.
Reiter, M. (2009): “Solving heterogeneous-agent models by projection and perturbation,” Journal of Economic Dynamics and Control, 33, 649–665.
Schmitt-Grohé, S. and M. Uribe (2004): “Solving dynamic general equilibrium models using a second-order approximation to the policy function,” Journal of economic
dynamics and control, 28, 755–775.
Zwick, E. and J. Mahon (2014): “Do financial frictions amplify fiscal policy? Evidence
from business investment stimulus,” .
19
A
Solution Method
A.1
Stationary Equilibrium
Denote s̄ as the aggregate state in stationary equilibrium.
Discretization
While calculating value function an individual firm has two idiosyn-
cratic states: shock s and capital k. Since s follows (2) which is an AR(1) process, we
use Rouwenhorst method as in Kopecky and Suen (2010) to set a grid of ns = 9 points.
For capital level k, we calculated steady state ex-post and choose a grid of nk = 50 points
exponentially distributed in interval [0.001, 0.8]. This interval ensures that all the capital
policy choices, in stationary equilibrium, is interior of this interval. We approximate incumbent firm’s value function with cubic splines. To pin down nk cubic spline parameters
for each idiosyncratic shock realization, we need a nk − 2 auxiliary capital grid. This grid
n
o
is arbitrary, for simplicity I use kjaux = {k1 , k3 , k4 , ..., .knk −3 , knk −2 , knk }.
Therefore, stationary distribution also depends on s and k. To keep track of it we use
m
g
discrete bins to characterize the histogram. Denote L ≡ {(si , ki )}i=1
as an outer product
of idiosyncratic shock Rouwenhorst grid and a dense capital grid with mk = 100 points,
exponentially distributed in the same interval. mg = ns · mk .
For entrant firm’s problem, we discretize signal space into nq = 100 discrete values.
Firm’s Value Function
We first approximate incumbent firm’s value function. Then
entrant firm’s value function can be represented as a function of incumbent firm’s value
function. Denote Tj3 (k), j = 1...nk as cubic spline interpolant for capital. Then for each
idiosyncratic shock level si ,
V (si , k; s) =
nk
X
θij (s) Tj3 (k)
j=1
We use De Boor’s algorithm to generate cubic spline interpolants. Define
(
aux , kaux
1
k
∈
k
j
j+1
Tj0 (k) =
0 otherwise
and
aux − k
ki+n+1
k − kaux
n−1
Tjn (k) = aux i aux Tjn−1 (k) + aux
aux Tj+1 (k)
ki+n − ki
ki+n+1 − ki+1
Therefore computing incumbent’s value function v is equivalent to calculating all θij .
20
We compute entrant firm’s value function on the discrete signal space.
Distribution
We define distribution as histogram on discrete grid L.
For all incumbent firms that share the same idiosyncratic state (s, k), there’s a probability Pre (s, k) to exit from the market, a probability Prn (s, k) to adjust their capital
to kn (s, k) without paying fixed cost , Pra (s, k) to adjust their capital to ka (s, k) while
paying fixed cost. To sum up, incumbent firm will choose a capital level and a probability
to adjust to that new capital level essentially. In distribution approximation we follow a
lottery rule, i.e., if some policy function k′ is not exactly on the discrete grid we chose
ki < k′ < kt+1 then firms will transfer to state ki with probability
state ki+1 with probability
ki+1 −k ′
ki+1 −ki
and transfer to
k ′ −ki
ki+1 −ki .
For all entrant firms who get signal q, they will choose an investment level today ke
which will be their capital level next period. This ke also follows the lottery rule we
defined above. Let the measure of potential firm entrants be Q (q).
Denote the discrete counterpart of stationary distribution µ as D (s, k) then
i
X
hX
Pr (s, k) Pr k′ |s, k D (s, k) +
Pr (s, k) Pr k′ |s, k D (s, k)
D s′ , k′ = Pr s′ |s
n
a
X
′
′
+
Pr s |q Pr k |q Q (q)
(17)
The stationary distribution is unique and exists when the support of all probability
transitions have the same support.
Solution Algorithm We are going to find equilibrium marginal utility m, wage w,
value function V which can be represented as {θij }, and the associated policy functions.
