Aggregate implications of firm dynamics:
Financing small versus young firms
In Hwan Jo* and Tatsuro Senga†
February 2016
Abstract
Access to external finance is a major obstacle for small and young firms. What are the
macroeconomic effects of policies that enhance access to finance for small and young firms? To
quantify the aggregate implications of such policies, we build a general equilibrium heterogeneous
firm model with endogenous entry and exit. Furthermore, forward-looking collateral constraints
are introduced so that the model can simultaneously replicate the financing behavior over the life
cycle of firms and the size and age distribution of firms. This model can match with salient
empirical regularities of firm dynamics, including a highly skewed firm size distribution, higher
exit rates among small and young firms, and the variability of firm growth rates across different
firm sizes and ages. We construct a new panel data of firms that contains real and financial data at
the firm level, and we use it to calibrate our model to evaluate policy counterfactuals: agedependent, size-dependent, and productivity-dependent policy. We find that age-dependent policy
is the most efficient targeted policy that could foster the aggregate productivity level. The key to
our main result is the possibility of self-financing over the life cycle of firms.
Keywords: firm dynamics, misallocation, productivity, financial frictions; firm size and age
JEL Codes: E22, G32, O16
*
†
National University of Singapore (e-mail: [email protected])
Queen Mary University of London (e-mail: [email protected])
1
1
Introduction
“The debate on productivity has mainly focused on the macroeconomic causes and
solutions. Small businesses are the lifeblood of the UK economy. If we are to improve our
productivity levels there needs to be a small business focused solution.” — UK All Party
Parliamentary Small Business Group
Are policies targeted to a small …rm the answer for accelerating the aggregate productivity level? Should we focus on young start-up …rms rather than on all small …rms?1
To address these questions, we construct a general equilibrium model with heterogeneous
…rms that are allowed to enter and exit but face collateral constraints, and we calibrate
it using the …rm-level panel data. We then conduct policy counterfactual exercises to ask
which policies could potentially nourish the …rm dynamics that result in enhancing aggregate productivity. Our results show that allowing easier access to …nance for small and
young …rms could potentially improve the aggregate productivity: the productivity gains
range from 0.2 percent to 1.5 percent. Especially, we …nd that targeting young startups
will be the most productivity-enhancing policy option, as it undoes capital misallocation
by the greatest amount. This policy option requires only the smallest mass of …rms targeted. On the other hand, targeting small …rms requires a large number of targeted …rms
but merely results in a smaller impact on aggregate productivity.
The key mechanism that we look at is capital misallocation in the presence of …nancial
frictions. That is, a …rm’s choice of capital relative to the e¢ cient level may be insu¢ cient
due to the lack of collateral. Our main …nding depends on where capital misallocation
is pervasive across the distribution of …rms. An age-dependent policy can undo capital misallocation by the greatest amount as capital is misallocated amongst young and
small …rms rather than amongst all small …rms. We …rst examine …rm-level data and
build a model to be able to replicate this dimension of the economy. Then, we conduct
quantitative policy counterfactual exercises to investigate several targeted policy options.
To quantify the aggregate implications of di¤erent policies, we need several model
ingredients that allow us to capture the realistic cross sectional distribution of …rms and
their dynamics. First, we employ a bounded Pareto distribution for the …rm-level productivity process. It is well-known that the distribution of …rm size in employment is highly
skewed, i.e., small …rms dominate the business population, whilst large …rms account for
1
Since January 2011, the Obama Administration has been promoting a set of entrepreneur-focused
policy, including Jumpstart Our Business Startups (JOBS) Act. The White House (2012) claim that these
policies allow “Main Street small businesses and high-growth enterprises to raise capital from investors
more e¢ ciently, allowing small and young …rms across the country to grow and hire faster.”
2
a large fraction of employment in the economy.2 Coupled with decreasing returns to scale
for production in our economy, the above productivity process speci…cation successfully
captures the empirical size distribution.3 Second, we introduce the endogenous entries
and exits of …rms.4 It is essential to have this setting to reproduce the empirical patterns
of …rm dynamics, including substantial lower survival rates among young …rms. This
also allows us to examine the importance of extensive margins in contributing to aggregate productivity.5 Third, forward-looking collateral constraints are added in the spirit of
Kiyotaki and Moore (1997), and this, together with the second ingredient of our model,
leads to the empirical age distribution of …rms.6 The presence of this collateral constraint
hinders the immediate growth of newborn …rms. This helps us not only to have a substantial number of small …rms in the economy as in the data but also to hit the right life
cycle pattern of …rms: newborn …rms gradually build up capital stock by relying on debt
issuance.
We study three di¤erent kinds of policies that relax collateral constraints: one for
small …rms (size-dependent policy), one for young …rms (age-dependent policy), and one
for …rms with low productivity (productivity-dependent policy). In each policy simulation, we analyze the policy impact on the …rm size and age distribution, the life cycle patterns of …rms and the relationship between …rm dynamics and the aggregate productivity.
Our main …nding is that targeting of young …rms will be the most e¢ cient productivityenhancing policy amongst the considered. Targeting small or low-productivity …rms is
less e¢ cient as they obtain little productivity gains in aggregate, even though the mass
of targeted …rms is large. This result is driven by the fact that capital misallocation
due to …nancial frictions is more pervasive among young small …rms in our model, rather
than among all small …rms. Particularly, the following features in our model environment
deliver our quantitative results. First, most startups in our economy are born with a
2
In 2012, the share of employment at large …rms is 51.6 percent while the employment share of small
…rms is 16.7 percent in the US (Statistics of U.S. Businesses, 2012).
3
Jo (2015) builds a business cycle model of such and shows the important role of the empirical …rm
size distribution in driving business cycle ‡uctuation following …nancial shocks. Guner, Venture and
Xu (2008) and Gourio and Roys (2014) study the aggregate implication of policies that distort the size
distribution of …rms.
4
See the Hopenhayn (1992) model and the Jovanovic (1982) model for a workhorse model of …rm
selections and dynamics.
5
Amongst others see Foster et al. (2008), Osotimehin (2013) and Bartelsman et al. (2013) for a
growing body of work on the study of resource allocation.
6
The importance of the …rm age have been studied by the following seminal papers: Davis et al.
(1996), Dunne et al. (1989), and Evans (1989). More recently, Fort et al. (2013) emphasize the …rm age
for the heterogeneity among …rms in response to the recent recession in the U.S. See Hsieh and Klenow
(2014) for the life-cycle of manufacturing plants for Mexico, India and the US.
