Firm Entry and Exit with Financial Frictions
Patrick Macnamara∗
University of Rochester
November 28, 2011
Abstract In this paper, I consider a model of firm dynamics to study the quantitative
importance of financial frictions for the entry and exit of firms over the business cycle. Firms
face a working capital constraint in that they must pay wages and choose labor in advance.
The firm finances wages with a defaultable one-period non-contingent bond. Financial frictions then introduce an external finance premium into the firm’s cost of borrowing, which
constrains the firm’s demand for labor and influences the firm’s entry and exit decisions
through its effect on future profits. From the data, I then measure a financial shock from
corporate bond spreads and feed it back into the model. I find that, historically, financial
shocks account for 43% of cyclical movements output in the United States, 37% for hours,
21% for exit rates and 7% for entry rates. In this model, aggregate fluctuations in output
and hours are driven by the largest firms who are less dependent on external finance, while
fluctuations in entry and exit rates are driven by firms who are smaller and more dependent
on external finance. Thus, given the distribution of debt financing needs observed in the
data, financial frictions can account for fluctuations not only in aggregate output and hours
but exit rates as well.
∗
Department of Economics, University of Rochester. Email: [email protected]. The author is grateful
to Yongsung Chang, Mark Bils and seminar participants at the Federal Reserve Bank of Richmond for advice
and suggestions. All errors and flaws are my own.
1
1
Introduction
In the Great Recession, real GDP fell by 5% from the fourth quarter of 2007 to the
second quarter of 2009. Over the same time period, the entry rate fell by 13% and the
exit rate increased by 10%. However, models of financial frictions, which have been used
to explain fluctuations in aggregate output, generally have not assumed that the entry and
exit of firms is endogenous. This modeling choice overlooks the potential role for financial
frictions in explaining movements in entry and exit rates over the business cycle. Financial
frictions forcing firms to exit and limiting entry could be further evidence of their quantitative
importance. Thus, I construct a model of firm dynamics with financial frictions in order to
address the question of how much credit market conditions matter for cyclical movements
in entry and exit rates, as well as standard variables like output and employment. Under
reasonable parameters, I find that historical financial shocks can account for 43% of cyclical
movements in output in the United States, 37% of movements in hours, 21% of movements
in exit rates and 7% of movements in entry rates.
Building on Hopenhayn (1992), I assume that firms are ex-ante heterogeneous, face idiosyncratic productivity shocks and must pay fixed costs both to enter and to operate. To
model financial frictions, I include a working capital constraint by assuming that firms must
hire labor and pay wages one period in advance, subject to a decreasing returns to scale
technology. This assumption is motivated by the observation that many production inputs
such as labor and materials have to be purchased ahead of time. The firm must partially
finance wages with a defaultable one-period non-contingent bond. Firms are not allowed
to carry assets across periods to relax the working capital constraint, but the effect of this
assumption is mitigated by allowing the firm to have costless access to external equity for
part of the wage bill. Financial frictions then introduce an external finance premium into
the firm’s cost of borrowing through two components: (1) a bankruptcy cost that is paid
whenever the firm defaults and (2) a credit shock which directly increases the interest rate
a firm must pay. In particular, I introduce the credit shock because bankruptcy costs alone
2
are not enough to explain observed interest rates.1 Interpretations for the credit shock include various factors not modeled here that may influence the firm’s interest rate, such as
a stochastic discount factor shock or a term premium. Then, the external finance premium
increases the marginal cost of hiring an additional worker. Together, these two financial
frictions depress labor demand by driving a wedge between the expected marginal product
of labor and the wage. Moreover, although entry and exit would occur in this model with
perfect credit markets, financial frictions will further influence the entry and exit decisions
of the firm through their effect on future profits.
I then construct a historical credit shock from observed credit spreads and recovery rates
on corporate bonds and feed it back into the model. As noted earlier, historically, credit
shocks can account for 43% of cyclical movements in output, 37% of movements in hours,
21% of cyclical movements in exit rates and 7% of movements in entry rates. Regarding the
Great Recession of the late 2000s, financial shocks can explain much of the initial decline
until the fourth quarter of 2008, but not the weak recovery that followed. However, according
to Henly and Sánchez (2009), the largest 5% of firms in the United States account for roughly
75% of employment. This suggests a disconnect between the firms which determine cyclical
movements in aggregate output and hours and the firms which determine cyclical variation
in entry and exit rates. Thus, in this paper, these results are obtained after accounting for
heterogeneity in firm size and financing needs observed in the data. In particular, in this
model, the exit rate of firms who are more dependent on external finance is very sensitive to
credit shocks. Moreover, the output and hours of these firms are also very sensitive to credit
shocks, but since they are smaller on average, they matter less for aggregate fluctuations in
output and employment. Conversely, the exit rate of firms who are less dependent on external
finance is less sensitive to credit shocks. Since they account for most of employment, the
decisions of these firms are relatively more important for aggregate fluctuations in output
and employment. It turns out that under reasonable parameters, despite this heterogeneity,
1
I show this in Section 4. However, others have argued that structural models of default significantly
underestimate credit spreads. See Collin-Dufresne et al. (2001) and Huang and Huang (2003).
3
this model predicts that observed credit shocks can explain significant cyclical variation in
not only output and employment, but also exit rates.
This paper is linked to two strands of literature. The first is the literature on financial frictions, which can further be divided into two parts. In the work of Bernanke and
Gertler (1989), financial frictions are modeled as an agency problem between lenders and
borrowers. In particular, Bernanke and Gertler (1989) assume the “costly state verification”
model of Townsend (1979) and Gale and Hellwig (1985), in that lenders can only observe
the entrepreneur’s project returns after paying an auditing cost. The optimal contracting
arrangement then requires a higher auditing probability for borrowers with lower net worth.
This introduces an “external finance premium” into the firm’s cost of borrowing, as now
the firm must compensate the lender for expected future auditing costs. Thus amplification
arises because in bad states aggregate net worth is lower and agency costs are higher, which
restricts investment. Carlstrom and Fuerst (1997) and Bernanke et al. (1999) build on this
model by embedding the agency problem in a general equilibrium model. Both assume that
an entrepreneurs face an idiosyncratic shock to investment returns, which are i.i.d. across
entrepreneurs and time. Both Carlstrom and Fuerst (1997) and Bernanke et al. (1999) need
to avoid the scenario where entrepreneurs eventually accumulate enough wealth so as to completely avoid the financial friction. To address this, Carlstrom and Fuerst (1997) assumes
that the infinitely-lived entrepreneurs have a lower discount factor than households2 , while
Bernanke et al. (1999) exogenously assumes that a constant fraction of entrepreneurs enter
and exit each period.
I follow these models in that I also assume a “bankruptcy cost” which must be paid when
the firm defaults. However, this is not an agency cost, because in this model the firm faces
an AR(1) productivity shock. Given that the lender needs to know the firm’s state before
the loan is made in order to determine the probability of default and the interest rate, the
bankruptcy cost should be interpreted as any cost associated with default. Moreover, in
2
In an earlier working paper version, Carlstrom and Fuerst (1997) did assume that a constant fraction
of entrepreneurs died each period.
4
contrast to these models, the external finance premium also includes a direct credit shock,
which is separate from the bankruptcy cost.
Another view of financial frictions began with Kiyotaki and Moore (1997), which emphasizes the role of collateral constraints in amplifying business cycle fluctuations. In particular,
the debt of firms is secured by collateral, so that firms can borrow any amount at a constant
rate, up to a certain limit, which is endogenously determined by the price of the collateral.
This mechanism amplifies shocks by generating a wedge in the marginal product of capital
between constrained and unconstrained firms. However, others such as Kocherlakota (2000)
and Cordoba and Ripoll (2004) have found that under standard assumptions these models
do not generate quantitatively significant amplification.
The second related strand of literature considers models of financial frictions with heterogeneous firms. Cooley and Quadrini (2001) and Cooley et al. (2004) both consider models
of firm dynamics and financial frictions, however they assume exogenous entry and exit.
Albuquerque and Hopenhayn (2004) and Clementi and Hopenhayn (2006) do consider theoretical models of firm dynamics with endogenous exit. Moreover, Arellano et al. (2011) also
consider a model of firm dynamics with endogenous entry and exit. Like this paper, the
financial friction operates by introducing a wedge between the expected marginal product
of labor and the wage. However, the key aggregate shock is an uncertainty shock, while the
financial friction creates a wedge because default is assumed to cause the firm to exit. Thus,
the value added of this paper is not only that it models both financial frictions and entry
and exit, but that it quantifies the importance of financial shocks measured directly from
corporate bond spreads.
This paper is organized as follows. Section 2 presents an overview of the model. Section 3
characterizes the model and discusses how financial frictions matter. Section 4 discusses the
construction of the credit shocks and Section 5 reviews the model’s calibration. Section 6
examines the main results and Section 7 concludes.
5
2
Model
The model considered here is a discrete-time general equilibrium model of firm entry and
exit with financial frictions. This model is based on Hopenhayn (1992) with important modifications. First, firms face aggregate financial shocks as well as idiosyncratic productivity
shocks. Second, entry is modeled as in Clementi and Palazzo (2010) and Lee and Mukoyama
(2010). In Hopenhayn (1992), there is an infinite mass of potential entrants who must pay a
fixed entry cost before learning their productivity. However, in this model, there is a finite
mass of potential entrants who already know their productivity. Entry is modeled this way
to guarantee that entry rates are procyclical and exit rates are countercyclical. With entry as in Hopenhayn (1992), there can be too much entry after a positive shock, producing
procyclical exit rates as new entrants fail.
In the following subsections, the components of the model are described in detail.
2.1
Firms
Firms are owned by risk-neutral households.3 They are perfectly competitive and produce
a single homogeneous good. Both capital and labor are inputs in the firm’s production
γ
function, f (s, k, n) = s [k α n1−α ] , where s is an idiosyncratic productivity shock, k is the
capital input and n is the labor input. It is assumed that γ ∈ (0, 1), implying that there
are decreasing returns to scale at the firm level. One way to interpret diminishing returns
to scale at the firm-level is to think of the “span of control” models of Rosen (1982) and
Lucas (1978). Here a “firm” can be interpreted as consisting of an entrepreneur, n units of
labor and k units of capital. The idiosyncratic productivity, s, can reflect heterogeneity in
the skill of managers and diminishing returns to scale is a consequence of the diminishing
returns of an entrepreneur in managing larger operations. However, management is not
being modeled directly and entrepreneurs earn positive profits only because of diminishing
returns to scale and entry is costly. Although there are decreasing returns to scale at the
3
Throughout this paper, the terms “firms” or “entrepreneurs” may be used interchangeably.
6
firm level, there still are constant returns to scale in the aggregate because the firm can be
replicated. With perfect competition, firm-level diminishing returns allow for heterogeneity
to exist in equilibrium and prevents the most productive firms from taking over the market
completely.
The process for the idiosyncratic shock is assumed to be
ln s0 = (1 − ρs ) as + ρs ln s + σεs εs
where s0 is the next-period idiosyncratic shock and εs is an i.i.d. innovation drawn from a
standard normal distribution. Let’s denote the conditional cumulative distribution and density functions for s0 by H(s0 |s) and h(s0 |s), respectively. Given this process for idiosyncratic
productivity, entry and exit occurs endogenously because firms must pay fixed costs both to
enter and to operate.
To investigate the effect of financial frictions, I include a working capital constraint,
an assumption motivated by the observation that inputs such as labor and materials have
to be purchased in advance. Specifically, the firm must pay wages and choose tomorrow’s
employment in the current period. As for the capital input, firms rent capital from the
household sector at rate u. I do not assume any frictions in the capital market. Therefore,
when the firm chooses how much capital to rent today, it also knows its productivity today.
However, given that capital and labor are complements in the firm’s production function,
the employment decision made yesterday will be relevant for capital demand today. This
creates a feedback mechanism which indirectly amplifies the effect of financial shocks that
directly influence labor demand.
