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Working Paper Series
Financing Development: The Role of
Information Costs
WP 08-08R
Jeremy Greenwood
University of Pennsylvania
Juan M. Sanchez
Federal Reserve Bank of Richmond
Cheng Wang
Iowa State University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/economic_
research/working_papers/index.cfm
Version: August, 2009
Financing Development: The Role of Information Costs
by
Jeremy Greenwood, Juan M. Sanchez and Cheng Wang
Working Paper No. 08-08R
Abstract
To address how technological progress in …nancial intermediation a¤ects the economy, a costlystate veri…cation framework is embedded into the standard growth model. The framework has two
novel ingredients. First, …rms di¤er in the risk/return combinations that they o¤er. Second, the
e¢ cacy of monitoring depends upon the amount of resources invested in the activity. A …nancial
theory of …rm size results. Undeserving …rms are over …nanced, deserving ones under funded.
Technological advance in intermediation leads to more capital accumulation and a redirection of
funds away from unproductive …rms toward productive ones. Quantitative analysis suggests that
…nance is important for growth.
Keywords: costly-state veri…cation, economic development, …nancial intermediation, …rm size
JEL Nos: E13, O11, O16
A- iations: University of Pennsylvania, Federal Reserve Bank of Richmond, and Iowa State University
An abridged version of this paper, containing the theory sections, is forthcoming in the American
Economic Review. A variant of the model with credit rationing is developed in NBER Working Paper No.
13104.
1
Introduction
Financial development “accelerates economic growth and improves economic performance
to the extent that it facilitates the migration of funds to the best user, i.e. to the place in
the economic system where the funds will earn the highest social return,” noted Goldsmith
(1969, p. 400) some thirty …ve years ago. Information production plays a key role in this
process of steering of funds to the highest valued users. If the costs of information production drop, then …nancial intermediation should become more e¢ cient with an associated
improvement in economic performance. A theoretical model is developed where reductions
in the cost of information processing allow for more e¢ cient capital allocation. Quantitative analysis suggests that cross-country di¤erences in the e¢ ciency of intermediation may
account for a substantial, but not the dominant, part of cross-country income di¤erentials.
Some stylized facts of the kind illustrated in Figure 1 were o¤ered by Goldsmith (1969)
to suggest that …nancial intermediation might be important: First, the ratio of business debt
to GDP has risen. In 1952 business debt was 30 percent of GDP.1 Today it is 65 percent.
Second, the value of …rms relative to GDP has also moved up. In 1951 …rms were worth
50 percent of GDP, while today they are valued at 176 percent. This may be evidence of
improved intermediation; now it is easier for …rms to enter stock and bond markets and raise
funds.
Direct measures of the impact of improved e¢ ciency in the …nancial system on the
economy are hard to come by. Improvements in the e¢ ciency of …nancial intermediation,
due to improved information production, are likely to reduce the spread between the internal
rate of return on investment in …rms and the rate of return on savings received by savers.
The spread between these returns re‡ects the costs of intermediation. This spread will
include the costs of ex ante information gathering about investment projects, the ex post
information costs of policing investments, and the costs of misappropriation of savers’funds
by management, unions, etc., that arise in a world with imperfect information. One may
1
Data sources are provided in Appendix 14.2.
1
2.0
2.0
0.7
Debt
Capital
0.5
1.0
0.4
0.5
1.8
Capital / Output
Business Debt / GDP
1.5
Market Value of Firms / GDP
0.6
1.6
1.4
Value
0.3
1940
1960
1980
1940
2000
1960
1980
Year
Year
Figure 1: Trends in …nancial intermediation
2
2000
observe little change in the rate of return earned by savers over time, because aggregate
savings will adjust in equilibrium so that this return re‡ects savers’rates of time preference.
If the wedge between the internal rate of return earned by …rms and the rate of return
received by savers falls, due to more e¤ective intermediation, then the capital stock in the
economy should rise. Indeed, there is some evidence that this may be the case. Figure 1
plots the capital-to-GDP ratio for the business sector of the economy, where the capital stock
includes intangible investments such as …rms’research and development, following the lead
of Corrado et al. (2006). It has increased signi…cantly over the postwar period. Of course, an
economy’s capital-to-GDP ratio may rise on other accounts, too. For example, lower taxes
on capital income should increase it, as should declines in the price of capital goods due to
investment-speci…c technological advance. To the extent that capital stock measures exclude
intangibles, or fail to incorporate improvements in quality due to embodied technological
progress, more e¤ective intermediation will be picked up as increases in productivity, instead.
There is evidence of the above phenomena in the cross-country data, too. Figure 2
plots the relationship between interest-rate spreads and output for a sample of 49 countries.
As can be seen, there is a negative association. Additionally, capital-to-output ratios and
output are positively related, in a sample of 48 countries. Hence, there is evidence of capital
deepening both in the US time-series data and in the cross-country data.
While the above facts are stylized, to be sure, it will be noted that empirical researchers
have used increasingly sophisticated methods to tease out the relationship between …nancial
intermediation and growth. This literature is surveyed masterfully by Levine (2005). The
upshot is that …nancial development has a causal e¤ect on economic development; speci…cally, …nancial development leads to higher rates of growth in income and productivity.
A general equilibrium model of …rm …nance, with competitive intermediation, is presented to address the impact that …nancial intermediation has on economic development.
At the heart of the framework developed here is a costly-state veri…cation paradigm that has
its roots in classic work by Robert M. Townsend (1979) and Stephen D. Williamson (1986).
The model here has two novel ingredients, though. First, in the standard costly-state veri…-
3
2.4
0.14
0.12
1.6
Interest-rate spread
k/o (capital to GDP)
2.0
1.2
0.8
0.4
0.10
0.08
0.06
0.04
0.02
0.00
0.0
0
5000
10000 15000 20000 25000 30000 35000
0
5000
10000
15000
20000
25000
30000
GDP per capita
GDP per capita
Figure 2: The cross-country relationship between per-capita GDP, on the one hand, and
interest-rate spreads and capital-output ratios on the other
cation paradigm, the realized return on a …rm’s investment activity is private information.
This return can be monitored, but the outcome of this auditing process is deterministic:
once monitoring takes place, the true state of the world is revealed with certainty. This is
true whether or not a deterministic decision rule for monitoring is employed, as in Townsend
(1979) or Williamson (1986, 1987), or a stochastic one is used, as in Ben Bernanke and Mark
Gertler (1989) or John H. Boyd and Bruce D. Smith (1994)— see also Townsend (1979, Section 4). By contrast, in the current setup the outcome of monitoring is random. Speci…cally,
the probability of detecting malfeasance depends upon the amount of resources devoted to
policing the returns on a project and the e¢ ciency of the monitoring technology. In the
model a project’s funding and level of monitoring will be jointly determined. The information asymmetry between the …rm and the intermediary gives the …rm an opportunity to
exploit its private knowledge about the realization of the investment return. In particular,
it can extract rents from the intermediary, and hence savers.
Second, a …rm’s production technology is subject to idiosyncratic randomness. This is
true in the standard costly-state veri…cation framework as well. Here, though, there is a
distribution across …rms over the distribution of these returns. In particular, some …rms
4
may have investment projects that o¤er low expected returns with little variance, while
others may have projects that yield high expected returns with a large variance.
Two key features in the analysis follow from these ingredients. First, the setup yields
a …nance-based theory of the equilibrium size distribution for …rms. This theory derives
from the facts that: (a) investment opportunities di¤er in the expected returns and levels of
risk that they o¤er and (b) producing information about these returns is costly. A simple
threshold rule for funding results. All …rms with an expected return at least as great as
the cost of raising capital are funded. Funding is increasing in a …rm’s expected return,
and is decreasing in its risk. Loan size is determinate for a given type of project because
the costs of …nancing a project are increasing and convex in the level of the monitoring
activity. Thus, high mean projects receive limited funding because the costs of monitoring
will rise disproportionately with loan size. The riskier a project is, the bigger is the di¤erence
between the returns in good and bad states. This increases the incentive for a …rm to lowball its earnings in good states. Hence, more diligent monitoring will be required, which
increases its …nancing cost. In an abstract sense, one could think that the diminishing
returns in information production modeled here provide a microfoundation for Robert E.
Lucas’s (1978) span of control model.
Second, the framework provides a link between the e¢ ciency of the …nancial system and
the level of economic activity. Such a tie was developed earlier in the models of Valerie
R. Bencivenga and Smith (1991), Greenwood and Boyan Jovanovic (1990), Levine (1991),
and Albert Marcet and Ramon Marimon (1992). The analysis here provides a crystal clear
delineation of the Goldsmith (1969) mechanism, however. It stresses the connection between
the state of technological development in the …nancial sector, on the one hand, and capital
accumulation, both along the extensive and intensive margins, on the other. If technological
improvement in the …nancial sector occurs at a faster pace than in the rest of the economy,
then …nancial intermediation becomes more e¢ cient. Loans are monitored more diligently
and the rents earned by …rms shrink. Additionally, lending activity will change along both
extensive and intensive margins. Projects with high (low) expected returns will now receive
5
more (less) funds. Those investments with the lowest expected returns will be cut. At high
levels of e¢ ciency in the …nancial sector the economy approaches the …rst-best equilibrium
achieved in a world without informational frictions. This reallocation e¤ect distinguishes
the analysis from earlier research by Shankha Chakraborty and Amartya Lahiri (2007) and
Aubhik Khan (2001) that also embed the standard costly-state veri…cation framework into
a growth model. In these frameworks all …rms receive the same amount of capital.
In order to assess the importance of …nancial intermediation for the level of economic
activity, the developed model is solved numerically. The model is calibrated to match some
stylized facts for the U.S. economy. It is found that the model can replicate the increase
in the U.S. capital/output ratios, shown in Figure 1. It does a very good job matching the
…rm-size distribution for the U.S. economy for the year 1972. The improvement in …nancial
sector productivity required to generate the observed rise in the U.S. capital/output ratio also
appears to be reasonable. In the baseline model, improvements in …nancial intermediation
account for 1/3 of U.S. growth.
The calibrated model is then taken to the cross-country data. Others have tried to
quantify the importance of factors such as limited investor protection [Castro et al. (2009)]
for economic development. The emphasis here is on taking a micro-foundation …nance model
to the data in order to quantify the importance of information production for economic
development. Townsend and Ueda (2006) estimate a version of the Greenwood and Jovanovic
(1991) model using Thai data. Their model emphasizes the role that …nancial intermediaries
play in producing ex ante information about the state of the economy at the aggregate or
sectorial levels. Financial intermediaries o¤er savers higher and safer returns. In the current
analysis, intermediaries produce ex post information about …rm-level returns. The return
earned by savers is …xed (and therefore safe) in the analysis. The gain to the economy
from improved intermediation derives from squeezing, via information production, the rents
appropriated by …rms on capital investment. This leads to a reduction in borrowing rates
and more e¢ cient capital allocation.
The model’s imputed measures for the e¢ ciency of …nancial intermediation in var-
6
ious countries match up very well with independent measures. It also does a reasonable
job predicting the di¤erences in cross-country interest-rate spreads or capital/output ratios
shown in Figure 2. Financial intermediation turns out to be important quantitatively. For
example, in the baseline model Sri Lanka would increase its GDP by 70 to 76 percent if it
could somehow adopt Switzerland’s …nancial system. World output would rise by 28 to 48
percent if all countries adopted Switzerland’s …nancial practice. Still, the bulk (or 81 to 87
percent) of cross-country variation in GDP cannot be accounted for by variation in …nancial
systems.
2
The Environment
Imagine a world resting in a steady state that is made up of three types of agents: consumer/workers, …rms and …nancial intermediaries. In a nutshell, …rms produce output using
capital and labor. The consumer/worker supplies the labor, and intermediaries, the capital.
All funding for capital must be raised outside of the …rm. Financial intermediaries raise
the funds for capital from consumer/workers. They also use labor in their lending activity.
Output is used for consumption by consumer/workers and for investment in capital by intermediaries. The behavior of …rms and intermediaries will now be described in more detail.
The consumer/worker plays a more passive role in the analysis, which is relegated into the
background by assuming that he supplies one unit of labor and saves at some …xed interest
rate, rb.2
3
Firms
Firms produce output, o, in line with the production function
o = k l1
2
;
Think about a representative consumer with time separable preferences over consumption. The steady
state interest rate will then be given by rb = 1=
1, where is his discount factor.
