Lending E¢ ciency Shocks
Kaiji Chen
Emory University
Tao Zha
FRB Atlanta, Emory Univ. and NBER
June, 2014
Abstract
This paper develops a theory in which shocks to the e¢ ciency of information acquisition
by …nancial intermediation translate into business cycle ‡uctuations via capital reallocation. In our theory, under costly veri…cation, the bank chooses to only monitor the returns
of those entrepreneurs with insu¢ cient net worth. This distorts the existing capital allocation among entrepreneurs of di¤erent sizes. A crucial ingredient of the model is that the
outcome of monitoring is random and depends on both the e¢ ciency of monitoring and
the resources devoted to policing the returns of a project. As a consequence, a negative
shock to monitoring e¢ ciency forces bank to increase monitoring intensity and reduce the
loan toward small entrepreneurs. This results in an increase in productivity dispersion and
a recession. Using the COMPUSTAT dataset, we …nd a signi…cant countercyclical pattern
for the relative capital productivity of small to large …rms, and a procyclical capital allocation between them. Such an empirical observation distinguishes the lending e¢ ciency
shocks from other aggregate shocks as the sources of business cycles.
JEL Classi…cation: E32, G34
Keywords: Lending e¢ ciency shocks, …nancial frictions, capital reallocation, TFP ‡uctuations.
Department of Economics, Emory University Atlanta, GA 30322.
kaiji.chen@emory.edu.
Phone: (404) 727-2944.
Email:
1
Introduction
The recent …nancial crisis suggests that shocks to …nancial intermediation might be an important driver of economic ‡uctuations. In particular, a disturbance in the e¢ ciency of information acquisition by banks, say under a volatile economic environment, may force them to
increase their business lending standards. The mechanism through which such disturbances
are transmitted into business cycles is, however, an open issue. One potential channel is resource reallocation across …rms of di¤erent sizes. Intuitively, …rms with di¤erent sizes might
be subject to di¤erent degrees of information asymmetry and, thus, …nancial constraints. Such
an asymmetry in …nancing constraints, moreover, is found to be crucial in driving productivity dispersion at the …rm level and its ‡uctuations over business cycles.1 These observations
indicate that variations in …nancial frictions, by reallocating capital at the disaggregate level,
might be important for shocks to …nancial intermediation to translate into aggregate TFP
‡uctuations and, thus, business cycles.
This paper develops a theory for disturbances in the e¢ ciency of …nancial intermediation to
translate into aggregate TFP ‡uctuations via capital reallocation. In our theory, due to costly
veri…cation, the bank chooses to only monitor those entrepreneurs with insu¢ cient net worth.
A key element of the model is that the outcome of monitoring is random and depends on both
the e¢ ciency of monitoring and the resources devoted to policing the returns of a project. As
a consequence, a negative shock to monitoring e¢ ciency (“lending e¢ ciency shocks”hereafter)
forces bank to increase monitoring intensity and reduce the loan toward small entrepreneurs.
This results in an increase in productivity dispersion between entrepreneurs of di¤erent sizes
and a recession. Such countercyclical productivity dispersion is strongly supported by our
evidence using COMPUSTAT data, and helps to distinguish the lending e¢ ciency shocks from
other aggregate shocks as the sources of business cycles.
Our main results are driven by two key model ingredients. The …rst is heterogeneity of
entrepreneurs in terms of net worth. Only entrepreneurs with insu¢ cient net worth, who rely
heavily on bank loans, are subject to bank monitoring. Accordingly, small entrepreneurs are
…nancially constrained, while large ones are not. The second element in our model is shocks
to the e¢ ciency of bank monitoring. Such shocks alter the probability of a bank to detect
malfeasance by entrepreneurs and, therefore, the monitoring intensity of a bank. This, in
turns, varies the tightness of …nancial constraints of the constrained entrepreneurs.
1
See Chen and Song (2013).
1
A combination of these two elements delivers the following mechanism for lending e¢ ciency
shocks to translate into TFP ‡uctuations. A reduction in the e¢ ciency of monitoring triggers
bank to monitor more intensively. This incurs a higher monitoring cost and, thus, a larger
wedge between the marginal product and the user cost of capital. Accordingly, the amount
of bank loan advanced to the constrained entrepreneurs is reduced. Capital, therefore, is
reallocated from constrained to unconstrained projects, leading to a widening of productivity
dispersion between these two types of projects. This increase in the productivity dispersion,
which represents a loss in allocative e¢ ciency, shows up as a fall in aggregate TFP.
Our theory deliver a key testable prediction that distinguishes the lending e¢ ciency shocks
from other shocks. That is, the dispersion of marginal product of capital between …nancially
constrained and unconstrained …rms is countercyclical. While other shocks, such as aggregate
technology shocks and shocks to marginal e¢ ciency of investments, may trigger similar boombust cycles in aggregate variables, in our economy they all exhibit procyclical variations in
the dispersion of marginal product of capital. Intuitively, the marginal cost for …nancially
constrained entrepreneurs involves not only the replacement cost, but also the monitoring
cost. Hence, given monitoring intensity, unconstrained entrepreneurs’ input demand is more
sensitive than constrained ones to those shocks that in‡uence the price of inputs. Accordingly,
those shocks imply a procyclical ratio of capital deployed in …nancially constrained …rms to
that in unconstrained …rms, contrast to the predictions by lending e¢ ciency shocks.
To test the above theoretical implication, we use the COMPUSTAT dataset to estimate the
ratio of capital productivity between constrained to unconstrained …rms. Firms are classi…ed
into constrained and unconstrained groups by asset sizes. We …nd that, on average, the constrained …rms are more productive than the unconstrained in terms of revenue-based capital
productivity. Moreover, consistent with the model prediction, the relative capital productivity
between the two groups has a correlation coe¢ cient with GDP of
0:44. The key factor under-
lying this countercyclical capital productivity dispersion is the procyclical capital allocation
between small and large …rms: the ratio of capital of constrained to unconstrained …rms is
positively correlated with GDP with a correlation coe¢ cient of 0:55. Therefore, our empirical
evidence supports lending e¢ ciency shocks as important drivers of U.S. business cycles.
Our model is closely related to Greenwood, Sanchez and Wang (2010). In both paper, the
probability of detecting malfeasance depends on the e¢ ciency and intensity of bank monitoring. Their paper, however, di¤ers from ours along several dimensions: First, in their paper,
entrepreneurs are shut down from savings and their asymmetric access to bank loans stems
from idiosyncratic productivity uncertainty. By contrast, entrepreneurs in our model are al-
2
lowed to save and subject to the same degree of idiosyncratic uncertainty. This results in a
completely di¤erent mechanism for the heterogeneous access to bank credit by entrepreneurs,
i.e. via their net worth. Second, in their paper, resource reallocation and aggregate productive
e¢ ciency gain occur along the extensive margins, i.e. via entry and exit, a margin empirically
more relevant for aggregate productivity growth over the medium or long run.2 By contrast,
capital reallocation in our model happens along the intensive margin, i.e. between …rms of
di¤erent sizes. Such a capital reallocation pattern and implied variations in productivity dispersion are in line with our evidence. Finally, Greenwood et. al (2010) explore the role of
information production in the …nancial sector in the context of economic development. Our
paper, instead, focus on how disturbances in the e¢ ciency of …nancial intermediation a¤ect
economic ‡uctuations.
Our work is closely related to the emerging literature on uncertainty-driven business cycles.
In particular, following the recent …nancial crisis, researchers point to …nancial frictions as a
transmission mechanism for uncertainty shocks to trigger the business cycles. The …rst attempt
to link uncertainty and …nancial frictions is Christiano, Motto and Rostagno (2010). They
explore the roles of uncertainty shocks in the framework of Bernanke, Gertler and Gilchrist
(1999, “BGG” henceforth). More recently, Gilchrist, Sim and Zakrajšek (2010) extend the
BGG framework to allow uncertainty shocks to a¤ect the capital allocation among investmentgood production …rms, as an ampli…cation mechanism for investment dynamics over business
cycles.3 Alternative frictions for uncertainty shocks to transmit into economic ‡uctuations
include …rm-level factor adjustment costs (see Bloom, Floetotto and Jiamovich, 2009 and
Bachmann and Bayer, 2009) or a combination of entry costs and investment irreversibility
(Sim, 2008). A common feature of the above studies is that variations in the dispersion
of idiosyncratic productivity or time-series volatility of productivity shocks are treated as
exogenous shocks to the economy. By contrast, in our model, we show that various in dispersion
of capital productivity at …rm levels can be itself driven by shocks to …nancial intermediation
sector. Hence, our work contributes to the understanding of the source of uncertainty shocks.
The lending e¢ ciency in our paper is closely related to the empirical literature on bank
monitoring. Mester, Nakamura and Renault (2006) …nd that the information on …rms’transaction account, mainly account receivable and inventory, help …nancial intermediaries monitor
borrowers and predict credit downgrades and loan write-downs. Moreover, the lender intensi2
Foster, Haltiwanger and Krizan (2000, 2002) …nd that new entrants tend to be less productive than surviving
incumbents, but exhibit substantial productivity growth over …ve or ten year horizons.