With the assumption of financial frictions (which will be specified later), the ModiglianiMiller theorem4 does not apply and thus the firm’s shareholders are not indifferent between
financing wages with equity or debt. However, if obtaining external equity is costless, the
firm would strictly prefer equity to debt. In that case, the firm would not borrow, the wage
4
Modigliani and Miller (1958)
7
bill would be financed entirely with external equity and financial frictions would have no
effect on labor demand. However, in reality, external equity is not costless but to simplify
the analysis, I assume that the firm finances a fraction 1 − κ of wages with equity and the
remaining fraction with debt. Moreover, the firm is not allowed to carry any assets (i.e.,
internal equity) across periods to alleviate the working capital constraint. However, the
effect of this assumption is mitigated by allowing the firm to have costless access to external
equity. Thus, while net worth plays no role in this model, it can still be modeled exogenously
by varying the parameter κ.
The firm then finances the fraction κ of the wage bill today by issuing a defaultable
one-period non-contingent bond. Only profits from output produced tomorrow can be used
as collateral for the bond and, thus, the firm’s equity cannot be used as collateral. To make
clear the role of financial frictions in this model, I assume that there is a perfectly competitive
financial intermediation sector. Each financial intermediary borrows from households and
lends to firms by buying their bonds. The firm promises to pay b0 units of the output good to
the financial intermediary tomorrow and in return the firm receives qb0 in the current period,
where q is the price of the bond. This price is set to guarantee the risk-neutral intermediary
an expected return equal to the risk-free interest rate, r. The possibility of default will lower
q, but this alone will not have any affect on the firm’s labor demand. Thus, the financial
frictions in this model consist of two components, a bankruptcy cost and a credit shock,
which both constrain the firm’s labor demand by lowering the bond price, q. To understand
precisely how the two financial frictions lower the bond price, or equivalently, the interest
rate firms pay, I must first explain when firms will default.
2.1.1
Default Decision
I assume that there is enough anonymity across periods so that the firm faces no additional penalty beyond losing its collateral if it chooses to default. Thus, the firm will only
default when the value of its profits (i.e., the collateral) is less than the required payment to
8
the lender. Next period, the firm’s realized profits, π(s0 , k 0 , n0 ; u0 ), are given by
π(s0 , k 0 , n0 ; u0 ) ≡ f (s0 , k 0 , n0 ) − u0 k 0
(1)
where u0 is the rental rate on capital tomorrow. Recall that tomorrow’s labor n0 is chosen
in the current period when the firm does not know s0 . I assume no frictions in the capital
market. Thus, the firm, which knows u0 in advance, can condition the amount of capital it
rents tomorrow on its realized idiosyncratic productivity, s0 . Then, since the firm’s optimal
capital demand will depend positively on s0 and n0 , credit shocks today which constrain labor
demand n0 will also constrain capital demand. Substituting for the optimal capital demand,
the firm’s profits can then be expressed as
π(s0 , n0 ) ≡ f (s0 , k 0 (s0 , n0 ; u0 ), n0 ) − u0 k 0 (s0 , n0 ; u0 ).
Note that π(s0 , n0 ) implicitly depends on the rental rate u0 , but since in equilibrium it will
always be constant, profits are not explicitly specified as a function of u0 . Then, the firm
will default if and only if π(s0 , n0 ) < b0 . For b0 > 0, n0 > 0 and given the assumed functional
form for f (s, k, n), there must exist a productivity cutoff, s̄(n0 , b0 ), such that the firm will
default if and only if s0 < s̄(n0 , b0 ). This cutoff is defined as the value of the idiosyncratic
productivity, s̄, such that
π(s̄, n0 ) = b0 .
(2)
Moreover, when the firm defaults, the lender receives the collateral minus a bankruptcy
cost. This bankruptcy cost is assumed to be a constant fraction of the collateral, ξπ(s0 , n0 ),
where ξ ∈ [0, 1]. This cost is motivated by various costs of bankruptcy, such as legal costs or
costs related to the liquidation of the firm’s assets. This is the same type of assumption used
in Bernanke and Gertler (1989) and is related to the “costly state verification” problem in
Townsend (1979). The key difference here is that the idiosyncratic productivity shock is per-
9
sistent and observable by the lender, where in Bernanke and Gertler (1989) the idiosyncratic
shock driving default was assumed to be i.i.d. and unknown to the lender without using
a costly monitoring technology. Therefore, in this model, the bankruptcy cost technically
cannot be viewed as an “auditing” cost, since idiosyncratic productivity is observable, but
merely as all the unavoidable costs associated with bankruptcy.
2.1.2
Bond Price
Since the default cutoff, s̄(n0 , b0 ), depends on the firm’s choice of n0 and b0 , the bond price,
q, will also depend on those choices. Then, given that the lender must earn an expected
return equal to the risk-free rate, the bond price is given by
R(n0 , b0 ; s)
1−φ
0 0
1 − H (s̄(n , b )|s) 1 − (1 − ξ)
q(n , b ; φ, s) =
1+r
b0
0
0
(3)
where
0
0
Z
R(n , b ; s) ≡
s̄(n0 ,b0 )
π(s0 , n0 )
0
h(s0 |s)
ds0 .
H(s̄(n0 , b0 )|s)
In addition to n0 and b0 , the bond price depends on φ, which is the credit shock, and s,
the firm’s current-period idiosyncratic productivity. The current period’s productivity is
relevant for H (s̄(n0 , b0 )|s), the probability that the firm will default, and R(n0 , b0 ; s), the
expected value of the collateral next period, given default.
Financial frictions influence q through two channels: ξ and φ. As discussed earlier, ξ is
the bankruptcy cost parameter and it lowers the price of the bond (or equivalently, raises
the interest rate). It does this by lowering the lender’s expected recovery rate given default,
which is given by (1 − ξ)R(n0 , b0 ; s)/b0 . However, the bankruptcy cost is limited in how much
it can influence the interest rate by the probability of default, which may be a small number.
Therefore, in addition to the bankruptcy cost, I include a credit shock, φ, which directly
lowers the price of the bond. This shock is motivated by the fact that observed default rates
10
and recovery rates are not enough to explain the level of interest rates. Interpretations for the
credit shock include various factors not modeled in this paper, such as a stochastic discount
factor shock or a term premium. Together, these two financial frictions contribute to the
“external finance premium”, which constrains the firm’s demand for labor and influences the
entry and exit decisions through its effect on future profits.
In what follows, I allow for aggregate shocks to φ. In particular, I assume that the process
for the credit shock is given by
ln φ0 = (1 − ρφ ) aφ + ρφ ln φ + σεφ εφ
(4)
where φ0 is the next-period credit shock and εφ is an i.i.d. innovation drawn from a standard
normal distribution. I follow Tauchen (1986) and approximate this process with a discrete
Markov process with finite support and denote by πφ (φ0 |φ) the probability of transitioning
from φ to φ0 .
Moreover, it is important to note that the possibility of default does not directly contribute to the external finance premium. If ξ = 0 and φ = 0, the external finance premium
would be exactly zero since both the firm and lender are risk neutral. While the firm may
be paying a high interest rate to compensate the lender for default, it is only paying that
rate in states of the world in which it does not default. Overall, in expectation, the firm
would still be paying an interest rate exactly equal to the risk-free rate. In this model, then,
default only increases the external finance premium through bankruptcy costs which are only
incurred when the firm defaults.
2.1.3
Exit Decision
Separate from the default decision is the exit decision. Each period the firm must pay a
fixed operating cost, cf > 0, denominated in terms of the output good. Given an assumed
outside option of zero, the firm will exit when its value of continuing is less than zero.
11
Let’s denote by V (x, s) the firm’s value of continuing in the current period, where x is the
aggregate state today and s is the current period’s idiosyncratic shock. This value function
is defined later in Equation 7. Therefore, as discussed, the firm will exit today if and only if
V (x, s) < 0. Then, there exists an exit cutoff productivity, s(x), such that the firm will exit
if and only if s < s(x). This cutoff is defined as the value s such that
V (x, s) = 0.
(5)
Since V (x, s) is strictly increasing in s, unbounded above and bounded below by −cf for a
given x, this cutoff is uniquely defined. Note that this cutoff only depends on the aggregate
state while the default cutoff tomorrow depends on the current-period productivity. Therefore, given that the default and exit decisions are not linked, default does not necessarily
mean that the firm will exit. It is possible for the firm to continue operating tomorrow but
default if tomorrow’s idiosyncratic shock is high enough. Moreover, while it is possible for
the firm to exit and default, it is also possible for the firm to exit and not default. If the
current period’s idiosyncratic shock is low enough, the firm could transition to a new state
tomorrow in which the value of its collateral covers its debt payment, but its productivity
is still so low that it must exit. In the absence of financial frictions, exit still occurs in this
model. Therefore, it is plausible that a firm will not default because the value of its collateral
is still greater than its debt, but still exits because its future prospects are so low.
2.1.4
Entry Decision
As in Clementi and Palazzo (2010), every period there is a finite mass M̄ of prospective
entrants, each of which receives a signal se about tomorrow’s productivity, where se is drawn
from a distribution with the cumulative distribution function, G(·). A potential entrant with
signal se faces the same probability distribution for tomorrow’s idiosyncratic productivity
as an incumbent with s = se . After a potential entrepreneur makes the decision to enter,
12
it pays the fixed entry cost, ce > 0. This implies that there is an entry cutoff for potential
entrants, s̄e (x), where s̄e is defined as the value of se such that
V (x, s̄e ) = ce .
(6)
Since V (x, s) is strictly increasing in s, unbounded above and bounded below by −cf for
a given x, s̄e (x) is uniquely defined. Therefore, a potential entrant will enter if and only
if se ≥ s̄e (x). Moreover, since ce > 0, it is guaranteed that V (x, s̄e ) > 0, implying that
s̄e (x) > s(x) or, equivalently, that all entrants will choose to operate.
2.1.5
Firm’s Problem
The firm’s problem can now be formulated recursively. Let x = (φ, µ) be the vector of
aggregate state variables, where µ(·) is defined to be a density function over the current
period’s idiosyncratic shock, s. This function represents the distribution of incumbent firms
at the beginning of the period before the entry and exit decisions are made. Then, let’s define
V (x, s) as the value of continuing for a firm in the aggregate state x whose idiosyncratic
productivity is s in the current period, after any dividends from the operations of the current
period have been issued. Then, V (x, s) is given by:
V (x, s) = max
{−(1 − κ)w0 (x)n0 − cf +
n0 ,b0
#)
"Z
Z ∞
∞
X
1
[π(s0 , n0 ) − b0 ] h(s0 |s)ds0 +
πφ (φ0 |φ)
V (x0 , s0 )h(s0 |s)ds0
(7)
1 + r s̄(n0 ,b0 )
0
s(x )
φ0
subject to
q(n0 , b0 ; φ, s)b0 = κw0 (x)n0
µ0 = T (x).
13
That is, the firm chooses tomorrow’s employment n0 and the number of bonds b0 to maximize
expected discounted profits, subject to its working capital constraint and subject to the
transition rule for the distribution of firms. In the current period, the firm must pay a
fraction 1 − κ of the wage bill, w0 n0 , and the fixed operating cost, cf , directly with external
equity. Then, next period, the firm receives π(s0 , n0 ) − b0 if it chooses not to default and zero
otherwise. Moreover, it receives the continuation value V (x0 , s0 ) if it chooses not to exit and
zero otherwise.
2.2
Households
Households are risk neutral, infinitely lived and maximize lifetime utility, discounting
the future with β ∈ (0, 1). There is a constant mass, N̄ , of households, or workers, who
supply labor to firms and lend to firms through an intermediary who earns zero expected
profits. Moreover, households own physical capital, make the investment decisions, and rent
out capital to firms. All households are identical and have preferences over consumption and
labor supply as defined by:
∞
X
β t [ct + ψ ln (1 − nt )]
(8)
t=0
where ct and nt are consumption and labor supply in period t, respectively, and ψ is a parameter governing the disutility of labor. Labor supply is indivisible in that each worker can
only choose between not working (n = 0) and working full time (n = n0 ). Following Hansen
(1985), households choose the probability of working and a lottery determines whether or not
the household actually works. This implies that the economy behaves as if the representative
household had preferences given by
∞
X
β t [ct − aNt ]
t=0
where
a≡−
ψ ln(1 − n0 )
n0
N̄
14
(9)
and Nt is aggregate employment. This implies that as long as there is not full employment,
the equilibrium wage is equal to a. Then, in response to aggregate credit shocks, the equilibrium wage will not change. This result follows not only from the household’s preferences but
also the entry condition. The assumption of Hansen indivisible labor eliminates the substitution effect in labor supply and risk-neutrality kills the income effect. However, modeling
entry as in Clementi and Palazzo (2010) allows for there to be positive unemployment in
the steady state equilibrium. If entry is modeled as in Hopenhayn (1992), full employment
would almost certainly be the outcome in the steady state. The free entry condition would
pin down the equilibrium wage and if this wage is greater than a, firms would continue to
enter until aggregate labor demand reaches the vertical part of the labor supply curve. This
would then imply that in response to any shock which shifts labor demand, the wage would
adjust so much that equilibrium employment would be unchanged.