7
where k and l represent the inputs of capital and labor used in production. The variable
gives the productivity level of the …rm’s production process. Productivity is a random
variable drawn from a two-point vector
and Pr( =
1 2( 1
2)
2
2) ,
=
2
1.
=1
respectively.3
( 1;
with
2)
1
The mean and variance of
<
2.
Let Pr( =
are given by
1 1
+
1)
=
2 2
1
and
Thus, for a given set of probabilities these statistics di¤er
in accordance with the values speci…ed for
and
1
2.
The realized value of
is the …rm’s
private information.
Now, the productivity vector, , di¤ers across …rms. In particular, suppose that …rms
in the economy are distributed over productivities in line with the distribution function
F : T ! [0; 1], where T
2
R+
and
F (x; y) = Pr(
1
x;
2
y).
Think of this distribution as somehow specifying a trade-o¤ between the mean and variance
of project returns. Due to technological progress in the production sector of the economy, this
distribution will evolve over time. Figure 3 plots the density function for F in mean/variance
space that is used in the quantitative analysis.
The …rm borrows capital, k, from the intermediary before it observes the technology
shock, . It does this with both parties knowing its type, . It can employ labor, at the
wage rate w, after it sees the realization for . In order to …nance its use of capital the …rm
must enter into a contract with a …nancial intermediary. Last, note that a …rm’s production
is governed by constant returns to scale. In the absence of …nancial market frictions no rents
would be earned on production. Additionally, in a frictionless world only …rms o¤ering the
highest expected return would be funded. With …nancial market frictions, deserving projects
are underfunded, while undeserving projects are simultaneously over funded.
3
Observe that 1
2
= (1
1 ) 1 1 + (1
2
1 + (1
2
1) 1 2
2
1) 2
2
[
1 1 + (1
1 2 1 (1
1)
2
1) 2]
= (1
8
=
2
1 1
1) 1[
2 2
1 1
2
2
1+ 2
2 1 2 1+2
2 1 2 ] = (1
2
2
1 2 1+ 2 1
1) 1( 1
2 2
2 1
2
2) .
density
0.0006
0.0004
0.0002
0.0000
5
mean, θ
10
15
20
2
4
6
8
10
12
standard deviation, θ
Figure 3: The distribution for F in mean/variance space that is used in the baseline quantitative model
3.1
Pro…t Maximization by Firms
Consider the problem faced by a …rm that receives a loan in terms of capital in the amount k.
The …rm hires labor after it sees the realization of its technology shock, . It will do this in
a manner so as to maximize its pro…ts. In other words, the …rm will solve the maximization
problem shown below.
R( ; w)k
maxf k l1
(P1)
wlg:
l
The …rst-order condition associated with this maximization is
(1
) k l
= w;
which gives
l=
(1
)
1=
(1)
k:
w
Substituting the solution for l into the maximand and solving yields the unit return function,
R( ; w), or
r = R( ; w) = (1
)(1
9
)=
w
(1
)=
1=
> 0:
(2)
Think about ri = R( i ; w) as giving the gross rate of return on a unit of capital invested in
the …rm given that state
while the variance reads
4
i
occurs. The expected gross rate of return will be
1 2 (r1
1 r1
+
2 r2 ,
r2 )2 .
Financial Intermediaries
There is a competitive intermediation sector that borrows funds from consumers and lends
capital to …rms. While the intermediary knows a …rm’s type it cannot observe the state of
a …rm’s business either costlessly or perfectly.4 That is, the intermediary cannot costlessly
observe , o and l. The …rm will make a report to the intermediary about its business
situation. The intermediary can devote some resources in order to assess the veracity of
this report. The payments, p, from a …rm to the intermediary will be conditioned both
upon the report made by former, and the outcome of any monitoring activity done by latter.
By channelling funds through …nancial intermediaries consumers avoid a costly duplication
of monitoring e¤ort that would occur in an equilibrium with direct lending between them
and …rms— see Diamond (1984) and Williamson (1986) for more detail. Likewise, in the
environment under study, it is optimal for a …rm to borrow from only one intermediary at a
time.
Suppose a …rm reports that the productivity on its project in a given period is
may di¤er from the true state
i.
j,
which
The intermediary can devote resources, mj , to verify this
claim. The probability of detecting fraud is increasing in the amount of resources devoted
to this activity. In particular, let Pij (mj =k) denote the probability that the …rm is caught
cheating conditional on the following: (1) the true realization of productivity is
…rm makes a report of
j;
i;
(2) the
(3) the intermediary spends mj in monitoring; (4) the total
4
Recall that the intermediary knows the …rm’s type, . One could think about this as representing the
activity, industry or sector that a …rm operates within. For instance, Castro et al (forthcoming, Figure 3)
present data suggesting that the capital goods sector is riskier than the consumption goods one. It would be
possible to have a screening stage where the intermediary veri…es the initial type of a …rm. The easiest way
to do this would be to have them pay a …xed cost to discover . If the …rm’s type can’t be undercovered
perfectly, as in the classic work of Boyd and Prescott (1986), then it may be possible to design the contract
to reveal it.
10
amount of borrowing is k (which represents the size of the project). The function Pij (mj =k)
is assumed to be monotonically increasing in mj =k. Additionally, let Pij (mj =k) = 0 if the
…rm truthfully reports that its type is
i.
Any lender to the …rm must monitor the whole
project to detect cheating, because his claim to pro…ts will depend on the total level of
receipts vis à vis the total amount of disbursements paid out to others. Borrowing through
a single intermediary then avoids a costly duplication of monitoring e¤ort.
A convenient formulation for Pij (mj =k) is
8
>
>
1 ( mj1=k) < 1; with 0 < < 1;
>
>
>
>
< for a report 6= and m =k > 1= ;
j
i
j
Pij (mj =k) =
>
>
0;
>
>
>
>
:
for a report j = i or mj =k 1= :
To guarantee that Pij (mj =k)
0, this speci…cation requires that some threshold level of
monitoring, mj > k= , must be exceeded to detect cheating. Note that this threshold level
of monitoring can be made arbitrarily small by picking a large enough value for ".5
Also,
an arbitrarily large value for " can be chosen so that the threshold level of monitoring is
very small. Figure 4 makes this clear, while illustrating the function Pij (mj =k).
Monitoring is a produced good, measured in units of consumption. The production of
monitoring is project speci…c. Monitoring produced for detecting fraud in one project cannot
be used in a di¤erent one. Let monitoring be produced in line with the production function
1=
m = zlm
; with 0
1=
1,
where lm represents the amount of labor employed in this activity. The cost function,
C(m=z; w), associated with monitoring is given by
C(m=z; w) = w(m=z) :
Costs are linear in wages, w. With diminishing returns to scale in production (1= < 1), the
cost function is increasing and convex in the amount of monitoring, m, and decreasing and
5
The choice of " can be thought of as normalization relative to the level of productivity in the production
of monitoring services–see footnote 7.
11
1.0
0.8
0.8
(m/z)γ, monitoring cost in labor
Pij(mj/k), probability of detection
1.0
0.6
0.4
0.2
0.0
0.6
for z = 1.0
0.4
for z = 1.25
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
mj/k
0.4
0.6
0.8
1.0
m/z
Figure 4: The functions Pij (mj =k) and C(m=z; w)=w, for " = 100,
(the values used in the quantitative analysis)
= 0:52, and
= 1:86
convex in the state of the monitoring production technology, z. Figure 4 portrays the cost
of monitoring in terms of labor, or plots C(m=z; w)=w.
Now, exactly which …rms are funded depends on three things: (1) the …rm’s type, ;
(2) the state of the monitoring production technology in the …nancial intermediation sector ,
z; (3) the expense of monitoring e¤ort as re‡ected by the wage, w. As will be seen, when
the variance of a …rm’s project becomes larger, the informational problems associated with
contracting become more severe. Therefore, high variance projects are less likely to get
funded, ceteris paribus.
5
The Financial Contract
A contract between a …rm and an intermediary is summarized by the quadruple fk; pj ; pij ; mj g.
Here k represents the amount of capital lent by the intermediary to the …rm, pj is the …rm’s
payment to the intermediary if it reports
bank if the borrower reports
j
j
and is not found cheating, pij is payment to the
and monitoring reveals that productivity is
12
i
6=
j,
and mj
is the intermediary’s monitoring e¤ort when
j
is reported. Denote the value of the …rm’s
outside option by v.
The intermediary chooses the details of the …nancial contract, fk; pj ; pij ; mj g, to maximize its pro…ts. The contract is designed to have two features: (1) it entices truthful
reporting by …rms; (2) it o¤ers …rms an expected return of v. The optimization problem is
I( ; v)
max
p1 ;p2 ;p12 ;p21 ;m1 ;m2 ;k
f 1 p1 +
rek
2 p2
subject to
1 w(m1 =z)
2 w(m2 =z)
g;
(P2)
p1
r1 k;
(3)
p2
r2 k;
(4)
p12
r1 k;
(5)
p21
r2 k;
(6)
[1
P12 (m2 =k)](r1 k
p2 ) + P12 (m2 =k)(r1 k
p12 )
r1 k
p1 ;
(7)
[1
P21 (m1 =k)](r2 k
p1 ) + P21 (m1 =k)(r2 k
p21 )
r2 k
p2 ;
(8)
and
1 (r1 k
p1 ) +
2 (r2 k
p2 ) = v:
(9)
Note that the cost of capital, re, is given by re = rb + ; i.e., the interest paid to investors
plus the depreciation on capital. The …rst four constraints just say the intermediary cannot
demand more than the …rm earns; that is, the …rm has limited liability. Equations (7) and (8)
are the incentive-compatibility constraints. Take (8). This simply states that the expected
return to the …rm from reporting state one when it actually is in state two, as given by the
left-hand side, must be less than telling the truth, as represented by the right-hand side.
Observe that the constraint set is not convex due to the way that m1 enters (8). Therefore,
the second-order conditions for the maximization problem are important to consider. The
last constraint (9) speci…es that the contract must o¤er the …rm an expected return equal to
v, its option value outside. A …rm’s outside option is the expected return that it could earn
13
on a loan from another intermediary. This will be determined in equilibrium. Finally, note
the solution for fpj ; pij ; mj ; kg is contingent upon the …rm’s type,
= ( 1;
2 ).
To conserve
on notation, this dependence is generally suppressed.
The lemma below characterizes the solution to the above optimization problem.
Lemma 1 (Terms of the Contract) The solution to problem (P2) is described by:
1. The size of the loan from the intermediary to the …rm, k, is
k=
v
2 (r2
r1 )[1
P21 (m1 =k)]
(10)
:
2. The amount of monitoring per unit of capital when the …rm reports a bad state, m1 =k,
solves the problem
I( ; v)
maxf( 1 r1 +
m1 =k
2 r2
where k can be eliminated using (10) above.
re)k
1w
z
k (
m1
)
k
vg;
(P3)
(a) Monitoring in the bad state is simply given by
m1 = (m1 =k)k;
where m1 =k solves (P3).
(b) The intermediary does not monitor when the …rm reports a good state so that
m2 = 0:
(11)
3. The payment schedule is
p1 = r1 k;
p 2 = r2 k
v= 2 ;
(12)
(13)
p12 = r1 k;
(14)
p21 = r2 k:
(15)
Proof. See Appendix 14.1.
It is intuitive that there are no bene…ts to the …rm from claiming a better outcome than
it actually realizes, since it will only have to pay the intermediary more. The intermediary
would like to reduce the …rm’s incentive to report being in the low state. So, suppose the
14
…rm reports a low state. If cheating is not detected, then the …rm pays all of its revenue
(minus labor cost) that would be realized in the low state— see (12). If the …rm is caught
cheating, then it must surrender all of the revenue (sans labor cost) that it earns in the high
state— see (15). Note that due to the incentive-compatibility constraints a false report will
never occur so that the payments shown by (14) and (15) do not occur in equilibrium.
