Credit Crunch, Persistence of Shocks, and Firm Entry

Shijun Gu

University of Minnesota

Lichen Zhang

University of Minnesota

Sept. 25, 2017

Abstract

During the Great Recession, the number of start-ups drops by over thirty percent and

barely recovers since then. In this paper, we study the impact and the aggregate dynamics

of a credit shock, disciplined by the time-series of the debt-output ratio from data, on firm

entry in an incomplete-market heterogeneous agent model with an occupational choice.

We test if this model can generate both the large decline and the slow recovery in firm

entry following the credit crunch. We find that the answer depends on the persistence

of firm-specific productivity shocks: if the idiosyncratic shocks are more persistent, an

unexpected credit shock leads to a larger drop in firm entry but a quicker transition

to the pre-crisis steady state. Conversely, with less persistent idiosyncratic shocks, the

decline in firm entry is smaller but the recovery is much more sluggish. Finally, we find

that the effect of a TFP shock, constructed to mimic the endogenous TFP dynamics from

the credit shock in the baseline economy, on firm entry and the stock of firms is very

minor. This may indicate that a credit shock more closely resembles the Great Recession.

Keywords: Firm Entry, Financial Friction, the Great Recession, Slow Recovery

JEL Classification: E13, E22, E44, L25, D21, D25

1 Introduction

One of the key features of the Great Recession is that firm entry declines dramatically but

barely recovers since then. In 2015, the number of start-ups is still thirty percent lower than

the pre-crisis level. Existing literature that studies the Great Recession driven by financial

shocks (e.g. Khan, Senga, Thomas (2014), Siemer (2016)), although able to generate the

significant decline in firm entry, fails to obtain the slow recovery of it: firm entry in those

models recovers too fast to match the data.

In our paper, we show that in a heterogenous agent model with an occupational choice

and financial constraints that a large financial shock results in the decline in firm entry, but

the magnitude of this decline and the recovering speed both depend on the persistence of id-

iosyncratic productivity shocks: less persistent shocks can make the recovery more protracted

but simultaneously dampen the initial drop of firm entry following the financial shock.

Consider a potential entrant (worker) with a business idea and some wealth that can serve

as collateral to borrow from banks. In order to develop his idea, he requires some capital

and labor. Workers are hired in a competitive labor market, but capital must be financed

through collateral. Potential entrants may be discouraged from entering the market if low

levels of own wealth and collateral borrowing constraints prevent them from raising sufficient

capital for production. A credit crunch, or a negative financial shock, defined as shocks

reducing the fraction of collateral that lenders can seize in the event of default, that tightens

the collateral constraints can persistently change the wealth distribution of potential entrants

by decumulating their wealth. This can significantly affect the transitional dynamics of firm

entry since it is harder for potential entrants with less wealth to enter the market even when

credit condition recovers to its pre-crisis state.

We find that the wealth accumulation behavior of potential entrants depends on the persis-

tence of idiosybcratic productivity shocks to entrepreneurs. Thus, we treat the idiosyncratic

shock persistence as a parameter, and ask about both the steady state and the transitional

dynamics of firm entry as the persistence varies.

Our main results are as follows. Consider first the steady state. If productivity shocks are

more persistent, self-financing is an effective substitute for credit access for entrepreneurs and

substantially help undo the capital misallocation, so the output and capital stock are higher

1

in steady state.1. We also find that the fraction of wealth held by potential entrants (workers)

is decreasing in the persistence of shocks. With more persistent shocks, potential entrants are

less likely to become entrepreneurs so they tend to hold less assets since assets, serving as

collateral to finance the capital for production, is useful for potential entrants only when they

become entrepreneurs. Conversely, when shocks are more transitory, entrepreneurs are more

likely to lose their entrepreneurial talent before accumulating sufficient assets, but workers

will have better chance to switch to entrepreneurs in the future. Consequently, entrepreneurs

tend to accumulate less assets while workers accumulate more.

Now consider the transition to steady state. Suppose there is a financial shock that tightens

the collateral constraints of firms as what happened in the Great Recession and then the credit

condition gradually recovers to the pre-crisis state. If productivity shocks are more persistent,

an unexpected credit shock leads to a larger drop in firm entry but a quicker transition to the

pre-crisis steady state (i.e. a quicker recovery). Conversely, less persistent shocks imply that

the decline in firm entry is smaller but the recovery is much more sluggish.

The magnitude of the decline in firm entry resulting from an unexpected credit shock is

determined by the interplay between the wealth distribution of potential entrants in steady

state and the change in the cutoff of wealth for entering the market. A negative credit shock

makes it harder for potential entrants to enter the market by increasing the cutoff of wealth

for entry given a productivity level. Lower idiosyncratic shock persistence leads to a much

smaller increase in the cutoff of wealth for entry, together with a smaller steady state cutoff

for entry and a more left-skewed wealth distribution of potential entrants, resulting in a much

smaller drop in firm entry.

As for the speed of the transition of firm entry, with more persistent productivity shocks,

potential entrants with high productivity know that they will be more likely to become en-

trepreneurs when credit condition recovers, so they tend to build up the stock of assets more

quickly to get rid of the collateral constraints. This speeds up the transition of firm entry to

pre-crisis steady state substantially. In contrast, if shocks are more transitory, agents with

high productivity would build up the stock of assets more slowly for smoothing consumption.