3
See also, Chugh (2010), which explores the roles of uncertainty shocks in the framework of Carlstorm and
Furest (1997).
3
…es monitoring as loan deteriorate-loan review become lengthier and are more frequent. The
bank’s valuation of account receivable and inventories (or internal collateral) is an important
ingredient in loan monitoring process. This valuation includes subjective discounts (haircuts)
from book values. These hiarcuts provide a comfort level for the lender and re‡ect the liquidity
and quality of accounts receivable and inventories. For example, as account receivable remain
uncollected, their quality may deteriorate. Similarly, the liquidity of inventory may vary the
demand changes. Therefore, the monitoring e¢ ciency, as captured by the precision of haircut
to measure the true quality and liqudity of accounts receivable and inventory, may deteriorate
in a more turbulent economic environment (say due to an increase in demand volatility and
an increase in default rates).
Our empirical evidence on capital allocation echoes previous empirical …ndings on resource
reallocation and productivity dispersion over business cycles. For example, Maksimovic and
Philips (2001) …nd that less productive …rms tend to be sold as prospects of the aggregate
economy improve. Correspondingly, aggregate output and the productivity dispersion across
…rms are found to be negatively correlated (Eisfeldt and Rampini, 2006). Using COMPUSTAT
dataset, Chen and Song (2013) further …nd that the dispersion of marginal (revenue) productivity of capital between constrained and unconstrained …rms are signi…cantly countercyclical.
Our evidence suggests that such counter-cyclicality in productivity dispersion originates from
procyclical capital allocation between small and large …rms.
The paper is organized as follows. Section 2 describes the evidence to motivate our theory
on lending e¢ ciency shocks. Section 3 presents the model environment. Section 4 introduces
the primitive shock and calibrates the benchmark model. In section 5, we report the simulation
results. Section 6 concludes. The Appendix provides proof of various propositions.
2
Evidence
This section establishes the empirical evidence that motivates our theory in the next section.
We …rst describe our data and the measurement of capital productivity. We then introduce
our empirical strategy to estimate the dispersion of capital productivity. After that, we report
the empirical results.
2.1
Data and Measurement
Our dataset consists of quarterly COMPUSTAT data from 1975Q2 to 2012Q3 for publicly
listed …rms, excluding foreign …rms (those with a foreign incorporation code), …nancial …rms
(SIC code 6000-6999) and utilities (SIC codes 4000-4949). The full sample includes 216187
4
observations, for an average of 1511 observations per quarter. The details of data sources and
construction are in the Appendix.
2.1.1
Constructing Firm Groups
The …nance literature provides various proxies for the severity of …nancial constraints a …rm
is subject to. What is mostly relevant to this paper is …rm size. Intuitively, small …rms rely
more heavily on bank credit than large …rms. Moreover, small …rms are more vulnerable to
bank monitoring. As a result, they are more likely to be …nancially constrained.
Therefore, …rms in our sample are classi…ed into two groups in each year, with size as proxy
for being …nancially constrained. The fraction of potentially/likely …nancially constrained …rms
in COMPUSTAT, accordingly to Hadlock and Pierce (2010, Table 1), is 26 percent, on average.
Therefore, in our sample, the group of …nancially constrained …rms includes those in the bottom
size quartile, while the group of unconstrained …rms is composed of those in the remaining size
quartiles.
2.1.2
Measuring Capital Productivity Dispersion
We now turn to the …rm-level productivity measure using COMPUSTAT data. The literature provides various approaches to estimate plant-level productive e¢ ciency (e.g., Olley and
Pakes, 1996 and Levinsohn and Petrin, 2003). These estimations are di¢ cult to apply here
since COMPUSTAT does not report …rm-speci…c wage compensation, nor does COMPUSTAT
have information on value-added. However, COMPUSTAT contains information on operating
income.4 Then, capital productivity (KP henceforth) can be measured by the ratio of Operating Income before Depreciation (OIBDP) to one-quarter-lag net Plant, Property & Equipment
(PPENT). We focus on all …rm-quarter observations with positive operating income before
depreciation and a non-missing value for capital stock.
We next compute the ratio of capital productivity between the two groups (KP ratio henceforth) as a proxy for the corresponding productivity dispersion caused by …nancial frictions.5
2.1.3
Estimating Capital Productivity Dispersion
We then address the empirical strategy of estimating the capital productivity dispersion between …nancially constrained and unconstrained …rms or, more precisely, the relative capital
4
In COMPUSTAT, operating income (before depreciation) is equal to sales minus the cost of goods sold and
selling, general and administrative expenses.
5
Ideally, we should use the ratio of marginal product of capital, which is not directly observable. However,
the ratio of marginal product of capital and the ratio of average product of capital are equal in our model
presented in the next Section.
5
productivity of constrained to unconstrained …rms. For each time t; the KP ratio is estimated
by regressing log of capital productivity, denoted as log KPit ; on a dummy variable, dit ; where
dit equals one for the constrained …rms and zero for the unconstrained.
log KPit = at + bt dit + "it :
(1)
The key coe¢ cient of bt in (1) corresponds to the log of the ratio of average capital productivity
of …nancial constrained …rms to unconstrained …rms. To reduce the in‡uence of outliers, we
Winsorize log KPit at the …rst and ninety-ninth percentiles. Our results hold qualitatively
without Winsorization.
2.1.4
Results
Table 1 reports the summary statistics of estimated exp(bt ), the estimated relative capital
productivity of constrained to unconstrained …rms. The …rst four columns report the timeseries mean, median, minimum and maximum of exp(bt ) between 1975Q2 and 2012Q3. The
estimated bt is statistically signi…cant at one percent throughout the sample years, suggesting
that …nancially constrained …rms are more productive than unconstrained …rms. As shown by
the …rst two columns, the estimated capital productivity of constrained …rms is, on average,
more than 45-percent higher than that of unconstrained …rms.
Estimated b
Table 1. Estimated KP Ratio
mean median max
min ST D
1.474
1.484
1.916 1.088 0.097
mean t-stat
7.00
Figure 1 plots the H-P …ltered estimated bt . The NBER recessions are highlighted with
the shaded bars. It is clear that the dispersion of capital productivity is countercyclical. The
correlation coe¢ cient between the H-P …ltered real GDP and the estimated bt is equal to
0:44. The p-value for testing the hypothesis of no correlation is virtually zero.
[Insert …gure 1 here]
We go a step further to explore the source of countercyclical variations in capital productivity dispersion. Note that the variation in the dispersion of capital productivity between the
two groups can be decomposed into two terms: changes in the ratio of operating income and
changes in the ratio of capital stock between two groups.
log
KPtc
=
KPtu
log
6
OItc
OItu
log
Ktc
Ktu
We adopt similar empirical strategy to estimate the ratio of operating income and the ratio of
capital stock between constrained and unconstrained …rms.
Figure 2 plot the cyclical component of the estimated ratio of capital between the two
groups. Obviously, the ratio of capital stock of constrained to unconstrained group is highly
procyclical. Furthermore, Table 2 shows the contemporaneous correlation of the ratio of capital and GDP is 0:55: By contrast, the ratio of operating income is essentially acyclical and, if
any, correlates positively with GDP. In other words, the countercyclicality of capital productivity dispersion between small and large …rms is entirely explained by the procyclical capital
allocation between the two groups.
[Insert …gure 2 here]
Table 2. Cross Correlation of Capital Productivity and its Components with GDP
GDPt 2 GDPt 1 GDPt GDPt+1 GDPt+2
Ratio KP
0:34
0:41
0:44
0:42
0:35
Ratio of K
Ratio of OI
(0:0000)
(0:0000)
(0:0000)
(0:0000)
0:33
0:49
0:55
0:55
(0:0000)
0:03
(0:7339)
(0:0000)
0:05
(0:5560)
(0:0000)
0:08
(0:3106)
(0:0000)
0:11
(0:1994)
(0:0000)
0:50
(0:0000)
0:13
(0:1162)
In summary, our exercise uncovers three empirical observations that motivate our theory
below: …rst, the capital productivity of …nancially constrained …rms is, on average, higher than
that of unconstrained …rm. Second, the dispersion of capital productivity between …nancially
constrained and unconstrained …rms is signi…cantly countercyclical. Finally, this countercyclicality is essentially driven by procyclical capital allocation between …nancially constrained and
unconstrained …rms.
3
The Benchmark Model
In this section, we describe the environment for the model in a general equilibrium economy.
The economy is inhabited by an in…nitely-lived representative household, a continuum of entrepreneurs and intermediate good producers with unit mass and a representative capital good
producer.
Each period, entrepreneurs purchase intermediate goods to produce the …nal output. There
are two types of entrepreneurs (type-c and type-u) ; di¤ering in their utility discount factors,
which governs entrepreneurs’ net worth at the steady state. A competitive bank exists to
provide working capital loan to entrepreneurs to …nance the purchase of intermediate goods.