2.2.1
Supply of Capital
Moreover, each period the household chooses next period’s capital stock. In particular, at
the beginning of period t, the household chooses the capital stock which will be rented out to
firms tomorrow. The key assumption is that the household knows the current period credit
shock, φt , when it makes its investment decision. This implies that the household knows
today exactly the level of aggregate capital demand tomorrow. Then, since the household is
risk-neutral, the household’s supply of capital will be infinitely elastic at
u = r + δ = 1/β − 1 + δ
where δ is the depreciation rate of capital.
15
(10)
2.3
Recursive Competitive Equilibrium
A recursive competitive equilibrium can then be defined as follows. Given µ0 and an
initial stock of capital, K0 , a recursive competitive equilibrium consists of (i) the value
function V (x, s) (ii) policy functions n0 (x, s), b0 (x, s) (iii) cutoff rules s̄(x, s), s(x), s̄e (x)
∞
(iv) bond price function q(n, b0 ; φ, s) (v) sequences of wages {wt }∞
t=0 , rental rates {ut }t=0 ,
∞
incumbents measures {µt }∞
t=0 , mass of entrants, {Me,t }t=0 such that
1. V (x, s), n0 (x, s), b0 (x, s) solve the firm’s problem given in Equation 7.
2. The equilibrium default cutoff, s̄(x, s), is given by
s̄(x, s) = s̄(n0 (x, s), b0 (x, s))
where s̄(n0 , b0 ) is the function given by Equation 2.
3. The equilibrium exit cutoff rule, s(x), is given by Equation 5.
4. The equilibrium entry cutoff rule, s̄e (x), is given by Equation 6.
5. For all t ≥ 0, the mass of entrants is given by
Me,t = M̄ (1 − G(s̄e (xt ))) .
6. The wage wt+1 clears the labor market for all t ≥ 0. First, define aggregate labor
demand in period t + 1, given wage wt+1 , as follows:
d
Z
∞
Z
0
∞
n (xt , st )µt (st )dst + M̄
L (xt ; wt+1 ) =
s(xt )
n0 (xt , se )dG(se ).
s̄e (xt )
If Ld (xt ; wt+1 = a) ≤ N̄ , then, wt+1 = a is the equilibrium wage. However, if
16
Ld (xt ; wt+1 = a) > N̄ , then the equilibrium wage is the value of wt+1 > a such that
Ld (xt ; wt+1 ) = N̄ .
7. The bond market clears, implying that the firm’s bond price is given by Equation 3.
8. The risk-free rate is given by
r=
1
−1
β
9. The rental rate ut clears the capital market for all t > 0. Since household supply of
capital is infinitely elastic, according to Equation 10, the equilibrium rental rate of
capital is always given by
ut = r + δ =
1
− 1 + δ.
β
10. The transition rule for incumbent firms distribution is given by
Z
∞
µt+1 (st+1 ) = T (xt ) ≡
Z
h(st+1 |st )µt (st )dst + M̄
s(xt )
3
∞
h(st+1 |se )dG(se ).
s̄e (xt )
Characterization of Equilibrium
To understand the role of financial frictions in this model, let’s first reduce the firm’s
two-dimensional maximization problem over n0 and b0 to a one-dimensional problem over
n0 . Substituting for b0 in Equation 7 by using the bond price in Equation 3 and the firm’s
working capital constraint, qb0 = κw0 n0 , the firm’s problem can be re-stated as:
φ
V (x, s) = max
− 1+κ
w0 (x)n0 − cf +
n0
1−φ
"Z
#)
Z ∞
∞
X
1
π(s0 , n0 )dH(s0 |s) − B(x, s, n0 ) +
πφ (φ0 |φ)
V (x0 , s0 )dH(s0 |s)
1+r 0
0)
s(x
φ0
(11)
17
where
0
Z
s̄(n0 ,b0 (x,s,n0 ))
π(s0 , n0 )dH(s0 |s)
B(x, s, n ) ≡ ξ
(12)
0
is the unconditional expected value of the bankruptcy cost the next period given a choice
of labor n0 . Note that the definition of B(x, s, n0 ) depends on b0 (x, s, n0 ), which is defined
to be the value of b0 that satisfies the firm’s working capital constraint. In other words,
b0 (x, s, n0 ) is the value of b0 such that q(n0 , b0 ; φ, s)b0 = κw0 (x)n0 . The first order condition for
Equation 11 implies that the optimal choice of labor must satisfy the following condition:
Z ∞
φ
1 dB(x, s, n0 )
1
dπ(s0 , n0 )
0
1+κ
w +
=
dH(s0 |s).
1−φ
1+r
dn0
1+r 0
dn0
(13)
In other words, the firm chooses tomorrow’s labor so that the discounted expected marginal
product of labor is equal to the marginal cost of labor plus the discounted marginal bankruptcy
cost. Furthermore, with the credit shock, the marginal cost of labor consists of two components: the wage plus a premium due to the credit shock. To hire an additional unit of
labor, the firm would need to borrow more, and since the credit shock forces the firm to pay
a premium for debt, it essentially increases the marginal cost of hiring an additional unit of
labor.
Figure 1 then illustrates how the two financial frictions impact the firm’s labor demand.
This figure plots various components of the optimality condition in Equation 13 against
tomorrow’s labor choice, n0 . In particular, the line with the label (1) is the right-hand side
of Equation 13, the discounted expected marginal product of labor. Line (2) is then the wage,
w0 , which means that the employment level n0 represents the frictionless labor demand (i.e.,
assuming ξ = 0 and φ = 0). Line (3) represents the marginal labor cost assuming φ > 0,
φ
0
or in other words, it is given by w 1 + κ 1−φ . Then, because the credit shock raises the
marginal cost of labor, the new employment level with a credit shock, n1 , is lower than the
efficient level, n0 . And finally, line (4) plots the left-hand side of Equation 13. Line (4) is
above line (3) because the marginal bankruptcy costs are always positive. See Appendix A.3
18
for a proof. Now the new labor demand is n2 and thus bankruptcy costs constrain labor
demand even more.
If bankruptcy costs are zero (i.e., ξ = 0), then Equation 13 can easily be used to derive a
functional form for the firm’s labor demand. However, with positive bankruptcy costs, the
firm’s labor demand still admits some key properties described below.
Proposition 1 Given the assumed functional form for f (s, k, n) and the process for the
idiosyncratic productivity, the firm’s policy rule for labor, n(x, s) satisfies the following:
ρs
∂ ln n(x, s)
=
∂ ln s
1−γ
∂ ln n(x, s)
1 − αγ
=−
∂ ln ŵ
1−γ
where ŵ = w
0
1+
φ
κ 1−φ
(14)
if κ = 1
(15)
is the effective wage.
For the proof, see Appendix A.4. That is, the productivity elasticity and effective-wage
elasticity of labor demand are both constant. When ξ = 0, the exact functional form for
n(x, s−1 ) is known and these elasticities can easily be derived. Moreover, for ξ = 0, the
effective-wage elasticity of labor demand is always equal to −(1 − αγ)/(1 − γ), even for
κ < 1.
The result that the productivity elasticity of demand is constant implies that while firms
with lower idiosyncratic productivity demand less labor, the effect of productivity on labor
demand is orthogonal to the effect of financial frictions. In other words, n/n∗ does not vary
with productivity, where n is the firm’s labor demand and n∗ is the frictionless labor demand.
Or, stated in a third way, financial frictions have no effect on the productivity-elasticity of
labor demand. In addition, the result that effective-wage elasticity of labor demand is
constant directly quantifies the effect of a credit shock on the firm’s labor demand.
Moreover, we also have the following proposition.
Proposition 2 The equilibrium default probability H(s̄(n(x, s), b0 (x, s))|s) does not vary
19
with s. Moreover it also does not vary with φ if κ = 1.
For the proof, see Appendix A.4. This implies that all firms face the same interest rates
and the same default probability in equilibrium. Even less-productive firms with a higher
probability of exiting next period face the same default probability as a highly-productive
firm. While these firms are more likely to exit, the firm’s bond is secured by its profits
next period and not by equity. Thus, while less-productive firms expect profits to be lower,
they also hire fewer workers and borrow less. Moreover, it also turns out that the default
probability does not depend on the credit shock if the firm finances the wage bill entirely
with debt. Yet, for κ < 1, it turns out in numerical solution that increases in φ lead to an
increase in default rates.
3.1
Effect of Credit Shocks on Aggregates
The effect of credit shocks on aggregate variables can now be quantified. For this purpose,
I derive a credit-shock semi-elasticity for four aggregate variables: output, employment, entry
rates and exit rates. In what follows, let’s define the effective credit shock as φ̂ = φ/(1 − φ).
• Aggregate Employment and Output
As long as there is unemployment, shocks to aggregate labor demand do not cause the
wage to adjust and shifts in aggregate labor demand translate directly into shifts in
aggregate employment. In this case, the equilibrium credit-shock semi-elasticities of
employment and output are given by:
1 − αγ
≈ −κ
1−γ
dφ̂
d ln Y
(1 − α)γ
≈ −κ
.
1−γ
dφ̂
d ln N
(16)
(17)
This implies that a quarterly 1 percent increase in the effective credit shock, φ̂, will lead
to a κ(1 − αγ)/(1 − γ) percent decrease in aggregate employment and a κ(1 − α)γ/(1 −
20
γ) decrease in aggregate output in equilibrium. These elasticities were calculated
assuming (1) that the effects of entry and exit on aggregate labor demand are small
and (2) that the bankruptcy costs do not significantly amplify nor dampen effects
of credit shocks. In fact, it does turn out that bankruptcy costs have little effect
on aggregate labor demand in response to credit shocks. While an increase in the
credit shock does tend to increase the expected bankruptcy costs for a given level of
employment, the credit shock also lowers employment which tends to have the opposite
effect on bankruptcy costs.
Therefore, only three model parameters (κ, α, γ) matter for the quantitative impact
of credit shocks on aggregate output and employment. This result is driven by how
these parameters matter for the elasticity of aggregate labor demand. While other
parameters do matter for the effect of credit shocks on entry, exit and the distribution
of firms, those effects are small when aggregating across all firms for employment and
output. In particular, the key parameters are κ and γ. When κ is higher, firms are
more dependent on external financing, and thus credit shocks have a larger percentage
change in the effective wage. Moreover, the higher the returns to scale at the firm level,
the larger the impact of credit shocks on aggregate output and hours. Intuitively, γ
matters for the elasticity of aggregate labor demand. As γ approaches 1, the elasticity
of aggregate labor demand approaches infinity and thus, given labor supply, credit
shocks have a larger impact on aggregate employment and thus output.
• Entry and Exit Rates
Similarly, credit-shock semi-elasticities can be derived for entry and exit rates as follows:
d ln me
dφ̂
d ln mx
dφ̂
=−
=
g(s̄e ) ds̄e
g(s̄e ) ∂V /∂ φ̂
=
1 − G(s̄e ) dφ̂
1 − G(s̄e ) ∂V /∂s̄e
µ(s) ds
µ(s) ∂V /∂ φ̂
=−
M (s) dφ̂
M (s) ∂V /∂s
21
where me is the entry rate, mx is the exit rate, and M (x) ≡
Rx
−∞
µ(s)ds is the mass of
incumbent firms at the beginning of the period with productivity less than x. For both
elasticities, there are two components which matter for the quantitative importance of
the credit shock. The first is a hazard rate, which reflects how many firms are near the
productivity cutoff threshold relative to all of the firms who enter or exit, respectively.
The second component reflects how much the productivity cutoff responds to the credit
shock. If the exit and entry thresholds respond similarly to the credit shock, then any
difference in the credit-shock elasticity for entry and exit rates reflects a difference in
hazard rates. Either way, this elasticity does not readily admit a functional form, which
requires the model to be solved numerically to see how much credit shocks matter for
entry and exit rates.