3
relatively high leverage ratio compared to that of incumbents. Second, the …rm-level productivity process is persistent, implying that …nancially constrained young startups need
to maintain their leverage ratio at a high level for a while by accumulating capital stock
until they reach the e¢ cient level of capital stock, which is consistent with interest rates
and their own productivity. Third, and related to the second feature, if …rms survive long
enough and become grown-up, they are more likely …nancially unconstrained, regardless
of the …rm size. At this stage, the …rm size is determined by productivity rather than
whether or not the …rms are …nancially constrained. Therefore, among small …rms in our
model economy, some are indeed …nancially constrained, and thus productivity gains are
expected when undoing capital misallocation by relaxing collateral constraints. However,
there are also …rms that are small just because their productivity is low, not because
they face collateral constraints. If the fraction of such …rms is large among small …rms,
then targeting small …rms does not deliver the e¢ cient results in enhancing productivity.
Instead, since young …rms tend to be …nancially constrained and, more importantly, some
of them are actually highly productive but without capital stock, targeting young …rms
leads to more e¢ cient results in increasing aggregate productivity level.
Related Literature In policy debates about small …rms and macroeconomic performance, there are lots of factors that are to blame for preventing small …rms from growing
and hiring fast.7 This paper focuses on the role of …nancial frictions for small and young
…rm dynamics. The main contribution is to build a general equilibrium model with …rm
dynamics and o¤er quantitative analysis of several policy alternatives that are often argued to be useful. The result of this paper sheds lights on the importance of the …rm
age when it comes to considering policy options for enhancing access to …nance for small
and young …rms. In particular, we show that age-dependent policy is more productivity
enhancing than size-dependent policy. This result is in line with what Fort et al. (2013)
argue by showing the importance of controlling the …rm age to determine …rm dynamics.
They also show that young and small …rms are more sensitive to shocks than old and
large …rms are. The high sensitivity of small and young …rms to shocks does not necessary imply that policy geared to them are the best recommendation for the aggregate
productivity. Our results con…rms, however, that targeted policies that unlock access to
capital for startups indeed is the best possible option.
To the best of our knowledge, this paper is the …rst quantitative policy simulation ex7
For example, targets of the Obama administration’s entrepreneur-focused policy initiatives include
…nance, education, red tapes, innovation, market opportunities for entrepreneurs. (White House, 2012)
The policy recommendations by Lord Young for Prime Minister David Cameron include helping small
businesses to secure loans, to get public procurements and education and support. (Young, 2015)
4
ercise that looks at several policy alternatives both for size-dependent and age-dependent
options in a comprehensive way. In terms of size-dependent policy, Guner et al. (2008)
studied the macroeconomic implications of restricting large …rms and found sizable e¤ects
on aggregate productivity. Gourio and Roys (2014) also looked at the quantitative impact
of French size-dependent policy and found an important role of such policy distortion in
explaining aggregate productivity. Nonetheless, age-dependent policy is only studied in
our paper.
Behind our results from policy simulations, there are two main mechanisms at work
about allocative e¢ ciency of resources: one among incumbents and one among entrants
and exiting …rms. The role of …nancial frictions generating misallocation of resources and
its aggregate implications has been studied by Khan and Thomas (2013), Buera and Moll
(2013), Buera et al. (2011), and Midrigan and Xu (2014) and others.
Our model builds on a family of the standard theory of …rm dynamics such as Hopenhayn (1992) and extended it into a general equilibrium environment with collateral frictions.8 In studying …rm dynamics with …nancial, Cooley and Quadrini (2002) is the closest
paper to ours. Our simpli…ed collateral constrains are well-suited for the large panel simulations like ours and the general equilibrium setting allows us to examine disproportionate
impact on small versus large, and young and old …rms following policy changes.9
2
Empirics: US and UK data
For the United States, the Business Dynamics Statistics (BDS) allows us to have an
overall picture of …rm dynamics and the …rm distribution aggregated by …rm characteristics such as size and age. The BDS is constructed from the Longitudinal Business
Database (LBD) and covers about 98 percent of U.S. private employment starting from
1977. Firms and establishments are categorized by size and age in addition to their annual
job creation and destruction. Figure 1 shows the …rm size distribution by its population
share and employment share. As can be seen in the …gure, small businesses with employment from 1 to 9 dominate the population share of the economy while large businesses
8
Amongst others see Clementi and Palazzo (2015), Siemer (2015), and Sedlacek (2015).
Cooley and Quadrini (2002) study non-contingent debt contracts between …rms and lenders and show
that their model can explain …rm dynamics conditional on the …rm age and size simultaneously. Khan,
Senga and Thomas (2014) build an equilibrium business cycle model with non-contingent debt contracts
and study how default risks shape …rm dynamics and aggregate ‡uctuations following real and …nancial
shocks.
9
5
Figure 1 : US Firm Size Distribution
0.8
Population share
Employment share
0.7
0.6
percent
0.5
0.4
0.3
0.2
0.1
0
00
10
to
49
99
00
50
00
25
10
00
to
to
24
99
9
99
9
0
to
50
25
0
to
0
10
to
49
9
24
99
to
50
50
to
9
10
to
1
99
99
0
E mployment Size
Notes: The data is taken from the BDS, averaging between 1977 and 2014.
with more than 10,000 employees account for more than 20 percent of employment in the
whole economy. We could track a cohort born since then and look at how employment
such as job creation and destruction grow over the lifecycle and the survival patterns
among the …rms in the same cohort. While employment, …rm startups, and closures
are available in the BDS, …nancial data is not available, and this is problematic for our
purpose of examining the role of …nancial frictions in shaping the …rm’s lifecycle.
2.1
Financing behavior by startups …rm cohort
To investigate the …nancial behaviours of …rms after startup, we take a look at the
UK’s …rm data and track a cohort after its birth. The data are from the ORBIS, which is
a …rm-level global dataset for public as well as private companies. The ORBIS database
contains …rms’real and …nancial data from balance sheets, income statements, and other
company information (e.g. ownership structure, geographical location) for over 130 million companies all over the world. The advantage of this dataset includes the fact that
researchers are allowed to get ahold of balance sheets data for small, unlisted enterprises,
which is impossible using public company datasets such as COMPUSTAT. Given the fact
6
Figure 2 : Financing behavior by UK startups <2006 cohort>
4
3.5
3
2.5
2
1.5
1
0
1
2
3
0
1
2
3
4
5
6
7
8
4
5
6
7
8
1.2
1
0.8
0.6
0.4
0.2
Age
Notes: All variables of the …rm age 0 group are normalised to 1. The sample contains a cohort of
UK …rms of which year of incorporation is 2006. Data is taken from Orbis Global. We keep samples
of …rms with at least one employee throughout 2006 and 2010. The resulting sample is 593 …rms.
that the group of …rms with the smallest employment accounts for the largest population
share in the economy, we lose a plenty of information on …rm dynamics. Furthermore,
the dataset includes information on the year of incorporation, which is presumably better
information with which to identify …rm births.
Figure 2 displays the median levels of employment, shareholders funds, and total asset
size levels for a cohort of startups born in 2006 incorporated in the UK. The median
surviving enterprise at the age of 8 is three times larger than the median startups in
employment. The median size of shareholders funds at the age of 8 is more than three
and half times larger than the median startups while the median size of a total asset at
the age of 8 is just above one and half times of that of startups. Looking at the leverage
ratio in several measures presented here exhibits the robust patterns of deleveraging over
the operating periods.