All entrepreneurs are subject to idiosyncratic shocks to production technology, which becomes
private information after they are realized. To shut down the extensive margin for resource
7
allocation, we assume that the share of each type of entrepreneurs is constant, with a fraction
of type- entrepreneurs.
Intermediate goods producers for each type of project choose capital and labor to minimize
the production cost, given the demand for intermediate goods.
3.1
The Household
We now turn to the household problem. The household has no access to production technology, but provides physical capital and labor to intermediate good producers each period.
In addition, the household is entitled to the pro…ts of the capital good producer. After the
production takes place, the household makes optimal decisions on consumption, hours to work,
and investment in physical capital. The representative household solves the following problem:
E0
1
X
t
u(Cth
$Cth 1 ; Ht ) with 0 <
< 1;
(2)
t=0
subject to
Cth + qt aht+1 = [qt (1
) + rt ] aht + wt Ht +
k
t
where aht+1 is end-of-period t physical capital purchased by the household, cht and Ht denote
the household’s consumption and the total hours supplied, qt is the price of physical capital.
k
t
is the pro…ts of the representative capital producer. The Euler equation is therefore
qt M Uth = Et [qt+1 (1
where M Uth = uch cht
t
3.2
$cht 1 ; Ht
$uch cht+1
t
) + rt+1 ] M Uth
$cht ; Ht+1 :
Technology
Each type of entrepreneur is endowed with the following technology to produce the …nal goods:
Ytj = Ajt ytj
for j = c or u;
where Ajt is production technology to a type-j project, ytj is the intermediate goods, Ytj is the
…nal output.
j
We assume that Ajt = A
t;
where
t
is the idiosyncratic technology shock. Consider the
following i:i:d shock process to idiosyncratic technology, t ;
8
q
1
< 1
with probability
q
=
t
: 1+
with probability 1
1
8
where the (unconditional) mean of this shock process is 1 and the unconditional variance is
2:
Denote
Aj1 = A
j
Aj2
j
=A
"
r
1
1+
r
j
Note that E Ajt = A : In addition, V ar Ajt j
#
1
;
:
1
=
(1
) Aj2t
Aj1t
2
.
There is a representative …nal goods producer with CES technology combining di¤erentiated
variety
Yt =
Z
1
1
Yti
0
(3)
di
For simplicity, we assume that for i 2 [0; ) ; all variety are subject to type-u technology, while
i 2 ( ; 1] ; all varieties are subject to type-c varieties. In particular, given the spci…cation of
idiosyncratic shocks, (3) can be rewritten as
Yt = f [ (Au1 ytu ) + (1
) (Au2 ytu ) ] + (1
) (Ac1 ytc ) + (1
) (Ac2 ytc ) g
1
The …rst-porder condition delivers the demand function for each di¤erentiated goods.
Pitj
=
Yt
Aji ytj
!1
; j = c or u; i = 1 or 2,
(4)
where Pitj is the price of type-j varieity at state i.
3.3
The Financial Contract
Before …nal good production takes place, entrepreneurs need to purchase the intermediate
goods at a cost ! t ytj , where ! t is the price of intermediate good to be determined later. Entrepreneurs can use both internal funds accumulated at the end of last period and external
borrowing to …nance working capital. The gap between the working capital and entrepreneurial’s net worth is borrowed from a …nancial intermediary.
Each period, both types of entrepreneurs enter a …nancial contract with a …nancial intermediary:
The contract is designed after the aggregate productivity shock, zt ; is realized, but before the
idiosyncratic productivity shock,
j
t;
is realized. Debt is repaid at the end of the period.
Since the realized idiosyncratic technology is private information, payments to the …nancial
intermediary at the end of the period will be made according to the report of the entrepreneur.
The …nancial intermediary utilizes a costly-state-veri…cation technology to verify the veracity
9
of the report. Following (Greenwood, Sanchez, and Wang 2010), the probability that the
entrepreneur of type-j is found cheating is speci…ed as
8
1
1
< 1, > 0
>
<
"t mjt =ytj )
(
j
j
P (mt =yt ) =
for a report jt 6= j2t ;
>
:
0; for a report jt = j2t :
where mjt denotes the monitoring input of a type-j project and is measured in consumption
goods.
Note that the …nancial intermediary will only have to check when a bad state is reported.
Also, under the assumption of
> 0; the probabilities of detecting malfeasance depends
positively on the amount of monitoring per unit of intermediate goods, mjt =ytj ; which we refer
to as monitoring intensity. In other words, as the demand for intermediate goods and thus
the size of loan increases, more monitoring inputs are needed to maintain the same probability
of detecting misreport. Moreover, the above speci…cation requires some threshold level of
monitoring, mjt > ytj ="t : The assumption of the threshold level of monitoring makes sure that
P
0: "t a¤ect the e¢ ciency of monitoring technology, and follows a stochastic process. Note
that "t is an aggregate shock to …nancial sector, capturing the fact that during a crisis, the
e¢ ciency of …nancial intermediation deteriorates due to, for example, a more volatile economic
environment. We refer "t as the lending e¢ ciency shocks.
The timing of …nancial contract at period t is as follows. Financial intermediary …rst decides
on the amount of working capital to be lent out, and resources devoted to detect cheating if
monitoring is needed. Then entrepreneurs use the working capital to purchase intermediary
goods before the realization of the technology shock. At the end of period, entrepreneurs of
type-j projects make a report on the production outcome. The …nancial intermediary decides
whether to conduct a monitoring action and how much monitoring resources to put in. Finally,
output is split between entrepreneurs and the …nancial intermediary based on the outcome of
monitoring. Since this is a within-period loan, the interest payment is 0.
Given the entrepreneurial net worth ajt at the beginning of each period and a contract value
vtj to be determined later, the optimal contract problem for a type-j project is
max
bj1t ;bj2t ;ytj ;mjt
f
j
1 b1t
+
j
2 b2t
! t ytj
qt ajt
j
1 mt g
(5)
subject to
bj1t
j
P1;t
Aj1 ytj
(6)
bj2t
j
P2;t
Aj2 ytj
(7)
10
h
j
P (mjt =ytj )] P1;t
Aj1 ytj
[1
j
bj1t + P2;t
Aj2
j
j j
1 (P1;t A1 yt
and (4) : ! t ytc
bj1t ) +
j
P1;t
Aj1 ytj
j
j j
2 (P2;t A2 yt
qt act is the external borrowing;
i
j
P2;t
Aj2 ytj
bj2t
bj2t ) = vtj
(8)
(9)
captures the unit cost of monitoring input
in terms of …nal goods. In equation (8) ; the left-hand side is the payo¤ to the entrepreneur
at state 2, when he is not caught cheating (it is optimal to set the payo¤ to zero when the
j
entrepreneur report the low state and is caught cheating). P1;t
Aj1 ytj
j
bank when reporting low state, while P2;t
Aj2
bj1t is the payo¤ from the
j
P1;t
Aj1 ytj is the bene…t for cheating when the
true state is 2.
To pin down type-c entrepreneur’s contract value vtj , we assume the …nancial sector is
competitive so that …nancial intermediary earns zero pro…t. The entrepreneur’s contract value
v j can be solved as:
vtj
Yt1
1
Aj1
+
2
ytj
Aj2
! t ytj
Proposition 1 There is no monitoring if and only if Yt1
ytj;f b
arg maxYt1
1
ytj
Aj1
+
2
Aj2
ytj
j
1 mt
(10)
! t ytj;f b
qt ajt ; where
qt ajt
Aj1 ytj;f b
! t ytj :
Proof: see Appendix.
The intuition is straight forward. At the …rst-best production scale, if the payo¤ in the
worst state is still larger than the borrowing cost for the …nancial intermediary, then there is
no incentive for intermediary to engage in costly monitoring. Proposition 1 implies that as the
…rm becomes su¢ ciently large, it becomes less relying on loan from …nancial intermediary to
…nance working capital. Accordingly, it is optimal for intermediary to have zero monitoring if
the borrowing cost is less than the output in the lowest state.
In reality, this asymmetry in monitoring intensity stems from the fact that large …rms rely
less on indirect …nancing as they can resort to self-…nancing or direct …nancing (bond, stock
market). On the other hand, small …rms rely largely on indirect …nance (bank loan) for working
capital …nance. As a result, they are subject to more intensive monitoring.
Proposition 2 Given that Yt1
Aj1 ytj;f b
< ! t ytj;f b qt ajt ; the limited liability constraint (6)
is binding, the limited liability constraint for (7) is not binding and the Incentive Compatibility
constraint is binding.
Proof: see Appendix.