The contract speci…es that the intermediary should only monitor the …rm when it reports
a bad outcome (state 1) on its project— see (11). Monitoring in the low state is done to
maximize the intermediary’s pro…ts, subject to the incentive-compatibility cum promisekeeping constraint (10), as problem (P3) dictates. Note that the higher is the value of the
…rm, v, the bigger must be the loan, k, to satisfy the incentive-compatibility cum promisekeeping constraint (10). This constraint (10) ensures that the contract provides the …rm an
expected return equal to what it would earn if it misrepresented the outcome in the good
state,
2 (r2
r1 )[1 P21 (m1 =k)]k. Furthermore, this expected return is set equal to the …rm’s
outside option, v. The size of the loan, k, is increasing in the amount of monitoring that
occurs in the low state, m1 =k. This happens because the probability of the …rm not getting
caught from misrepresenting its revenues, 1
P21 (m1 =k), is decreasing in the intermediary’s
monitoring activity.
The theory of intermediation presented here is at an abstract level, as in Boyd and
Prescott (1986). It is not intended to explain the presence of real world forms of intermediation, such as the existence of banks, bond markets, or stock markets. Levine (2005)
notes that, empirically speaking, it is of secondary importance whether the source of the
development in …nancial systems arises from improvements in banks, stock markets or bond
markets.
Now, a …nancial contract will be o¤ered by an intermediary to a …rm only if it yields
the former nonnegative pro…ts, I( ; v)
0. Suppose that r1 < re. A necessary condition
for a contract to yield nonnegative pro…ts is for the intermediary to devote more than the
minimal level of resources per unit of funds lent, 1= , to monitoring a report of a bad state.
If this is not done, the …rm will always claim that it is in the low state, and the intermediary
15
can only earn a loss on the contract. Brie‡y consider the case where r1
re. Here the …rm’s
return on capital in its worst state of nature is at least as large as the cost of capital, re.
Firms would desire to borrow an in…nite amount of capital. An equilibrium will not exist.
Assumption: r1 = R( 1 ; w) < re = rb + , for all …rm types.
Later, a lower bound on the level of productivity in the …nancial sector, , will be imposed
that guarantees that this assumption holds whenever z > . This lower bound ensures that
the equilibrium wage, w, is high enough so that the assumption will always hold–see (2).
Lemma 2 (Interior solution for monitoring) m1 =k > 1= , for all v > 0.
Proof. The argument is presented in Appendix 14.1.
6
Competitive Financial Intermediation
In the economy there is perfect competition in the …nancial sector. Consequently, an intermediary must o¤er a contract that maximizes a …rm’s value, subject to the restriction that
the former does not incur a loss. If an intermediary failed to do so it would be undercut by
others. The upshot is that intermediaries will make zero pro…ts on each type of loan. Furthermore, for a …rm to produce, it must make non-negative pro…ts too. It is intuitive that for
all this to happen a project must o¤er some potential for a surplus or that
1 r1
+
2 r2
re;
i.e., the expected return on capital must exceed its cost. Assume this. The value of a …rm,
v, is then determined by the condition
v =V( )
arg maxfx : I( ; x) = 0g:
x
(16)
The implications of perfect competition will now be analyzed. Key questions are: (ii) What
will be the loan size? (ii) Which …rms will get funded?
The size of a loan for a project, k, can now be determined. To do this, substitute (10)
in Problem (P3) and solve for the optimal level of monitoring, m1 =k. Plug this solution for
16
monitoring in the objective function for (P3) to obtain a formula for I( ; v).6
Next, solve
for v using condition (16). Plugging the obtained formulae for m1 =k and v into (10) yields
k = (
+
)
( 1 r1 +
2 r2
=(
)
re)(
1
( )
=(
+ )=(
)
)
(
1
)(
+
( 1 w)
=(
+ )=(
)
)
(17)
( z)
]
r1 )
2 (r2
[
=(
)
:
Equation (17) gives a determinate loan size for each type of funded project. Furthermore,
funding is increasing in a project’s expected return and is decreasing in its volatility.7
Lemma 3 (Loan size) The level of investment in a …rm, k, is increasing in its expected
net return, 1 r1 + 2 r2 re, and the state of technology in the …nancial sector, z, and is
decreasing in the variance of the return, r2 r1 (holding the wage rate, w, …xed).
Proof. See Appendix 14.1.
Attention will now be directed toward determining which projects will be funded. Consider the set of …rms, A(w), de…ned by
A(w)
f :
1 r1
+
2 r2
re > 0g:
(18)
Intuitively, one might expect that this set of projects will be funded in equilibrium because
they o¤er an expected return on capital,
1 r1
+
2 r2 ,
that is greater than its user cost, re.
This turns out to be true. The contracting problem (P2) implies that the …rm will never
make negative pro…ts, given (3) to (6). The construction of equation (17) suggests that the
intermediary will be able to make a loan in this situation, and not incur a loss.
6
The solution obtained for I( ; v) is
I( ; v) = (
1
)( )
+
(
1 w)
=(
+
=(
+
1
)( + )=( +
+
( z)
)
[
] =( +
r1 )
2 (r2
)
)
(
)
v
(
=(
1 r1
+
+
2 r2
)
v:
re)(
+ )=(
+
)
This solution presumes that r1 < re, so there is an interior solution for monitoring, and that 1 r1 + 2 r2 re.
It is easy to see that the intermediary’s pro…t function, I( ; v), is \ shaped with the following properties:
(i) I( ; 0) = 0; (ii) limv!0 @I( ; v)=@v = 1; (iii) @ 2 I( ; v)=@v 2 < 0; (iv) limv!1 I( ; v) = 1. Therefore,
there is only one v > 0 that solves (16).
7
Observe that loan size is a function of z. In this sense, the choice of the constant
detection function can be thought of as normalization relative to z.
17
in the odds of
Lemma 4 (The set of funded …rms) A necessary and su¢ cient condition for a type- …rm
to be active or funded, or for k( ) > 0; I( ; V ( )) = 0 and V ( ) > 0, is that 2 A(w).
Proof. Those interested should go to Appendix 14.1.
Therefore, as was mentioned in the introduction, a simple threshold rule exists for funding,
as characterized by (18). Call A(w) the set of active …rms. From (2) it is easy to see that
1 r1
+
2 r2
)(1
re = (1
)=
w
(1
)=
[
1=
1 1
+
1=
2 2
]
re
0:
Observe that the …rm’s pro…ts are decreasing in wages. A type- …rm will operate when
w
W ( ), and will not otherwise, where the cuto¤ wage, W ( ), is speci…ed by
W( )
=(1
)
(1
)(
1=
1 1
+
re
1=
2 2
)
=(1
)
:
(19)
So, the set of active projects A(w) can be expressed equivalently as
A(w) = f : w < W ( )g.
(20)
The active set depends on the wage because ri = R( i ; w). It contracts (expands) with a
rise (decrease) in the real wage, since R( i ; w) is decreasing in w. In equilibrium the wage
rate, w, turns out to be increasing function of the state of technology in the …nancial sector,
z. Hence, the active set will shrink with technological improvement in the …nancial sector,
or a rise in z. This will become clearer in Section 8.
Figure 5 summarizes the discussion on funding. As the expected return on a project rises
more capital is allocated to it, as is illustrated in the …rst panel. (Note that the direction
of the x and y axes is speci…c to each panel.) The increase in funding (or the scale of the
…rm) is associated with higher monitoring costs given the increasing, convex form of the
cost function (fourth panel). The amount of monitoring done per unit of capital is then
economized on (third panel). As a consequence, the odds of detecting fraud drop (second
panel). In response the expected rents earned by the …rm rise (…fth panel).
As the risk associated with a project rises its funding is slashed (…rst panel). When the
di¤erence between the good and bad state widens there is more incentive for the …rm to
18
falsify its earning. The intermediary therefore monitors more per unit of capital lent (third
panel). Total monitoring costs fall with risk, because the size of the loan is smaller (fourth
panel). The probability of detecting malfeasance therefore moves up since monitoring per
unit of capital is now higher (second panel). A …rm’s rents will fall with a rise in risk (…fth
panel), because it receives a smaller loan and faces more vigilant policing.
7
Stationary Equilibrium
The focus of the analysis is on stationary equilibria. First, the labor-market-clearing condition for the model will be presented. Second, a de…nition for a stationary equilibrium will
be given. Third, it will be demonstrated that a stationary equilibrium for the model exists.
On the demand side for labor, only …rms with
2 A(w) will be producing output. On
the supply side, recall that the economy has one unit of labor in aggregate. The labormarket-clearing condition will then appear as
Z
[ 1 l1 ( 1 ; 2 ) + 2 l2 ( 1 ; 2 ) +
1 lm1 ( 1 ; 2 )]dF ( 1 ; 2 )
= 1.
(21)
A(w)
It is now time to take stock of the situation so far by presenting a de…nition of the
equilibrium under study. It will be assumed that the economy rests in a stationary state
where the cost of capital is re = rb + .
De…nition 1 Set the steady-state cost of capital at re. A stationary competitive equilibrium is
described by a set of labor allocations, l and lm , a …nancial contract, fp1 ; p2 ; p12 ; p21 ; k; m1 ; m2 g,
a set of active monitored …rms, A(w), …rm values v, and a wage rate, w, such that:
1. The …nancial intermediary o¤ers a contract, fp1 ; p2 ; p12 ; p21 ; k; m1 ; m2 g, which maximizes its pro…ts, I, in accordance with (P2), given the cost of capital and wages, re
and w, and the value of …rms, v. The intermediary hires labor for monitoring in the
amount lm = (m=z) .
2. The …nancial contract o¤ered by the intermediary maximizes the value of a …rm, v, in
line with (16), given the prices re and w.
3. A …rm is o¤ered a contract if and only if it lies in the active set, A(w), as de…ned by
(18), given re and w. It hires labor, l, so as to maximize its pro…ts in accordance with
(P1), given, w, and the size of the loan, k, o¤ered by the intermediary.
19
(i)
k
P12(m1/k)
(ii)
0.015
0.025
0.030
0.020
0.040
0.035
Mean, r
0.030
0.025
Mean, r 0.035
Std dev, r
0.020
0.040
0.015
Std dev, r
(iv)
m1/k
w(m1/z)γ/k
(iii)
0.015
0.030
0.025
0.035
Mean, r
0.040
0.015
0.020
0.040
Std dev, r
Mean, r
(v)
0.040
Mean, r
0.030
0.035
0.030
0.025
Std dev, r
v
v/k
(vi)
0.015
0.015
0.020
0.035
0.020
0.040
0.025 Std dev, r
0.020
0.035
Mean, r
0.030
0.025
Std dev, r
Figure 5: The determination of …rm size as a function of the mean and standard deviation
of r = R( ; w)
20
4. The wage rate, w, is determined so that the labor market clears, in accordance with
(21).
When will an equilibrium exist for the economy under study? To address this question,
let
1
maxf
1
: ( 1;
2)
2 T , over all
2 g.
Next, de…ne the constant ! by the equation
R( 1 ; !) = re:
(22)
The constant ! speci…es a lower bound on the feasible equilibrium wage rate.8 When w = !
for a type-( 1 ;
2)
project it will happen that r1 = re. In this situation the intermediary could
simply ask for a payment of r1 k in both states of the world and engage in no monitoring. It
would earn zero pro…ts. The …rm’s pro…ts would be
2 (r2
r1 )k. It would desire a loan of
in…nite size. Therefore, as the equilibrium wage, w, approaches ! from above the equilibrium
de…ned above will eventually become tenuous.
The situation is portrayed in Figure 6, which graphs the demand and supply for labor.
The demand for labor is portrayed by the solid line labeled L. The properties of this demand
schedule are established during the course of the proof for Lemma 5. Demand is downward
sloping in w. The question is whether or not it will cross the vertical supply schedule for
labor. Now, at a given wage rate, loan size increases with more e¢ cient intermediation. This
leads to more labor being demanded when intermediation improves. In other words, it can
be shown that the demand for labor schedule shifts rightward with an upward movement in
z. Now, de…ne
as the level of z such that the demand curve intersects the supply curve at
the point (1; !).
Lemma 5 (Existence of an equilibrium) There is a constant
exists a stationary equilibrium for the economy.
such that for all z >
there
Proof. See Appendix 14.1.
8
The function R( ; w) is continuous and strictly decreasing in w, with limw!0 R( ; w) = 1, and
limw!1 R( ; w) = 0; hence, ! is well de…ned.