We also check the transitional dynamics of the major macroeconomic aggregates including

output and capital stock. With more persistent idiosyncratic shocks, both output and capital1This is consistent with Buera and Shin (2011) and Moll (2014)

2

stock drop less and recover more slowly.

Finally, we explore the behavior of a calibrated economy in response to an aggregate TFP

shock and then compare it with its response to a credit shock. We find that the implications

for the dynamics of firm entry, employment of startups, and the stock of firms with the TFP

shock are starkly different from those with the financial shock. While the credit crunch driven

by the financial shock can generate a large decline in entry and âa missing generation of firmsâ,

the effect of the TFP shock, constructed to mimic the endogenous TFP dynamics from the

financial shock, on firm entry as well as the stock of firms is very minor. This may imply that

a negative credit shock more closely resembles the 2007-2009 U.S. recession.

Related Literature Our paper is related to an extensive literature on the effect of financial

frictions in macroeconomics, starting with Bernanke and Gertler (1989), Carlstrom and Fuerst

(1997), Kiyotaki and Moore (1997), and Kocherlakota (2000), and later, Kiyotaki and Moore

(2008) and Jermann and Quadrini (2012). They use, however, a representative framework,

so our mechanism generated through the change in wealth distribution cannot be obtained.

More recent quantitative papers relates financial frictions to macroeconomic aggregates in an

Aiyagari-Bewley styled heterogenous agent model (e.g. Buera and Moll (2015), Gopinath et al.

(2017), Khan and Thomas (2013), Liu and Wang (2014), Midrigan and Xu (2014), Shourideh

and Zetlin-Jones (2016)), but none of the models mentioned above involve firm dynamics to

learn firm behavior during and after the Great Recession.

The model used in our paper is a natural extension of Evans and Jovanovic (1989) to

dynamic environments. Buera and Shin (2011, 2013) first develop this dynamic quantita-

tive incomplete-market heterogenous agent model with an occupational choice. This model

(or the extension of this model) was used to study various topics: the relationship between

aggregate/sector-level TFP and financial development across countries (Buera, Kaboski, Shin

(2011)), the effect of financial market imperfection on capital misallocation and cross-country

TFP differences (Buera and Shin (2013)), as well as how a credit crunch in the Great Reces-

sion leads to protracted unemployment (Buera, Fattal-Jaef, Shin (2015)). But no one has ever

used this model to explore the effect of financial frictions on firm entry and stock of firms. We

are the first to use Buera and Shin (henceforth, BS) model to study the transitional dynamics

of firm entry following a credit crunch. We believe BS model is a good candidate to explore

3

the response of firm entry to a temporary credit shock. The reason is that for the existing

models in the spirit of Hopenhayn (e.g. Khan, Senga, Thomas (2014), Siemer (2016)), firm

entry only depends on the state in current periods but not in previous periods and moreover,

exiting firms cannot re-enter the market, but for BS model, firm entry does not only depend on

the current state but also previous states. For example, a credit shock that hits the economy

five periods ago can affect potential entrants’ entry decision in current period by affecting his

wealth accumulation since five periods ago.

Our paper is most closely related and complementary to two recent papers (Buera and

Shin (2011) and Moll (2014)) that also highlight the role played by the idiosyncratic shock

persistence. Buera and Shin (2011) only check steady states, so little is known about the

transitional dynamics. They also concentrate more on the different motives for assets accu-

mulation. When shocks are more persistent, on the one hand, it becomes harder to self-insure

idiosyncratic shocks, but on the other hand, entrepreneurs have a stronger motivation for

self-financing, which leads to better allocation of production factors. Since the focal point

of our paper is firm entry, which is more related to whether a would-be entrepreneur will

have sufficient assets to finance an optimal level of capital, we only focus on the self-financing

motive. More importantly, we check the transitional dynamics of aggregates including firm

entry responding to a path of credit shocks.

Moll (2014) studies the differential implications of shock persistence for both steady states

and transitional dynamics using a model whose structure is very similar to but more tractable

than the model used in our paper (i.e. BS model). The most significant difference lies that

his model does not feature an occupational choice, so there is no firm dynamics and workers

are hand-to-mouth agents. Since a worker will never become an entrepreneur, provided that

the interest rate is less than the rate of time preference, holding assets does not bring any

benefit, and workers hence hold zero assets in the long run steady state. In BS model, even

when the persistence of shocks are very high, as long as the probability that workers become

entrepreneurs is non-zero, workers are still accumulating some assets. Thus, we can use this

model to check the transitional dynamics of firm entry to better understand how a credit

crunch affect firm entry via its effect on wealth distribution of potential entrants. Due to

this major difference between Moll (2014) and BS model, the transitional dynamics of output

and capital stock behaves differently as well. In Moll (2014), more persistent shocks leads to

4

a larger drop due to an unexpected aggregate shock, while in our paper, with higher shock

persistence, both output and capital stock decline less.

We join a growing literature that attempts to understand the reasons for lack of entry

during and after the Great Recession. Ayres and Raveendranathan (2016) and Clementi et al.