Later, we will show that for a type-c entrepreneur, the IC constraint is binding. Hence the
optimal contract problem for a type-c entrepreneur can be rewritten
11
max Yt1
bc2t ;ytc ;mct
(
c
1 (A1t )
c
2 (A2t )
+
) (ytc )
(! t ytc
qt act )
c
1 mt
vtc
(11)
subject to
[1
P (mct =ytc )]Yt1
(Ac1t ) ) (ytc ) = Yt1
((Ac2t )
(Ac2t ytc )
bc2t
(12)
and
1
2 (Yt
(Ac2t ytc )
bc2t ) = vtc
(13)
Combine (12) and (13), we can pin down the demand of intermediate goods by a type-c
entrepreneur
ytc
=
"
vtc
1
2 Yt
((Ac2t )
(Ac1t ) )
("t mct =ytc )
#1
(14)
Equation (14) shows how lending e¢ ciency shocks a¤ect the scale and thus resources to be
allocated to a type-c project. Given vtc ;a reduction in lending e¢ ciency shock "t intensi…es
the degree of information asymmetry, which then reduces the demand for the intermediate
goods by the type-c entrepreneur. On the other hand, a reduction vtc reduces the incentive
for the entrepreneur to tell the truth, which also reduces the working capital advanced by the
intermediary. From a di¤erence perspective, we see that given the demand for the intermediate
good, a fall in the lending e¢ ciency increase monitoring intensity. Besides, a fall in the contract
value for type-c entrepreneur increases the monitoring intensity, as the entrepreneur’s incentive
to cheat increases.
Substituting (14) and the expression of P (mct =ytc ) into the objective function and maximizing with respect to mct =ytc , we can get the …rst-order condition with respect to mct =ytc
Yt1
bc
where A
c
1 (A1t )
+
b c (y c )
A
t
c
2 (A2t )
1
!t =
c c
1 mt =yt
1+
:
(15)
: Equation (15) suggests that the presence of monitoring cost
drives a wedge between marginal product and marginal cost of intermediate goods: Obviously,
countercyclical variation in mct =ytc will cause the wedge be countercyclical.
A combination of equation (14) and (15) solves ytc and mct =ytc simultaneously. To see the
intuition of how various shocks a¤ect ytc and mct =ytc simultaneously, we plot ytc and mct =ytc
governed by (14) and (15) in Figure 1. (14) leads to a positive relationship between ytc and
mct =ytc : Intuitively, under information asymmetry, a higher demand for intermediate input ytc
12
request a higher monitoring intensity. Equation (15), on the other hand, implies a negative relationship between ytc and mct =ytc , as a higher monitoring intensity involves a higher minitoring
cost and thus a lower demand for ytc : The two schedules determine the demand for intermediate goods and monitoring intensity simultaneously (Point A). A shock to lending e¢ ciency
making the (14) schedule steeper or ‡ater, thus, push monitoring intensity and the demand
for intermediate input in the opposite direction (Point B). A shock that increases ! t , on the
other hand, shifts the (15) schedule towards the origin. Accordingly, ytc and mct =ytc moves in
the same direction (Point C). This property provides a key identi…cation scheme to distinguish
the lending e¢ ciency shock from other shocks, which a¤ect ! t .
As in the literature of …nancial frictions, the net worth of type-c entrepreneurs serves as
…nancial accelerator. It is easy to see from (10) that qt act has a one-to-one direct impact on
vtc . In addition to this direct e¤ect, the presence of …nancial friction will amplify the impact
of act on vtc . This is because as vtc increases, the …nancial constraint is relaxed. As a result, ytc
will increase, which further increases the contract value, vtc . Note that the interest rate for the
within-period loan is 1. Hence
vtc0 (act )
qt
is the internal rate of returns.
vtc0 (act )
=
qt
1
1
c
1 mt
vtc
:
(16)
(16) shows that the more the intermediary monitors (relative to the project value), the
higher is the internal rate of return. The external …nancing premium can be de…ned as
St =
=
Yt1
vtc
(Ac2t ytc )
! t ytc
2
qt act
;
bj2t
! t ytc
qt act
which is the ratio between the lending rate of the intermediary, measured as the ratio of payo¤
to the bank in good state to the amount lent to the entrepreneur, and the interest rate that
the intermediary borrows from the household, which is unit.
The external …nancing premium can be determined by the zero pro…t condition of the bank
j
1 b1t
+ (1
j
1 ) b2t
Dividing both side of the equation by ! t ytj
bj1t
1
! t ytj
mjt
qt ajt
j
1 mt
= ! t ytj
qt ajt
qt ajt and use the de…nition of S; we have
+ (1
13
1 ) St
=1
(17)
which gives
1
1
St =
bj1t
mjt = ! t ytj
1
qt ajt
(18)
1
Given that the condition in Proposition 1 is satis…es, bj1t = ! t ytj
qt ajt and mjt = 0: As a result,
we goes back to the …rst-best case, St = 1: However, if the condition for Proposition 1 is not
j
satis…ed, according to Proposition 2, we have bj1t = P1;t
Aj1 ytj < ! t ytj qt ajt : Hence
h
i
j
j
j j
j
1
1
m
=
P
A
y
P1;t
Aj1 ytj = ! t ytj qt ajt
1
t
1;t 1 t
St =
>1
1
1
(19)
j
Note that in BGG, mjt = P1;t
Aj1 ytj ; the so-called bankcrupcy cost per unit of output, is
constant. However, in our model, it is time varying. Now, we explore the impact of various shocks on St : First, a decrease in lending e¢ ciency shocks will increase the monitoring intensity, therefore tends to increase the external …nancing premium.
Both a nega-
tive shock to aggregate technology and MEI shocks tends ot reduce the demand for credit,
! t ytj
h1
1
qt ajt ; which tends to reduce the external …nancing premium. Finally, an increase in
(the default probability), tends to increase the external …nancing premiu,. This is because
i
j
j
Aj1 ytj = ! t ytj qt ajt < 1.
P1;t
Aj1 ytj
mjt = P1;t
3.4
The Consumption-Saving Problem of Entrepreneurs
Now we switch to the consumption-saving problem of a type-j entrepreneur. Each period, after
the production takes place, a type-c entrepreneur decides how much to consume and to invest in
physical capital ajt . To simplify our problem, we assume perfect consumption insurance within
the group of type-c entrepreneurs after they receive the idiosyncratic productivity shock. As a
result, the consumption-saving problem of a type-j entrepreneur can be aggregated and written
as a problem faced by a representative type-j entrepreneur
E0
1
X
j t
u(cjt
$cjt
1)
with 0 <
<1
) + rt ] ajt + vtj
qt ajt
(20)
t=0
subject to
cjt + qt ajt+1 = [qt (1
The Euler equation becomes
We assume that
u
qt M Utj =
j
c
=
=
,
h
j
Et M Ut+1
rt+1
; where
j0
qt+1 + vt+1
ajt+1
< 1: Note that
i
< 1 implies that the type-c
entrepreneur to discount future consumption more that the representative household. This is
14
to make sure that a type-c entrepreneur is borrowing constrained at the steady state. On the
other hand, a type-u entrepreneur is more patient than a type-c entrepreneur. At the steady
state, v u0 (aut ) =qt = 1; which means that the type-u entrepreneur is not …nancially constrained.
Then, according to Proposition (1) it must be that at steady state, type-u entrepreneurs
accumulate su¢ cient assets such that
Yt1
(Au1t ytu )
! t ytu
qt aut
(21)
Lemma 1 At steady state, for a type-u entrepreneur, neither (6) nor (8) is binding. The
demand of intermediate good by a type-u entrepreneur is determined by
Yt1
Proof: see Appendix.
b u (y u )
A
t
1
= !t
(22)
To see how resources are allocated between the two types of projects, note that the variety
hjt
of a type j project can be expressed as Ytj = Ajt ezt ktj
(1
)
ltj
1
: Accordingly,
the average marginal product of capital of a type j project is
b j e zt k j
A
t
j
M RP K t = Yt1
1
hjt
(1
)
ltj
1
; j = c or u:
We measure of the degree of capital misallocation with ratio of average marginal revenue
product of capital between Type-c and Type-u project
c
1
b c kc
M RP K t
A
t
M P KRt =
>1
u =
u
b
ktu
M RP K t
A
Here we use the feature of 28. In the …rst best where there is no asymmetric information,
c
u
M P KRt = 1 even if A 6= A :
Assuming that (22) holds around the steady state, a combination of (15) and (22) implies
that
M P KRt =
c c
1 mt =yt
!t
1+
+1
Hence, the higher is the marginal monitoring cost relative to marginal production cost, the
larger is the dispersion of capital productivity.
15
Equation (15) and (22) deliver the price elasticity of intermediate goods for the two types
of entrepreneurs.