Algorithm is the following:
1. Guess marginal utility of consumption m0 .
2. Guess wage w0 .
3. Take an initial guess on {θij }. Given {θij }, w0 and m0 search optimal policy functions for ka and kn , as well as cut-off thresholds ĉf and ĉ0 .
4. Put the policy functions we get from step 2 back into value function (??). Take left
n o
′
hand side of value function interpolants θij
as unknown. Then value functions
n o
n o
n o
′ . Solve θ ′
′ . If
form an equation system for θij
and compare {θij } and θij
ij
they are different, value function doesn’t converge and we go back to step 2 but put
n o
′
new θij
inside until value function converges.
21
5. With w0 , m0 and {θij }, compute entrant problem (12). Then compute stationary
distribution according to () (technically, (17)).
6. With w0 and stationary distribution, calculate labor demand from firm side (4) and
labor supply from household side (14-15). If supply is greater than demand, guess a
new equilibrium wage w′ < w0 . If demand is greater than supply, take a new guess
w′ > w0 . Take the new guess and go back to step 3 until demand equals supply.
7. Now we can calculate the equilibrium m from (??). Since marginal utility of consumption will only change levels of value function but not the policy functions, we
do step 4 again to solve equilibrium {θij }.
From step 1 - 7 we get stationary equilibrium. Convergence criterion is 10−8 among all
the exercises.
A.2
Dynamics
Following Reiter (2009), we are going to find a system with state variables x and control
variables y to form a system a la Schmitt-Grohé and Uribe (2004),
Et H y ′ , y, x′ , x = 0
Equilibrium Conditions State variables are s = (z, b−1 , c−1 , µ). After discretization
we have state variables x = (z, b−1 , c−1 , D) hence there are 3 + ns mk variables.
a
Control variables are y = θ (s, k) , ka (s, k) , ĉe (q) , ke (q) , kdense
(s, k) , w, m . We
could add more control variables inside, but since all the other control variables are implicit
functions of x and y, we only need to take these many for the sake of dimensionality. There
are 2 + 2ns nk + 2nq + ns mk variables. We construct equations to match the same number
of variables in building the equation system.
Count the equations:
1. To calculate ka , which is capital choice on the grid that we employ to approximate
incumbent firm’s value function, we use first order condition with respect to ka ,
λ 1 + 2c1
ka (si , kj )
− (1 − δ)
kj
−β
nk
ns X
ns X
X
o=1 k=1 l=1
which includes ns nk equations.
22
′
Π (so |si ) θkl
∂Tl3 (ka (si , kj ))
=0
∂ka (si , kj )
(18)
2. To calculate θ, which is interpolants of value function, we use definition of value
function (??) as consistency condition. First, given ka , we can compute kn following
definition


(1 − δ + a) kj
kn (si , kj ) = ka


(1 − δ − a) kj
ka > (1 − δ + a) kj
ka < (1 − δ − a) kj
and then ĉ0 from (??),
(
"
2
a
1
k (si , kj )
a
n
ĉ0 (si , kj ) =
max −λ (k − k ) + c1
− (1 − δ) kj
λkj
kj
2 #
n
k (si , kj )
− (1 − δ) kj
−c1
kj
)
nk
ns X
ns X
X
3 a
′
n
3
+β
Π (so |si ) θkl Tl (k ) − Tl (k ) , 0
o=1 k=1 l=1
and exit scrap value Vx (si , kj ) from (11), then ĉf from (??),
"
!
( (
2
a
k
(s
,
k
)
1
i
j
G0 (ĉ0 (si , kj )) −λ ka + c1
− (1 − δ) kj + E (c0 ) kj
ĉf (si , kj ) = max
λ
kj
#
nk
ns X
ns X
X
′
+β
Π (so |si ) θkl
Tl3 (ka )
o=1 k=1 l=1
"
kn (si , kj )
− (1 − δ)
+ (1 − G0 (ĉ0 (si , kj ))) −λ k + c1
kj
#)
)
nk
ns X
ns X
X
n
x
3
′
− V ,0
Π (so |si ) θkl Tl (k )
+β
n
2
kj
!
o=1 k=1 l=1
With all these calculated we can put back into Bellman equation
V (si , kj ) = π (si , kj ) + Gf (ĉf ) [G0 (ĉ0 ) Va (si , kj ) + (1 − G0 (ĉ0 )) Vn (si , kj ) − E (cf < ĉf )]
+ (1 − Gf (ĉf )) [Vx − E (c0 )]
(19)
There are also ns nk equations.