(2017), both using a Hopenhayn styled model of firm dynamics featuring aggregate uncertainty

and firm heterogeneity, show that demand shock is a major contributor to the significant drop

in firm entry. Clementi et al. (2014), Khan, Senga, Thomas (2014), and Siemer (2014) develop

a quantitative heterogenous firm model featuring endogenous borrowing constraint, showing

that lack of entry is a consequence of credit crunch since credit tightening directly affects

small firms the most. However, all of the works mentioned above cannot explain the slow

recovery of firm entry. Our work, using BS framework, shows that for the same credit shock,

the drop in firm entry and the speed of the recovery of firm entry depends on the persistence

of idiosyncratic shocks, which is complementary to their works.

Layout The paper is organized as follows. Section 1 describes the model set-up and defines

the competitive equilibrium; Section 2 presents our quantitative results in which we emphasize

the persistence of idiosyncratic shocks to entrepreneurial productivity. Section 3 concludes the

paper.

2 Model

We model an economy populated by a continuum of individuals, who are heterogenous with

respect to their wealth (or assets) and entrepreneurial productivity. In each period, an in-

dividual chooses whether to work for a wage or to operate an individual-specific technology

(entrepreneurship).

Access to capital is determined by entrepreneurs’ wealth through a simple collateral con-

straint, motivated by the imperfect enforceability of capital rental contracts. One entrepreneur

can operate only one production unit (establishment) in a given period. Entrepreneurial ideas

are inalienable, and there is no market for managers or entrepreneurial talent. We use an

entrepreneur and a firm interchangeably.

5

Heterogeneity and demographics

Individuals live infinitely and are heterogenous in their asset a and entrepreneurial productivity

z ∈ Z. The entrepreneurial productivity follows a AR(1) process. The population size of the

economy is normalized to one, and there is no population growth. There is no aggregate

uncertainty either.

Preferences

Individual preferences are described by the following expected utility function over sequences

of consumption, ct:

U (c) = E

[ ∞∑t=0

βtu (ct)

](1)

where β is the discount factor. The expectation is taken over the realizations of the en-

trepreneurial productivity z.

Technology and Occupational Choice

At the beginning of each period, an individual chooses whether to operate his own business or

not. If not, he works for the market wage wt. An entrepreneur with talent z produces using

capital k and labor n using decreasing-return-to-scale technology according to:

AzF (k, n) = Az(kαn1−α)1−ν (2)

where 1− ν is the span-of-control parameter. A share of 1− ν of output goes to factor of

inputs. Out of this, a fraction of α is going to capital and 1− α going to labor.

Financial Markets

Productive capital is the only asset in the economy. There is a perfectly competitive financial

intermediary that receives deposits and rents out capital to entrepreneurs. The return on

deposited assets—i.e. the interest rate in the economy—is rt. The zero-profit condition of the

intermediary implies that the rental price of capital is rt+ δ, where δ is the depreciation rate.

We assume that entrepreneurs’ capital rental k is limited by a collateral constraint kt ≤ λtat.

6

2.1 Individual’s problem

In each period t, she chooses numeraire consumption ct and the amount of wealth to bring to

the next period, at+1. In this paper, just as in Buera and Shin (2011), we focus on within-

period borrowing, or capital rental, for production purposes. We do not allow borrowing

for intertemporal consumption smoothing, which translates into a ≥ 0. This no-borrowing

constraint has no direct bearing on the behavior of entrepreneurs and workers: entrepreneurs

will also need to hold assets to finance capital for production and workers whose probability

of becoming an entrepreneur is non-zero also need to hold asset for the same reason.

At each period, an agent’s wealth is represented by at . If the agent chooses to be an

entrepreneur, he needs capital kt for production. Part of the capital comes from financial

asset directly. The rest of the capital requires borrowing bt = kt − at.

At each period, an individuals chooses between being a worker or an entrepreneur given

her productivity and wealth. The value function of an individual is given by:

v (z, a) = max{vW (z, a) , vE (z, a)

}(3)

worker’s problem

vW (z, a) = maxcu (c) + βE[v(z′, a′)]

(4)

s.t.

ct + at+1 ≤ wt + (1 + rt) at (5)

entrepreneur’s problem

vE (z, a) = maxc,k,n,a′u (c) + βE[v(z′, a′)]

(6)

s.t.

ct + at+1 ≤ AtztF (kt, nt)− wtnt − (rt + δ) kt + (1 + rt) at (7)

kt ≤ λtat. (8)

7

The capital and labor demand functions are given by

k (z, a; r, w, λ,A) =

[

rAαz

(Aνzw

) νν−1

] ν−11−α−ν

λa.

if k ≤ λa.

otherwise

(9)

n (z, a; r, w, λ,A) =

(Azνk (z, a; r, w, p, λ)

w

) 11−ν

(10)

That is, capital demand equals its unconstrained level if that level is consistent with the

borrowing constraint. Labor demand depends indirectly on the borrowing constraint because

it is a function of capital demand. When the borrowing constraint binds, both factor demand

functions depend on personal wealth and the collateral constraint parameter. The constraint

on capital and labor demand implies a constraint on both entrepreneurial output and profits

any time the entrepreneurs’ optimal scale is larger than that allowed by the borrowing limit.