@ log ytc
@ log ! t
1
=
=
log ytu
@
@ log ! t
!t
1
!t +
(1
1
) M P KRt
(23)
c c
1 mt =yt
1+
1
=
(24)
1
c
c
1 mt =yt
where the second equality comes from M P KRt =
1+
+ 1: (23) and (24) implies
!t
that the price elasticity of intermediate goods by type-u entrepreneur is larger than that
for type-c entrepreneurs, given the capital misallocation between the two types of project,
i.e. M P KRt > 1: Intuitively, the marginal cost for type-c entrepreneurs involves not only
replacement cost, !; but also expected monitoring cost, captured by
c c
1 mt =yt
1+
. Hence,
with given monitoring intensity, unconstrained entrepreneurs’input demand is more sensitive
to changes in the price of intermediate goods. As will be shown later, this feature is crucial
to explain why others shocks that we examine below, such as aggregate technological shocks,
implies counterfactual predictions regarding the cyclicality of the productivity dispersion:
3.5
The Intermediate Good Producer’s Problem
We assume that each intermediate good producer face the same Cobb-Douglas technology
combining capital labor and …nal output
ezt ktj
hjt
(1
)
ltj
1
ytj
(25)
The intermediary good producers for both type of projects rent capital and labor to minimize
the cost of producing intermediate goods, given its demand ytj :
n
o
min rt ktj + wt hjt + ltj
(26)
ktj ;hjt ;ltj
subject to (25) :
The …rst order conditions imply the capital and labor demand for each type of project
ktj =
hjt
ytj
e zt
(1
ytj
= zt
e
rt
ytj
e zt
rt
ltj =
(1
wt
)
1
rt
)
1
1
1
1
;
1
(1
wt
(1
1
)
)
16
;
1
(1
wt
(1
)
)
1
1
1
:
(27)
Accordingly we have
ktc
ktu
wt
=
=
c
u
ht
ht
(1
) rt
(28)
Equation (28) indicates that the capital-labor ratio between the two types of projects is the
same, though resources might be misallocated across projects. In other words, we shut down
within-in project resource misallocation as a potential source of misallocation.
Assume that the intermediate good market is competitive. The price of intermediate goods
is thus equal to the marginal cost.
!t = e
zt
rt
(1
wt
(1
)
1
1
)
(29)
1
Assuming that aggregate technology shock zt follows AR(1) process:
zt+1 =
where
z ~N (0;
t+1
2
z
z
zt +
z
t+1
):
In Appendix 7.1, we show that the solution to (5) is equivalent to that of an alternative
problem where producers of variety j rent factor inputs directly from the factor market and
produce Ytj according to Ytj = Ajt ezt ktj
ing capital is
rt ktj
+ wt hjt
+ lti
hjt
(1
)
ltj
1
. Accordingly, the total work-
and the producer of variety j borrows rt ktj + wt hjt + lti
qt ajt from
the intermediary: The …nancial contract is signed before (after) the idiosyncratic (aggregate)
productivity shock Ajt (zt ) is realized. Hence, our benchmark setup can been seen as the recursive form of this alternative problem: …rst, given ytj
hjt
ezt ktj
the cost minimization problem (26) : Second, solve the optimal
ytj ;
(1
)
ltj
together with
1
; solve
bj1t ; bj2t ; mjt ;
for the problem (5) : The advantage of solving the …nancial contract in the recursive form is to
shed light on the transmission mechanism of various shocks in a more intuitive way.
3.6
The Capital Producer’s Problem
To pin down the price of physical capital, we assume that in this economy, the is a representative
capital producer. The capital producer purchases It units of consumption goods from the
…nal good producer (and (1
) Kt units of physical capital from the household and type-c
entrepreneur), and produces the new capital stock to be sold to the household and type c
entrepreneur at the end of the period.
The technology to transform new investment into installed capital involves installation
cost, S (It =It
1) ;
which increases in the rate of investment growth. Since the marginal rate
of transformation from previously installed capital (after it has depreciated) to new capital is
unity, the price of new and used capital are the same.
17
The capital producer’s period-t pro…t can be expressed as
k
t
t
= qt [(1
) Kt +
t (1
S (It =It
1 )) It ]
qt (1
) Kt
It
is the marginal e¢ ciency shocks to investment (MEI shocks) and follows log AR(1) process.
log
where
N 0;
t
t+1
= (1
) log
+
log
t
+
t
2
Dynamically, the capital producer solves the following optimization problem
8
9
1
<X
=
j
k
maxEt
t+j t+j
:
;
It+j
(30)
j=0
where
t
is the multiplier on the household’s budget constraint. Note that if S (It =It
1)
= 0;
qt = 1 and the model goes back to our previous case. The …rst order condition delivers
1
qt =
Et
h
t 1
t+1
t
t+1 S
qt+1
S 0 (It =It
0 (I
t+1 =It )
It
1 ) It 1
It+1
It
S (It =It
2
i
)
1
In order for qt to be equal to 1 at the steady state and procyclical we assume S (1) = S 0 (1) = 0:
More importantly, S 00 (1) > 0: As a result, when It =It
S 0 (I
k
t
t =It 1 )
1
falls relative to the steady state,
also falls. This leads to a decline in qt : It is easy to see that at the steady state,
= 0:
3.7
Competitive Equilibrium
A competitive equilibrium consists of …nancial contract fbj1t ; bj2t ; ytj ; mjt ; vtj g1
t=0 for each type-j
project, a set of factor inputs,fktj ; hjt ; ltj g for each intermediate good producer producing ytj ; a
n
o1
1
set of allocation cjt ; ajt+1
for each type-j entrepreneur, a set of allocation cht ; aht+1 ; Ht t=0
t=0
for the representative household, and the prices fqt ; rt ; wt ; ! t g, such that:
1. Given ! t ; v j ; the …nancial intermediary of type-j project o¤ers a contract fbj1t ; bj2t ; ytj ; mjt g
by solving (5) (??).
2. vtj is such that the intermediary earns zero pro…t in a competitive environment.
n
o
3. Given ytj ; rt ; wt ; the intermediate good producer minimizes its production cost, by
solving the problem (26).
4. Given prices frt ; wt g; the representative household solves the problem .
18
5. Given rt ; a type-c entrepreneur solves the problem (20) :
6. Given qt ; the capital producer solves the problem (30)
7. All markets clear:
(a) Wage rate wt clears labor market: hct + (1
) hut = Ht .
(b) Rental rate rt clears capital market: act + aht = Kt = ktc + (1
) ktu :
(c) The price of physical capital; qt ; clears the capital markets: Kt+1 = (1
(1
S (It =It
) Kt +
1 )) It .
(d) Good market clears:
h
Yt = (1
c
b ezt (k c ) (hc )1
)A
t
t
= Ct + It +
where Ct
c
1 mt
cht +(1
(ltc )(1
)
+ Lt
b u ezt (k u ) (hu )1
+ A
t
t
) cut + cct denote the aggregate consumption, Lt
(ltu )(1
(1
)
) ltu +
ltc denotes the …nal goods used for intermediate gods production:
Note that the aggregate value added is
Yet = Yt
c
1 mt
Lt
T F Pt (Kt ) (Ht )1
4
Numerical Results
To explore how …nancial frictions and capital allocation responds to various shocks, in this
section, we provide numerical examples to illustrates the impulse responses of di¤erent variables
to various shocks in our model: lending e¢ ciency shocks, aggregate technology shocks and other
shocks commonly used in the literature.
Note that at the steady state, the asset distribution between the household and type-u
entrepreneurs is indeterminate, since both agents are unconstrained. Therefore, we assume
that type-u entrepreneur pocess the minumum asset that allow him to be unconstrained.
According to (21) ; the minimum asset of type-u entrepreneur at the steady state is
au = !y u
Y1
19
(Au1 y u )
=q
(31)
i1
where y u is such that equation (22) is satis…ed at the steady state. Then, given (??) ; equation
(10) gives
vu = Y 1
(
u
1 (A1 )
+
u
2 (A2 )
) (ytu )
(!ytu
qaut )
(32)
Note that at the steady state, mu = 0: Finally, we can solve for cu according to the type-u
entrepreneur’s budget constraint
cu = [q (1
) + r] au + v u
(33)
Table 3a summarizes the calibrated parameter values.
Parameter
A
u
"
c
A
Table 3a: Calibrated Parameters
Target
Exogenously Chosen
Labor income share of type-u project
normalization
Share of …nal goods in intermediate goods
Bloom Floetotto and Jiamovich (2010)
Akteson and Kehoe (1995)
Endogenously Chosen
Inter-quartile range of M RP K; 1.56
Aggregate capital output of 10
Employment Ratio
Investment-capital ratio of 0.02
Interest rate of 3%
Hours worked of 1/3
Value
0.4
1
0.8
0.41
0.85
4.76
1.35
1.67
0.02
0.99
3.84
Table 3b summarizes the values of parameters to be estimated.
Parameter
c
$
z
z
"
"
Table 3b: Parameters to be Estimated
Description
Frisch elasticity of labor
Type-c Discount Factor
Share of type-u project
Capital Adjustment Cost
Consumpion habit
Two-state approx. to normal distribution
Quarterly persistence of tehcnology shocks
SD of aggregate tehcnology shocks
persistence of lending e¢ ciency shock
SD of lending e¢ ciency shock
mean of MEI Shock
persistence of MEI shocks
SD of MEI shocks
20
Value
0.4
0.75
0.5
0.1
0.5
0.5
0.98
0.01
0.98
0.01
1
0.98
0.01
4.1
Lending E¢ ciency Shocks
Figure 3 plots the responses of various macro variables to a one percentage decrease in lending
e¢ ciency, "t . As shown by the top panel, such a shock generates a recession, as aggregate
output, consumption, investment and hours worked all fall on impact. Moreover, aggregate
TFP decreases, though the aggregate technology, z; is unchanged. Such a fall in aggregate TFP
results from an increase in the M RP K gap between …nancially constrained and unconstrained
projects, as evident by the middle right panel.