7
3
Model
There is a large number of heterogeneous …rms producing a homogeneous good. In addition to physical capital accumulation, each …rm is allowed to externally borrow subject
to a collateral constraint. Individual decisions of employment, investment, and borrowing
across …rms generate substantial di¤erences along with persistent individual productivities. In addition to the rich heterogeneity, incumbent …rms are allowed to endogenously
exit in each period, and potential entrants decide their entry to the production sector.
Along with exogenous exit, both endogenous entry and exit decisions characterize the …rm
dynamics in the model. We close the model economy by describing the household problem, and then de…ne a recursive competitive equilibrium. We summarize the …rm-level
state of capital and borrowing in a state-variable called cash-on-hand, which allows us to
derive individual …rm’s decisions in a tractable way. Our model environment is based on
that of Jo (2015), but we depart from it by introducing the endogenous entry and exit by
…rms.
3.1
3.1.1
Firms
Production and Financial Friction
The economy is consist of a continuum of …rms. Each …rm accumulates its capital
stock, k, and hires labor, n, in a competitive labor market. The production technology
follows y = z F (k; n), where F ( ) exhibits decreasing-returns-to-scale (DRS) property.
Productivity level of each …rm has two independent exogenous components, z and . z
is the aggregate level of productivity common across …rms, and represents …rm-speci…c
idiosyncratic productivity. We assume that z follows a Markov chains with zf g Pr(z 0 =
P z z
2 E
f 1 ; :::; N g
zg jz = zf )
0 and N
g=1 f g = 1 for each j = 1; :::; Nz . Also,
PN
0
independently follows a Markov chain with ij Pr( = j j = i ) 0 with j=1 ij = 1
for each i = 1; :::; N .
In this model, not all …rms are able to …nance their desirable investment due to
…nancial frictions. Each …rm faces an individual borrowing constraint for obtaining oneperiod discount debt at the price of q. In our speci…cation, the borrowing constraint
limits the amount of new debt today, b0 , by a …rm’s collateral in the future period. This is
based on the idea of limited enforceability of …nancial contracts, and we assume that the
amount of collateral is the …rm’s future period capital, k 0 . Thus, for each unit of newly
issued debt b0 which needs to be repaid in the next period, …rms only receive q units from
a competitive …nancial market. Therefore, the collateral constraint is given by b0
k0.
8
Notice that the …nancial parameter, , partly represents the leverage ratio, and captures
the economy-wide level of …nancial frictions. Even though is common across …rms, their
endogenous decisions depend on each …rm’s state. When is close to 1, the …nancial
market always allow …rms to invest their desired level. Then a sudden drop in value
corresponds to a credit shock. Note the above collateral constraint is forward-looking in
the spirit of Kiyotaki and Moore (1997). Because we do not restrict the sign of b, …rms
can also accumulate …nancial assets in terms of a negative value of b. This implies that
the individual borrowing constraint is occasionally binding.
3.1.2
Entry and Exit
We model endogenous entry and exit by …rms following the standard approach in the
literature.10 One distinctive feature of our model is the presence of …nancial frictions
with the standard capital accumulation by …rms. Therefore, …rms’ individual states of
productivity, capital, and borrowing jointly a¤ect the decision of exit and entry.
For simplicity and tractability, we assume that …rms have to pay a …xed operation
cost in order to remain for production. At the beginning of each period, …rms need to pay
11
This …xed cost of operation
o in terms of output goods before conducting production.
creates a binary decision of endogenous exit. If a …rm does not pay the …xed cost, it has
to exit the industry with zero value. Thus, only …rms with enough pro…tability continue
to produce within a given period.
In addition to the above individual exit decisions by …rms, we assume that the incumbent …rms are also exposed to exogenous risk of exit in each period. This re‡ects the other
possible channels of …rm exit behavior while preventing immediate …rm growth over time,
combined with the …nancial friction. Speci…cally, …rms in our model are subject to a …xed
probability of exogenous exit in each period which is given by d . In any given period,
after …rms decide to remain by paying the …xed cost o , the exogenous exit information
arrives. Unlike the endogenous exit decision, however, …rms are supposed to exit at the
end of the period after production.12 These exiting …rms liquidate all their remaining
earnings and assets, and only the continuing …rms to the next period make intertemporal
10
Hopenhayn (1992) is the seminal work providing the modeling approach of the endogenous entry and
exit with industry dynamics.
11
This happens after both individual and aggregate productivity shocks are realized, but before their
exogenous exit is known. Below, we will discuss the timing of decisions in a period.
12
Notice that the exogenous exit happens at the end of a period, while the endogenous exit at the
beginning of a period. Thus, this di¤erence in the timing of exiting can be also viewed as representing
almost simultaneous exit behavior between two adjacent periods.
9
decisions of investment and borrowing.
Regarding the …rm entry behavior in the model, we assume that there is a …xed
measure of potential entrants in each period. Each potential entrant’s initial capital, k0 ,
is assumed to be randomly drawn from a uniform distribution U (0; k 0 ), while its initial
debt level is a fraction, b , of the initial capital stock re‡ecting the average leverage ratio of
entrants. We assume that the upper bound of the initial capital, k 0 , is also a fraction, k ,
of the e¢ cient level of the incumbents’capital stock at the medium productivity level.13
When the potential entrants decide to enter, they need to pay a …xed entry cost, e , in units
of output goods. In this way, we are able to simply generate the problem of endogenous
entry decision by …rms, and it allows only potential …rms with su¢ cient pro…tability to
actually enter the production sector by paying the entry cost. This endogenous entry
happens at the end of a period, and the actual entrants start producing in the next
period. Each entrant’s initial idiosyncratic productivity, "0 ; is drawn from the ergodic
distribution of , and follows the same Markov transition probability as the incumbents
once the entry is complete.
3.1.3
Timing and Firm Distribution
Before setting up the …rm’s problem in detail, we illustrate the timing of exogenous
shocks and decisions made by …rms in a given period of the model. This is because of the
complexity of our model environment involving the …rm entry and exit.
At the beginning of a period, a …rm is identi…ed by its individual state vector (k; b; );
the predetermined capital, k 2 K R+ , the amount of debt carried from the previous
period, b 2 B R, and the current period idiosyncratic productivity level, 2 E. We
summarize the distribution of …rms in the model by a probability measure (k; b; ) which
is de…ned on a Borel algebra S K B E. For simplicity, we use s (z; ) to denote
the aggregate exogenous state with its transition probability slm 0. Then the aggregate
state of the model is given by (s; ), and the evolution of follows a mapping such that
0
= (s; ).
After the exogenous shocks are realized on s and , a …rm decides whether to remain
in the industry by paying the …xed cost, o . If it decides to exit, the …rm immediately
disappears. If remaining, on the other hand, it then realizes its exogenous exit status at
the end of the period. Given the aggregate and individual states, (k; b; ; s; ), the …rm
13
This e¢ cient level of capital is derived without considering the collateral constraints as will be
discussed later. However, the value of k is set to represent the relatively small size of entrants than that
of incumbents.