4
Corporate Bond Spreads and Default Rates
To assess quantitatively the importance of financial frictions, a credit shock is measured
from the historical corporate bond credit spreads and fed it into the model to assess how
much it can account for historical movements in output, employment, entry and exit rates.
For the source of the data, see Appendix A.1.
4.1
Bankruptcy Costs and Interest Rates
Bankruptcy costs are included in the model to match observed recovery rates. However,
to motivate the inclusion of a credit shock, let’s first decompose the model-predicted credit
spread into its component parts. Using the bond price function constructed in Equation 3,
the firm’s credit spread can be approximated as
rk − r ≈ φ + ξpD RR∗ + pD (1 − RR∗ )
22
(18)
where rk ≡ 1/q − 1 is the firm’s interest rate, pD is the expected probability of default and
RR∗ is lender’s the expected recovery rate given default, assuming no bankruptcy costs.
Thus, there are three components to the credit spread, only two of which are part of the
external finance premium. The first term is φ, the exogenous credit shock which will be
measured from the data. The second term is the premium which compensates the lender for
expected bankruptcy costs. The third term is the lenders expected loss rate from default,
before any bankruptcy costs. Only the first two terms are part of the external finance
premium. Thus, in this model, if there are no financial frictions (i.e., ξ = 0 and φ = 0), the
credit spread will still be positive since the recovery rate will be less than one. Nevertheless,
despite a positive credit spread, the firm’s labor demand will be unaffected since the firm’s
interest rate in expectation is still equal to the risk-free rate.
Therefore, from Equation 18, the credit shock can be interpreted as the portion of the
credit spread which is not explained by observed default and recovery rates. Thus, to motivate the inclusion of the credit shock, I construct the components of the credit-spread
generated by the final two components. Figure 2 plots the Baa - 10 year Treasury spread
against the spread predicted by the final two components of Equation 18. This assumes a
recovery rate given default equal to 41.34%, the historical average corporate bond recovery
rate. Without credit shocks, very little of the Baa spread is explained by observed default
rates and recovery rates. Moreover, consider the model-predicted spread under a worstcase scenario, in which the recovery rate is assumed to be zero. Even under the worst-case
scenario, default rates explain very little of the observed spread.
In Figures 3 and 4, the same exercise is repeated for the investment-grade and speculative
grade credit spreads, respectively. As before, with the investment-grade spread, very little of
the spread is explained by observed default rates. However, for the speculative-grade spread,
observed default rates do a better job explaining the spread, but the model-predicted spread
is still consistently smaller than the actual spread by an average of 200 basis points.
23
4.2
Credit Shock Measurement
Thus, in this model, bankruptcy costs alone are not enough to explain observed credit
spreads. To accurately measure the credit shock, Equation 3 can be rearranged to obtain
φt = 1 −
qt (1 + rt )
1 − pD,t (1 − RRt )
where 1/qt −1 is the interest rate firms pay, rt is the risk-free interest rate, pD,t is the expected
probability of default, and RRt is the expected recovery rate given default. Data on these
variables can then be used to construct a historical credit shock. In particular, three separate
credit shocks are constructed: a Baa credit shock, an investment-grade credit shock, and a
speculative-grade credit shock. For all these shocks, the same risk-free rate and recovery rate
are assumed, but a different interest rate and default rate are used. The 10 year Treasury
was used for the risk-free rate, rt . The recovery rate, RRt , was assumed to be 0.4134,
which is the average historical recovery rate on all defaulted bonds between 1982 and 2010,
according to Moody’s5 . The same recovery rate is used for all credit shocks because, while
recovery rates are observed to be countercyclical, Moody’s reports that there is no systematic
relationship between recovery rates and bond ratings, either at origination or immediately
prior to default. Moreover, historical recovery rates are observed to be countercyclical, but
a constant recovery rate is assumed when calculating the credit shock. This is is done
because limited data is available for historical recovery rates. Nevertheless, the effect of this
assumption on the calculated credit shock is small for the Baa and investment-grade credit
shocks, given the low default rates observed.
4.2.1
Baa Credit Shock
The Baa Credit Shock is constructed for seasoned corporate bonds with a Baa rating
from Moody’s. The bond price, qt , was calculated from the Baa interest rate. The expected
5
To calculate the average recovery rate, Moody’s uses the post-default trading prices and weights each
issuer equally.
24
probability of default, pD,t , was taken to be the actual default rate on Baa corporate bonds,
as reported by Moody’s. The default rate is only reported annually by Moody’s, while the
interest rates are monthly. Therefore, φt was calculated at a monthly frequency, using the
same probability of default, pD,t , for each month of the same year. Then, this series was
averaged to obtain a historical credit shock at a quarterly frequency.
Figure 5 plots the resulting credit shock along with the NBER recession dates. In addition
to the credit shock calculated using the 10 year Treasury rate as the risk-free rate, this figure
also includes a credit shock calculated using the Aaa interest rate as the risk-free rate. Data
for the 10 year Treasury rate only go back to 1953. However, using the Aaa rate as the
risk-free rate, the credit shock goes as far back as 1920. According to this measure, the
credit shock was highest during the Great Depression, with the second-highest credit shock
occurring during the recent late 2000s recession. The following analysis only uses the credit
shock calculated using the 10 year Treasury rate and the term “Baa credit shock” will be
used to refer to the shock constructed using the Treasury rate.
4.2.2
Investment-Grade and Speculative-Grade Credit Shocks
Moreover, to properly gauge the credit shocks faced by all firms in the economy, credit
shocks for investment-grade and speculative-grade firms are also constructed. The investmentgrade credit shock is constructed using the Merrill Lynch U.S. Corporate Master index for
qt and the historical default rates of investment-grade bonds, as reported by Moody’s, for
pD,t . Analogously, the speculative-grade credit shock is constructed using the Merrill Lynch
U.S. Cash Pay High Yield index for qt and the historical default rates of speculative-grade
bonds for pD,t . Figure 6 plots the resulting credit shocks. The credit shocks do not appear to
increase significantly for the early 1990s and 2001 recessions, but do for the Great Recession
in the late 2000s. For most of the sample period, the two shocks were roughly equal, but
at the peak of the Great Recession, the speculative-grade shock was more than twice the
investment-grade shock, even after accounting for the higher default and interest rates of
25
speculative-grade bonds. Moreover, the fact that even the investment-grade credit shock
increased significantly during the Great Recession indicates that while some firms are facing
larger credit shocks than others, all firms were affected to some extent.
5
Calibration
The model can now be calibrated. Table 1 lists the calibrated parameters. In the bench-
mark calibration, the parameters are chosen to be consistent with the firms in the 2003
Survey of Small Business Finances (SSBF). The SSBF covers businesses with fewer than
500 employees. According to Census Bureau’s Statistics of U.S. Businesses (SUSB) in 2008,
99.7% of all firms had fewer than 500 employees, yet these firms only accounted for approximately half of employment. Thus, the decisions of small firms are most important for
understanding movements in entry and exit rates. However, the decisions of the top 0.3% of
firms by construction must have a negligible impact on entry and exit rates, but do matter
a lot for aggregate output and employment. Thus, later I consider a calibration where I
exogenously allow for heterogeneity in κ.
G(·), the cumulative distribution from which potential entrants draw the signal about
their idiosyncratic productivity, is assumed to be the invariant distribution of the stochastic
process for s. This implies that G(·) is the log normal cumulative distribution function with
2
mean as and variance σεs
/(1 − ρ2s ). Furthermore, firms are assumed to finance its wage bill
only partially with debt. In particular, the benchmark calibration assumes that κ = 0.4.
In the 2003 SSBF, the median ratio of the firm’s liabilities to its assets was approximately
0.4. Therefore, thinking of the ratio of liabilities to assets as a rough analogue to κ, I set
κ = 0.4, implying that the firm finances only 40% of its wages with debt. The bankruptcy
cost parameter, ξ, is set to match the historical recovery rate on all corporate bonds, which
was 41.34%. Then, using ordinary least squares, an AR(1) process was estimated for the Baa
credit shock constructed in Section 4.2. Table 2 reports the results of the OLS estimation
26
for the Baa credit shock. For comparison purposes, the same estimation is performed for the
credit shock assuming the Aaa interest rate is the risk-free rate, and the results are similar.
Using the results of this estimation in Table 2, aφ was set to −5.534, the persistence of the
credit shock was set to ρφ = 0.929 and the standard deviation of the innovation to the credit
shock was chosen to be σεφ = 0.165.
The depreciation rate of capital, δ, was set to 0.025. The parameter, α, which determines
the capital share in the firm’s production function, was chosen to be 0.35. These values are
consistent with values commonly used in the literature. The parameter γ, which determines
the degree of returns to scale for the firm, was chosen to be 0.95. Based on 3-digit industry
data, Burnside et al. (1995) report returns to scale between 0.81 and 0.92 (see Table 5 of
their paper). Basu and Fernald (1997) report a weighted average returns to scale of 0.97 at
the industry level. Thus, the choice of γ = 0.95 falls toward the upper end of these estimates.
Given that the parameter γ reflects returns to scale at the firm level, and not the plant level,
this value is reasonable, especially given that a single firm can have multiple plants. The
household’s discount factor, β, was chosen to be 0.99 to generate an annual real interest rate
of about 4 percent. The quantity of labor that an employed worker supplies to the firm,
n0 , was chosen to be 1/3, implying that an employed worker spends one-third of his time
working. The labor disutility parameter, ψ, was then set to normalize the wage to 1. From
Equation 9, this requires that ψ equal the following:
ψ=−
n0 N̄
.
ln(1 − n0 )
This implies that the equilibrium wage will be exactly equal to 1 as long as there is positive
unemployment.
As for the number of workers, N̄ , note that this parameter does not matter for equilibrium
outcomes as long as there is not full employment. For instance, let’s suppose N̄ is doubled.
Then, ψ is doubled to maintain the normalization of the wage to one. The firm’s problem
27
is unaffected, and thus the equilibrium number of firms which survive in the steady state
is unaffected. This implies that aggregate labor demand is unaffected, and thus neither the
average size of firms nor the equilibrium number of employed workers is affected. Thus, the
only consequence of doubling N̄ is that the unemployment rate has increased. Therefore,
N̄ is simply chosen to target an unemployment rate of 0.05, but be aware that it does not
affect any other equilibrium outcome.
In contrast, the number of potential entrants, M̄ , does matter for the number of firms
which operate in the steady state. If M̄ is doubled, the equilibrium number of firms which
operate in the steady state is doubled as well. For a given wage, aggregate labor demand
is doubled as well as aggregate output. As long as there is still unemployment after M̄ is
doubled, the equilibrium wage rate is unaffected. Then unemployment falls, but all that
has really changed is the scale of the economy. Thus, M̄ technically cannot be pinned down
given how N̄ is chosen. However, given that M̄ merely changes the scale of the economy, M̄
is set to normalize the mass of incumbent firms to 0.997, which is the percentage of all firms
in the economy which have less than 500 employees.
The parameters ρs and σεs could potentially be estimated from data directly in procedure
described by Hopenhayn and Rogerson (1993). The firm’s policy rule for labor and the
process for the firm’s idiosyncratic process imply that the firm’s employment obeys the
following law of motion:
ln nt+1 = Ct + ρs ln nt +
ρs
εs,t .
1−γ
Using cross-sectional data on surviving firms, ln nt+1 could be regressed on ln nt . The coefficient estimate on ln nt would be an estimate of ρs and the residual variance would be
2
an estimate of ρ2s σεs
/(1 − γ)2 . However, due to selection induced by endogenous exit, the
ordinary least squares estimate of ρs will be negatively biased. Lee and Mukoyama (2010)
perform this regression on manufacturing data from the Annual Survey of Manufactures
(ASM). Thus, instead of inferring ρs and σεs from the results of this regression, these two
parameters were set to match coefficient on ln nt and the variance of the error in the OLS
28
regression reported in Lee and Mukoyama (2010). This implies that at a quarterly frequency,
ρs = 0.998 and σεs = 0.0129.