The intuition is as follows. A reduction in lending e¢ ciency reduces the probability for the
bank to identify the misreport of the entrepreneurs. This tightens the incentive compatibility
constraints. As a result, the bank increases the monitoring intensity, as shown by the bottom
left panel of …gure 3. Since monitoring is costly, an increase in monitoring intensity tightens
the …nancial constraint of type-c projects, forcing them to reduce the demand of intermediate
goods. This widens the gap of marginal product of capital between the two types of projects.
As a consequence, aggregate TFP falls on impact.
The above mechanism is further ampli…ed by the …nancial accelerator channel. The lower
demand for intermediate goods causes the production scale and the pro…t of the type-c project
to be shrunk. Accordingly, the contract value for the type-c entrepreneur falls, which further
tightens the bank loan and shrinks the production scale to a type-c project. Associated with
a tightening of the …nancial constraint is an increase in the external …nancing premium and a
fall in the price of capital, shown in the bottom middle and right panels of …gure 3.
In summary, a fall in lending e¢ ciency not only triggers a recession for aggregate variables,
but also generates countercyclical dispersion of marginal product of capital between …nancially
constrained and unconstrained projects. As a consequence, aggregate TFP falls.
[Insert …gure 3 here]
4.2
Aggregate Technology Shocks
We now turn to the impulse response to a reduction in aggregate technology, zt : Figure 4 plots
the impulse responses of various variables to such a shock. As in a standard RBC model,
a fall in aggregate technology leads to a fall in aggregate output, investment, consumption
and hours worked. However, it also generates a counterfactual increase in aggregate TFP.
The reason is that the dispersion of M RP K between constrained and unconstrained projects
shrinks, contrast to what we have observed in the data. Moreover, the bottom panels show
that both monitoring intensity and external …nancing premium falls in recessions. Again, these
21
results are counterfactual.
The key underlying the counterfactual predictions of aggregate technological shocks on
variations in the dispersion of M P K and aggregate T F P is the asymmetric price elasticity
of input demand for the two types of entrepreneurs. A fall in aggregate technology, by construction, discourages the input demand of entrepreneurs via a higher replacement cost. Since
the input demand by unconstrained …rms is more sensitive to an increase in price, such a
shock reallocates resources from the unconstrained to constrained entrepreneurs. Financially
constrained projects, therefore, expands initially, despite a fall in aggregate technology. This
results in a narrowing of the the gap of M P K and an increase in aggregate T F P: In addition,
the initial expansion of …nancially constrained project increases the project value, which helps
to relax the incentive compatibility constraint and reduce the monitoring intensity. These predictions run counter to the coutercyclical dispersion of M P K and procyclical capital allocation
between small and large …rms as we found in Section II.
[Insert …gure 4 here]
4.3
Other Aggregate Shocks
In this section, we further explore the predictions of other aggregate shocks regarding the
cyclicality of M P K dispersion and aggregate TFP. These shocks include marginal e¢ ciency
shocks to investment or so called “MEI shocks”(e.g. Justiano, Primiceri and Tambalotti, 2011),
shocks to labor disutility, shocks to utility discount factors and shocks to capital depreciation
rates.
The impulse responses to each of the above shocks are plotted in Figure 5, 6, 7, and 8
respectively. It is clear that though these shocks can generate a recession in terms of aggregate
variables such as aggregate output, hours worked, investment and/or consumption, they all
predict that the dispersion of M P K narrows in recessions, contrast to what is observed in the
data. Accordingly, each of these shocks predicts a counterfactual increase in aggregate TFP.
Again, similar to aggregate TFP shocks, the key underlying the counterfactual predictions
of these shocks on the cyclicality of M P K dispersion is the asymmetric price elasticity of input
demands by the two types of entrepreneurs. Note that all these shocks in‡uence the demand of
intermediate goods by entrepreneurs via the price of intermediate goods alone. Speci…cally, a
MEI shock a¤ects the capital supply and therefore the rental price of capital. A labor disutility
shock, on the other hand, a¤ects the aggregate labor supply and, thus, the wage rate. A shock
to utility discount factor alters the saving incentives and therefore the interest rate. Finally, a
capital depreciation shock a¤ects the supply of installed capital and again the rental price of
22
capital. Therefore, these shocks trigger a larger response of the input demand by unconstrained
entrepreneurs than that of constrained entrepreneurs. This leads to a reallocation of capital in
the opposite direction to what we found in the data, and therefore, counterfactual predictions
on dispersion of M P K and aggregate TFP.
[Insert …gure 5, 6, 7 and 8 here]
5
Conclusion
This paper develops a theory in which shocks to the e¢ ciency of …nancial intermediation
translate into business cycle ‡uctuations via capital reallocation. In our theory, the realized
idiosyncratic productivity is subject to asymmetric information between the bank and the entrepreneurs. However, due to costly veri…cation, the bank will monitor only the cash ‡ows
to those entrepreneurs with insu¢ cient net worth. This creates a gap of marginal product of
capital between entrepreneurs of di¤erent size. Shocks to the e¢ ciency of bank monitoring, by
a¤ecting the probability to detecting misreport by entrepreneurs, alter the monitoring intensity of the bank. Accordingly, capital is reallocated between entrepreneurs of di¤erent sizes,
creating countercyclical variations in the dispersion of marginal productivity of capital and
procyclical TFP ‡uctuations. By contrast, in our model other shocks commonly adopted in
the literature predict a procyclical dispersion of marginal product of capital and countercyclical TFP ‡uctuations. Using the COMPUSTAT dataset, we …nd a signi…cant countercyclical
pattern for the relative capital productivity of …nancially constrained to unconstrained …rms,
which is driven by the procyclical capital allocation between these two types of …rms. Therefore, both our theory and evidence suggest that shocks to bank lending e¢ ciency might be
important drivers of business cycles.
23
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[16] Greenwood, J., Sanchez, J. M. and Wang, C. (2010), “Financing Development: The Role
of Information Costs”, American Economic Review, 100, 1875-1891.
[17] Sim, J.(2008), “Uncertainty driven business cycle”, mimeo.
[18] Stock, J., and M. Watson (1998), “Business Cycle Fluctuations in US Macroeconomic
Time Series”, in Handbook of Marcoeconomics, ed. by J. Taylor, and M.Woodford. Elsevier.
25
6
Appendix
We now prove the propositions in Section III. To simplify the notation, we drop the time
subscript.
6.1
The Alternative Setup
In this subsection, we consider an alternative setup of the environment, which provides more
intuition regarding the capital allocation among …rms of di¤erent types.
Here, the total working capital is rt ktj + wt hjt + lti : The entrepreneur …nance the working
capital using both internal funds, qt ajt ; and bank loan, which is the gap between internal
funds and the working capital. After …nancial contract is signed, the idiosyncratic shock, Ajt ;
is realized. And the production of variety i follows Ytj = Ajt ezt ktj
Denote ytj
(1
hjt
ezt ktj
)
ltj
1
hjt
(1
)
ltj
1
:
such that Ytj = Ajt ytj : The price of Ytj in terms
of ytj is, therefore, 1=Ajt and thus the cost in terms of …nal goods is ! t =Ajt : (In our original
problem the variety j producer purchase intermediate good ytj in a competitive market and
assmemble the intermediate goods into the variety good according to Ytj = Ajt ytj ).