10
maximizes the sum of current and future expected discounted dividend payments. After
production, the …rm pays its wage bill and its existing debt b is cleared. Conditional on its
survival to the next period, the …rm conducts the intertemporal decisions of investment;
i, and borrowing, b0 , while deciding the current period dividends D to its shareholders.
The capital accumulation of each …rm is standard, i = k 0 (1 )k. Markets are perfectly
competitive, so …rms take the wage rate w(s; ) and the discounted debt price q(s; ) as
given.
3.1.4
Firm’s Problem
We use di¤erent notations of a …rm’s value depending on the timing of exit and entry
described above. Let v0 (k; b; ; s; ) be the value of a …rm with (k; b; ) at the beginning of
the current period before making the decision of paying the operation cost. Accordingly,
de…ne v1 (k; b; ; s; ) as the value of a remaining …rm after the endogenous exit decision
within the period, but before the exit status is known. Finally, if a …rm is surviving to
the next period, its within-the-period value is given by v(k; b; ; s; ). Then the …rm’s
optimization problem can be recursively de…ned by using v0 , v1 , and v.
(1)
v0 (k; b; i ; sl ; ) = maxf0; v1 (k; b; i ; sl ; )g
v1 (k; b; i ; sl ; ) =
d
+(1
max [zl i F (k; n)
n
d)
w(sl ; )n + (1
)k
b
o]
(2)
v(k; b; i ; sl ; )
With the current realizations of (s; ) at the beginning of the period, the …rm makes
a binary decision of exit over the values of zero (endogenous exit) and v1 (k; b; i ; sl ; )
(continue) in (1) above. After paying o , the …rm takes expectations over the possibility
of exogenous exit in (2). In case of exogenous exit, the …rm still maximizes its liquidation
value at the end of the period without considering the dynamic decisions. The value of
…rms surviving to the next period is then given by v(k; b; ; s; ) as below.
11
v(k; b; i ; sl ; ) =
max
n;k0 ;b0 ;D
"
D+
subject to
0
s
lm dm (sl ;
m=1
D = zl i F (k; n)
b0
0
Ns
X
)
N
X
ij v0 (k
0
0
0
; b ; j ; sm ; )
j=1
w(sl ; )n + (1
)k
#
b
o
(3)
k 0 + q(sl ; )b0
k0
=
(sl ; )
In (3), the continuing …rm optimally chooses its labor demand n, future capital k 0 , and
new debt level b0 to maximize the sum of the …rm’s current dividend payments D and the
beginning-of-the-period value v0 (k 0 ; b0 ; j ; sm ; 0 ) at the future period. The current period
dividends are the residual in the …rm’s budget constraint, and we restrict the dividends
to be non-negative. The …rm’s future value is discounted by using the stochastic discount
factor dm (sl ; ). The …nancial friction of the …rm is included as a collateral constraint.
ve = maxf0;
e+
Ns
X
s
lm dm (sl ;
)v0 (k0 ; b0 ; 0 ; sm ; 0 )g
(4)
m=1
Lastly, we de…ne the value of the potential entrants by ve . In (4) above, a potential
entrant with its initial capital stock, k0 , and the initial debt level, b0 , makes a binary
decision of entry by whether paying the …xed entry cost e . Once it enters, the entering
…rm starts producing in the next period and its future value is given by v0 (k0 ; b0 ; 0 ; sm ; 0
with its individual productivity 0 .
3.2
Households and Equilibrium
Representative Household We assume that there is a unit measure of identical
households in the economy. In each period, the households earn their labor income by
supplying a fraction of the time endowment. The period utility function is given by
U (C; 1 N ), and they discount the future utility by the subjective discount factor, 2
(0; 1). The representative household holds a comprehensive portfolio of assets; one period
…rm shares with measure and non-contingent discount bonds . The representative
household maximizes its lifetime expected discounted utility by choosing the quantities
of aggregate consumption demand, C h , and labor supply, N h , while adjusting the asset
portfolio in each period. The household value is de…ned as below.
12
h
V ( ; ; sl ; ) =
max
C h ;N h ; 0 ;
0
"
h
U (C ; 1
0
C + q(sl ; )
+
Z
1 (k
S
h
w(sl ; )N +
0
N )+
s
h 0
lm V ( ;
0
0
+
Z
0
; b0 ; 0 ; sl ; ) 0 (d[k
0 (k; b;
; sl ; ) (d[k
#
(5)
; sm ; )
m=1
subject to
h
Ns
X
h
b
])
b
])
S
= (sl ; )
We apply the conventional notation for the price of …rm shares regarding dividends.
In (5), 1 (k 0 ; b0 ; 0 ; sl ; ) denotes the ex-dividend prices of …rm shares, and 0 (k; b; ; sl ; )
is the dividend-inclusive values from the current shareholding. Let ( ; ; s; ) be the
household’s decision for bond holding, and h (k 0 ; b0 ; 0 ; ; ; s; ) be the new choice of …rm
shareholding with future state (k 0 ; b0 ; 0 ).
Equilibrium and Prices We de…ne a recursive competitive equilibrium (RCE) as
follows. For convenience, we denote the distribution of producting …rms as p , while the
measure of entrants as e and the measure of endogenously exiting …rms as ex :
s
A RCE is a set of functions: prices w; q; (dm )N
m=1 ;
and values v0 ; v; V h
, quantities N; K; B; D; C h ; N h ;
0; 1
h;
that solve the optimization problems, clear each market, and the asso-
ciated policy functions are consistent with the aggregate law of motion as in the following
conditions.
1. v0 , v1 , and v respectively solve Equation (1) to (3), and (N; K; B; D) are the associated
policy functions for …rms.
2. V h solves Equation (5), and (C h ; N h ;
h;
h)
are the associated policy functions for house-
holds.
3. The labor market clears, N h =
4. The goods market clears.
Ch =
R
S N (k;
p (d[k
; s; )
Z h
z F (k; N ) (1
(1
d ) K(k; b; ; s; )
S
Z
Z
e
+
(k0
(d[k0 b0 ])
k
e)
U (0;k0 )
S
13
b
)k +
ex
(d[k
]).
d (1
b
)k
])
i
p
(d[k
b
])
h
,
5. The law of motion for the …rm distribution is consistent with the value functions (v0 ; ve )
and the measures (
p;
the mapping from
to
e;
ex ),
0
with
and with the associated policy functions, where
d,
de…nes
K(k; b; ; s; ), and B(k; b; ; s; ).
As noted by Khan and Thomas (2013), it is convenient to modify the …rm’s values
(v0 ; v1 ; v; ve ) by using the equilibrium prices at the market clearing condition in the above.