This is the key mechanism for this model: a household with little financial and housing wealth

can receive a large productivity draw but still be forced to operate at a scale that is well below

the unconstrained optimal level, or may even choose to be a worker instead.

2.2 Competitive equilibrium

Given an initial distribution of individual wealth and entrepreneurial productivityG0 (z, a) and

a sequence of exogenous housing prices {pt}∞t=0 , a competitive equilibrium comprises prices

{wt, rt}∞t=0 and allocations {ct (z, a) , at+1 (z, a) , kt (z, a) , nt (z, a) , ot (z, a)}∞t=0 such that:

1. Given prices {wt, rt}∞t=0 , the allocations are solutions to the individuals problems;

2. Capital market clears:

∫k (z, a) dG (z, a) =

∫adG (z, a) (11)

3. Labor market clears: ∫n (z, a) dG (z, a) =

∫dG (z, a) (12)

4. The joint distribution of (non-housing) wealth and entrepreneurial productivity {Gt (z, a)}∞t=0

8

evolves according to the following equilibrium mapping:

Gt+1 (z, a) = ψ

∫at+1(z,a)≤a

dGt (z, a) + (1− ψ)∫at+1(z,a)≤a

dGt (z, a) (13)

3 Quantitative exploration

In quantifying our theory, we interpret our entrepreneurial technology as an establishment in

the data. We assume that the US economy before the Great Recession is the benchmark, and

pin down parameter values using relevant moments of the US data. Section 2.1 discusses the

calibration strategy. Section 2.2 reports the results of steady state of major macro variables

including output and capital stock as well as wealth share as we vary the persistence of

idiosyncratic shocks. In section 2.3, we highlight the role of shock persistence played in both

the magnitude of drop in firm entry and major macro variables and the transition speed of

them as well when a negative credit shock hits. We also provide a detailed explanation for

the results. We then compare the response of the calibrated economy to a credit shock with

its response to a TFP shock in section 2.4.

3.1 Calibration

We first describe the parametrization of the model, and then discuss how to calibrate the

parameters to match the data.

We use the standard constant-relative-risk-aversion utility function, and σ is the coefficient

of risk aversion :

u(c) =c1−σ

1− σ.

The entrepreneurial productivity z follows a discretized version of an AR(1) process with

normal innovations:

logz′ = ρlogz + ε, ε ∼ N(0, σ2).

We approximate this autoregressive process with a 11-state Markov chain following the the

procedure of Rouwenhorst (1995). As discussed by Kopecky and Suen (2010), Rouwenhorst

9

Parameters Value Moment Model DataPanel A: CalibratedDiscount factor β 0.94 Real interest rate 0.04 0.04Persistence of TFP ρz 0.95 Establishment exit rate 0.10 0.10Std. of TFP σz 0.20 Entry to existing firm size 0.46 0.47Collateral constraint λ 5.9 External finance to output ratio 1.75 1.75Capital share α 0.37 Capital-output ratio 2.9 2.9Panel B: AssignedRisk aversion σ 2Depreciation rate δ 0.06Span-of-control 1− ν 0.83

Table 1: Parameter Values in Benchmark Economy

method is more reliable than others in approximating highly persistent processes and generat-

ing accurate model solutions. In our benchmark model, we require a process with high serial

correlation in order to match the firm-level data. This makes the Rouwenhorst method very

suitable in our numerical exercises.

The calibration strategy intends to match the steady-state moments to the relevant aggre-

gate and firm-level data in the US economy. We assume a time period in the model to be one

year.

Table 1 Panel B shows the parameters that we exogenously assign following the standard

literature. Specifically, we choose the relative risk aversion σ = 2, depreciation rate δ = 0.06,

and span-of-control 1− ν = 0.83.

Table 1 Panel A shows the parameters that we parametrize to match the moments in data.

We need to specify five parameter values: the technological parameters α; the parameters

governing the persistence and dispersion of entrepreneur’s productivity ρ, σz; the subjective

discount factor β; the parameters governing the tightness of collateral constraint λ. We target

the following moments in the US data: the annual exit rate of establishments, the real interest

rate, and the ratio of external finance to the total output ratio, the relative size of entering

and existing establishment, and US capital-output ratio.

The persistence of entrepreneurial productivity process ρ is chosen to match the 10 percent

annual exit rate of establishment. The standard deviation of productivity σz can be mapped

into the relative size of entering and existing establishment The discount factor β is calibrated

to match the 4 percent annual real interest rate. There is a direct link from the collateral

constraint parameter λ and the external finance to output ratio. We choose capital share α

10

Figure 1: The relationship between macro aggregates and idiosyncratic shock persistence

to match the US capital-to-output ratio. All the parameters are jointly determined and we

discuss the most related matching between parameters and data.

3.2 Steady State

In this section, we explore the impact of the persistence of shocks, holding fixed their uncon-

ditional distribution, on steady-state variables. We allow the persistence of the shocks vary

only within a the range (0.85, 0.95) as explained in the last section. We fix λ = 5.9, which

matches the pre-crisis external finance to asset ratio in the data.