This then leaves three parameters to calibrate in the steady state: the constant term for
the idiosyncratic productivity process, as , the fixed operating cost, cf , and the fixed entry
cost, ce . The constant term as is set to target the average size of firms to 15 employees. In
the 2003 SSBF, the average size of firms was about 30 employees. However, in the Census
Bureau’s Statistics of U.S. Businesses between 1992 and 2008, the historical average size of
firms with less than 500 employees was reported to be approximately 10 employees. Thus,
the choice of 15 employees is roughly consistent with the data. The fixed costs cf and ce are
then chosen to match the historical average exit rate and the yearly survival rate of entrants
from the data. The entry rate is on average larger than the exit rate, due to the fact that
the number of operating firms exhibits an upward trend. For this reason, the exit rate, and
not the entry rate, was targeted. The target exit rate was the average establishment exit
rate between 1992-III and 2009-IV from the BLS’s Business Employment Dynamics (BED)
survey, which was 2.9% quarterly. The target average yearly survival rate of entrants was
78.39%, which is the historical average yearly survival rates of establishments who entered
between 1994 and 2009, as reported in the BED survey.
6
Results
The Baa credit shock from Section 4.2 can now be fed into the model. See Appendix A.2
for details on the methods used to solve the model numerically. The model’s predictions
for output, hours, entry and exit rates are then compared to those observed in the data.
Figure 7 plots the model fluctuations against the data for the benchmark model. The top
left panel shows the cyclical deviation in aggregate output, the top right aggregate hours, the
bottom left the entry rate and the bottom right the exit rate. The model time series is the
percentage deviation from the mean level. The data time series is the percentage deviation
29
from the Hodrick-Prescott (HP) filter trend.
Figure 7 shows that under this calibration, the model performs fairly well in explaining
historical movements in output, hours and the exit rate, but is less successful for explaining
movements in the entry rate. Moreover, Table 3 reports the relative standard deviation of
each of the four variables considered. In other words, it reports σm /σd , where σm is the the
standard deviation of the relevant model-predicted series and σd is the standard deviation
observed in the data. In the benchmark model, observed credit shocks can explain over
half of cyclical movements in output, 47% of cyclical movements in hours, 17% of cyclical
movements in exit rates and 5% of cyclical movements in entry rates. Even though the credit
shocks are temporary, this model shows that credit shocks can explain not only movements
in output and hours, but also historical movements in exit rates. Moreover, observe that
these results are obtained without making strong assumptions about the consequences of
default for the exit decision. For example, it is not assumed that default necessarily means
the firm must exit nor is it assumed that the firm must borrow to finance the fixed operating
cost. Nevertheless, even though the observed credit shocks are relatively temporary, they
can explain 17% of cyclical variation in exit rates solely through their effect on future profits.
However, this simulation may underestimate the true effect of credit shocks on entry and
exit rates because the Baa corporate bond is less likely to enter default than the average
corporate bond and those firms who can issue corporate bonds tend to be the most productive
and least financially constrained. According to Moody’s, the historical average default rate
on Baa bonds between 1920 and 2010 was 0.270%, while for all bonds it was 1.146%.6
Given that difference between the investment-grade and speculative-grade credit shocks,
this implies that the average firm in the economy may have faced larger swings in its credit
shock over the time period considered.
6
This is an issuer-weighted average.
30
6.1
Great Recession
In addition to the analysis of the historical effect of credit shocks, I also consider how
much observed credit shocks contributed to the Great Recession in the late 2000s. For this
purpose, consider the model’s predictions beginning in the fourth quarter of 2007, which
the NBER identifies as the peak of the recession. As for the data, however, the HP filter
is not used to detrend output, hours, entry and exit rates. The recovery has been so weak
that the HP filter shows a substantial reduction in trend growth. Therefore, rather than
using the HP trend, the output trend and hours trend in 2007-IV is assumed to be equal to
the actual value in 2007-IV, respectively. For the following quarters, it is assumed that the
output trend and the hours trend grows at the historical average of their respective series.
For the trend of entry and exit rates, it is assumed that each trend is always equal to the
actual entry and exit rate in 2007-IV, respectively.
Figure 8 plots the benchmark model’s predictions for output, hours, entry and exit rates
against those actually observed in the data. Each series has been normalized so that all are
equal to 1 in 2007-IV. The actual data series is the percentage deviation from trend, while
the model prediction is the percentage deviation from the 2007-IV value. Up until 2008-IV,
the Baa credit shock can account for roughly 55% of the decline in output, 73% of the decline
in hours, 17% of the initial increase in the exit rate and 7% of the initial decline in the entry
rate. However, after 2008-IV, the credit shock predicts a much stronger recovery than that
which actually occurred. While credit spreads increased during the Great Recession, by
2008-IV interest rates had returned to normal. Thus, if financial frictions are to account for
the weak recovery, they cannot be manifested in credit spreads.
6.2
Heterogeneity in Debt Financing
While the benchmark calibration predicts that measured credit shocks can explain 17%
of cyclical variation in exit rates and 5% for entry rates, the model points to a larger role for
financial frictions in explaining entry and exit rates and a smaller role for output and hours.
31
According to Henly and Sánchez (2009), the largest 5% of firms in the United States account
for roughly 75% of employment. This is a pattern that has been relatively stable between
1974 and 2006. The same inequality is apparent in the SSBF, where 99.7% of firms only
account for 50% of employment. This suggests that the employment decisions of a small
number of firms are largely responsible for cyclical variation in aggregate output and hours.
However, entry and exit rates are mostly determined by the smallest 95%. To motivate this,
Table 6 reports two statistics for different firm age groups from the 2003 SSBF. Here the
firm’s “age” is technically the number of years since the business was established, purchased
or acquired by the current owner(s). The first statistic is the average size of the firm within
the specified age group, and the second statistic is the median ratio of the firm’s liabilities
over its assets. In this model, the firm’s ratio of its liabilities to its assets can be thought
of as a rough analogue to κ. Then, according to Table 6, younger firms are more dependent
on debt (i.e., have a higher κ) and are smaller on average. Intuitively, the older the firm,
the more time the firm has had to retain earnings and accumulate net worth to relax the
working capital constraint.
This model then suggests that it is precisely these young and debt-dependent firms who
are more likely to exit when faced with credit shocks. However, since these younger firms are
smaller on average, the behavior of the older and larger firms will be more important for the
cyclicality of aggregate output and employment. Older and larger firms are less dependent
on external finance, but this model suggests that the employment and output of these firms
are still sensitive to credit shocks. Thus, to more accurately quantify the impact of credit
shocks, I consider an additional calibrations in which I exogenously allow for heterogeneity
in κ. In particular, I consider ng different groups of firms, indexed by i, each with its own
κ = κi , as = ais and M̄ = M̄i . Each group has its own signal distribution, Gi (·), which is
assumed to be the invariant distribution of each group’s idiosyncratic productivity. All other
parameters are assumed to be the same as in the benchmark calibration.
I then consider two variants of this calibration, labeling them “Heterogeneity Model
32
1” and “Heterogeneity Model 2”, respectively. In particular, for Heterogeneity Model 1, I
assume ng = 2, while κ1 , a1s and M̄1 are assumed to be the same as in the benchmark model.
In other words, group 1 is calibrated to be consistent with firms in the 2003 SSBF. The
top group can then be viewed as the large firms not in the SSBF. Thus, I conservatively
choose κ2 = 0.2, and calibrate a2s and M̄2 so that the mass of group 2 firms is 0.003 and
the average size of these firms is such that group 2 firms account for 60% of employment7 .
In Heterogeneity Model 2, I assume ng = 11. In this version, I divide group 1 of Model
1 into 10 different groups. Then, I assume the parameters of group 11 are the same as in
group 2 in Model 1. The κ and as of groups 1 through 10 are then calibrated to match the
corresponding average size and median liabilities to assets ratio in Table 6. The number
of potential entrants, M̄i , for each group is calibrated to target the actual number of firms
observed in each group in Table 6, where the total number of firms in groups 1 through 10
is calibrated to be 0.997.
These calibrations are an attempt to approximate exogenously heterogeneity in κ, which
is important for the quantitative impact of credit shocks, without having to endogenously
model the causes of the heterogeneity. The resulting parameters for κ, as and M̄ for Model
2 are reported in Table 7. Remember, for Model 1, group 1’s parameters are taken from the
benchmark calibration and group 2’s parameters are taken from group 11 in Model 2. All
other parameters are the same as in the benchmark calibration. Since the groups with lower
average sizes also have higher exit rates on average, the number of potential entrants for these
groups must be higher in order to maintain an equal number of firms in each group. Thus,
in this calibration, there is a large number of small potential entrants with high financing
needs but small number of large potential entrants with low financing needs.
Table 3 reports the resulting relative standard deviations of output, hours, entry and
exit rates for both models with heterogeneity. The results are similar to those found in the
7
Since for group 11, the exit rate is close to zero, to make the problem computationally feasible, I needed
to assume that there is a small probability of exogenously exiting (say, 1%). This was only done for group
11.
33
benchmark model. After accounting for heterogeneity in financing needs and firm size, the
model predicts that entry and exit rates will be slightly more sensitive to credit shocks, while
output and hours are slightly less sensitive. Including large firms in Model 1 has a negligible
effect on the sensitivity of entry and exit rates to credit shocks. However, in Model 1, since
40% of employment is determined by firms with κ = 0.2, the sensitivity of output and hours
to credit shocks falls to 43% and 37%, respectively. After accounting for heterogeneity within
the SSBF in Model 2, the sensitivity of output and hours to credit shocks falls even more.
Moreover, Table 8 reports statistics for each group. In particular, the large firms with
low financing needs tend to have lower exit rates and higher yearly survival rates in the
model. Moreover, I also report the relative standard deviations. In other words, for output,
i
i
is the standard deviation of the group
/σd , where σm
hours, entry and exit rates, I report σm
i model-predicted series, while σd is the standard deviation of the same series in the data.
Not surprisingly, this model then predicts that output, employment, entry and exit rates are
more sensitive to credit shocks when the firms have higher debt needs.
6.3
Sensitivity of Results to Parameters
The quantitative importance of credit shocks for output and hours depends critically on
two parameters: κ and γ. Using Equations 16 and 17, Tables 4 and 5 report the creditshock semi-elasticities for hours and output assuming different values for κ and γ. In the
benchmark calibration, the credit-shock elasticity for hours was −5.34 and for output it was
−4.94. In other words, the model predicts that a 100 basis point quarterly increase in the
credit shock will lead to a 5.34% fall in hours and a 4.94% fall in output. However, low values
of κ can still generate a large response of hours and output to a credit shock if firm-level
returns to scale are sufficiently high.
34
6.4
Alternative Entry Condition
If firms had viewed the credit shock each period as permanent, then this model would
have predicted a much larger response in entry and exit rates but would have had no effect
on the response of output and hours. However, when the firm views the shock as AR(1),
entry and exit rates respond less. While the firm is paying a larger external finance premium
in the current period and may have negative cash flows, it finds it optimal to wait out the
shock because it recognizes that the credit shock is temporary. However, the same logic does
not work with the entry margin. Entry rates respond less with AR(1) credit shocks only
because a temporary credit shock has a smaller effect on the firm’s expected lifetime profits
than a permanent one.
However, a simple modification of the entry condition could generate a larger response
in entry rates. Suppose there is an increase in the credit shock. Entry rates could fall not
just because of the fall in lifetime profits, but because potential entrants optimally decide
to wait out the shock. However, in this model, the assumption was that there was a fixed
pool of potential entrants each period. For this story to work, assume that there is a pool
of potential entrants each period, each with a signal for its productivity if it chooses to
operate. However, now let the number of potential entrants each period be endogenous.
Each period, a fixed number of new potential entrants are born, and their signal is drawn
from the distribution G(·). However, if a potential entrant chooses not to enter, it survives
to the next period only with some exogenous probability less than one. Thus, a potential
entrant who faces a small fall in expected lifetime profits due to a temporary increase in the
credit shock now faces a tradeoff between entering today or entering tomorrow.
7
Conclusion
This paper provides a basic model of firm dynamics to study the quantitative importance
of financial frictions for the entry and exit of firms over the business cycle, as well as aggregate
35
output and hours. Historical credit shocks are constructed from observed credit spreads,
default rates and recovery rates on corporate bonds and then fed into the model. Historically,
financial shocks can account for much of the cyclical movements in aggregate output, hours,
and exit rates, but is less successful for entry rates. To explain these historical fluctuations,
this model stresses the importance of heterogeneity in firm size and the use of debt financing.