Let C and N be the market clearing quantities for aggregate consumption and labor
supply from the perspective of households. Then the equilibrium value of output goods
can be expressed by using the marginal utility of consumption at C, D1 U (C; 1 N ).
Likewise, the real wage w(s; ) is equal to the marginal rate of substitution between
leisure and consumption. Next, the inverse of the discounted bond price, q 1 , equals the
expected gross real interest rate. Lastly, the stochastic discount factor dm (sl ; ) is equal to
the household’s intertemporal marginal rate of substitution across states. The following
equations summarize the equilibrium prices in terms of the marginal utilities.
D2 U (C; 1 N )
D1 U (C; 1 N )
PNs s
0
m=1 lm D1 U (Cm ; 1
q(s; ) =
D1 U (C; 1 N )
0
0
D1 U (Cm
; 1 Nm
)
dm (s; ) =
D1 U (C; 1 N )
w(s; ) =
0
Nm
)
By using the above price notations, we solve the equilibrium allocations directly from
the …rm’s problem in a consistent manner with the households’ optimal decisions. Let
p(s; ) = D1 U (C; 1 N ) be the marginal utility of consumption at C. Then …rms value
their current output and dividends by p(s; ), and the value functions v0 , v1 , v, and
ve can be re-written in terms of p(s; ) without carrying the stochastic discount factor.
In particular, we de…ne the new values V0 , V1 , V , and Ve by respectively multiplying
p(s; ) to the original value fucntions. Then the following equations summarize the …rm’s
problem.
(6)
V0 (k; b; i ; sl ; ) = maxf0; V1 (k; b; i ; sl ; )g
V1 (k; b; i ; sl ; ) =
d
+(1
max p(sl ; ) [zl i F (k; n)
n
d)
V (k; b; i ; sl ; )
14
w(sl ; )n + (1
)k
b
o]
(7)
V (k; b; i ; sl ; ) =
max
n;k0 ;b0 ;D
"
p(sl ; )D +
D = zl i F (k; n)
b0
0
s
0 0
lm ij V0 (k ; b ; j ; sm ;
m=1 j=1
subject to
0
Ns X
N
X
w(sl ; )n + (1
)k
b
o
0
#
)
(8)
k 0 + q(sl ; )b0
k0
=
(sl ; )
Ve = maxf0; p(sl ; )
e
+
Ns
X
s
lm V0 (k0 ; b0 ; 0 ; sm ;
0
)g
(9)
m=1
For the remainder of this paper, we occasionally suppress the aggregate state (s; ) in
the price functions and the decision rules for notational simplicity.
3.3
Firm Types
To characterize the …rm-level decision rules in the model, it is convenient to …rst de…ne
a subset of …rms in the distribution whose decisions are not a¤ected by the collateral
constraints in any possible individual state. We follow the approach in Khan and Thomas
(2013) to distinguish …rm types in our model economy. Note that this distinction is solely
for deriving all the intertemporal decisions conducted by …rms, and that a …rm may alter
between di¤erent types depending on its individual state across periods. We de…ne a …rm
as unconstrained when it has already accumulated su¢ cient wealth such that it never
worries about facing a binding collateral constraint in any possible future state. In this
case, the unconstrained …rm’s all multipliers on its borrowing constraints become zero,
and the …rm is indi¤erent between paying dividends and internal saving. On the other
hand, the rest of …rms in the distribution is de…ned as constrained. Constrained …rms
may or may not experience binding borrowing constraints in the current period, and they
put non-zero probability of having a binding constraint in the future. These constrained
…rms then choose to pay zero dividends in the current period, because the shadow value
of retained earnings is greater than that of paying dividends.
In this subsection, we begin with deriving the decision rules of the unconstrained …rms.
We then characterize the entire decision rules in the next subsection, after reducing …rms’
state-space. Because we do not assume any explicit adjustment friction in labor demand,
…rms are allowed to optimally hire their desired labor. Thus, all …rms with the same (k; )
15
choose n = N w (k; ; s; ) which solves the static labor condition, z D2 F (k; n) = w(s; ).
Next, we consider the choice of future capital stock k 0 by the unconstrained …rms.
The collateral constraint is irrelevant for this type of …rms, so we can easily derive their
optimal level of k 0 = K w ( ; s; ) as follows. Let w (k; ; s; ) z F (k; N w ) wN w be the
current earnings of a …rm with its optimal labor hiring N w . Given the Markov property
for the assumed stochastic processes and the lack of any capital adjustment cost, K w is
the solution to the following problem.
max
0
k
"
p(sl ; )k 0 +
Ns
X
s
lm p(sm ;
0
m=1
)
N
X
ij
(
w
(k 0 ; j ; sm ; 0 ) + (1
j=1
#
)k 0 )
With the policy functions N w and K w , we make use of the de…nition of an unconstrained …rm to derive the optimal borrowing (or, saving) policy for b0 . The minimum
savings policy b0 = B w ( ; s; ) is de…ned recursively by the two equations below.
B w ( i ; sl ; ) =
min
(sm ; j )j;m
e (K w ( i ; sl ; ); j ; sm ; 0 )
B
w
e
B(k;
i ; sl ; ) = zl i F (k; N )
wN w
o
+ (1
)k
(10)
K w ( i)
(11)
+q min fB w ( i ; sl ; ); K w ( i )g
e w ; j ; sm ; 0 ) is the maximum level of debt (or the minimum level of saving) that an
B(K
unconstrained …rm can hold at the beginning of the next period in which the exogenous
states (sm ; j ) are realized. Having chosen the unconstrained choice of capital K w at the
current period, the …rm will remain unconstrained by de…nition. The minimum savings
policy, B w , ensures that the …rm’s debt never exceeds this threshold level, given all possible
e can be retrieved by using B w
realizations of (s; ). Moreover, the threshold function B
and K w at the current period state with (k; i ; sl ; ). Notice that the minimum operator
at the bottom of (11) re‡ects the collateral constraint in the …rm’s problem, so that B w
truly is the optimal borrowing policy for these …rms. B w in (10) again is de…ned to have
the unconstrained …rms una¤ected by the constraint over any future path of (s; ).
3.4
Cash-on-hand and Decision Rules
The incumbent …rm’s problem in (8) is a challenging object because of the occasionally
binding constraints. In addition, a …rm’s individual state vector includes two endogenous
16
variables (k; b) with their continuous supports, in addition to the idiosyncratic productivity . Following the approach in Jo (2015), we collapse the two continuous individual state
variables (k; b) into a newly de…ned variable called cash-on-hand. This method does not
alter the equilibrium allocation of the model because the new individual state variable is
a su¢ cient statistic for the information contained in k and b. By reducing the state-space,
the …rm-level decision rules can be fully identi…ed depending on the level of cash-on-hand.
First, de…ne the cash-on-hand, m(k; b; ), of a …rm with (k; b; ) by using the optimal
labor demand policy N w .
z F (k; N w )
m(k; b; ; s; )
wN w + (1
)k
b
The cash-on-hand m itself is a function of (k; b; ), and represents the …rm’s net worth
after production.14 Notice that the decisions of k 0 and b0 , along with the realization of
future and s, immediately determine the level of future cash-on-hand, m(k 0 ; b0 ; 0 ; s0 ; 0 ).