21
Wages
wˆ≡ max
W (τ )
τ
L, for z > ζ
w
ω
L, for z = ζ
1
Demand, Supply
Figure 6: Existence, demand and supply for labor
22
8
The Impact of Technological Progress on the Economy
The primary goal of the analysis is to understand how technological advance in the …nancial
sector a¤ects the economy. To this end, the impact that technological progress, in either the
…nancial or production sector, has on the portfolio of funded projects will be characterized.
To develop some intuition for the economy under study, some special cases will be examined.
8.1
Balanced Growth
In the …rst special case, technological progress in the …nancial sector proceeds in balance with
the rest of the economy. Speci…cally, assume that the economy is moving along a balanced
growth path where the
Therefore, Ft+1 ( 1 ;
2)
1=
i
’s grow at the common rate g 1= and z grows at rate g 1=(1
= Ft ( 1 =g;
2 =g).
) 9
.
The salient features of this case are summarized
by the proposition below.
1=
Proposition 1 (Balanced growth) Let the i ’s grow at rate g 1= and z increases at rate
g 1=(1 ) . There exists a balanced growth path where the capital stock, k, wages, w, and rents,
v, will all grow at rate g 1=(1 ) . The amount of resources devoted to monitoring per unit of
capital, m1 =k, remains constant.
Proof. Refer to Appendix 14.1.
In this situation the …nancial sector is not becoming more e¢ cient over time, relative to
the rest of the economy. The amount of monitoring done per unit of capital invested remains
constant over time. Thus, the probability of a …rm getting caught by misrepresenting a high
level of earnings, P21 (m1 =k), is constant over time too. For any particular project type, the
spread between the return on capital (net of labor costs) and the interest earned by investors,
1 r1 + 2 r2
re, is …xed over time. One could easily allow the production of monitoring services
1=
to require capital as well labor, say as given by m = zkm lm with 0 < 1= ; ; 1= +
Now, for a balanced growth path to obtain, z would have to grow at rate g
9
=(1
)
< 1.
. The
In order to get a …xed interest rate assume that the consumer/worker has isoelastic preferences over
consumption. Then, in standard fashion along a balanced growth path, rb = g =(1 ) =
1, where is the
coe¢ cient of relative risk aversion.
23
existence of a balanced growth path results from the fact that the probability of detection,
P21 (m1 =k), depends on the employment of monitoring services relative to the size of the loan.
If loan size did not enter this function, then technological advance in the non-…nancial sector
would lead to a drop in the cost of monitoring as the economy’s capital stock rose. That is,
there would be a positive feedback loop from the state of development in the non-…nancial
sector to the state of development in the …nancial one, as is modeled in three di¤erent ways
in de la Fuente and Marin (1996), Greenwood and Jovanovic (1990), and Harrison, Sussman
and Zeira (1999).
8.2
Unbalanced Growth
The above case suggests that for technological progress in the …nancial sector to have an
impact it must outpace advance in the rest of the economy. Suppose that this is the case.
Then, one would expect that as monitoring becomes more e¢ cient those projects o¤ering
the lowest expected return will be cut.
Proposition 2 (Technological progress in …nancial intermediation) Consider z and z 0 with
z < z 0 . Let w and w0 be the wage rates associated with z and z 0 , respectively. Then,
A(w0 ) A(w). Additionally, if = ( 1 ; 2 ) 2 A(w) A(w0 ) and 0 = ( 01 ; 02 ) 2 A(w0 ) then
1=
+ 2 ( 2 )1= < 1 ( 01 )1= + 2 ( 02 )1= .
1( 1)
Proof. See Appendix 14.1.
An increase in z makes …nancial intermediation more e¢ cient. For any given wage rate, w,
the aggregate demand for labor will increase for two reasons. First, more capital will be
lent to each funded project. Second, more labor will also be hired by the intermediary to
monitor the project. Since the demand for labor rises, the wage rate must move up to clear
the labor market. This increase in wages causes the set of active projects, A(w), to shrink,
with the projects o¤ering the lowest expected return being culled.
Alternatively, technological advance could occur in the production sector and not the
…nancial one. Here, the lack of development in the …nancial sector will hinder growth in
the rest of the economy. Speci…cally, technological advance in the production sector of the
economy will drive up wages. This leads to the costs of monitoring rising. Therefore, less
24
is done. This lack of scrutiny by intermediaries now allows …rms with marginal projects
o¤ering low expected returns to receive funding.
1=
Proposition 3 (Technological progress in production) Suppose all the i ’s increase by the
factor g 1= , holding z …xed. Then, the set of active projects, A(w), expands with the new
projects o¤ering lower expected returns than the old ones.
Proof. See Appendix 14.1.
8.3
E¢ cient Finance
An extreme example of Proposition 2 would be to assume that z grows forever. Then, the
…nancial sector will become in…nitely e¢ cient relative to the rest of the economy. This leads
to the fourth special case.
2
, with
Proposition 4 (E¢ cient …nance) Suppose T is a compact and countable subset of R+
a positive measure of projects for each type, = ( 1 ; 2 ). Then,
1. lim A(w) = A
1=
1 (x 1 )
arg max[
z!1
=(
2. lim m1 =z = 0; for
z!1
1=
2 (x 2 )
+
2A ,
3. lim m1 =k = 1 and lim P12 (m1 =k) = 1, for
z!1
z!1
4. lim p2 = r2 k; for
z!1
5. lim v = 0; for
];
1 ; 2 )2T
2A ,
2A ,
2A ,
z!1
6.
lim w = w
=(1
z!1
)
(1
)f
max[
=(
1=
1( 1)
+
1=
2( 2)
1 ; 2 )2T
]=e
rg
=(1
)
=(1
)
;
(23)
7.
lim
z!1
Z
A(w)
kdF = k
( )1=(1
re
)
f
max[
=(
25
1 ; 2 )2T
1=
1( 1)
+
1=
2( 2)
]g
:
(24)
Proof. Refer to Appendix 14.1.
As the cost of monitoring borrowers drops, the intermediation sector becomes increasingly e¢ cient. The …nancial intermediary can then perfectly police loan payments without
devoting a signi…cant amount of resources in terms of labor to this activity, as points (2)
and (3) in the proposition make clear. Since …rms are operating constant-returns-to-scale
production technologies, no rents will accrue on their activity— see point (5). Firms must
pay the full marginal product of capital to the intermediary— point (4). That is, the spread
between a …rm’s internal rate of return (before depreciation),
1 r1
+
2 r2
, and the user
cost of capital, re = rb + , vanishes, where the latter is made of the interest paid to savors, rb,
and rate of depreciation, . In this world only projects with the highest return are …nanced,
as point (1) states, even though they may be the most risky. In the aggregate any idiosyncratic project risk washes out. Therefore, in the absence of a contracting problem, only
the mean return on investment matters. And, with constant-returns-to-scale technologies
everything should be directed to the most pro…table opportunity. The wage rate, w , and
aggregate capital stock, k , in the e¢ cient economy are determined in standard fashion by
the conditions that the marginal product of capital for the most pro…table projects must
equal the user cost of capital, re, and the fact that the labor market must clear. These two
conditions yield (24) and (23). (By comparison, consider the standard deterministic growth
model with the production technology o = k l1
w
=(1
)
(1
)[
1=
=e
r]
=(1
)
and k
and one unit of aggregate labor. Here
[ =e
r]1=(1
)
(
1=
)
=(1
)
. The di¤erences in the
formulae are due to two facts that pertain to the current setting: (i) the best projects from
a portfolio T are chosen; (ii) there is uncertainty in .)
9
9.1
Fitting the Model to the U.S. Economy
Procedure
The quantitative analysis will now begin. To start with, assume that …rms now produce
output in line with the production function
26
o = x k l1
;
where x represents a known level of aggregate productivity. To simulate the model, values
must be assigned to its parameters. This will be done by calibrating the framework to match
some stylized facts for the U.S. economy. Some parameters are standard. They are given
conventional values. Capital’s share of income,
, is chosen to be 0:33, a very standard
number. Likewise, the depreciation rate, , is set to 0:07, again a very common number.
The chosen value for the discount factor,
= 1=(1 + rb), implies that the interest rate earned
by savers is 6:3 percent. This is in conformity with Cooley and Prescott’s (1995, p. 19)
estimate of 6:9 percent for the real return to capital over the postwar period. The concept
used for the capital stock is much narrower here, though; i.e., it is just the stock of business
capital. Therefore, matching the low observed capital-output ratio will be harder.
Nothing is known about the appropriate choice for parameters governing the intermediary’s monitoring technology, or ; ; and . The selection of a value for
a normalization (relative to some baseline level of z). Therefore, set
little is known about the distribution of returns facing …rms. Let
R
…rms of expected total factor productivity (TFP); i.e., m = (
wise,
v
=
R
v
m
amounts to
= 100. Similarly,
be the mean across
1 1
+
2 2 )dF .
Like-
will denote the mean over …rms of the logarithm of the volatility of TFP; i.e.,
ln[
1 2( 2
2
1 ) ]dF .
In a similar vein,
will represent the correlation between the
means and (ln) volatilities of …rm-level TFP, while
2
m
and
2
v
will denote the variance of
these …rm-level variables. Assume that these means and (ln) volatilities of …rm-level TFP
are distributed according to a bivariate truncated normal, N (
need to be selected for the parameters ; ;
m;
v;
2
m;
2
v;
m;
v;
2
m;
2
v;
). Thus, values
and .
Let Targetsj represent the j-th component of a n-vector of observations that the model
should match. Similarly, Output(param) denotes the model’s prediction for this vector.
The model’s solution will be a function of the list of calibrated parameters, param. These
parameters are picked to minimize a weighted sum of the squared deviations between the
27
data targets and the model’s output:
min
param
n
X
weightsj [Targetsj -Outputj (param)]2 ;
(25)
j=1
where weightsj represents the weight attached to j-th target. The data targets will now
be discussed.
The distribution of returns across …rms will be integrally related to the distribution of
employment across them. Firms with high returns will have high employment, other things
equal. Therefore, the size distribution of …rms for the year 1972 is chosen as a data target.
Eight points on this distribution are picked. As was mentioned in the introduction, to the
extent that …nancial intermediation has become more e¢ cient over time, relative to the
non…nancial sector, one would expect that the capital/output ratio would rise. If a nation’s
productivities in the …nancial and non-…nancial sectors increase in a balanced way then just
output will increase. Denote an aggregate quantity in bold. The model provides a mapping
between the aggregate level of output (per person), o, and the capital/output ratio, k=o, on
the one hand, and the state of technology in its production and …nancial sectors, x and z, on
the other. Represent this mapping by (o; k=o) = M (x; z)–the dependence of this mapping
on the model’s parameters has been suppressed for notational convenience. Now, while the
states of the U.S.’s …nancial and non-…nancial technologies are unobservable directly, this
mapping can be used to make an inference about (x; z), given an observation on (o; k=o),
by using the relationship
(x; z) = M
1
(26)
(o; k=o):
Therefore, TFP’s in the …nancial and non-…nancial sectors are picked to match GDP and the
capital/output ratio for the years 1972 and 2000. This determines x1972 , z1972 ; x2000 , and z2000 ,
as functions of the other estimated parameter values and, of course, the target values for o
and k=o. The calibrated parameter vector is param
(; ; ;
m;
v;
2
m;
2
v;
; x1972 ; z1972 ; x2000 ; z2000 ).
Thus, the calibration procedure is picking 12 parameters to match 12 observations, as best
as can be done subject to the restriction (26). Table 1 presents the parameter values for the
model.
28
Payroll, Cumulative Share
1.0
0.8
0.6
0.4
Data, 1972
Model, 1972
0.2
0.0
0.6
0.7
0.8
0.9
1.0
Establishments, Percentile
Figure 7: Firm-size distribution, 1972
9.2
Results
The model’s prediction for the 1972 …rm-size distribution is shown in Figure 7. It …ts
remarkably well. The …rm-size distribution shifts somewhat between 1972 and 2000. Now,
monitoring and the provision of …nancial services are abstract goods, so it hard to know what
a reasonable change in z should be. One could think about measuring productivity in the
…nancial sector, as is often done, by k=lm , where k is the aggregate amount of credit extended
by …nancial sector and lm is the aggregate labor that it employs. By this traditional measure,
productivity in the …nancial sector rose by 3.8 percent between 1973 and 2000. Berger
(2003, Table 5) estimates that productivity in the commercial banking sector increased
by 2.2 percent a year over this same period (which includes the troublesome productivity
slowdown) and by 3.2 percent from 1982 to 2000. The model’s ability to match cross-country
di¤erences in the e¢ ciency of …nancial intermediation and di¤erences in the interest-rate
spreads between borrowers and lenders will be addressed in Section 10.