In particular, the exit rate of firms least constrained by the financial friction are also the least
sensitive to credit shocks. Because these firms are also the largest, their employment decisions
are most important for determining fluctuations in aggregate output and employment. Thus,
for reasonable parameters, while the exit decisions of these firms are affected less than
more debt-dependent firms, observed credit shocks still can account for much of historical
movements in output and hours. Conversely, the exit rate of the more debt-dependent firms
will be much more sensitive to credit shocks. Moreover, since entering firms tend to use more
debt, the debt-dependent firms are most important in explaining cyclical fluctuations in total
entry and exit rates. The output and hours of these firms are also very sensitive to credit
shocks, but since they are smaller on average, they matter less for aggregate fluctuations in
output and employment.
Furthermore, an obvious extension to this research would be to allow firms to accumulate
net worth. One potential issue of the quantitative financial accelerator model of Bernanke et
al. (1999) is that the entry and exit of entrepreneurs was exogenous. This is a key assumption
in that it prevents entrepreneurs from accumulating too much net worth so that they avoid
the financial frictions altogether. This paper attempts to account for the distribution of
net worth in explaining aggregate fluctuations. However, by endogenously modeling net
worth, entry and exit, it would be possible to determine how much endogenous entry and
exit actually mitigates the impact of credit shocks on aggregate output and employment.
36
References
Albuquerque, Rui and Hugo A. Hopenhayn, “Optimal Lending Contracts and Firm Dynamics,” The Review of Economic Studies, April 2004, 71 (2), 285–315.
Arellano, Cristina, Yan Bai, and Patrick Kehoe, “Financial Markets and Fluctuations in
Uncertainty,” January 2011.
Basu, Susanto and John G. Fernald, “Returns to Scale in U.S. Production: Estimates and
Implications,” Journal of Political Economy, April 1997, 105 (2), 249–283.
Bernanke, Ben and Mark Gertler, “Agency Costs, Net Worth, and Business Fluctuations,”
American Economic Review, March 1989, 79 (1), 14–31.
,
, and Simon Gilchrist, “The Financial Accelerator in a Quantitative Business
Cycle Framework,” in John B. Taylor and Michael Woodford, eds., Handbook of
Macroeconomics, Vol. 1, Elsevier Science B.V., December 1999, chapter 21, pp. 1341–
1393.
Burnside, Craig, Martin Eichenbaum, and Sergio Rebelo, “Capital Utilization and Returns to
Scale,” in Ben S. Bernanke and Julio J. Rotemberg, eds., NBER Macroeconomics Annual,
January 1995, pp. 67–124.
Carlstrom, Charles T. and Timothy S. Fuerst, “Agency Costs, Net Worth, and Business
Fluctuations: A Computable General Equilibrium Analysis,” American Economic Review,
December 1997, 87 (5), 893–910.
Clementi, Gian Luca and Dino Palazzo, “Entry, Exit, Firm Dynamics, and Aggregate Fluctuations,” September 2010.
and Hugo A. Hopenhayn, “A Theory of Financing Constraints and Firm Dynamics,”
Quarterly Journal of Economics, February 2006, 121 (1), 229–265.
37
Collin-Dufresne, Pierre, Robert S. Goldstein, and J. Spencer Martin, “The Determinants of
Credit Spread Changes,” Journal of Finance, December 2001, 56 (6), 2177–2207.
Cooley, Thomas F. and Vincenzo Quadrini, “Financial Markets and Firm Dynamics,”
American Economic Review, December 2001, 91 (5), 1286–1310.
Cooley, Thomas, Ramon Marimon, and Vincenzo Quadrini, “Aggregate Consequences of
Limited Contract Enforceability,” Journal of Political Economy, August 2004, 112 (4),
817–847.
Cordoba, Juan-Carlos and Marla Ripoll, “Credit Cycles Redux,” International Economic
Review, November 2004, 45 (4), 1011–1046.
Gale, Douglas and Martin Hellwig, “Incentive-Compatible Debt Contracts: The One-Period
Problem,” Review of Economic Studies, October 1985, 52 (4), 647–663.
Hansen, Gary D., “Indivisible Labor and the Business Cycle,” Journal of Monetary
Economics, November 1985, 16 (3), 309–327.
Henly, Samuel E. and Juan M. Sánchez, “The U.S. Establishment-Size Distribution: Secular
Changes and Sectoral Decomposition,” Economic Quarterly, Fall 2009, 95 (4), 419–454.
Hopenhayn, Hugo A., “Entry, Exit and Firm Dynamics in Long Run Equilibrium,”
Econometrica, September 1992, 60 (5), 1127–1150.
Hopenhayn, Hugo and Richard Rogerson, “Job Turnover and Policy Evaluation: A General
Equilibrium Analysis,” Journal of Political Economy, 1993, 101 (5), October.
Huang, Jing-Zhi and Ming Huang, “How Much of Corporate-Treasury Yield Spread Is Due
to Credit Risk?: A New Calibration Approach,” May 2003.
Kiyotaki, Nobuhiro and John Moore, “Credit Cycles,” Journal of Political Economy, April
1997, 105 (2), 211–248.
38
Kocherlakota, Narayana R., “Creating Business Cycles Through Credit Constraints,” Federal
Reserve Bank of Minneapolis Quarterly Review, 2000, 24 (3), 2–10.
Lee, Yoonsoo and Toshihiko Mukoyama, “Entry, exit, and plant-level dynamics over the
business cycle,” December 2010.
Lucas, Robert E., “On the Size Distribution of Business Firms,” The Bell Journal of
Economics, Autumn 1978, 9 (2), 508–523.
Modigliani, Franco and Merton H. Miller, “The Cost of Capital, Corporation Finance and
the Theory of Investment,” American Economic Review, June 1958, 48 (3), 261–297.
Rosen, Sherwin, “Authority, Control, and the Distribution of Earnings,” The Bell Journal
of Economics, Fall 1982, 13 (2), 311–323.
Tauchen, George, “Finite state markov-chain approximations to univariate and vector autoregressions,” Economics Letters, 1986, 20 (2), 177–181.
Townsend, Robert M., “Optimal Contracts and Competitive Markets with Costly State
Verification,” Journal of Economic Theory, 1979, 21 (2), 265–293.
A
Appendix
A.1
Data Sources
In this paper, I use data on interest rates, default rates, output, hours, entry and exit
rates. For convenience, all the data sources are listed here.
• Interest Rates: Historical interest rates for seasoned corporate bonds rated Aaa and
Baa by Moody’s, as well as 10-year Treasury notes, were obtained from the Federal
Reserve’s historical H.15 release. Moreover, for interest rates on investment-grade
corporate bonds, the Merrill Lynch U.S. Corporate Master (C0A0) index was used.
39
For interest rates on speculative-grade bonds, the Merrill Lynch U.S. Cash Pay High
Yield (J0A0) index was used.
• Aggregate Output: For aggregate output, quarterly seasonally adjusted real GDP
from BEA’s National Income and Product Accounts (NIPA) was used.
• Aggregate Hours: Aggregate hours were obtained from BLS’s Current Employment
Statistics (CES). The specific hours statistic is the aggregate weekly hours of production and nonsupervisory employees in the private sector. This variable is reported
monthly, so the quarterly average is used.
• Entry and Exit Rates: For entry and exit rates, establishment birth and death
rates from the BLS’s Business Employment Dynamics (BED) survey were used. In
the BED survey, an establishment death is defined to occur when an establishment
reports zero employment in the third month of a quarter and did not report positive
employment in the third month of the next four quarters. The establishment birth
rate is defined analogously. This strict definition of entry and exit eliminates most
temporary or seasonal entry and exit.
A.2
A.2.1
Computational Method
Approximation of Value Function
The firm’s value function is approximated by value function iteration. Assuming that
the model is calibrated such that there is never full employment, the firm distribution, µ,
can be omitted from the set of state variables. In that case, the firm’s value function has
two state variables, current-period productivity s and credit shock φ, and the wage is always
equal to one.
1. First define the grid points for the state variables s and φ.
40
2. Starting with the initial guess for the value function, Vn (φ, s), calculate the set of
liquidation cutoffs, s(φ), by solving Vn (φ, s̄) = 0 for each φ. Schumaker’s shapepreserving quadratic spline is used to interpolate the value function when solving for
the liquidation cutoffs.
3. For each (s, φ) on the grid, calculate the optimal n0 and b0 as follows:
(a) Calculate [n, n̄], the range of feasible choices for n0 . In this model n = 0. However,
to calculate n̄, search for the n̄ such that maxb0 q(n̄, b0 ; φ, s)b0 = κw0 n̄.
(b) Search for the n0 such that the first order condition is sufficiently close to zero.
That is, for each potential n0 , compute
1
df = −(1 − κ)w0 +
1+r
"Z
∞
s̄
dπ
h(s0 |s)ds0 −
0
dn
∂q 0
b
∂n0
∂q 0
b
∂b0
κw0 −
q+
!
#
(1 − H(s̄|s))
where b0 is calculated to be the debt value such that qb0 = κw0 n0 . Since n ∈ [n, n̄],
it will always be feasible to find this b0 . Search for the value of n0 such that
df ≤ 1.0−6 . Use Brent’s method to determine the next guess for n0 .
4. For each (s, φ) on the grid, given the optimal choice of n0 and b0 , approximate the firm’s
value. The key difficulty is the need to calculate the firm’s continuation value:
X
0
Z
∞
πφ (φ |φ)
V (φ0 , s0 )h(s0 |s)ds0
s(φ0 )
φ0
Given that s(φ0 ) is the key variable in this model, for accuracy reasons, the process
for idiosyncratic productivity is not approximated with a discrete Markov process as
in Tauchen (1986). Thus, for each s and φ0 , the following integral
Z
∞
V (φ0 , s0 )h(s0 |s)ds0
s(φ0 )
is numerically approximated using an adaptive integration routine from the GNU Sci41
entific Library. Schumaker’s shape-preserving quadratic spline is used to interpolate
the value function in the calculation of this integral.
5. Keep iterating until |Vn+1 (φ, s) − Vn (φ, s)| ≤ 1.0−6 [Vn (φ, s) + cf ] for all grid points on
(φ, s). Use Howard’s improvement algorithm to speed up convergence.
A.2.2
Approximation of Steady State Firm Distribution
Let µ(s) be the density representing the productivity distribution of incumbent firms
at the beginning of the period before the entry and exit decisions are made. Given an
approximated value function, V (φ, s), the steady state µ is calculated by iterating on the
transition rule. Given that there are aggregate credit shocks, the steady state is calculated
assuming φ = as for all periods, where as is the mean of the invariant distribution for φ.
The details of the algorithm can then be described as follows.
1. Define a grid for productivity.
2. Calculate the entry cutoff, s̄e (as ), by solving V (s̄e , φ) for φ = as . When solving for s̄e ,
interpolate V (s, φ) over s using Schumaker’s interpolation algorithm for a given φ.
3. Starting with an initial distribution, µn (s), compute µn+1 (s), assuming that φ = as .
Specifically, for each s0 on the grid, calculating µn+1 (s0 ) as follows:
0
Z
∞
Z
0
∞
h(s |s)µn (s)ds + M̄
µn+1 (s ) =
s(as )
h(s0 |s)g(se )dse .
s̄e (as )
Given that the cutoff rules s and s̄e are key variables in this model, the process for
idiosyncratic productivity is not approximated with a discrete Markov process as in
Tauchen (1986). Instead, the first integral is numerically approximated using an adaptive quadrature routine from the GSL Scientific Library while the second can be computed exactly. When calculating this integral, the prior distribution, µn (s), is interpolated using linear interpolation.
42
4. Keep iterating until |µn+1 (s) − µn (s)| ≤ 1.0−6 µn (s) for all s on the grid.
A.2.3
Simulation
To simulate the model by feeding in a history of credit shocks, {φt }, the following approach was used.
1. Approximate the firm’s value function V (φ, s) using the algorithm in Appendix A.2.1.
2. Approximate an initial steady state distribution of firms over productivity, µ0 (s), using
the algorithm outlined in Appendix A.2.2.