By using this new state variable, We transform the incumbent …rm’s problem in (7) - (8)
and the potential entrant’s problem in (6), by de…ning the corresponding values W0 , W1
and W as follows.
W0 (m; i ; sl ; ) = maxf0; W1 (m; i ; sl ; )g
(12)
W1 (m; i ; sl ; ) =
W (m; i ; sl ; ) =
d
max
m0 ;k0 ;b0 ;D
p (m
"
o)
Ns X
N
X
pD +
D=m
b0
0
m0jm
d)
o
(13)
W (m; i ; sl ; )
s
0
lm ij W0 (mjm ; j ; sm ;
m=1 j=1
subject to
0
+ (1
0
#
)
(14)
k 0 + q(sl ; )b0
k0
=
(sl ; )
m(k 0 ; b0 ; j ; sm ; 0 )
= zm j F (k 0 ; N w (k 0 ; j ))
w0 N w (k 0 ; "j ) + (1
)k 0
b0
Given the unconstrained policies K w and B w solved at each realization, we can
de…ne a threshold level of m to exactly distinguish the unconstrained …rms out of the
14
We can also de…ne m to include the period operation cost, but this does not make any di¤erence in
deriving the decision rules.
17
…rm distribution . From the …rm’s budget constraint in (14), the dividend payments
by an unconstrained …rm is given by Dw = m
K w + qB w . This implies that
o
the …rm’s cash-on-hand is greater or equal to a certain threshold level m(
e ; s; )
o +
K w ( ; s; ) qB w ( ; s; ). This threshold function also depends on the aggregate state
and the individual productivity . Therefore, any …rm with m(k; b; ; s; ) < m(
e ; s; )
can be identi…ed as constrained.
Before we analyze the decisions by the constrained …rms, recall that, in any given
period, some constrained …rms experience binding collateral constraints while the others
do not. We call the constrained …rms without currently binding borrowing constraints as
Type-1 …rms, while the other with binding constraints as Type-2 …rms. Note that Type-1
…rms still can adopt the unconstrained capital policy K w , but not B w . Their debt policy
can be easily determined by the zero-dividend policy after substituting k 0 = K w in the
budget constraint. On the other hand, Type-2 …rms can only invest to the extent their
borrowing limits allow. This constrained choice of future capital is derived from the level
of cash-on-hand, m, held by this type of …rms. Speci…cally, D = 0 implies that Type-2
…rms’ decisions on k 0 and b0 must be feasible, given their available cash-on-hand after
paying the operation cost. From the binding collateral constraint and m, we de…ne the
upper bound of capital choice by Type-2 …rms as K(m) m1 q o . Firms with more cashon-hand, therefore, can relax this upper bound until they can choose the unconstrained
policy K w . Notice also that the upper bound, K(m), approaches to in…nity as the …nancial
parameter becomes closer to 1=q, which is the case of a perfect credit market. In sum,
the decision rules of k 0 and b0 by each …rm type (unconstrained, Type-1, and Type-2) with
(m; ) are as follows.
Firms with m
m(
e ; s; ) are unconstrained, and adopt K w ( ) and B w ( ).
For constrained …rms with m < m(
e ; s; ), the upper bound for future capital choice is
K(m)
m o
.
1 q
– Firms with K w ( )
m).
o
K(m) are Type-1, and adopt k 0 = K w ( ) and b0 = 1q (K w ( )+
– Firms with K w ( ) > K(m) are Type-2, and adopt k 0 = K(m) and b0 = 1q (K(m) +
m):
o
18
4
Model Parameters
We parameterize the model to quantitatively investigate the implications of macroeconomic policies aimed at speci…c groups of …rm size and age. We set the model parameter
values to jointly match the key macroeconomic moments and the rich heterogeneity at
…rm-level observed in the U.S. data.
4.1
Calibration
The period utility function of the representative household is standard as in the literature. We speci…cally assume the functional form of indivisible labor following Rogerson
(1988), U (C; 1 N ) = log C + (1 N ). The decreasing-returns-to-scale (DRS) production technology is in Cobb-Douglas form, F (k; n) = k n with ; > 0 and + < 1.
The model is annual, and we calibrate the model parameters to be consistent with the
U.S. data moments at both macroeconomic- and micro-levels. Most parameter values are
jointly determined because the size and age distributions of …rms in our model economy
respond sensitively with any changes in aggregate-level parameters. Table # summarizes
the calibrated parameter values.
The value of the households subjective discount factor, , is set to imply the long-run
real interest rate of 4 percent. The preference parameter of labor disutility, , is set to
match the average hours of worked at 0.33. The value of annual depreciation rate, , is
chosen to match the average aggregate investment-to-capital ratio of the postwar U.S.
economy. The curvature of capital input in the production function, , is set to imply the
average capital-to-output ratio of 2.3. The other production function parameter, , is set
to imply the average share of labor income of 0.6 following Cooley and Prescott (1995).
For the data series of aggregate investment, output, and private capital stock, we use the
Fixed Asset Tables and National Income and Product Accounts (NIPA) in the Bureau of
Economic Analysis (BEA).
We jointly determine the rest of model parameter values, ( ; o ; e ; d ; m ; M ; ; ),
to match the aggregate debt-to-asset ratio, the total exit rate of …rms, the size and age
distribution of …rms.15 In addition, we control the distribution of producing …rms to have
a unit measure at the steady state of the model. The leverage parameter in the collateral
constraint, , is set to be 0.9 to be consistent with the average debt-to-asset ratio of the
non-farm, non-…nancial businesses in the Flow of Funds from 1954 to 2007. The period
15
To our best knowledge, this paper is one of the …rst attempt to match both …rm size and age
distributions endogenously from a model of heterogeneous …rms with …nancial frictions.
19
operating cost, o , and the …xed entry cost, e , are set to imply about 10 percent of total
exit rate, together with the exogenous exit rate, d .16 We use the data tables from the
Business Dynamics Statistics (BDS) to calculate the average total exit rates of the private
…rms in the U.S.17 BDS also reports the …rm size and age distributions in detailed size/age
groups, with which we match in 6 employment size bins and 11 age bins.
We assume that the idiosyncratic productivity, , is drawn from a time-invariant distribution G( ; m ; M ; ) which takes the form of a bounded Pareto distribution.18 In each
period, a …rm in the model retrains its individual productivity level with a …xed probability . We set = 0:75 to be consistent with the evidence on the average persistence
of …rm-level productivity in the data. (Foster, Haltiwanger, and Syverson (2008)) The
bounds of support, ( m ; M ), and the curvature parameter, , are chosen to have both
the employment share and the population share at each …rm size bin are aligned with the
average values reported in BDS from 1977 to 2007. We discretize the values of by N in
our numerical solution of the model.