29
Table 1: Parameter Values
Parameter
De…nition
Basis
= 0:33
Capital’s share of income
Standard value
= 0:07
Depreciation rate
Standard value
= 1=(1 + rb) = 0:94
Discount factor
Cooley and Prescott (1995)
= 100
Pr of detection, constant
Normalization
= 0:52
Pr of detection, exponent Calibrated to …t targets
= 1:86
Monitoring cost function
Calibrated to …t targets
means
Calibrated to …t targets
variances and correlation
Calibrated to …t targets
TFP’s
Calibrated to …t targets
m
= 1:54;
v
= 0:80
2
m
= 0:49;
2
v
= 0:66;
= 0:88
x1972 = 60:364; z1972 = 1:083e5;
x2000 = 88:549; z2000 = 6:785e5
Table 2 presents results for some other variables of interest. Here, the aggregate value
R
for a variable is again indicated in bold, so that x = A(w) xdF for x = m1 , w 1 (m1 =z) , k,
R
etc. The expected value for x is given by x
dF . Monitoring becomes less expensive
A(w)
as z rises (relative to x). This results in the amount of monitoring per unit of capital
rising, as re‡ected in the larger values for m1
k for 2000 versus 1972. As a consequence,
the likelihood of intermediaries detecting fraud increases. The fraction of a …rm’s output
dissipated in pure rents, v
o, declines. This can be seen another way. The internal rate
of return, i, earned by a …rm on its investment is given by i =
1 r1
+
2 r2
. The average
internal return earned by …rms, weighted by their level of investment, will then be de…ned
R
R
by i
kidF
[k
dF ]. Likewise, denote the average rate of return earned by
A(w)
A(w)
R
R
the intermediary on its lending activity by ei
( 1 p1 + 2 p2
k)dF [k A(w) dF ].
A(w)
The gap between these two returns, e
i ei, measures the average excess return earned by
…rms due to rents. This excess return is squeezed as rents shrink. In similar fashion, the
average spread between the rates of return that intermediaries and savers earn, s
ei
rb,
re‡ects the costs of intermediation incurred by the necessity to monitor borrowers. This
interest rate spread declines as the costs of intermediation fall due to technological progress
30
in information production. Rousseau (1998, Figure 4) presents evidence suggesting that
…nancial innovation reduced loan-deposit spreads in the U.S. between 1872 to 1929. Li and
Sarte (2003, Table 3), using a structural VAR, present evidence suggesting that drops in the
cost of …nancial intermediation account for a signi…cant part of long-run ‡uctuations in U.S.
manufacturing output.
A rise in the probability of detecting fraud relaxes the incentive constraint (8), and makes
it easier to lend more capital to …rms. This has two e¤ects. First, as borrowing rates decline
renting more capital becomes pro…table. This results in a higher aggregate amount of capital
being invested per unit of output produced, k
o. Second, for a given amount of lending,
funds are redirected toward those …rms o¤ering the highest rate of return. Along with the
…rst e¤ect, this increases GDP, o. Denote the levels of capital and output that would obtain
in the …rst-best economy by k and o . As can be seen, capital and output steadily rise,
relative to their …rst-best outcome, as z moves up (relative to x).
Model TFP is quite volatile across plants, as can be seen from Table 2. Hsieh and Klenow
(2008, Tables 1 and 2) report (weighted) standard deviations of 0.45 and 0.85 for 1977. The
(weighted) number found here lies in the middle of their range. For the U.S., they show little
or no increase in this number over time. This is similar to what is found here. Hence, the
volatility in plant-level TFP required to match the 1972 U.S. …rm size-distribution appears
to be reasonable.
Finance is important in the model. This can be gauged by undertaking the following
counterfactual question: By how much would GDP have risen between 1972 and 2000 if
there had been no technological progress in the …nancial sector? As can be seen from the
third column of Table 2, output would have risen from $22,097 to $34,590 or by about 1.6
percent a year (when continuously compounded). This compares with the increase of 2.4
percent ($22,097 to $43,268) that occurs when x rises to its 2000 level. Thus, about one
third of the increase in growth is due to innovation in the …nancial sector. Likewise, the
model predicts that about 15 percent of TFP growth was due to improvement in …nancial
intermediation. The …nancial system actually becomes a drag on development when z is not
31
allowed to increase. Wages rise as the rest of the economy develops. This makes monitoring
more expensive. Therefore, less will be done. As a consequence, interest rates rise and the
economy’s capital/output ratio drops. Without an improvement in the …nancial system, the
…rm-size distribution actually moves slightly in the wrong direction.
Table 2: Impact of Technological Progress
in the Financial Sector
1972
2000
Counterfactual
Production Sector, x
60.364
88.549
88.549
Financial Sector, z
1.083e5
6.785e5
1.083e5
0.1018
0.1738
0.0809
0.9651
0.9734
0.9608
1.029e6
3.004e6
1.431e6
0.1010
0.0768
0.1132
530.43
807.30
760.86
0.6953
0.6763
0.7131
0.1443
0.1174
0.1611
0.0787
0.0737
0.0818
0.0656
0.0436
0.0783
0.0149
0.0099
0.0180
1
0.0638
0.0638
0.0638
o
1.5400
1.7613
1.4279
Monitoring-to-capital, m1 k
R
Pr of detecting fraud, A(w) P12 dF
Financial sector productivity, k
lm
R
A(w)
dF
Rents to output, v o
R
R
TFP x A(w) ( 1 1 + 2 2 )dF
dF
A(w)
R
p
Std ln(TFP),
1 2 A(w) k( ln 2 ln 1 )dF
Internal return (weighted), i
Lending rate, ei
Excess return, e = i
ei
Interest rate spread, s = ei
Return to savers, rb = 1=
Capital-to-output ratio, k
rb
R
A(w)
dF
Capital relative to …rst best, k
k
0.4452
0.5628
0.3903
Output relative to …rst best, o
o
0.7128
0.7879
0.6740
$22,097 $43,268
$34,590
GDP, o
32
10
Fitting the Model to the World Economy
Ever since Goldsmith (1969), economists have been interested in the cross-country relationship between …nancial structure and economic development. An implication of the current
model is that as the state of technology in the intermediation sector advances, the spread
between borrowing and lending rates in an economy will shrink, while its capital-to-output
ratio and level of aggregate output increases. The cross-country data is suggestive of such a
relationship, as Figure 2 shows.
For the cross-country analysis assume that production in a nation is undertaken in the
manner described earlier, but let x and z now represent country-speci…c productivity factors
for the non-…nancial and …nancial sectors. While the state of a country’s productivities
in the non-…nancial and …nancial sectors is unobservable directly, once again the mapping
(26) can be used to make an inference about (x; z), given an observation on (o; k=o). This
is done for a sample of 40 countries, using the parameter values listed in Table 1. This
implies that the distribution of potential projects di¤ers across countries by the factor of
proportionality, x, an assumption needed both for discipline and tractability. The results
are reported in Table 7 in Appendix 14.2. By construction the model explains all the
variation in output and capital/output ratios across countries.10
Still, one could ask how
well the measure of the state of technology in the …nancial sector that is backed out using
the model correlates with independent measures of …nancial intermediation. Here, take the
ratio of private credit by deposit banks and other …nancial institutions to GDP as a measure
of …nancial intermediation, as reported by Beck et al. (2001). (Other measures produce
similar results but reduce the sample size too much.) Additionally, one could examine how
well the model explains cross-country di¤erences in interest-rate spreads, s.
10
The model predicts a positive association between a country’s rate of investment and its GDP. Castro
et al. (2009, Figure 1) show that this is true. It is stronger when investment spending is measured at
international prices, as opposed to domestic ones. They resolve this di¤erence within the context of a twosector model where the relative price of capital goods is endogenously determined. In their framework,
capital goods are more expensive to produce in poor countries. This happens because this sector is risky,
implying that the costs of …nance are high in countries with poor investor protection. Thus, this puzzle
could be resolved here by adopting aspects of their two-sector analysis.
33
Table 3 reports the …ndings. The correlation between the imputed state of technology
in the …nancial sector and the independent measure of …nancial intermediation is quite
high. Thus, it appears reasonable to use the constructed values of z for investigating the
relationship between output and …nancial development. Interestingly, Finland and Peru both
have a capital-to-output ratio of about 1.6. The model predicts Finland’s z is about 160
percent (continuously compounded) higher than Peru’s–the former’s ln(z) is 12.99, compared
with 11.40 for the latter; again, see Table 7 in Appendix 14.2. But, recall that the units for
ln(z) are meaningless, since monitoring is abstract good. If one measures productivity in the
…nancial sector by the amount of credit extended relative to the amount of labor employed
in the …nancial sector, as was done earlier, then the analysis suggests that intermediation
in Finland is about 145 percent (continuously compounded) more e¢ cient than in Peru.
Why? Finland has a much higher level of income per worker and hence TFP than does
Peru ($40,603 versus $10,200). Therefore, given the higher wages, monitoring will be more
expensive in Finland. To give the same capital/output ratio, e¢ ciency in Finland’s …nancial
sector must be higher. As can be seen, the interest-rate spreads predicted by the model are
positively associated with those in the data. The correlation is reasonably large. That these
two correlations aren’t perfect, should be expected. There are other factors, such as the big
di¤erences in public policies discussed in Parente and Prescott (2000), which may explain a
large part of the cross-country di¤erences in capital/output ratios. Di¤erences in monetary
policies across nations may in‡uence cross-country interest rate spreads. Additionally, there
is noise in these numbers given the manner of their construction— see Appendix 14.2.
Table 3: Cross-Country Evidence
interest-rate spread …nancial intermediation
Correlation(model, data)
0.37
34
0.72
11
The Importance of Financial Development for Economic Development
It is now possible to gauge how important e¢ ciency in the …nancial sector is for economic
development, at least in the model. To this end, note that the best …nancial and industrial
practices in the world are given by x = maxfxi g and z = maxfzi g, respectively. Represent
country i’s output, as a function of the e¢ ciency in its industrial and …nancial sectors, by oi =
O(xi ; zi )–this is really just the …rst component of the mapping M (x; z). If country i could
somehow adopt the best …nancial practice in the world it would produce O(xi ; z). Similarly,
if country i used the best practice in both sectors it would attain the output level O(x; z).
The shortfall in output from the inability to attain best practice is O(x; z)
O(xi ; zi ).
Luxembourg turns out to have the highest value for x, and Switzerland for z.
The percentage gain in output for country i by moving to best …nancial practice is given
by 100
[ln O(xi ; z)
ln O(xi ; zi )]. The results for this experiment are plotted in Figure 8.
As can be seen, the gains are quite sizeable. On average a country could increase its GDP
by 31 percent, and TFP by 10 percent. The country with the worst …nancial system, Sri
Lanka, would experience a 76 percent rise in output. Its TFP would increase by 26 percent.
While sizeable, these gains in GDP are small relative to the increase that is needed to move
a country onto the frontier for income, O(x; z). The percentage of the gap that is closed by
a movement to best …nancial practice is measured by 100
O(xi ; zi )]
100
the sample.11
[O(xi ; z)
O(xi ; zi )]=[O(x; z)
G(xi ; zi ). Figure 9 plots the reduction in this gap for the countries in
The average reduction is this gap is only 17 percent. For most countries
the shortfall in output is accounted for by a low level of total factor productivity in the
non-…nancial sector.
Therefore, the importance of …nancial intermediation for economic development depends
on how you look at it. World output would rise by 28 percent by moving all countries to
the best …nancial practice— see Table 4. This is a sizeable gain. Still, it would only close 12
11
Luxembourg has been deleted from the graph. The reduction in its gap is 100 percent.
35
percent of the gap between actual and potential world output. Dispersion in cross-country
output would fall by about 17 percentage points from 77 percent to 58 percent.12 Financial
development explains about 28 percent of cross-country dispersion in output by this metric.