We first look at the effect of different idiosyncratic shock persistence on steady-state out-

put, capital stock and external finance to asset ratio. As figure 1 shows, output and capital

stock are increasing in the persistence of idiosyncratic productivity shocks, while external

finance to asset ratio is decreasing in the persistence of shocks. When shocks are more per-

sistent, entrepreneurs have stronger motivation for self-financing to get rid of the collateral

constraint, thus always being able to operate at the optimal scale, so the capital stock must

be higher, which directly leads to higher output in steady state. The negative correlation

between external finance to asset ratio and shock persistence is an obvious consequence of the

results that both capital stock and output are increasing in persistence of shocks.

We then plot both the fraction of assets and the average assets held by workers and

entrepreneurs respectively against the idiosyncratic shock persistence. In figure 2, we can

see that both the fraction of assets and the average assets held by workers are decreasing in

persistence of shocks. Correspondingly, both the fraction of assets and the average assets held

by entrepreneurs are increasing in persistence of shocks.

11

Figure 2: The relationship between wealth held by agents and idiosyncratic shock persistence

The reason is that varying the persistence of idiosyncratic productivity shocks result in

an asymmetric effect on the saving motives of entrepreneurs and workers. When shocks are

more persistent, entrepreneurs tend to increase their asset holdings in long run steady state

because self-financing can effectively help get rid of the collateral constraint. On the other

hand, workers tend to accumulate less wealth because now they are less likely to become en-

trepreneurs and assets are only useful for entrepreneurs. Conversely, when shocks are more

transitory, entrepreneurs are more likely to lose their entrepreneurial talent before accumu-

lating sufficient assets, but workers instead have better chance to switch to entrepreneurs

in the future. Therefore, when the persistence of shocks is reduced, entrepreneurs save less

and workers save more. This intuition is verified by figure 3 where we can see that with lower

shock persistence, wealth distribution of workers is more left-skewed. This means that a larger

fraction of workers tend to hold more assets. Since workers are essentially potential entrants,

this steady-state result has direct effect on both firm entry in steady state and in transitional

dynamics. The latter will be discussed in next section.

Based on the results generated from figure 2 and 3, it is no surprising to see that both

firm entry and the average size of entrants (in terms of employment) are decreasing in shock

persistence , as shown in figure 4. With more persistent shock, a larger fraction of potential

entrants (i.e. workers) have less assets at hand, which generates a smaller number of entrants

in steady state.

12

Figure 3: The wealth distribution of potential entrants (workers)

Figure 4: The relationship between firm entry and idiosyncratic shock persistence

3.3 Transitional Dynamics

Having examined how the change of steady-state variables depends on the persistence of

productivity shocks, we now turn to the transitional dynamics of the model. We simulate

the aggregate dynamics of the model following a tightening of the collateral constraint—i.e.,

a lower λt. We discipline the the path of the aggregate shocks to λt in three steps. First, we

choose λ0 = 5.9 to match the steady state external finance to output ratio. Second, we choose

[λ1, λ2, λ3, λ4, λ5] = [5.4, 4.8, 4.2, 4, 4.3] to match the time series of external finance to output

ratio. Third, for 80 ≥ t ≥ 6, we have λt = 0.75λt−1 + 0.25λ0.

We consider two cases: a high persistence case and a low persistence case. In high per-

sistence case, we choose ρ = 0.95 is to match the establishment exit rate before the Great

Recession. In low persistence case, we choose ρ = 0.85 to match the fraction of entrepreneurs

13

Figure 5: Ratio of External Finance to Output

exiting the entrepreneurship. We illustrate our key results in figure 6-8. The magnitude of

the decline in firm entry (and consequently, total employment of entrants and the stock of

firms) and their recovering speed both depend on the persistence of idiosyncratic productivity

shocks. If idiosyncratic shocks are more persistent, an unexpected credit shock leads to a

larger drop in firm entry but a quicker transition to the pre-crisis steady state. Conversely,

less persistent shocks imply that the decline in firm entry is smaller but the recovery is much

more sluggish.

Based on our key results, we need to answer two questions: why are more persistent

productivity shocks associated with a more sizable drop in firm entry after the economy is hit

by a credit crunch? Why is the transition of firm entry slower when the productivity shocks

are relatively transitory? In order to answer the two important questions, we first need to

more closely examine the agents’ decision rule and distribution.

Before turning to address the two questions, we briefly recap the results from steady state

analysis which are particularly related to this part. Varying the persistence of idiosyncratic

productivity shocks has a direct impact on the self-financing motives for both entrepreneurs

and workers. Low persistent shocks are more favorable to workers because they are more likely

to draw a new and large enough productivity that allows them to switch occupation. Con-

14

Figure 6: The transitional path of firm entry, total employment of entrants, and the stock offirms under two cases

versely, high persistent shock are more favorable to entrepreneurs, in particular for those with

low assets, as it enables them to overcome the collateral constraint through asset accumulation.

To address the first question, it is helpful to check the occupation switching rule of agents

and understand how credit shock affects occupation choices. From the perspective of this

model, the firm entry declines after the credit shock because more potential entrants find that

staying at the current working status gives them higher income than starting a new business.