max
bj1t ;bj2t ;ktj ;hjt ;lti ;mjt
f
j
1 b1t
rt ktj + wt hjt + lti
j
2 b2t
+
j
1 mt =
qt ajt
g
(34)
subject to
[1
bj1t
j
P1;t
Aj1 ezt ktj
hjt
bj2t
j
Aj2 ezt ktj
P2;t
hjt
2
6
P (mjt =ytj )] 4
j
P2;t
Aj2 ezt ktj
j
j zt
1 (P1;t A1 e
ktj
hjt
(1
)
ktj
j
P1;t
Aj1 ezt
+
hjt
ltj
j
P2;t
Aj2
(1
1
)
j
P1;t
Aj1
1
ltj
bj1t )+
(1
)
(1
)
ltj
(1
hjt
ezt
ltj
)
ktj
1
(35)
1
(36)
ltj
hjt
1
(1
3
bj1t
)
ltj
7
5
1
bj2t
j
j zt
2 (P2;t A2 e
ktj
hjt
(1
)
ltj
1
(37)
bj2t ) = vtj
(38)
For notation concision, we drop the superscript j and the subscript i in the following
analyis, as our proof applied to both types of entrepreneurs and …nancial contracts in all
periods. Denote fb1 ; b2 ; k ; h ; l ; m g as the optimal solution to (34) : Also by de…nition, we
have y
ez (k )
(h )(1
)
(l )1
. From the …rst-order conditions of (34) ; we can obtain
26
k =h and k =l and h =l : For any of the two ratios, together with the de…nition of y , we
obtain
(1
w
k =
y
ez
h =
y
ez
r
l =
y
ez
r
(1
)
1
r
)
1
1
1
1
;
1
(1
w
(1
)
1
)
(1
w
(1
;
1
)
1
1
)
:
1
Hence, we have
rk + wh + l = !y
where ! is de…ned as in (29) :
n
o
eb1 ; eb2 ; ye; m
e , denoted with superscript “e”
n
o
from the problem (5) : And given ye; we derives the optimal factor input e
k; e
h; e
l according to
Similarly, we can solve the optimal contract
(27) : Clearly we have
re
k + we
h+e
l = !e
y:
Now we would like to prove that y = ye: Denote
Q
=
1 b1
+
2 b2
e =
Q
1 b1
e +
2 b2
e
(rk + wh + l
(!e
y
qa)
qa)
1m
=
e
1 m=
e Step 2, we show that the solution to (5)
Our proof has two steps: Step 1, we show that Q = Q;
n
o
e implies eb1 ; eb2 ; e
and (34) is unique, such that Q = Q
k; e
h; e
l; ye; m
e = fb ; b ; k ; h ; l ; y ; m g :
1
e Then,
Proof: Suppose Q > Q:
Q
=
1 b1
+
2 b2
(rk + wh + l
=
1 b1
+
2 b2
(!y
qa)
qa)
1m
1m
2
=
=
e
> Q
n
o
e
e
Since fb1 ; b2 ; y ; m g satis…es (6) ; (7) ; (8) and (9) ; we conclude that b1 ; b2 ; ye; m
e is not
the optimal solution for (5) : contradition.
e
Suppose Q < Q:Then,
e =
Q
=
e +
1 b1
e +
1 b1
> Q
e
2 b2
e
2 b2
(!e
y
qa)
re
k + we
h+e
l
27
e
1 m=
qa
e
1 m=
n
o
Note that eb1 ; eb2 ; e
k; e
h; e
l; m
e satis…es the constraints (35) ; (36) ; (37) and (38) : Hence, fb1 ; b2 ; k ; h ; l ; m g
is not the optimal solution for (34) : Contradition.
e
Therefore, Q = Q:
Step 2: For the problem (5) ; the objective function is concave in fb1 ; b2 ; y; mg, and the choice
set is strongly convex in fb1 ; b2 ; y; mg due to the strict concavity of the demand function for
the variety good, Yi ; (4) and the strict concavity of the monitoring technology in m=y: Hence,
the optimal problem (5) has a unique solution.
Similarly, for the problem (34) ; the objective function is concave in fb1 ; b2 ; k; h; l; mg and
the choice set is strongly convex in fb1 ; b2 ; k; h; l; mg ; due to, again, the strict concavity of
the demand function for the variety good, Yi ; (4) and the strict concavity of the monitoring
technology in m=y: Hence, the optimal problem (34) has a unique solution.
n
o
e and fb ; b ; k ; h ; l ; m g and eb1 ; eb2 ; ye; m
Therefore, with Q = Q
e
the optimal solutions
1 2
n
o
for (34) and (5), respectively, we have eb1 ; eb2 ; e
k; e
h; e
l; ye; m
e = fb1 ; b2 ; k ; h ; l ; y ; m g : End of
Proof.
6.2
Proof for Proposition 1
We …rst drive the necessary condition for mjt = 0 or, equivalently, P = 0: From the IC
constraint (8) ; we have
Aj2 ytj
pj2t
also, combining (9) with (10) ; we get
h
j
j
j
j
j
y
+
A
A
1 (A1 yt
2 2
1 1
t
pj1t + Aj2
Aj1 ytj
pj1t ) +
j
j
2 (A2 yt
Plugging (39) (with equality) into (40) ; and given that
ytj
Aj1
pj2t )
mjt
i
(39)
j
1 mt =
! t ytj
qt ajt
(40)
= 0; we obtain the necessary
condition
j
1 A1
or
+
j
2 A2
ytj
h
j
j
2 A1 yt
pj1t
Aj1 ytj
Aj1 ytj
Aj2
+
! t ytj
Aj1
ytj
pj1t + ! t ytj
i
j
1 (A1
ytj
pj1t )
! t ytj qt ajt
qt ajt
qt ajt
where the second inequality obtains from the limited liability condition (6) : To prove the
su¢ ciency, the …nancial contract can be simply designed as
pj1t = pj2t = ! t ytj;f b
28
qt ajt
(41)
Note that, by assumption, the payo¤ at the low state, Aj1 ytj
! t ytj
qt ajt is non-negative,
and thus is feasible. Plugging (41) into the IC constraint (8) ;
[1
h
P (mjt =ytj )] Aj2 ytj;f b
! t ytj;f b
qt ajt
i
Aj2 ytj;f b
! t ytj;f b
qt ajt
Obviously, the above IC constraint is always satis…ed, even if no monitoring resource is used
and P = 0. So IC constraint can be dropped from the intermediary’s problem (5). Also,
the zero pro…t condition of the intermediary, (40) ; is satis…ed. Hence, it is optimal to set
mjt = 0 and P = 0: Intuitively, since the intermediary does not monitor in either states, given
that entrepreneur has incentive to lie, it is optimal to set the payo¤ at both states to be the
borrowing cost of the intermediary.
6.3
Proof for Proposition 2
The Lagrangian of the problem (5) is
j
1 p1
L=
+
+
j
1 [ 1 (A1
j
2 [A2
+
j
2 p2
+
!y j
yj
pc2
j
31 [A1
yj
1m
j
=
pj2 )
j
c
2 (A2 (y )
pj1 ) +
yj
qaj
vj ]
P (mj =y j ))(Aj2 y j
(1
j
32 [A2
pc1 ] +
pj1 )]
pj2 ]
yj
The …rst-order conditions are:
@L
=
@p1
1 (1
1)
@L
=
@p2
@L
@y j
=
+
2 (1
!+
1
1)
(
c
31 A1
c
32 A2 )
+
(y c )
1
2
j
1 A1
1
j
j
2 [ A2 y
+(
P (mj =y j ))
2 (1
+
+
32
j
2 A2 )
31
=0
=0
yj
(43)
1
+
@P (mj =y j ) j j
(A2 y
@y j
= 0
Proof. From the …rst-order conditions, we will have the following results:
Result 1 :
1
2 (0; 1)
29
(42)
pj1 )
(1
1
P (mj =y j )) Aj2 y j (44)]
Proof: From (43), we have
Now we turn to prove
32
1
2 [0; 1]. Since promise-keeping constraint is binding,
1
2 (0; 1].
6= 1. Suppose not, then from (42) and (43), we will have
2
=
= 0. Then from (44),
Result 2 :
1
1
31
=
= 0, a contradiction.
> 0, limited liability constraint for pc1 is binding
31
Proof: from result 1 and (42), we can directly have this result.
Result 3 :
2
> 0;
32
= 0, so IC constraint is binding and limited liability constraint for
pc2 is not binding.
Proof: suppose not, from result 1 and (43), we have
Case 1:
2
Aj2
yj
Case 2:
2
Aj2
pj2
pj2
<
6.4
yj
= 0;
32
P (mj =y j ))(Aj2
(1
> 0;
> 0 Then we have pj2 = Aj2 y j
32
> 0 Then we have
P (mj =y j ))(Aj2
= (1
yj
yj
. However, result 2 and IC imply
Aj1 y j ) < Aj2
pj2 = Aj2 y j .
Aj1 y j ) > 0 )
yj
, a contradiction.
However, result 2 and IC imply
Aj2 y j
> pj2 , a contradiction.
Proof of Lemma 1
To prove Lemma 1; it is useful to show a dual problem of (5) ; which is to design contract
o
n
j
j
; ytj ; mjt to maximize the expected payo¤ of the entrepreneur, subject to the limited
; C2t
C1t
liability conditions and the IC and PC constraint for intermediary.
max
j
j
C1t
;C2t
;ytj ;mjt
f
1 (1
j
P1 ) C1t
+
j
2 C2t g
(45)
subject to
[1
h
P (mjt =ytj )] Aj1 ytj
j
1 A1
+
j
2 A2
(ytc )
pj1t + Aj2
j
1 mt =
Aj1
j
1 C1t
ytj
+
j
C1t
0
j
C2t
i
0
j
2 C2t
Aj2 ytj
! t ytj
pj2t
qt ajt
It is easy to show that proposition (1) still holds under this dual problem. As a result, when
Au1 (ytu )
! t ytu
qt aut
j
C1t
= Aj1 ytj
! t ytj
qt ajt
(46)
j
C2t
= Aj2 ytj
! t ytj
qt ajt
(47)
Plug (46) and (47) into the objective function, the original problem for type-u becomes
max
(
u
yt
u
1 A1
+
u
u
2 A2 ) (yt )
30
! t ytu + qt aut
The …rst order condition is
u
1
A (ytu )
6.5
= !t
A Micro-fundation for the Monitoring Technology
In our model, the monitoring technology P , that is, the probability that a type-j entrepreneur
who receives Aj2 , but reports Aj1 ; is found cheating by the intermediary, is positively a¤ected
by two variables: the lending e¢ ciency shock "t ; and the monitoring intensity, mj =y i : In this
section, we provide a micro-foundation for this monitoring technology.