3. To calculate ke , which is entrant capital choice for next period, we use first order
condition with respect to ke ,
λ−β
nk
ns X
ns X
X
′
Π (so |qi ) θkl
o=1 k=1 l=1
∂Tl3 (ke (qi ))
=0
∂k e (qi )
for each discrete realization of q. Hence there are nq equations.
23
(20)
4. To calculate ĉe , the threshold is the same as value function for entrants for them to
break-even. Then following (12),
0 = −ce (qi ) − k′ + β
ns
X
Π (so |qi )
o=1
λ′
V so , k ′
λ
(21)
These equations have the same dimensionality as ke . There are nq equations.
a
5. We use a dense grid to keep track of the distribution. The way to calculate kdense
is the same as that in calculating Euler of incumbent’s value function,
!!
nk
ns X
ns X
a
a
3
X
kdense
(si , kj )
′ ∂Tl (k (si , kj ))
=0
− (1 − δ)
−β
Π (so |si ) θkl
λ 1 + 2c1
kj
∂ka (si , kj )
o=1 k=1 l=1
(22)
There are ns mk equations.
6. To keep track of the distribution,
′
D (si , kj ) =
+
ns
X
o=1
mk
X
Π (si |so ) Gf (ĉf )
"m
k
X
l=1
(1 − G0 (ĉ0 )) Pr (kj |so , kl ) D (so , kl )
n
#
G0 (ĉ0 ) Pr (kj |so , kl ) D (so , kl ) +
a
l=1
nq
X
Π (si |qo ) Pr (kj |qo ) Q (qo )
o=1
(23)
first line describes the survival firms and second line describes the potential entrants.
There are ns mk equations.
7. Labor market clears, following (??)
α

mk
ns X
X
D (si , kj ) l (si , kj ) = w
χ
(24)
i=1 j=1
where l (si , kj ) follows static maximization problem (4).
8. Good market clears,
− η1
λ
=
mk
ns X
X
D (si , kj ) [π (si , kj ) + (1 − Gf (ĉf )) V x (si , kj )
i=1 j=1
− Gf (ĉf ) (G0 (ĉ0 ) (ka − (1 − δ) kj + Eg (ka , kj ))
n
n
+ (1 − G0 (ĉ0 )) (k − (1 − δ) kj + g (k , kj )))] −
nq
X
o=1
24
Q (qo ) ke (qo )
(25)
9. Habit formation following (13),
log b′−1
≡ log b = (1 − ρb ) log b̄ + ρb log b−1 +
1
c
− 1 log
c−1
b̄
10. Pass current consumption as one aggregate state next period, c′−1 = c.
11. There’s law of motion of aggregate state z ′ = ρz + σǫ.
The number of equations above match the number of variables.
Linear Perturbation
This step is the same as solving a typical DSGE model. Denote
y = g (x) for policy function and x′ = h (x) for law of motion of states.
First order Taylor expansion gives
Hy′ Ey ′ − ȳ + Hy (y − ȳ) + Hx′ x′ − x̄ + Hy′ (x − x̄) + Hǫ ǫ = 0
which can be put in matrix form
′ x
x
+B
+ Gǫ = 0
A
Ey ′
y
Following Klein (2000) we use Schur decomposition to solve this system. We can
write A = QT Z ′ and B = QSZ ′ where Q and Z are unitary squares and T , S are
upper-triangular. Since Eǫ = 0, we can write the system as
′
′
′ ′
′
Z11 Z12
Z11 Z12
T11 T12
S11 S12
x
x
+
=0
′
′
′
′
′
Ey
0 T22
Z21 Z22
0 S22
Z21 Z22
y
and a converging solution will be
′
y = − Z22
−1
′
Z21
x
then we can back out law of motion of state variables as well.
All derivatives that Taylor expansion used is approximated with finite difference
method. We use the maximum of 10−6 or 10−4 % of the steady state as infinitesimal
when doing numerical differentiation.
25