In other words, some workers characterized by their wealth holding a and entrepreneurial

productivity z should have become entrepreneurs without the credit crunch. However, when

they are constrained due to the shock, profits drop below the wage. So they instead choose to

stay.

The figure 9 depicts the occupation switching rule of agents who has a 85th percentile

productivity. We plot the income of agents from working or producing against their wealth

status. The solid blue lines describe the profits from entrepreneurial activities at steady state

and the dashed blue lines plots its counterpart at the first period after the shock. The solid

red lines is the wage at steady state while the dashed red line is the wage at the first period

after the shock. The intersection point between two solid lines gives the occupation switching

cutoff at the steady state. The agent with an asset level to the left of the left dash-dot line

choose to be a worker and the agent with an asset level to the right of the left dash-dot line

choose to be an entrepreneur. Similarly, the intersection point between two dashed lines gives

the occupation switching cutoff at the first period after the credit shock hits. The occupation

choice also depends on the relative position between assets and the cutoff, the right dash-dot

15

Figure 7: Firm Entry - High Persistence: Model v.s. Data

Figure 8: Firm Entry - Low Persistence: Model v.s. Data

16

Figure 9: Shift in Wealth Cutoff of Occupation Switch

line.

The left panel of the figure 9 describes the case with highly persistent idiosyncratic pro-

ductivity shocks and the left panel shows the low persistence case. An important observation

from the figure is that the size of shift in cutoff following a credit shock when the productivity

shocks are highly correlated is significantly greater than that when shocks are more transitory.

Therefore, holding the wealth distribution fixed, more persistent shocks cause more potential

entrants, who should have entered without a credit shock, fail to make the occupation switch

due to the tightening of collateral constraint.

To understand what causes this result, we need to switch our attention to entrepreneurs.

As we discussed in the previous section, entrepreneurs in an economy with more persistent

idiosyncratic productivity shocks will hold more assets. When the wealth holding is big,

entrepreneurs are less vulnerable to the credit shock because they are less reliant on credit.

Hence a larger fraction of them will be unaffected in the choice of capital and labor. As the

capital and labor demand go down by less, the wage and interest rate will fall by less. On the

other hand, more transitory productivity shocks are associated with low-level asset holding

by entrepreneurs. This will cause capital and labor demand drop more, which leads to a

more significant drop in the factor prices. Figure 9 shows the different responses of wage and

interest rate following the credit shock in two cases.

The occupation switching rule depends both on wage and profits. In the high persistence

case, the wage and interest rate fall by less, and the tightening of collateral constraint will cause

17

Figure 10: Transition Paths of Wage and Interest Rate

the binding entrepreneurs’ profits fall by more. Meanwhile, since the wage is not adjusted too

much, agents characterized by a wide range of wealth holdings will make a different occupation

choice after the credit shock. Conversely, in the low persistence case, the wage and interest

rate fall by more, the tightening of collateral constraint will cause the binding entrepreneurs’

profits fall by less. Since the wage drops substantially, agents characterized by a narrow range

of wealth holding will choose to become workers instead of entrepreneurs after the credit shock.

Notice that in our previous analysis we assume that we have a fixed wealth distribution in

the two cases. Essentially we are showing that a potential entrants (workers) are less likely to

switch occupation when the shocks are more transitory. As the right panel of figure 8 shows,

the range of asset level which can support the entrepreneurial activities is shrinking. There

exists another force may affect our result. That is, the population affected by the shift in

cutoffs in high persistence case is actually less than that in the low persistence case. In our

numerical experiments, however, this force is less important.

To pin down our analysis, we also provide the expressions of probability of being an

entrepreneur in the next period, and the number of entrants.

Let η(a, z) denotes at time t, the probability of being an entrepreneur in the next period.

Using the decision rule of asset accumulation a′(a, z), the occupation switch rule o(a, z), and

transition probability p(z, z′), we can compute the probability of being an entrepreneur in

t+ 1 given state (a, z) by

18

η(ai, zj) =∑j

o′(a′(ai, zj), z′)p(zj , z

′),

where o(ai, zj) is a dummy variable indicating an agent’s occupation choice, 1 denotes en-

trepreneur and 0 denotes worker. o′(a′(ai, zj), z′) is the occupation choice of agents charac-

terized by wealth level ai and current period productivity zj in the next period. We know

an agent’s next period occupation given next period productivity shock by substitute in the

asset decision rule.

The number of entrants can be computed as

NE =∑j

∑i

η(ai,zj){o(ai,zj)=0}µ(ai,zj),

where µ(ai, zj) is the population distribution of agents. Since {o(ai, zj) = 0} removes the

current period entrepreneurs, so we are computing the population of agents who are workers

at t, but switch to be entrepreneurs at t+ 1.

In our analysis, we show that due to the change of o′(a′, z′) after a credit shock, in high

persistence case, it is more likely to see that o(ai, zj) = 1 at steady state, while o′(a′i, z′

j) = 0

in the first period after shock. So the probability of being an entrepreneur in t+1 given state

(a, z) at t falls by more with more persistent idiosyncratic productivity. This effect dominates

the force that the number of potential entrants who are affected by the change in probability

of entry is more in low persistence case. So the number of entrants falls by more in the high

persistence case.