3. For each t, given an initial distribution µt (s) at the beginning of the period, do the
following:
(a) Interpolate a value function Vφt (s) = V (φt , s) for each s on the grid using bilinear
interpolation.
(b) Given value function Vφt (s), calculate the entry cutoff s̄e (φt ) and exit cutoff s(φt )
by solving Vφt (s̄e ) = ce and Vφt (s(φt )) = 0, respectively. When solving for the
cutoffs, interpolate Vφt (s) using Schumaker’s interpolation algorithm.
(c) Calculate the mass of entrants in period t:
Me,t = M̄ (1 − G(s̄e (φt ))) .
(d) Calculate the mass of exiting firms in period t by approximating the following
integral:
Z
s(φt )
Mx,t =
µt (s)ds
−∞
(e) Calculate the mass of incumbent firms at the beginning of period t by approxi-
43
mating the following integral:
Z
∞
Mt =
µt (s)ds
−∞
(f) Calculate the period-t entry and exit rates:
me,t = Me,t /Mt
mx,t = Mx,t /Mt .
(g) Approximate the labor policy rule for each s, n0φt (s) = n0 (φt , s) using bilinear
interpolation.
(h) Calculate period-t + 1 employment by numerically approximating the following
integrals:
Z
∞
n0φt (s)µt (s)ds
Nt+1 =
∞
Z
n0φt (s)g(se )dse .
+ M̄
s(φt )
s̄e (φt )
(i) Calculate period-t + 1 output by numerically approximating the following integrals:
Z
∞
Yt+1 =
E f (s0 , n0φt (s))|s µt (s)ds + M̄
s(φt )
Z
∞
E f (s0 , n0φt (s))|s g(se )dse
s̄e (φt )
where
0
0
E [f (s , n )|s] =
αγ
αγ 1−αγ
u
(n0 )
(1−α)γ
1−αγ
E [s0 |s]
is the expected output tomorrow of a firm in period-t with productivity s.
44
A.3
Positive Marginal Bankruptcy Costs
In order for bankruptcy costs to depress the firm’s labor demand, at the margin, hiring
an additional unit of labor must increase the total expected bankruptcy costs. In particular,
it needs to be the case that
dB(x, s, n0 )
>0
dn0
for any n0 > 0, given x and s. From the firm’s optimality condition for labor in Equation 13,
this would then imply that bankruptcy costs depress labor demand. Then, recall that for a
given choice of n0 , the expected bankruptcy costs are given by:
Z
s̄
B=ξ
π(s0 , n0 )h(s0 |s)ds0
0
Differentiating this expression with respect to n0 and using the definition of the default cutoff
in Equation 2, it follows that:
dB
=ξ
dn0
Z
0
s̄
dπ
d ln s̄ π(s̄, n0 )
0
0
h(s |s)ds +
s̄h(s̄|s) .
dn0
d ln n0 n0
Then, given the assumed functional form for f (s, k, n), the profit function π(s, n) is given
by
h
π(s, n) = ζ sn
β̂
i 1−1 α̂
where ζ, α̂, and β̂ are defined as follows
α̂ ≡ αγ
β̂ ≡ (1 − α) γ
1−α̂α̂
α̂
ζ ≡ (1 − α̂)
.
u
45
(19)
With this functional form, it follows that dπ/dn > 0 for all n > 0. This then implies that
dB/dn0 > 0 if d ln s̄/d ln n0 > 0.
Taking logs of π(s̄, n0 ) = b0 , the default cutoff s̄ can be written as a function of b0 and n0 :
"
β̂
ln n0 − ln ζ
ln s̄ = (1 − α̂) ln b0 −
1 − α̂
#
(20)
Differentiating with respect to ln n0 , it follows that:
#
"
d ln s̄
d ln b0
β̂
= (1 − α̂)
−
d ln n0
d ln n0 1 − α̂
(21)
Then, d ln s̄/d ln n0 > 0 iff
d ln b0
β̂
>
.
0
d ln n
1 − α̂
Thus, to show that marginal bankruptcy costs are always positive at the margin, it needs
to be shown that when the firm increases n0 by 1%, it must increase its promised payment
b0 by more than β̂/(1 − α̂). To calculate d ln b0 /d ln n0 , the firm’s working capital constraint,
qb0 = κw0 n0 , can be implicitly differentiated to obtain
∂q 0
q − ∂n
d ln b0
0n
=
.
∂q 0
d ln n0
q + ∂b
0b
Note that it must be the case that q +
∂q 0
b
∂b0
(22)
> 0. This expression can be interpreted as
the marginal change in borrowed funds today for a marginal change in tomorrow’s promised
payment b0 , given a choice of n0 . However, for each n0 > 0, there is a maximum amount that
the firm can borrow, which is larger than κw0 n0 . Let b̄0 be the debt payment at which qb0
is maximized. Then, for b0 < b̄0 , q +
choose b0 < b̄0 , it follows that q +
∂q 0
b
∂b0
∂q 0
b
∂b0
> 0. Since it must be the case that the firm will
> 0.
Then, before continuing, the partial derivatives of the bond price q need to be computed.
46
Recall that from Equation 3, q can be expressed as:
1−φ
R̄
q=
1 − H(s̄|s) + 0
1+r
b
where R̄ is the unconditional expected recovery value, defined as:
s̄
Z
π(s0 , n0 )h(s0 |s)ds0
R̄ ≡ (1 − ξ)
(23)
0
Taking the partial derivatives of q with respect to n0 and b0 , it follows that:
Z
∂q 0 1 − φ 1 − ξ s̄ 0 dπ
0
0
n =
n 0 h(s |s)ds + ξ β̂s̄h(s̄|s)
∂n0
1+r
b0
dn
0
∂q
1−φ
q + 0 b0 =
[1 − H(s̄|s) − ξ(1 − α̂)s̄h(s̄|s)] .
∂b
1+r
dπ
Then, substituting into Equation 22 and using that π − n0 dn
0 =
0
d ln b
=
d ln n0
1 − H(s̄|s) +
1−γ 1−ξ
1−α̂ b0
hR
s̄
0
1−γ
π,
1−α̂
it follows that:
i
πh(s0 |s)ds0 − ξ β̂s̄h(s̄|s)
1 − H(s̄|s) − ξ(1 − α̂)s̄h(s̄|s)
(24)
Since γ ∈ (0, 1), it follows that β̂ < 1 − α̂. This implies that
1−γ1−ξ
1 − H(s̄|s) +
1 − α̂ b0
Given that q +
∂q 0
b
∂b0
Z
s̄
0
0
πh(s |s)ds − ξ β̂s̄h(s̄|s) > 1 − H(s̄|s) − ξ(1 − α̂)s̄h(s̄|s).
0
> 0, this implies that
d ln b0
β̂
>1>
.
0
d ln n
1 − α̂
Using Equation 21, this implies that d ln s̄/d ln n0 > 0, which in turn implies that dB/dn0 > 0.
47
A.4
A.4.1
Effect of Productivity and Credit Shocks
Effect of Idiosyncratic Productivity
First, I will jointly show that
ρs
∂ ln n0 (x, s)
=
∂ ln s
1−γ
and the equilibrium default probability, H(s̄(n0 (x, s), b0 (x, s))|s) does not vary with s. To see
this, let’s suppose that both are true and then check whether the firm’s optimality condition
and working capital constraint are still satisfied for marginal changes in productivity. Then,
since the solution to the firm’s problem is unique, both of the premises must be true.
First, note that H(s̄|s) does not vary with s iff
∂ ln s̄
= ρs .
∂ ln s
Then, differentiating the expression for ln s̄ in Equation 20 with respect to ln s, this requires
that
∂ ln s̄
∂ ln b0
∂ ln n0
= (1 − α̂)
− β̂
= ρs .
∂ ln s
∂ ln s
∂ ln s
Since
∂ ln n0
∂ ln s
=
ρs
,
1−γ
this also requires that
∂ ln b0
∂ ln s
=
ρs
.
1−γ
Next, it needs to be shown that the
firm’s working capital constraint is still satisfied for marginal changes in productivity. In
particular, it needs to be shown that
∂(qb0 )
∂n0
= κw0
.
∂ ln s
∂ ln s
In other words, given the marginal change in the firm’s labor demand, the marginal change
in the firm’s wage expenditure exactly equals the marginal change in borrowed funds, given
the assumption that the probability of default does not change. Then, from the definition
48
of q in Equation 3, the firm’s working capital constraint requires that:
1−φ
1+r
(1 − H(s̄|s))b0 + R̄ = κw0 n0
where R̄, the unconditional expected recovery value, is defined in Equation 23. Before
implicitly differentiating the working capital constraint, let’s first calculate
∂ ln R̄
.
∂ ln s
Using the
functional form for π(s, n) in Equation 19, R̄ can be expressed as:
0 β̂/(1−α̂)
R̄ = (1 − ξ)ζ (n )
β̂/(1−α̂)
= (1 − ξ)ζ (n0 )
s̄
Z
1/(1−α̂)
(s0 )
0
1
exp µ +
2
h(s0 |s)ds0
2
1 2
σ + µ − ln s̄/(1 − α̂)
√
σ erfc
2
σ 2
(25)
where
(1 − ρs )as + ρs ln s
1 − α̂
σεs
σ≡
1 − α̂
µ≡
(26)
(27)
Since H(s̄|s) does not vary with s by assumption, µ − ln s̄/(1 − α̂) is constant. Taking logs
and differentiating, this implies that
∂ ln R̄
β̂ ∂ ln n0
ρs
ρs
=
+
=
.
∂ ln s
1 − α̂ ∂ ln s
1 − α̂
1−γ
Implicitly differentiating the working capital constraint with respect to ln s, it follows that:
Since
∂ ln b0
∂ ln s
=
∂ ln R̄
∂ ln s
1−φ
1+r
=
0
∂ ln R̄
∂ ln n0
0 ∂ ln b
(1 − H(s̄|s))b
+ R̄
= κw0 n0
∂ ln s
∂ ln s
∂ ln s
∂ ln n0
∂ ln s
=
does not vary with s and that
ρs
,
1−γ
∂ ln n0
∂ ln s
this equation is satisfied. Thus, assuming that H(s̄|s)
=
ρs
,
1−γ
the firm’s working capital constraint is always
satisfied for marginal changes in productivity. However, it still remains to be shown that
49
the firm’s optimality condition in Equation 13 is satisfied as well for marginal changes in
productivity. The firm’s optimality condition is repeated below:
Z
0
∞
dπ
h(s0 |s)ds0 = (1 + r)ŵ + Bn
dn0
(28)
φ
where ŵ = 1 + κ 1−φ w0 is the effective wage and Bn ≡ dB/dn0 is the marginal bankruptcy
cost. Given the functional form for π(s, n) from Equation 19, the expected marginal product
of labor is given by:
Z
0
∞
β̂
β̂
1 2
dπ
0
0
0 1−α̂ −1
h(s |s)ds =
ζ (n )
exp µ + σ
dn0
1 − α̂
2
where µ and σ are defined in Equation 26 and 27, respectively. If ξ = 0 and there are no
bankruptcy costs, then the default cutoff, s̄, does not enter into the the firm’s optimality
condition and it is trivial to show that
for
∂ ln n0
∂ ln s
=
ρs
1−γ
∂ ln n0
∂ ln s
=
ρs
1−γ
is consistent with Equation 28. If ξ > 0,
to be consistent with the firm’s optimality condition, it needs to be the case
that the equilibrium Bn does not vary with s. It turns out that this is in fact the case. To see
this, consider a decomposition of Bn into two components, where Bn = Bn1 + Bn2 , as follows:
Z
s̄
dπ
h(s0 |s)ds0
0
dn
0
d
ln s̄ b0
s̄h(s̄|s)
Bn2 ≡ ξ
d ln n0 n0
Bn1
≡ξ
(29)
(30)
Thus, to show that Bn does not vary with s, it is sufficient to show that both Bn1 and Bn2 do
not vary with s.