Table 4: World-Wide move to financial best practice, z
Increase in world output (per worker)
28%
Reduction in gap between actual and potential world output
12%
Fall in dispersion of ln(output) across countries
17% ( ' 77% - 59%)
Fall in (pop-wghtd) mean of (cap-wghtd) distortion
15% ( ' 17% - 2%)
Fall in (pop-wghtd) mean dispersion of (cap-wghtd) distortion 4.65% ( ' 5% - 0.35%)
The presence of informational frictions causes the expected marginal product of capital,
1 r1
d=
+
2 r2 ,
1 r1
+
to deviate from its user cost, re. De…ne the induced distortion in investment by
2 r2
re. For a country such as Sri Lanka these deviations are fairly large. The
(capital-weighted) mean level of this distortion is 29 percentage points. It varies across plants
a lot, as indicated by a coe¢ cient of variation of 32 percent. This is the type of resource
misallocation e¤ect emphasized by Restuccia and Rogerson (2008). Here, the distortion is
modelled endogenously. If Sri Lanka adopted the Swiss …nancial practices the average size
of this distortion would drop to 1.5 percentage points. Its standard deviation across plants
collapses from 9 percentage points to just 0.3 percentage points. The elimination of this
distortion results in capital deepening among the active plants. Average TFP would rise by
26 percent in the model, as ine¢ cient plants are culled. For the world at large, the average
size of the distortion is 17 percentage points, with an average coe¢ cient of variation of 28
percent. The mean distortion drops to 2 percentage points with a world-wide movement to
…nancial best practice. The average standard deviation across plants falls from 5 percentage
12
The impact of …nancial intermediation on income will be larger if the former is allowed to a¤ect TFP in
the production sector more directly. Erosa and Hidalgo-Cabrillana (2008) undertake a theoretical analysis
where entrepreneurs produce an intermediate good that is important for the production of …nal output. They
employ a Lucas (1978) style span of control model. A limited ability to enforce …nancial contracts leads to
a poor selection of entrepreneurs in the economy. This channel of e¤ect may be important because Levine
(2005) documents that …nancial development has a causal impact of productivity. A related quantitative
model is in Amaral and Quintin (2005). They emphasize the capital accumulation channel.
36
points to a mere 0.35.
Last, it will be noted that the model could be used to make an inference about productivities in the production and …nancial sectors, x and z, by using interest rate spreads, s, instead
f 1 (o; s).
of the capital output ratio, k=o; i.e., by using the mapping of the form (x; z) = M
Erosa (2001) uses interest-rate spreads to quantify the e¤ects of …nancial intermediation on
occupational choice. The correlation between the model’s prediction between the state of
technology in the …nancial sector and the independent measure of …nancial development is
again high–Table 5. The model does a reasonable job predicting cross-country capital-tooutput ratios. Financial development is now more important, but it still does not explain
the bulk of cross-country variation in output–see Table 6. The di¤erence in the quantitative
signi…cance obtains from the fact that the percentage variation in cross-country interest rate
spreads is much larger than in capital/output ratios. The mean distortion in the world is
now 21 percentage points, with an average coe¢ cient of variation of 30 percent.
Table 5: Cross-Country Evidence
capital-output ratio …nancial intermediation
Correlation(model, data)
0.49
0.70
Table 6: World-wide move to best fin. prac., z
f 1 (o; s)]
[alternative results obtained when using interest-rate spread match, (x; z) = M
Increase in world output (per worker)
48%
Drop in shortfall between actual and potential world output
21%
Fall in dispersion of ln(output) across countries
12% (' 75% - 62%)
Fall in (pop-wghtd) mean of (cap-wghtd) distortion
18% ( ' 21% - 3%)
Fall in (pop-wghtd) mean dispersion of (cap-wghtd) distortion 5.44% ( ' 6% - 0.56%)
37
Sri Lanka
India
Bolivia
Morocco
Mauritius
Nicaragua
Colombia
Philippines
Costa Rica
Uruguay
Honduras
Brazil
Turkey
Mexico
Argentina
Ireland
UK
Panama
US
Portugal
Italy
Spain
Iceland
Peru
Finland
Netherlands
France
Denmark
Canada
New Zeland
Australia
Belgium
Luxembourg
Austria
Japan
Israel
Thailand
Norway
Switzerland
GDP per worker, % change
120
100
80
60
40
20
0
Figure 8: The impact of a move to …nancial best practice on GDP per worker
38
Sri Lanka
India
Bolivia
Morocco
Mauritius
Nicaragua
Colombia
Philippines
Costa Rica
Uruguay
Honduras
Brazil
Turkey
Mexico
Argentina
Ireland
UK
Panama
US
Portugal
Italy
Spain
Iceland
Peru
Finland
Netherlands
France
Denmark
Canada
New Zeland
Australia
Belgium
Austria
Japan
Israel
Thailand
Norway
Switzerland
Reduction in gap to the frontier, %
40
20
0
Figure 9: The impact of a move to …nancial best practice on the gap in GDP per worker
39
12
12.1
Robustness Analysis
Intangible Investments and Capital’s Share of Income
Suppose part of investment spending is undertaken in the form of intangible capital. As
a result, measured investment may lie below true investment. This will lead to measured
income, GDP, falling short of true output, o. This injects an upward (a downward) bias
in the measurement of labor’s (capital’s) share of income. Speci…cally, in context of the
standard neoclassical model, with a Cobb-Douglas production function, measured labor’s
share of income, LSI, will appear as
LSI =
o
GDP
(1
) > (1
):
Corrado, Hulten, and Sichel (2007) estimate the amount of intangible investment that was
excluded from measured GDP from 1950 to 2003. They show that when output is adjusted
to include these unrecognized intangibles, true output, o, is 12 percent higher than measured
output, GDP, for the period 2000-2003. As a consequence, it is easy to calculate that
=1
GDP
o
LSI = 1
1
(1
1:12
0:33) = 0:41:
How does this larger estimate for capital’s share of income a¤ect the analysis?
The calibration procedure described by (25) is redone for the case where
= 0:41. The
results are in accord with those obtained earlier. The model again …ts the U.S. data well.
In particular, it matches the …rm-size distribution for 1972 very well. With no …nancial
innovation, GDP would have risen by about 1.77 percent a year, compared with its actual
rise of 2.4 percent. Hence, …nancial development accounts for about one quarter of the
growth in GDP. About 18 percent of measured TFP growth is due to improvements in
…nancial intermediation.
Financial intermediation is now a little more important for economic development, at
least when the model is used to match up GDP and capital/output ratios across countries.
World output would increase by 33 percent, as opposed to the 28 percent found earlier,
40
if all countries moved to the best …nancial practice. When interest-rate spreads are targeted instead of capital/output ratios, …nancial development is slightly less important than
before–world output would rise by 47 percent. Therefore, all in all, the results obtained
earlier are quite robust to a change in capital’s share of income (when the model is suitably
recalibrated).
12.2
Varying the Degree of Substitutability between Capital and
Labor
Let output be produced according to a CES production function of the form
o = [ k + (1
1
)(x l) ] ; with
1.
This production function will have have implications for how labor’s share of income, LSI,
will vary across countries. To see this, think about the one sector growth model. Here labor’s
share of income can be written as LSI= (w=l)=(w=l + rk) = 1=[1 + (r=w)(k=l)]: Therefore,
labor’s share will rise whenever (r=w)(k=l) falls. With the above production function,
1=(1
=
) represents the elasticity of substitution between capital and labor. Hence, in
response to a shock in some exogenous variable, z, it will happen that d ln(r=w)=dz =
(1= )d ln(k=l)=dz. If the shock induces capital deepening [d ln(k=l)=dz > 0] then labor’s
share will rise or fall depending on whether the elasticity of substitution is smaller or bigger
than one. In the cross-country data, labor’s share either rises slightly or remains constant
with per-capita income.13
restricted so that 1=(1
This suggests that for the quantitative analysis
) < 1, which implies
should be
< 0; i.e., capital and labor are less
substitutable than Cobb-Douglas.
Let
=
0:38, roughly in line with Pessoa, Pessoa and Robb (2005). The calibration
procedure described above is redone for this value for . The CES framework does not …t
the …rm-size distribution for 1972 as well as the Cobb-Douglas case. It does worse predicting
the shift in the 2000 distribution, although the movement is still in the right direction. In
13
That is, r=w will decrease by more (less) than k=l rises when the elasticity of substitution is smaller
(greater) than one.
41
fact, if one allowed for
0 to be freely chosen in the calibration procedure then a value
close to zero (Cobb-Douglas) would be picked. For the U.S. economy, the CES speci…cation
predicts a rise in labor’s share from 0.73 to 0.75 as the capital stock deepens. The model
with a CES production function has a di¢ cult time matching the observed variation in crosscountry capital/output ratios. Labor’s share varies from 0.70 to 0.75. All in all, both the
U.S. and cross-country data prefer the Cobb-Douglas speci…cation. With a CES production
structure world output would increase by 18 percent, if all countries move to the best …nancial
practice. This is lower than the Cobb-Douglas case. This occurs because the potential for
capital deepening is more limited the higher the degree of complementarity between capital,
which is reproducible, and labor, which is …xed, in production.
13
Conclusions
What is the link between the state of …nancial intermediation and economic development?
This question is explored here by embedding a costly-state veri…cation framework into the
standard neoclassical growth paradigm. The model has two novel ingredients. As in the
standard costly-state veri…cation paradigm, the ex post return on a project is private information and an intermediary can audit the reported return. The …rst ingredient is that
likelihood of a successful audit is increasing and concave in the amount of resources devoted
to monitoring. The cost of auditing is increasing and convex in the amount of resources
spent on this activity. Second, there is a distribution over …rm type, each type o¤ering a
di¤erent combination of risk and return.
Two key features follow from these ingredients. First, a …nancial theory of …rm size
results. All …rms are funded that earn an expected return greater than the cost of raising
capital from savers. Funding is increasing in a project’s expected return and decreasing in
its variance. The size of a …rm is limited by diminishing returns in information production.
Second, a Goldsmithian (1969) link is created between the state of …nancial development
and economic development. The presence of informational frictions leads to a distortion
between the expected marginal product of capital and its user cost, the interest paid to
42
savers plus capital consumption. This distortion is modelled endogenously here. As the
e¢ cacy of auditing increases, due to technological progress in the …nancial sector, the size
of this distortion shrinks. The upshot is an increase in the economy’s income. Intuitively,
the rise in income derives from three e¤ects: (a) as the spread shrinks there is more overall
capital accumulation in the economy; (b) capital is redirected toward the most productive
investment opportunities in the economy; (c) less labor is required to monitor loans, which
frees up resources for the economy.
The developed model is taken to both U.S. and cross-country data. It is calibrated to
…t the U.S. …rm size distribution for 1972 and the rise in the U.S. capital-to-output ratio
between 1972 and 2000. It captures these features of the data well. The model’s predictions
for the e¢ ciency of …nancial intermediation in a cross-section of 40 countries matches up
well with independent measures. It does a reasonable job mimicking cross-country capitaloutput ratios and interest-rate spreads. The mechanism outlined above has quantitative
signi…cance. The average measured distortion in the world between the expected marginal
product of capital and its user cost falls somewhere between 17 and 21 percentage points.
The average coe¢ cient of variation in the distortion within a country is 28 to 29 percent.
World output could increase by 28 to 48 percent if all countries adopted the best …nancial
practice in the world. Still, this only accounts for 13 to 19 percent of the gap between actual
and potential world output. This happens because the bulk of the di¤erences in cross-country
GDP are explained by the huge di¤erences in the productivity of the non-…nancial sector.
There are two natural extensions of the above framework. The …rst would be to allow
for long-term contracts. On this, Smith and Wang (2006) embed a long-term contracting
framework into a model of …nancial intermediation. Clementi and Hopenhayn (2006) and
Quadrini (2004) have also examined the properties of dynamic contracting for …rm …nance
in worlds with private information. The use of dynamic contracts could mitigate the informational problems discussed here. How much is an open question. In a competitive world,
such contracts may be severely limited by the ability of each party to leave the relationship
at any point in time and seek a better partner. Second, …rms often use internal funds to
43
…nance investment. Incorporating internal …nance may a¤ect the importance of …nancial development. Perhaps accessing external funds involves a …xed cost, and hence is economized
on. Then, internal funds may be hoarded in case good investment opportunities come along.