Now we move on to our second question: Why is the transition of firm entry slower when

the productivity shocks are relatively transitory?

To answer this question, we implement two counterfactual experiments when calculating

the number of entrants in the transition dynamics (Low persistence case). First, we fix the

probability of being an entrepreneur in next period at t = 3, the last period before the recovery

of credit condition. Second, we vary the probability of being an entrepreneur in next period,

but fix the population distribution and occupation switching rule at t = 3. Figure 11 depicts

a comparison among the normal calculation (blue solid line), the counterfactual 1 (red dashed

line), and the counterfactual 2 (black dot-dash line).

The counterfactual analysis indicates that the probability of being an entrepreneur in

19

Figure 11: Firm Entry Counterfactual Analysis

next period is the most important determinants of the transitional dynamics of firm entry.

For workers, it is equivalent to the probability of entry. In the counterfactual experiment 1,

the number of entrants does not stop declining immediately after an improvement of credit

condition as the wealth distribution keep shifting left. Moreover, it never goes back to the

steady state. However, if we only fix the other two components, firm entry recovers quickly

and overshoots.

This result suggests that we should focus on the probability of entry. Because we assume

that the collateral shock remains at the lowest degree for three periods and let it recover very

quickly since then, o(a, z) , the occupation switching rule of agents, returns to steady state

rapidly. Therefore, the speed of recovery in the probability of entry depends on how fast

workers will rebuild asset stock determined by a′(ai, zj).

When the credit shock hits the economy, agents are forced to run down their assets for

smoothing consumption. In turn, an increasing number of agents, whose productivity is

eligible to run a business, choose to become workers because their asset level is too low. When

the credit condition is relaxed, the persistence of idiosyncratic productivity shocks shapes the

asset accumulation behavior of aforementioned agents. When the shocks are highly persistent,

they know that it is very likely for them to remain the current productivity level. So they

will substantially increase saving because they internalize that it can help them to overcome

20

Figure 12: Aggregate implication of a credit crunch vs. exogenous TFP shock

collateral constraint and switch occupation. However, when the shocks are relatively more

transitory, they know that they are less likely to become entrepreneurs by running up assets

slowly as they are more likely to lose the current entrepreneurial ability in the near future.

3.4 Credit shock v.s. TFP shock in the calibrated economy

Our goal in this section is to contrast the impact of the credit crunch to that of a more

standard source of business cycle fluctuations in the calibrated economy: a negative shock to

the aggregate TFP. To implement this experiment, we start with the calibrated economy in

its stationary equilibrium and hit it with an unanticipated drop and recovery in the aggregate

TFP - At in equation (2). This exogenous TFP series is constructed to mimic the endogenous

TFP dynamics from the credit crunch.

Figure 12 depicts the dynamics of aggregate quantities in response to the credit crunch

21

(solid line) and to the exogenous aggregate TFP shock (dashed line). Despite the identical

TFP dynamics of the two experiments, the TFP shock exercise shows a lightly sharper and

more protracted contraction in output and capital stock. Investment rate seem comparable in

both cases.

The most significant feature that differentiates a credit shock from an exogenous decline in

TFP is its reallocative nature. Even though all entrepreneurs face the same tighter collateral

constraint (i.e., lower λt ) during the credit crunch, only a subset of them—those who are

productive but have little financial wealth as collateral—becomes more constrained in their

choices of capital and labor inputs. Although constrained firms will decumulate assets fol-

lowing the shock, unconstrained firms will accumulate even more assets due to the general

equilibrium effect - lower interest rate. Therefore, capital stock (and thus output) drops less

and recovers more quickly, compared to the TFP shock. In contrast, an exogenous aggregate

TFP shock induces a contraction in capital demand of all firms symmetrically. Consequently,

firms all need less collateral to finance capital, so they all tend to save less. Since capital

demand and supply both drop, equilibrium interest rate does not change much. Capital stock

(and thus output) drops more and recovers more slowly since building up sufficient assets to

operate at the optimal scale when TFP is recovering takes time.

The factor prices—wages and interest rates—also respond differently to the two shocks.

The responses of interest rates are more starkly different between the two cases (as figure 12).

The credit crunch has a direct negative effect on many entrepreneurs’ capital demand, forcing

affected firms to scale down operations to the level allowed by the tighter collateral constraint.

To bring the capital rental market back to an equilibrium, the interest rate has to fall. Because

only a subset of entrepreneurs, who are unconstrained and more likely to be unproductive, can

increase their capital demand in response to lower rental rates, the market clearing interest

rate must fall by more (solid line) than when all entrepreneurs can symmetrically respond

to lower rental rates (i.e., the aggregate TFP shock case, dashed line). Comparing the wage

dynamics between the two experiments, we find that wages fall by slightly more with the credit

crunch (solid line) than with the exogenous TFP shock (dashed line). The reason is that for

credit shock, the reallocation of capital toward unconstrained, unproductive entrepreneurs

necessitates that the wage must fall further to clear the labor market.