Again, the intermediary veri…es the true state of the entrepreneur only when the entrepreneur announce the low state, i.e., Aj1 ; since entrepreneurs has no incentive to misreport
Aj2 when the actual realized idiosyncratic technology is Aj1 : To simplify notation, we drop the
superscript j, and assumes A = 1:
The key assumption is that …nancial intermediary only receives a noisy signal about the
realized idiosyncratic technology once it decides to verify the true state of the entrepreneur.
Speci…cally, the signal st follows
st = at
where at
(48)
t
log At is the unobserved realized idiosyncratic technology. Again, to be consistent
with the main text, at follows a binary i:i:d distribution
8
q
1
< 1
a1 with probability
q
at =
: 1+
a2 with probability 1
1
t
is i:i:d distributed noise and independent of at .
t
=
0
a2
t
follows a binary distribution.
with probability pt
a1 with probability 1
pt
pt can be interpreted as the precision of the signal. Note that the variance of the noise
2
;t
= pt (1
pt ) (a2
a1 )2
is time-varying. As pt decreases from 1 to 1/2, the variance of the noise increases.
We assume that if the intermediary observe a signal st = a2 ; he will launch a lawsuit
to entrepreneur and ask the judge to verify the true state. If the intermediary observe a
signal st 6= a2 ; it will not launch a lawsuit and only charge b1t = A1 (yt ) :6 In this case, the
entrepreneur who receives at = a2 ; but misreports to at = a1 can bene…t from cheating.7
6
Note that when at = a1 ; the intermediary receives a signal with value of either a1 or 2a1 a2 < 0: In either
case, the intermediary will not launch a lawsuit and only charge p1t = A1 (yt ) :
7
The entrepreneur with a = a1 always gets nothing, no matter whether a lawsuit is launched or not.
31
Therefore, the probability that the entrepreneur who actually receives at = a2 , but found
cheating is equal to
Pr (st = a2 j at = a2 ) = Pr (
t
= 0) = pt
where the …rst equality is derived from the assumption that
each other. Note that if the variance of the signal
;t
t
and at are independent of
! 0; Pr (st = a2 j at = a2 ) ! 1: In
other words, when there is no noise, once the intermediary verify, entrepreneur will be found
cheating.
Obviously, the more noisy is the signal, the chance for entrepreneur to be caught cheating
is smaller. The intuition is as follows: The probability the intermediary receives a high signal
about the entrepreneur’s realized technology, which given at = a2 , is equal to Pr (
However, a larger noise, i.e a higher
;t ;
t
(say, in recession) causes the probability Pr (
t
= 0).
= 0)
with at = a2 be smaller. This implies that with larger noise, the chance for the lender to
observe st = a2 (i.e. a high signal); given the entrepreneur’s true realized technology is a2
becomes smaller: As a result, it is less likely that the entrepreneur is caught misreport.
Now we would like to pin down the degree of the noise of the (idiosyncratic) signal. Assume
pt = 1
1
("t mt =yt )
;
>0
where "t is the lending e¢ ciency shocks in the benchmark model and
(49)
mt
yt
is the monitoring
intensity. (49) simply says that the precision of the signal depends positively on the level of the
lending e¢ ciency shocks and the monitoring intensity. Therefore, we have the probability of
being found cheating is positively a¤ected by both lending e¢ ciency shocks and the monitoring
intensity. Since pt is negatively related to the variance of the noise, equation (49) implies that
the variance of the noise shock have two components: one is exogenous, depending negatively on
the lending e¢ ciency shocks; the other is endogenous, depending negatively on the monitoring
intensity.
6.5.1
Rational Inattention
Our micro-fundation for the probability of detecting the misreport corresponds to the concept
of rational inattention. Putting in words, the precision of signal depends on both e¤ort and
the some exogenous shocks, which we called lending e¢ ciency shocks. To see this point, we
measure the precision of signal as the ratio of true variance of a to the condition variance of
at when the bank observe a signal st = a1 ; denoted as
8
Note that
2
ajs6=a1
2 = 2
8
at
at jst =a1 .
More generally, if at
= 0; since the bank can infer with certainty what’s the true state if s = a2 or 2a1
32
a2 :
and
t
are normal distributed,
measured by entropy.
2 = 2
at
at jst =a1
is positively related to the information s contains
I (at ; st ) = H (at )
H (at j st ) :
2
ajs=a1
We now compute
2
at jst =a1
= E a2t j st = a1
E [at j st = a1 ]2
(50)
The …rst argument on the right side of (50) is
E a2t j st = a1
= a21 Pr a2t = a21 j st = a1 + a22 Pr a2t = a22 j st = a1
Pr st = a1 j a2t = a21 Pr a2t = a21
Pr (st = a1 )
Pr st = a1 j a2t = a22 Pr a2t = a22
+a22
Pr (st = a1 )
2
2
a1 pt + a2 (1 pt ) (1
)
=
pt + (1 pt ) (1
)
= a21
(51)
Similarly, the second argument on the right side of (50) is
Pr (st = a1 j at = a1 ) Pr (at = a1 )
Pr (st = a1 )
Pr (st = a1 j at = a2 ) Pr (at = a2 )
+a2
Pr (st = a1 )
a1 pt + a2 (1 pt ) (1
)
=
pt + (1 pt ) (1
)
E [at j st = a1 ] = a1
(52)
Plugging (51) and (52) into (50) ; we have
2
at jst =a1
Obviously,
2
at jst =a1
=
pt (1 pt ) (1
) (a2 a1 )
[pt + (1 pt ) (1
)]2
decreases with pt when pt
2
at
2
at jst =a1
=
1=2: Finally we have
[pt + (1 pt ) (1
pt (1 pt )
where f (pt ) is increasing in pt ; given that
2
at
2
at jst =a1
2
a
(53)
)]2
f (pt )
is constant. Finally, with (49), we have
g ("t ; mt =yt )
(54)
where g is increasing in both arguments. (54) is the typical information constraint a rational
inattentional agent is subject to, which is equivalent to our monitoring technology (49).
33
0.25
0.2
0.15
log ratios of MRPK
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
1975
1980
1985
1990
1995
2000
2005
2010
Figure 1: Ratio of MRPK between Financially Constrained and Unconstrained Firms
34
0.2
0.15
Log ratio of capital stock
0.1
0.05
0
-0.05
-0.1
-0.15
-0.2
1975
1980
1985
1990
1995
2000
2005
2010
Figure 2: Ratio of Capital Stock Between Financially Constrained and Unconstrained Firms
0
x 10
-3
log_Y
log_mt
0.01
6
-2
x 10
-3
log_mpk_r
4
0.005
-4
2
-6
5
10
15
20
0
kc_ku
0
0
-0.01
-0.02
-0.03
0
5
x 10
-3
10
15
20
5
x 10
-3
10
15
20
0
log_tfp
8
-1
6
-2
4
-3
2
-4
5
log_H
10
15
20
-1
-0.005
-2
-0.01
x 10
0
log_I
0
5
0
-3
5
x 10
-3
10
15
20
15
20
15
20
ext_p
10
log_C
-1
-2
-3
5
10
15
20
-0.015
-3
5
10
15
20
-4
5
10
Figure 3: Response of Variables on Financial Friction to Lending E¢ ciency Shocks
35
log_Y
0
5
x 10
-4
log_mt
4
-0.005
x 10
-4
log_mpk_r
2
-0.01
0
0
-0.015
-0.02
5
1
x 10
-3
10
15
20
10
15
20
-2
5
-3
10
-2
5
log_tfp
0
-1
x 10
5
kc_ku
0
0
-5
15
20
10
-0.005
5
-0.01
0
-0.015
5
log_H
10
15
20
x 10
-4
-5
5
log_I
15
20
15
20
15
20
ext_p
10
log_C
0
-2
10
0
-0.02
-0.005
-0.04
-0.01
-4
-6
-8
5
10
15
20
-0.06
5
10
15
20
-0.015
5
10
Figure 4: Response of Macro Variables to Aggregate Technology Shocks
0
x 10
-3
log_Y
2
-2
x 10
-4
log_mt
2
x 10
-4
log_mpk_r
1
0
0
-4
-2
-6
1
5
x 10
-3
10
15
20
-1
-4
kc_ku
1
5
x 10
-4
10
15
20
-2
log_tfp
4
0.5
0.5
2
0
0
0
-0.5
-0.5
-1
0
5
x 10
-3
10
15
20
-1
10
15
20
-0.02
5
10
-4
log_I
-4
-6
5
0
-0.01
15
20
-0.03
x 10
10
15
20
15
20
15
20
ext_p
-2
log_H
-2
5
-4
2
5
x 10
-3
10
log_C
0
-2
5
10
15
20
-4
5
10
Figure 5: Responses of Macro Variables to Marginal E¢ ciency Shocks to Investment
36