Table 1 : Calibration
Model Parameters
Value
d
o
e
N
m
M
Description and Target Value
0.96
0.60
2.408
0.069
0.272
Annual gross real interest rate
Labor share of income
Average hours worked
Investmet to capital ratio (BEA)
Capital to output ratio (BEA)
0.90
0.0874
0.0138
0.1522
Debt to Asset ratio (Flow of Funds)
Average …rm exit rate (BDS)
Steady state measure of …rms
11
0.75
0.431
1.150
4.16
Bounded Pareto
16
Model
1.04
0.60
0.33
0.069
2.30
0.35
0.10-0.13
1.0
1=q = 1:04
N = 0:334
I=K = 0:069
K=Y = 2:24
D=A = 0:34
Exit
R rate = 0:101
d = 1:00
These parameters also jointly shape the …rm age distribution of the model economy.
BDS database is based on the Census data, and covers more than 90 percent of the U.S. private
employment starting from 1977.
18
Jo (2015) shows that the conventional log AR(1) identi…cation of is not enough to generate the
highly-skewed …rm size distribution as in the data.
17
20
5
5.1
Results
Steady State
Firm Heterogeneity and Decisions We begin with the stationary distribution of
…rms in the model. Figure 5 shows the entire …rm distribution over capital and debt
level (k; b). It displays all three types of …rms that we considered in the previous section;
unconstrained, Type-1, and Type-2 …rms. The unconstrained …rms are located in the area
on negative debt starting from the front mass near zero capital stock in the …gure. Their
levels of b re‡ect the minimum savings policy B w ( ). They are apart from the body of
the distribution with most …rms, and the connected areas toward the unconstrained …rms
indicate Type-1 …rms accumulating assets to be unconstrained. About 50 percent of all
…rms in our model are distinguished as Type-2 with currently under binding constraints.
They are mainly on the diagonal straight line that represents the inverse relationship
between these …rms’capital and …nancial savings.
The above …rm types are more easily distinguishable when we look at the decision
rules on capital k 0 , debt b0 , and dividends D, based on a …rm’s the cash-on-hand m.
Figure 6 shows the decision rules of the …rms with the same median value of . The level
of cash-on-hand, m(k; b; ) is denoted on the horizontal axis, and we add the two vertical
lines to distinguish each …rm type where the one near m = 7 represents the threshold
of being unconstrained, m(
e ). When a …rm has accumulated enough wealth such that
m
m(
e ), it is considered as unconstrained. The …rm then adopts the unconstrained
choice of capital K w ( ) as well as its minimum savings policy B w ( ), and pays positive
dividends. Constrained …rms with m less than the threshold value, on the other hand,
follow the zero-dividend policy to accumulate their wealth toward being unconstrained, as
discussed earlier. Type-1 …rms in the middle part of the …gure, between the two vertical
lines, can adopt the optimal level of capital, and gradually save more as their m increase.
Lastly, Type-2 …rms with small m are only able to invest until their borrowing limits
allow.
We compare the model’s …rm size distribution with that observed in data. To generate
the corresponding …rm distribution from the model, we simulate a large panel of …rms
at the steady state. Both of the empirical and model-generated size distributions are
reported in Table 2.
The …rst column of the table shows the average share of …rms in each size group
between 2003 and 2011. The empirical distribution exhibits the well-known shape, where
more than 75 percent of …rms hire less than 10 employees in a given year. The share
21
Table 2
Firm Size Distribution
Employment Share
Population Share
Employees
Data
Model
Data
Model
1 to 4
5 to 19
20 to 99
100 to 499
500 to 2499
2500+
0.0584
0.1455
0.1814
0.1395
0.1179
0.3573
0.0584
0.1455
0.1814
0.1395
0.1179
0.3573
0.5506
0.3342
0.0964
0.0153
0.0026
0.0009
0.5310
0.3369
0.0913
0.0226
0.0093
0.0089
of …rms in each size group decreases in the number of employees, and only 4.4 percent
in the database can be considered as relatively large with more than 50 employees. The
steady state size distribution of the model is on the next column of the table. With
our asymmetric treatment of using a Pareto distribution, the model matches the BDS
distribution relatively well, with small …rms are slightly more concentrated on the lower
tail. This realistic …rm size distribution comes from the combination of the model’s
…nancial friction with a fat-tailed distribution of …rm-level productivities. It becomes
clear when we re-calibrate our model after introducing the standard identi…cation of
process with a log-normal distribution as in the literature. In the last column of Table 2,
the size distribution from the model is solely driven by the borrowing constraint. With
the symmetric values of , we now have relatively more shares of mid-sized …rms in the
distribution. Figure 7 also plots the model generated …rm size distribution and and the
empirical couterpart.
The model also well captures the age distribution of …rms in BDS. Especially, endogenous entry and exit helps us to capture the disproportionately high exit rates among young
…rms.19 Table 3 reports the age distribution of …rms in our model and in data.20 Figure
7 also plots the model generated …rm age distribution and and the empirical couterpart.
19
Since the exogenous probability of exit commonly applies to all …rms regardless of size and age in
the model, we can indirectly identify that most of the exiting …rms are small, once a skewed …rm size
distribution is generated.
20
Due to the availability of …rm age data, we exclude the censored …rms in the data and the empirical
age distribution is also the average between 2003 and 2011.
22
Table 3
5.2
Age group
BDS database
Model
(Age)
0
1
2
3
4
5
6 to 10
11 to 15
(% shares)
12.36
9.50
8.06
6.97
6.13
5.48
19.39
13.02
13.67
10.56
8.80
7.59
6.56
5.72
19.86
11.04
Policy simulation exercises
The policy experiment that we consider includes several policies that attempt to relax
collateral constraints faced by the di¤erent characteristic of …rms. Speci…cally, we simulate
50,000 …rms for 1,000 years under the di¤erent policy regime and compute the average
life cycle patterns of …rms. Figure 3 shows several policy simulation results and compares
them to the one from the frictionless economy without collateral constraints. Here, we
plot the …rm life-cycle patterns of capital stock accumulation, employment and …nancing
behavior for three policy options. Age-dependent policy is to relax collateral constraints
by 10 percent for …rms with their age between 0 and 2. Size-dependent policy is the same
but for …rms with their employment size below 250 employees. Productivity-dependent
policy is also relaxing collateral constraints by 10 percent for …rms with their productivity
levels below the average of the entire population. Age-, size-, and productivity-dependent
policy counterfactuals ended up with supporting 13,804 …rms, 29,399 …rms, 35,202 …rms,
respectively. Overall, despite the smallest number of …rms that are subsided, the agedependent policy is the most productivity enhancing policy choice, increasing capital
stock by 0.12 percent, output by 0.09 percent, employment by 0.09 percent, and TFP by
0.04 percent. On other hand, the productivity-dependent policy is the least e¤ective in
increasing aggregate TFP. The bottom-right panel summarizes the impact of each policy
on the aggregate economy.