• Bn1 does not vary with s
50
Given the functional form for π(s, n) in Equation 19, Bn1 can be expressed as follows:
Bn1
β̂
β̂
−1
=ξ
ζ (n0 ) 1−α̂
1 − α̂
=ξ
β̂
β̂
−1
ζ (n0 ) 1−α̂
1 − α̂
Z
s̄
1/(1−α̂)
(s0 )
0
1
exp µ +
2
h(s0 |s)ds0
2
1 2
σ + µ − ln s̄/(1 − α̂)
√
σ erfc
2
σ 2
where µ and σ are defined in Equation 26 and 27, respectively. Then, taking logs and
taking the partial derivative with respect to ln s, it follows that:
∂ ln Bn1
=
∂ ln s
!
β̂
∂ ln n
ρs
−1
+
=0
1 − α̂
∂ ln s 1 − α̂
Thus, Bn1 does not vary with s.
• Bn2 does not vary with s
First, let’s substitute for d ln s̄/d ln n0 in the definition of Bn2 with Equation 21, to get:
Bn2
b0
d ln b0
− β̂
= ξs̄h(s̄|s) 0 (1 − α̂)
n
d ln n0
(31)
Then, s̄h(s̄|s) is constant because s is log normally distributed. Moreover, since
∂ ln b0
∂ ln s
=
∂ ln n0
,
∂ ln s
b0 /n0 is constant as well. Therefore, this implies that Bn2 is constant
if d ln b0 /d ln n0 is constant. From Equation 24, d ln b0 /d ln n0 is given by:
1−γ R̄
1 − H(s̄|s) + 1−
− ξ β̂s̄h(s̄|s)
d ln b0
α̂ b0
=
d ln n0
1 − H(s̄|s) − ξ(1 − α̂)s̄h(s̄|s)
Then, since H(s̄|s), s̄h(s̄|s) and R̄/b0 are constant, d ln b0 /d ln n0 is constant as well.
This then implies that Bn2 is constant.
Given that both Bn1 and Bn2 do not vary with s, Bn does not either. Thus, this implies
that a constant default probability and
∂ ln n0
∂ ln s
=
ρs
1−γ
are both consistent with not only the
firm’s optimality condition but the working capital constraint. Given that the solution is
51
unique, it must be the case that both premises hold.
A.4.2
Effect of Credit Shock
I now show that both
1 − α̂
∂ ln n0 (x, s)
=−
∂ ln ŵ
1−γ
and the equilibrium default probability, H(s̄(n0 (x, s), b0 (x, s))|s) does not vary with the credit
φ
shock φ (or ŵ) when κ = 1, where ŵ ≡ 1 + κ 1−φ
w0 is the effective wage. Let’s suppose
both are true, and check whether the firm’s optimality condition and working capital constraint are both satisfied. Then, since the solution to the firm’s problem is unique, both
premises must be true.
Note that H(s̄|s) does not vary with ln ŵ iff s̄ is constant. Then, differentiating the
expression for ln s̄ in Equation 20 with respect to ln ŵ, it follows that:
∂ ln s̄
∂ ln b0
∂ ln n0
= (1 − α̂)
− β̂
=0
∂ ln ŵ
∂ ln ŵ
∂ ln ŵ
This requires that
β̂ ∂ ln n0
∂ ln b0
=
∂ ln ŵ
1 − α̂ ∂ ln ŵ
(32)
Moreover, if H(s̄|s) is constant, then taking logs of the expression for R̄ in Equation 25 and
differentiating with respect to ln ŵ, it follows that:
∂ ln R̄
β̂ ∂ ln n0
=
∂ ln ŵ
1 − α̂ ∂ ln ŵ
Then, to see the effect of a credit shock, let’s implicitly differentiate the firm’s budget
constraint, qb0 = κw0 n0 , with respect to φ, to obtain:
ln b0 1 dŵ
∂ ln R̄ 1 dŵ
1 + r 0 0 ∂ ln n0 1 dŵ
1
(1 − H(s̄|s))b
+ R̄
=
κw n
+
∂ ln ŵ ŵ dφ
∂ ln ŵ ŵ dφ
1−φ
∂ ln ŵ ŵ dφ 1 − φ
0∂
52
Then, since
β̂ ∂ ln n0
∂ ln b0
∂ ln R̄
=
=
∂ ln ŵ
∂ ln ŵ
1 − α̂ ∂ ln ŵ
it follows that
∂ ln n0 1 dŵ
1
β̂ ∂ ln n0 1 dŵ
=
+
1 − α̂ ∂ ln ŵ ŵ dφ
∂ ln ŵ ŵ dφ 1 − φ
Rearranging, this expression becomes:
∂ ln n0 1 dŵ
1 1 − α̂
=−
∂ ln ŵ ŵ dφ
1−φ1−γ
Therefore, if
1
1 dŵ
=
ŵ dφ
1−φ
the firm’s working capital constraint will be satisfied. Since
κw0
dŵ
=
dφ
(1 − φ)2
it follows that
1 dŵ
1
κ
=
ŵ dφ
1 − φ 1 − (1 − κ)φ
Therefore, if κ = 1, this expression equals 1/(1 − φ), implying that the firm’s working capital
constraint is satisfied. Moreover, if κ ∈ [0, 1) and φ ∈ [0, 1), it will be the case that
κ
< 1.
1 − (1 − κ)φ
Thus, the firm’s working capital constraint is satisfied when κ = 1, but it remains to
be shown that the firm’s optimality condition for employment is also satisfied. The firm’s
53
optimality condition from Equation 13 is repeated below:
∞
Z
0
dπ
h(s0 |s)ds0 = (1 + r) ŵ + Bn
dn0
Therefore, if
∂ ln Bn
=1
∂ ln ŵ
the premises satisfy the optimality condition. To see that this is true, let’s decompose Bn as
before into Bn1 and Bn2 , where Bn1 and Bn2 are defined in Equations 29 and 30, respectively.
Then, let’s separately take the partial derivatives of Bn1 and Bn2 with respect to ln ŵ.
• Bn1
As for Bn1 , since s̄ is constant, it follows that
∂ ln Bn1
1 − γ ∂ ln n0
=−
∂ ln ŵ
1 − α̂ ∂ ln ŵ
• Bn2
From Equation 31, Bn2 can alternatively be expressed as
Bn2
b0
d ln b0
= ξ 0 s̄h(s̄|s) (1 − α̂)
− β̂
n
d ln n0
Since the default probability is constant, s̄h(s̄|s) is constant as well. Moreover, from
Equation 24, d ln b0 /d ln n0 can be written as:
1−γ
1 − H(s̄|s) + 1−
(1 − ξ) R̄
− ξ β̂s̄h(s̄|s)
d ln b0
α̂
b0
=
d ln n0
1 − H(s̄|s) − ξ(1 − α̂)s̄h(s̄|s)
Then, since H(s̄|s), s̄h(s̄|s) and R̄/b0 are constant, d ln b0 /d ln n0 is constant as well.
54
Therefore, this implies that:
∂ ln Bn2
∂ ln b0
∂ ln n0
1 − γ ∂ ln n0
=
−
=−
∂ ln ŵ
∂ ln ŵ ∂ ln ŵ
1 − α̂ ∂ ln ŵ
Then, using the partial derivatives of Bn ’s components, the partial derivative of Bn is
given by:
∂ ln Bn
=−
∂ ln ŵ
1−γ
1 − α̂
∂ ln n0
=
∂ ln ŵ
1−γ
−
1 − α̂
1 − α̂
−
=1
1−γ
This implies that the premises satisfy the both the firm’s working capital constraint and the
optimality conditions for n0 . Given that there is a unique solution, the premises must hold
as well.
Figure 1: Effect of Financial Frictions on Optimal Labor Demand
55
Figure 2: Baa-10y Treasury Spread Explained by Default
Figure 3: Investment-Grade Credit Spread Explained By Default
56
Figure 4: Speculative-Grade Credit Spread Explained By Default
Figure 5: Baa Credit Shocks
57
Figure 6: Investment-Grade and Speculative-Grade Credit Shocks
Parameter
Baa Notes
κ
0.400 Match median liabilities/assets in 2003 SSBF
0.581 Target recovery rate = 41.34%
ξ
aφ
-5.534 Estimate from data
0.929 Estimate from data
ρφ
0.165 Estimate from data
σεφ
δ
0.025 Standard
α
0.350 Standard
0.950 Estimates from literature
γ
β
0.990 Standard
n0
0.333 Standard
ψ
0.026 Normalize wage to 1
N̄
15.750 Target unemployment rate = 5%
M̄ 496.466 Normalize mass of incumbent firms to 0.997
ρs
0.998 Match employment process in Lee and Mukoyama (2010)
σεs
0.013 Match employment process in Lee and Mukoyama (2010)
as
-1.268 Target average firm size = 15 employees
cf
0.070 Target quarterly exit rate = 2.9%
ce
0.036 Target average yearly survival rate of entrants = 78.39%
Table 1: Parameter Values (Benchmark Calibration)
58
Baa vs Treasury
Constant
-0.390
(0.138)
ln φt−1
0.929
(0.024)
Root MSE
0.165
N
230
Baa vs Aaa
-0.570
(0.135)
0.907
(0.022)
0.189
363
Table 2: Estimated Credit Shock Process
Figure 7: Historical Fluctuations Explained By Benchmark Model
Benchmark
Model
Output
54.58%
Hours
46.99%
Entry Rate
5.19%
Exit Rate
17.22%
Heterogeneity
Model 1
43.47%
37.40%
5.19%
17.22%
Heterogeneity
Model 2
42.65%
36.68%
7.41%
21.35%
Table 3: Relative Standard Deviations
59
Figure 8: Great Recession with Baa Credit Shock in Benchmark Model
κ
0.2
0.3
0.4
0.5
γ
0.90 0.95
-1.37 -2.67
-2.06 -4.01
-2.74 -5.34
-3.43 -6.67
0.97
-4.40
-6.61
-8.81
-11.01
Table 4: Hours Credit-Shock Semi-Elasticities
κ
0.2
0.3
0.4
0.5
0.90
-1.17
-1.76
-2.34
-2.93
γ
0.95
-2.47
-3.71
-4.94
-6.17
0.97
-4.20
-6.30
-8.41
-10.51
Table 5: Output Credit-Shock Semi-Elasticities
60
Age
(Years)
1-3
4-5
6-8
9-11
12-14
15-17
18-21
22-26
27-32
33+
Average Size
(Employees)
16.00
20.33
22.03
28.68
27.42
37.07
40.15
30.62
42.58
56.62
Median
Liabilities/Assets
0.55
0.48
0.40
0.46
0.36
0.41
0.39
0.32
0.35
0.30
Percent
of Firms
11.0%
8.6%
11.8%
10.3%
9.4%
8.8%
10.8%
10.5%
8.8%
10.0%
Table 6: Firm Age, Size and Financial Dependence
Group
1
2
3
4
5
6
7
8
9
10
11
κi
0.55
0.48
0.40
0.46
0.36
0.41
0.39
0.32
0.35
0.30
0.20
ais
-1.589
-1.439
-1.398
-1.284
-1.302
-1.197
-1.174
-1.261
-1.157
-1.085
-0.501
M̄i
3.579 × 105
3.571 × 103
1.588 × 103
7.566 × 101
1.050 × 102
8.309 × 100
6.019 × 100
4.339 × 101
3.327 × 100
8.200 × 10−1
3.270 × 10−5
Table 7: Calibrated Parameters for Heterogeneity Model 2
Group
1
2
3
4
5
6
7
8
9
10
11
Yearly Entry
Survival Rate
71.31%
74.50%
75.40%
78.03%
77.63%
80.13%
80.67%
78.57%
81.15%
83.01%
95.58%
Exit Output
Rate σm /σd
4.90% 80.57%
3.92% 67.48%
3.66% 55.57%
2.97% 62.53%
3.07% 49.36%
2.47% 55.97%
2.34% 52.93%
2.83% 43.85%
2.25% 47.60%
1.87% 40.55%
1.05% 27.07%
Employment
σm /σd
68.93%
57.99%
47.80%
53.91%
42.53%
48.09%
45.57%
37.74%
40.93%
34.90%
23.29%
Entry Rate
σm /σd
14.99%
9.96%
7.05%
6.04%
5.15%
4.15%
3.81%
4.19%
3.03%
1.88%
0.00%
Exit Rate
σm /σd
36.92%
28.85%
20.43%
20.29%
17.21%
15.75%
14.24%
14.79%
12.14%
8.39%
0.08%
Table 8: Heterogeneity in Debt Financing: Statistics by Group
61