An extension along these lines calls for a dynamic theory of the …rm–Cooley and Quadrini
(2001) or Gomes (2001). In situations where the …nancial stakeholders of …rms delegate its
operation to others, however, the returns on any investment activity (whether internally or
externally …nanced) will need to be monitored. Ideally, any predictions from such a model
should be matched up with a cross-country data set on the sources of …rm …nance.
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14
14.1
Appendix
Theory
Proof for Lemma 1. First, substitute the promise-keeping constraint (9) into the objective
function to rewrite it as
( 1 r1 +
2 r2
re)k
1 w(m1 =z)
47
2 w(m2 =z)
v:
Next, it is almost trivial to see that optimality will dictate that p12 = r1 k and p21 = r2 k,
since this costlessly relaxes the incentive constraints (7) and (8). Next, drop the incentive
constraint (7) from problem (P2) to obtain the auxiliary problem now displayed:
subject to
e ; v)
I(
max f( 1 r1 +
p1 ;p2 ;m1 ;k
[1
re)k
2 r2
1 w(m1 =z)
vg;
(P4)
p1
r1 k;
(27)
p2
r2 k;
(28)
P21 (m1 =k)](r2 k
p1 )
r2 k
(29)
p2 ;
and
1 (r1 k
p1 ) +
2 (r2 k
(30)
p2 ) = v:
The strategy will be to solve problem (P4) …rst. Then, it will be shown that (P2) and (P4)
are equivalent. Problem (P4) will now be solved. To this end, note the following points:
1. The incentive constraint (29) is binding. To see why, suppose not. Then, reduce m1
to increase the objective.
2. The constraint (28) is not binding. Assume, to the contrary, it is. Then, (29) is
violated. This happens because the right-hand side is zero. Yet, the left-hand side is
positive, given that p1
r1 k < r2 k, so that [1
P21 (m1 =k)](r2 k
p1 ) > 0.
3. The constraint (27) is binding. Again, suppose not, so that p1 < r1 k. It will be shown
that exists a pro…table feasible deviation from any contract where this constraint is
slack. Speci…cally, consider increasing k very slightly by dk > 0 while adjusting p1 and
p2 in the following manner so that (29) and (30) still hold. Also, hold m1 …xed. The
implied perturbations for p1 and p2 are given by
2
3
2
3 1
dp
[1 P21 (m1 =k)] 1
4 1 5 = 4
5
dp2
1
2
2
3
2
r P (m =k) (r2 k p1 )(m1 =k )dP21 =d(m1 =k)
4 2 21 1
5 dk:
1 r1 + 2 r2
48
Note that such an increase in k will raise the objective function.
It will now be demonstrated that the optimization problems (P2) and (P4) are equivalent.
e ; v)
First, note that I(
I( ; v), because problem (P4) does not impose the constraint (7).
e ; v)
It will now be established that I(
I( ; v). Consider a solution to problem (P4). It
will be shown that this solution is feasible for (P2). On this, note that point 1 implies that
r2 k
p2 = [1
P21 (m1 =k)](r2 k
p1 )
r2 k
p1 ;
so that
p1
p2 :
Now, set m2 = 0 in (P2), which is feasible but not necessarily optimal. Then, constraint (7)
becomes p1
e ; v)
p2 , which is satis…ed by the solution to (P4). Therefore, I(
I( ; v).
Last, with the above facts in hand, recast the optimization problem as
I( ; v)
max f( 1 r1 +
p2 ;m1 ;k
re)k
2 r2
subject to
r2 k
p2 = (r2
r1 )k[1
1 w(m1 =z)
P21 (m1 =k)];
vg;
cf (29);
and
r2 k
p2 = v= 2 , cf (30):
The above two constraints collapse in the single constraint (10), by eliminating r2 k
p2 ,
that involves just m1 and k. The problem then appears as (P3).
Proof for Lemma 2. Suppose that the solution dictates that m1 =k
1= . Then, from
(P3) it is clear that the optimal solution will dictate that m1 =k = 0. This happens because
P21 (m1 =k) = 0 for all m1 =k
1= , yet monitoring costs are positive for all m1 =k > 0. Next,
by substituting (10) into (P3) it is easy to deduce that the intermediary’s pro…t function
can be written as
+ 2 r2 re
r1 )
2 (r2
Q 0 as r1 Q re.
I( ; v) = [
1 r1
49
1]v =
r1 re
v
r1 )
2 (r2
(31)
Therefore pro…ts are negative if r1 < re and v > 0. Hence, a contract will not be o¤ered
when m1 =k
1= .
Proof for Lemma 3. Clear from equation (17).
Proof for Lemma 4. Necessity: From problem (P3) it is clear that the intermediary
will incur a loss when
1 r1
+
2 r2
by Lemma 2.
Su¢ ciency: Suppose that
1 r1
re
+
0 and v > 0, for any
2 A(w), because m1 =k > 1=
re > 0 for some
2 r2
2 T . By equation (17) it is
immediate that the intermediary will issue a loan k > 0. By construction it will earn zero
pro…ts on this loan. Recall that the derivation of (17), discussed in the text, used a solution
for v. This solution is
v =V( )=(
)(
+
+
)=(
)
1
( )
=(
)
( 1 r1 +
2 r2
(
1
)(
+
re)
Therefore, the …rm will earn positive rents too.
+
+ )=(
[ 2 (r2
)
(32)
r1 )]
( 1 w)
(z= )
Proof for Lemma 5. To begin with let k = K(w; ; z) represent capital stock that will
be employed by a type- …rm when productivity in the …nancial sector is z and the wage
rate is w > !. Similarly, let m1 =z = M (w; ; z) denote the amount of monitoring services,
relative to z, that will be devoted to this project. Given this notation, the expected demand
for labor by both the …rm and intermediary for a project of type
L(w; ; z)
(
1=
1 1
+
1=
2 2
)[
(1
)
w
Note that this demand is only speci…ed for
1
] K(w; ; z) +
is
1 M (w;
(33)
; z) :
2 A(w) and w > !, where ! is de…ned by
(22). In order to characterize L(w; ; z); the properties of its components K(w; ; z) and
M (w; ; z) must be developed when
2 A(w) and w > !. First, take K(w; ; z). On this,
rewrite equation (17) as
(34)
k = K(w; ; z)
=
(
1
1
+
2
2
rew(1
)=
)(
+ )=(
)
50
[
2(
2
1)
]
=(
)
z
=(
1)
w
1=[ (
1)]
;
:
where
i
(1
)(1
)=
1=
i
and
(
+
)
) 1
=(
=(
( )
)
(
1
+
)(
+ )=(
)
.
Note the following things about this solution for k: (i) The level of investment in a …rm, k , is
continuous and strictly decreasing in w; (ii) k ! 0 as w ! W ( ) = [(
1
1+
2
r]
2 )=e
=(1
)
(when z is …nite); (iii) k is continuous and strictly increasing in z.
Now, switch attention to the second term in L(w; ; z). A formula for M (w; ; z) can be
derived in the same manner as the one for K(w; ; z). It is
m1 =z = M (w; ; z) =
where
[(
+
(
1
)=(
1
+
2
2
1=
+ )]
rew(1
)=
)(
f =[ 2 (
2
+1)=(
1 )]g
)
w
1=(
1=[ (
)
1)] 1=(
z
1)
;
(35)
. Note the following
things about this solution for m1 =z: (i) m1 =z is continuous and strictly decreasing in w;
(ii) m1 =z ! 0 as w ! W ( )
[(
1
1
+
r]
2 )=e
2
=(1
)
(when z is …nite); (iii) m1 =z is
continuous and strictly increasing in z.
Thus, the demand for labor by a type- project has the following properties: (i) L(w; ; z)
is continuous and strictly decreasing in w; (ii) limw!W ( ) L(w; ; z) = 0; (iii) L(w; ; z) is
continuous and strictly increasing in z. De…ne the function
8
< L(w; ; z); for w W ( ),
e
L(w; ; z) =
:
0;
for w > W ( ),
(36)
R
for w > !. The aggregate demand for labor can be expressed as A(w) L(w; ; z)dF ( ) =
R
R
e
e
L(w;
; z)dF ( ). Now, determine the constant by limw#! T L(w;
; )dF ( ) = 1;where
T
again the lower bound on wages ! is given by (22). From the simple closed-form solutions
for (34) and (35) it is easy to deduce that such a must exist. To summarize, the aggregate
R
e
demand for labor, T L(w;
; z)dF ( ), has the following properties:
1.
2.
3.
4.
R
R
R
R
T
T
T
T
e
L(w;
; z)dF ( ) is continuous and strictly decreasing in w for w 2 (!; 1);
e
L(w;
; z)dF ( ) is continuous and strictly increasing in z for z 2 ( ; 1);
e w;
L(
b ; z)dF ( ) = 0, where w
b = max
e
L(!;
; )dF ( ) = 1.
2T
51
W ( );
Therefore by the intermediate theorem for all z >
there will exist a single value of w that
sets labor demand equal to labor supply (or 1)— see Figure 6.
Proof for Proposition 1. Express the labor-market-clearing condition as
Z
(1
) 1=
1=
1=
f( 1 1 + 2 2 )[
] K(w; ; z) + 1 M (w; ; z) gdF = 1:
w
A(w)
Let the
1=
i
’s grow at the common rate g 1= and z grow at rate g 1=(1
)
(37)
. Recall that exists a
solution to the model without growth, as demonstrated by Lemma 5. It is easy to construct a
balanced growth path using this solution. The solution implies that there will be a wage rate
that solves (37). Conjecture that along a balanced growth path wages, w, will grow at rate
g 1=(1
)
. From (34) it can be deduced that K(w; ; z) will grow at rate g 1=(1
)
. Therefore,
the …rst term in braces in (37) will be constant. Equation (35) implies that M (w; ; z), or
the second term, will be constant too. The active set A(w) will not change— equation (20).
Therefore, labor demand remains constant. Hence, the conjectured solution for the rate of
growth in wages is true. Using (32) is easy to calculate that v will grow at rate g 1=(1
)
.
Last, since M (w; ; z) is constant it must be the case that m1 is growing at the same rate
as z, or g 1=(1
)
. Therefore, m1 =k will remain unchanged along a balanced growth path.
Proof for Proposition 2. First, point 2 in the proof of Lemma 5 established that
the aggregate demand for labor is continuous and strictly increasing in z. Therefore, at a
given wage rate the demand for labor rises as z moves up. In order for equilibrium in the
labor market to be restored, wages must increase, since the demand for labor is decreasing
in wages— point 1. Last, recall from (19) that a type- project will only be funded when
w < W( ) =
set of
=(1
)
(1
)[(
1=
1 1
+
1=
2 2
)=e
r]
=(1
)
. It’s trivial to see that as w rises the
2 T satisfying this restriction, or A(w), shrinks; if
= ( 1;
2)
ful…lls the restriction
for some wage it will meet it for all lower ones too, yet there will exist a higher wage that
will not satisfy it. Furthermore, observe that W ( ) is strictly increasing in
1=
1 1
+
1=
2 2
.
Therefore, those ’s o¤ering the lowest expected return will be cut …rst as w rises, because
they have the lowest threshold wage.
52
Proof for Proposition 3. Let the
1=
i
’s increase by the common factor g 1= > 1.
Suppose that wages increase in response by the proportion g 1=(1
)
. Will the labor-market-
clearing condition (37) still hold? The answer is no, because the demand for labor will fall.
Take the …rst term behind the integral, which gives the demand for labor by a …rm. From
(34) it is clear that K(w; ; z) will rise by a factor less than g 1=(1
)
, when z is held …xed.
Therefore, the …rst term in braces in (37) will decline. Turn to the second term. From (35) it
is easy to see that M (w; ; z) will drop under the conjecture solution. Therefore, wages must
rise by less than g 1=(1
)
, since the demand for labor is decreasing in w (as was established
in the proof of Lemma 5). The active set, A(w), will therefore expand, because
1 r1
+
2 r2
increases–see (18).