More importantly, we check how firm entry, employment of startups, and the stock of firms

22

Figure 13: Transition Paths of Wage and Interest Rate (TFP shock)

respond to the two shocks. From figure 13, we can see that in this calibrated economy, a credit

shock can indeed generate decline in firm entry and employment of startups as well as “the

missing generation of firms” and the subsequent slow recovery of the stock of firms. However,

it is still unable to attain the persistent response of firm entry. It is clear to see that the effect

of the TFP shock on firm entry is minor, so its effect on employment of startups and the stock

of firms. The reason is as follows. For the credit shock, the drop in constrained entrepreneurs’

profit and in wage are asymmetric, so for a potential entrant that draws a high productivity,

the profit that he can earn if he chooses to be an entrepreneur is smaller than the wage. This

leads to a much larger decline in firm entry. Conversely, for the TFP shock, the decrease in

the both the profit and wage decrease so that they become comparable, which induces very

limited decline in firm entry. From the perspective of firm behavior, it seems that a negative

credit shock more closely resembles the 2007-2009 U.S. recession.

4 Conclusion

The slow recovery in firm entry after the Great Recession has received much attention from

empirical research. However, few quantitative explorations succeed in reproducing this pattern

without using a permanent aggregate shock. In this paper, we propose a possible solution to

this problem under the framework of Buera and Shin (2013). Our results show that by

lowering the persistence of idiosyncratic productivity shocks, the recovery of firm entry can

23

be significantly protracted. When idiosyncratic shocks are more transitory, workers with high

productivity tend to build up assets more slowly as credit condition is recovering after a credit

crunch. This leads to a more sluggish recovery in the probability of entering the market which

is manifested as the cutoff of assets/wealth for entering the market given a fixed productivity

level. Since the transition speed of firm entry is mainly determined by the transition speed

of the probability of entering the market rather than that of the wealth-productivity joint

distribution of potential entrants (as shown in Section 2.2), the transition of firm entry to

pre-crisis steady state must be more protracted as idiosyncratic shock persistence is lower.

Our results may provide insight to further quantitative research attempting to generate a

persistent response of firm entry to a non-permanent aggregate shock like the credit shock

defined in our paper and many other papers.

24

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Appendix : Computational details

Computing the steady state

We solve for the stationary equilibrium of this economy using the nested fixed-point algorithm

of Aiyagari (1994). The difference is that we have to iterate on the wage w, interest rate r,

until the labor and capital markets clear are cleared.

1. Guess the interest rate in the stationary equilibrium, ri.

2. Guess the wage in the stationary equilibrium, wi,j .

3. Given the interest rate, and wage, solve the individuals’ problem using value function

iteration. Given the optimal decision rules, the stochastic process for entrepreneurial

ability and an uniform initial joint distribution of wealth and ability, iterate on the laws

of motion for the joint distribution until |µt+1(a,z)-µt(a,z)|<10−9, t denotes the number

of iteration.

4. Check the labor market clearing condition, aggregating labor demand and supply using

the stationary distribution of wealth and entrepreneurial productivity. If there is excess

labor demand (supply), choose a new wage wi,j+1. that is greater (smaller) than wi,j .

Use bisection.

5. Repeat steps 3–4 until the labor market clears under the invariant distribution.

6. Check the capital market clearing condition, aggregating capital demand and supply

using the stationary distribution of wealth and entrepreneurial productivity. If there is

excess capital demand (supply), choose a new interest rate ri+1. that is greater (smaller)

than ri.

7. Repeat steps 3–6 until the capital market also clears under the invariant distribution.

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Computing the transition dynamics under perfect foresight

To compute the entire transition dynamics during the credit crunch and the recovery (i.e., the

exogenous λt sequence), we have to iterate on the wage and interest rate sequences, following

the algorithm proposed by Guerrieri and Lorezoni (2017). Taking these sequences as given, we

solve for the individuals’ problem and then check whether the labor and capital markets clear

for the entire periods. We fix T , the period by which all transitions are completed, at 80. In

our exercise, the final stationary equilibrium is the same as the initial stationary equilibrium,

since there is no permanent exogenous permanent change.

1. Guess an interest rate sequence{rit}T=80

t=0, with rit equal to the initial stationary-equilibrium

interest rate for t ≥ T .

2. Guess a wage sequence{wi,jt

}T=80

t=0, with wi,jt equal to the initial stationary-equilibrium

wage for t ≥ T .

3. Let vT (a, z) = v0(a, z) where v(·) is the individual value function in the initial stationary

equilibrium. By backward induction, taking the wage and interest rate as given, compute

the value function vt(a, z) for t = T − 1, ..., 0.

4. Using the optimal decision rules, the stochastic process for entrepreneurial ability, and

the initial joint distribution of wealth and entrepreneurial ability iterate forward the

joint distribution over t. Check whether the labor market clears in every period. If not,

construct a sequence{wi,jt

}T=80

t=0, that clears the labor market period by period. Update

the wage sequence: wi,j+1t = ηww

i,jt + (1− ηw)wi,jt for all t, with ηw∈(0, 1).

5. Once the wage sequences converge, check whether the capital market clears in all periods.

If not, compute a sequence{rit}T=80

t=0, that clears the capital market period by period.

Update the interest rate sequence with ri+1t = ηrr

it + (1− ηr)rit for all t with ηr∈(0, 1).

6. Repeat steps 3–5 until the interest rate sequence also converges.

29