As in the bottom-left panel, the age-dependent policy enables …rms with their age 0,
1 and 2 borrow more with the same collateral value, and hence these …rms can increase
their capital stock compared to the baseline case as in the top-left panel of the …gure.
23
Figure 3
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0
0.1
0
3
6
9
12
15
18
1
0
3
6
9
12
15
18
0.15
0.5
0.1
0
0.05
-0.5
-1
0
0
3
6
9
12
15
18
capital
output
employment
TFP
Age
Notes: The …gure plots the life cycle patterns of …rms under the di¤erent policies. Each policy
counterfactual exercise simulates 50,000 …rms for 1,000 years and discard the …rst 100 years to
eliminate the e¤ects of initial conditions of the model economy. The upper left panel gives capital
stock per operating …rm versus …rm age. The upper right panel shows employment per operating
…rm versus …rm age. The lower left panel plots net borrowing per operating …rm versus …rm age.
The lower right panel shows percentage changes in aggregate capital stock, output, employment and
TFP for each di¤erent policy experiment.
24
Given that there needs to be one-period to build capital stock, the di¤erence between the
age-dependent policy case and the baseline one in capital stock accumulation appear for
the …rm age 1, 2 and 3. At the age 3, collateral constraints are set back to the baseline
level and therefore their capital stock accumulation pattern exhibits the peak at the age
3 and gradually reverts to the level of the baseline case. For the size- and productivitydependent policy cases, the policy support are provided regardless of the …rm age. Thus,
the policy impact can be seen more evenly across the di¤erent …rm age.
The importance of the …rm age for a targeted policy
Why is an age-dependent policy the most e¤ective in increasing the aggregate TFP?
The number of the targeted …rms is the smallest for the age-dependent policy amongst
other policy alternatives considered here. Why are a size-dependent and a productivitydependent policy less e¢ cient? The main reason is that the largest potential gains can
be obtained by undoing misallocation of resources among young, but productive …rms. In
this model, as in the other standard models of …rm dynamics, the …rm size is correlated
with the …rm e¢ ciency due to the decreasing returns to scale production function. There,
misallocation means that capital stock for each …rm is too small relative to the e¢ cient
level that is consistent with interest rates and the …rm level productivities. The severity of
capital misallocation is potentially larger for more productive …rms as their e¢ cient level
of capital stock is relatively large but collateral constraints adversely distort their capital
stock. That is, the gap between the actual and the e¢ cient level of capital stock tend to
be larger for productive …rms. If we undo collateral constraints for productive but small
…rms the capital stock increases and productivity gains are large, whilst the capital stock
increases less and thus the productivity gains are smaller if we relax collateral constraints
for less productive or small …rms.
What makes small …rms small? In our model environment, it could be due to the low
productivity or collateral constraints. If the latter is the case, then removing distortions for
small …rms are powerful, but if the former is the case it is not a good idea to put resources
there. Small …rms include small but old …rms that are more likely to be e¢ cient. By the
time …rms become grownups, they tend to have achieved the e¢ cient level of capital stock
by self-…nancing. This is why a policy that is geared towards to young …rms is more likely
to achieve the most e¢ cient reallocation of resources.
25
6
Concluding Remarks
Capital misallocation due to the presence of …nancial frictions is at the center of our
discussion, and our main …nding depends on where capital misallocation is pervasive across
the …rm distribution. We showed suggestive evidence on the early stage of the …rm lifecycle, businesses grow by hiring employees and accumulating equity and deleveraging, but
their total asset size, in particular, tangible assets, does not grow so much. This implies
the importance of intangible investment and its need for external …nance alongside with
the need for hiring to the growth of businesses.
The accumulation of intangible assets such as new technologies with related know-hows
and new markets with their customer bases have been pointed out as a key determinant
of …rm growth (Atkeson and Kehoe (2005), Foster, Haltiwanger, and Syverson (2013),
and Hsieh and Klenow (2014)). How could industrial policies with a focus on …nancial
hurdles for small and young …rms foster …rm growth and macroeconomic performance?
Our future work aims to investigate the impact of targeted industrial policies on intangible
asset investment at the …rm-level and aggregate productivity in theoretical and empirical
detail.
26
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29
Figure 4 : UK Firm Size Distribution
0.5
0.45
Population share
Employment share
0.4
0.35
percent
0.3
0.25
0.2
0.15
0.1
0.05
00
10
to
0
99
99
50
00
to
24
00
25
00
to
to
10
0
50
49
99
9
99
9
49
9
0
25
10
0
to
to
24
99
to
50
50
to
9
10
to
1
99
0
E mployment S ize
Notes: The sample contains UK …rms with at least 1 employee. Data is taken from Orbis Global
for year 2014 and the resulting sample is 83,334 …rms.
30
Figure 5 : Stationary Distribution of Firms
0.02
0.015
0.01
0.005
0
50
60
0
50
40
30
-50
20
10
-100
0
Notes: The …gure shows the stationary distribution of …rms over capital stock and debt.
31
Figure 6
Decision Rules at
median
2
1.5
capital k
debt b >0, saving b <0
dividends D
1
0.5
0
-0.5
-1
-1.5
-2
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
4
cash-on-hand m(k,b, )
Notes: The …gure displays the decision rules for capital, debt, and dividends as functions of the
cash-on-hand for the median value of idiosyncratic productivity.
32
Figure 7 : Comparison of Model Firm Size Distribution
0.6
BDS Data
Model
0.5
0.4
0.3
0.2
0.1
Employ ment Size
Notes: The data is taken from the BDS, averaging between 1977 and 2014.
33
+
e0
2m
010
m
e0
-2
49
9
49
9
9
-9
20
519
14
0
Figure 8 : Comparison of Model Firm Size Distribution
0.2
BDS Data
Model
0.18
0.16
0.14
0.12
0.1
0.08
0.06
0.04
0.02
Age
Notes: The data is taken from the BDS, averaging between 1977 and 2014.
34
+
26
5
-2
21
16
-2
0
5
-1
11
610
5
4
3
2
1
0
0
Figure 9
1
0.5
0.8
0.4
0.6
0.3
0.4
0.2
0.2
0
0.1
0
3
6
9
12
15
18
0
3
6
9
12
15
18
0.25
1
0.2
0.5
0.15
0
0.1
-0.5
0.05
-1
0
0
3
6
9
12
15
18
capital
output
employment
TFP
Age
Notes: The …gure plots the life cycle patterns of …rms under the di¤erent policies. Each policy
counterfactual exercise simulates 50,000 …rms for 1,000 years and discard the …rst 100 years to
eliminate the e¤ects of initial conditions of the model economy. The upper left panel gives capital
stock per operating …rm versus …rm age. The upper right panel shows employment per operating
…rm versus …rm age. The lower left panel plots net borrowing per operating …rm versus …rm age.
The lower right panel shows percentage changes in aggregate capital stock, output, employment and
TFP for each di¤erent policy experiment.
35