Proof for Proposition 4. The set of projects in T o¤ering the highest expected return
is given by
A = arg max[
By assumption
2 A then
R
=(
A
1;
1=
1( 1)
+
1=
2( 2)
]:
2 )2T
dF > 0. Take any equilibrium wage w. From (20) it is immediate that if
2 A(w), since
(b
r + ) is increasing in
1=
1 1
1 r1
+
+
2 r2
1=
2 2
re = (1
. Hence, A
)(1
)=
w
(1
)=
(
1=
1 2
+
1=
2 2
)
A(w) for all w. In equilibrium the
wage will be a function of z, so denote this dependence by w = W (z). Now, let z ! 1. It
will be shown that w = W (z) ! w , where
w
=(1
)
(1
)f
1=
1( 1)
max[
=(
1 ; 2 )2T
+
1=
2( 2)
]=e
rg
=(1
)
:
(38)
To see why, suppose alternatively that w ! w
e 6= w . First, presume that w
e < w . Then, by
(19) all projects of type
2 A will be funded since their cuto¤ wage is W ( ) = w > w.
e
ed (W (z); ; z) = 1, for 2 A .
From equations (33), (34) and (36) it is clear that limz!1 L
R
R d
e (W (z); ; z)dF = 1. Therefore, such an
Since, A dF > 0, this implies that limz!1 T L
equilibrium cannot exist because the demand for labor will exceed its supply. Second, no …rm
R d
e (W (z); ; z)dF = 0.
can survive at a wage rate bigger than w , by (19). Here, limz!1 T L
This establishes (23). Last, note that A(w ) = A .
53
It is immediate that A
limw"w A(w), because
2 A is viable for all wages w
w = W ( ) by (19). It is also true that limw"w A(w)
A , since from (19) any project
2
= A requires an upper bound on wages W ( ) < w to survive; that is, for any
2
=
A there will exist some high enough wage w such that W ( ) < w < w . Therefore,
limw"w A(w) = A = A(w ). This establishes point 1 of the Proposition.
To have an equilibrium it must be the case that m1 =z < 1 for
2 A , otherwise the
demand for labor would be in…nitely large. From equation (35) this can only happen when
rew(1
)=
!
1
1
+
2
2,
or equivalently when
1 r1
+
2 r2
! re. Solve problem (P3) for
the optimal level of monitoring, m1 =k, and then use (32) to solve out for v to obtain
m1 =k = (
+
+
)
1=
It is apparent that lim m1 =k = 1; because
z!1
[
( 1 r1 +
2 (r2
1 r1
+
2 r2
2 r2
r1 )
re)
]
1=
:
! re. Consequently, a false report
by a …rm will be caught with certainty, or lim P12 (m1 =k) = 1. The contracting problem
z!1
(P2) then requires lim p2 = r2 k and lim v = 0. A comparison of (32) and (35) leads to
z!1
z!1
the conclusion that in fact lim m1 =z = 0, when lim v = 0. Using this result and (38), in
z!1
z!1
R
e
conjunction with the labor-market-clearing condition, T L(w;
; z)dF = 1, then gives (24).
14.2
Data
Figure 1: (Left panel) The numbers represent total debt outstanding for businesses
(excluding …nancial ones) relative to gross domestic business value added (excluding
gross farm value added). This data derives from the Flow of Funds Accounts of the
United States (Table D.3: Debt Outstanding by Sector). The source of the data for
the value of …rms relative to GDP is Hobijn and Jovanovic (2001, Figure 1, p. 1204).
It refers to the total market capitalization of all securities contained in the CRSP data
set.
(Right Panel) A series for the intangible stock of capital is constructed by backing
out the implied data series on investment in intangibles that is reported in Corrado,
54
Hulten and Sichel (2006, Figure 1). Speci…cally, a capital stock series for intangibles is
constructed by iterating on the law of motion ki0 = (1
i )ki +ii ,
stock of intangible capital, ii is investment in intangibles, and
i
where ki is the current
is the depreciation rate
on intangibles. Two issues arise with this procedure. First, what is the depreciation
rate on intangible capital? A weighted average of the rates reported in Corrado, Hulten
and Sichel (2006, p. 23) suggests that it should be 33 percent. McGrattan and Prescott
(2006, p. 782) feel that an upper bound of 11 percent is appropriate. Taking a simple
average of these two numbers gives 22 percent, the value used here. Second, what
starting value for the intangible stock of capital should be used? Along a balanced
growth path the stock of intangible capital is given by ki = ii =(g + i ), where g is the
growth rate of GDP. This formula is used to construct an initial capital stock for 1947,
where g is assigned a value of 0:015. The stock of intangible capital is then simply
added to private nonresidential nonfarm …xed assets. The resulting series is divided
through by nonfarm business GDP.
Figure 7: The data is for establishments. The horizontal axis orders establishments
(from the smallest to highest) by the percentile that they lie in for employment. The
vertical axis shows the cumulative contribution of this size of establishment to the total
payroll in the U.S. economy. The use of payroll data controls for worker heterogeneity;
i.e., measures employment in e¢ ciency units. Note that …rms without paid employees
(sole-ownership …rms and joint proprietorships) are included in 1972 data. The source
for the raw data used is Statistics for U.S. Businesses, U.S. Census Bureau.
Figure 2 and Section 10: The data for the interest-rate spread is taken from Beck,
Demirguc-Kunt and Levine (2001). It is de…ned as the accounting value of banks’net
interest as a share of their interest-bearing (total earning) assets averaged over 1990
to 1995. The numbers for the ratio of private credit to GDP (the measure used for
…nancial intermediation) are reported in Beck, Demirguc-Kunt, and Levine (2000) as
the sum of deposit money in banks and related institutions, stock market capitalization,
55
and private bond market capitalization. These are averaged over the period 1990 to
1999. The other numbers derive from the Penn World Tables, Version 6.1— see Heston,
Summers and Aten (2002). The capital stock for a country, k, is computed for the
1990-2000 sample period using the formula k = i=(g + ), where i is gross investment, g
is the growth rate in investment, and
is rate of depreciation. This formula heroically
assumes that an economy is on a balanced growth path. The depreciation is taken
to be 0.06. The growth rate for investment is calculated from the investment data
reported in the tables for the period 1950 to 2000. Investment is recovered by using
data on investment’s share of GDP and GDP. A country’s total factor productivity,
T F P , was computed using the formula T F P = (y=l)=(k=l) , where y is GDP, l is
aggregate labor, and
is capital’s share of income. A value of 0.30 was picked for
. Aggregate labor is backed out using data on per-capita GDP, GDP per worker,
and population. The numbers in the analysis are reported in Table 7; the reported
values are generally the mean for 1990-2000. Most of the headings in the table are
obvious. Here ‘Spread’refers to the Beck, Demirguc-Kunt and Levine (2001) interestrate spread discussed above, while ‘Fin. Dev.’ represents the measure of …nancial
development from Beck, Demirguc-Kunt and Levine (2000). Recall that s denotes the
average spread in the model between the rates of return that intermediaries and savers.
Likewise,
o(x; z)
[O(x; z) O(x; z)] is the increase in a country’s income if it moved
to …nancial best practice, while G(x; z)
[O(x; z)
O(x; z)]=[O(x; z)
O(x; z)] refers
to the fraction of the gap between a country’s actual and potential output that would
be closed. Last, the mean (capital-weighted) distortion for a country is de…ned by
R
R
d(x; z)
(
r
+
r
r
e
)kdF=
dF . The standard deviation of this distortion
1
1
2
2
A(w)
A(w)
qR
R
( 1 r1 + 2 r2 re d)2 kdF= A(w) dF .
is then given by
A(w)
56
Table 7: Cross-country numbers, data and baseline model
(see Section 14.2 for an explanation of the data and headings)
Country
Sri Lanka
India
Bolivia
Morocco
Mauritius
Nicaragua
Colombia
Philippines
Costa Rica
Uruguay
Honduras
Brazil
Turkey
Mexico
Argentina
Ireland
UK
Panama
US
Portugal
Italy
Spain
Iceland
Peru
Finland
Netherlands
France
Denmark
Canada
New Zealand
Australia
Belgium
Luxembourg
Austria
Japan
Israel
Thailand
Norway
Switzerland
GDP
p.w.
7013
5121
6779
11419
23705
5923
12332
7864
13913
20251
6823
18001
14340
22100
25056
46945
39908
15255
57151
30350
50569
40138
39834
10200
40603
46929
45317
44024
45933
36422
45907
50839
80702
45560
37061
40777
11632
47845
45706
Data
K/GDP Spread
0.774
0.787
0.839
0.884
0.892
0.915
0.935
1.054
1.085
1.099
1.120
1.170
1.179
1.257
1.291
1.355
1.416
1.532
1.578
1.596
1.610
1.639
1.639
1.655
1.695
1.758
1.758
1.759
1.781
1.794
1.824
1.862
1.881
1.884
1.917
1.933
1.995
2.112
2.122
0.051
0.030
0.035
0.036
0.032
0.064
0.042
0.052
0.056
0.069
0.120
0.094
0.053
0.082
0.016
0.020
0.020
0.039
0.035
0.036
0.038
0.072
0.016
0.015
0.035
0.049
0.018
0.025
0.019
0.023
0.007
0.019
0.018
0.033
0.030
0.031
0.016
Fin.
Dev.
0.339
0.512
0.441
0.576
0.683
0.234
0.402
0.890
0.158
0.266
0.282
0.538
0.336
0.556
0.335
0.977
2.481
0.692
3.297
1.046
1.104
1.279
0.940
0.312
1.551
2.430
1.592
1.741
1.706
1.363
1.470
1.487
2.064
1.143
3.043
0.979
1.786
1.403
3.464
ln(z)
7.60
6.91
7.60
8.29
9.10
7.60
8.52
8.52
9.21
9.68
8.70
9.80
9.62
10.34
10.62
11.51
11.63
11.18
12.72
12.18
12.76
12.68
12.67
11.40
12.99
13.49
13.45
13.43
13.60
13.45
13.86
14.21
14.80
14.24
14.26
14.48
13.69
16.16
16.22
57
Model [ = 0:33; (x; z) = M 1 (o; k=o)]
s
x
o(x; z) G(x; z) d(x; z)
0.054
0.053
0.048
0.044
0.044
0.042
0.041
0.033
0.032
0.031
0.030
0.027
0.027
0.024
0.023
0.020
0.018
0.015
0.014
0.014
0.013
0.013
0.013
0.012
0.011
0.010
0.010
0.010
0.010
0.009
0.009
0.008
0.008
0.008
0.007
0.007
0.006
0.004
0.004
38.77
31.17
36.54
50.59
82.19
32.08
51.91
36.32
52.50
67.11
32.09
60.20
51.50
66.72
71.64
106.6
93.40
47.22
112.7
73.35
102.8
87.29
86.80
34.69
86.50
93.60
91.43
89.65
91.66
78.18
90.55
95.96
130.1
88.65
76.53
81.26
34.52
86.58
83.77
0.756
0.752
0.703
0.656
0.629
0.648
0.616
0.546
0.512
0.493
0.507
0.453
0.454
0.397
0.375
0.320
0.293
0.266
0.205
0.217
0.194
0.189
0.190
0.219
0.165
0.134
0..135
0.135
0.125
0.127
0.108
0.090
0.066
0.085
0.079
0.070
0.080
0.002
0.000
0.100
0.071
0.087
0.141
0.332
0.067
0.142
0.073
0.129
0.196
0.057
0.151
0.115
0.168
0.186
0.452
0.293
0.065
0.447
0.132
0.305
0.182
0.179
0.033
0.159
0.171
0.160
0.151
0.152
0.099
0.130
0.135
1.000
0.099
0.062
0.065
0.013
0.002
0.000
0.293
0.286
0.260
0.240
0.236
0.227
0.219
0.179
0.170
0.167
0.161
0.148
0.146
0.129
0.122
0.110
0.099
0.082
0.075
0.073
0.071
0.068
0.068
0.066
0.061
0.054
0.054
0.054
0.051
0.050
0.047
0.043
0.042
0.041
0.038
0.037
0.033
0.022
0.022
Std Dev
d(x; z)
0.093
0.090
0.080
0.073
0.072
0.068
0.065
0.052
0.049
0.047
0.045
0.041
0.041
0.035
0.033
0.029
0.026
0.020
0.019
0.018
0.017
0.016
0.016
0.016
0.014
0.012
0.012
0.012
0.012
0.011
0.011
0.010
0.009
0.009
0.008
0.008
0.007
0.004
0.004
@FedFRASER