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DEWEY
HB31
.M415
\/2^.
9s-i
@
working paper
department
of economics
A GENCY COSTS IN THE PROCESS OF DEVELOPMENT
Daron Acemoglu
Fabrizio Zilibotti
96-8
Feb.
1996
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
J;
AGENCY COSTS IN THE PROCESS OF DEVELOPMENT
Daron Acemoglu
Fabrizio Zilibotti
96-8
Feb.
1996
MASSAWU?'"
m^
13 1996
96-8
Agency Costs in the Process of
Development
Dciron Acemoglu
Fabrizio Zilibotti*
Massachusetts Institute
Universitat
of Technology
Pompeu Fabra
February
16,
1996
Abstract
We analyze an economy where production is subject to moral hazard. The
degree of the incentive (agency) costs introduced by the presence of moral
hazard naturally depends on the information structure in the economy;
cheaper
is
it is
to induce correct incentives in a society which posesses better ex
post information.
The degree
of ex post information
projects and entrepreneurs in the economy; the
depends on the number of
more
projects, the better the
information. This implies that at the early stages of development, the range
and the amount of information are limited and agency costs are
high. Since the information created by a project is an externality on others,
the decentralized economy is constrained inefficient; in particular, it does not
of projects
'experiment' enough.
The
analysis of the role of information also opens the
tigation of the development of financial institutions.
We
way
to an inves-
contrast the infor-
mation aggregation role of stock markets and information production role of
banks. Because the amount of available information increases with development, our model predicts the pattern of financial development observed in
practice; banks first and stock markets later.
JEL
Classification: D82, E44, G20.
KeyAvords: Agency Costs, Development, Information, Financial
Institutions,
Social Experimentation.
*We thank Andres Almazan, Alberto Bisin, Russ Cooper,
Peter Diamond, John Hassler, Torsten
Jaume Ventura and seminar particUPF, Wharton for helpful comments and
Persson, Sevi Rodriguez-Mora, Mfiria Saez-Marti, Ilya Segal,
ipants at Copenhagen, Harvard-MIT, IIES-Stockhohn,
Benjamin Levin
for excellent research assistance.
International Institute of
Economic
the Ministry of Education of Spain
Zilibotti
acknowledges the hospitality of the
and financial support from
Studies, University of Stockholm,
(DGICYT
PB93-0388).
Introduction
1
The
efficient allocation of resources
who
are not the
reqmres that
many
tasks be delegated to agents
residual claimants of the returns they generate. It
full
natural that agency relations play an important role in
development
[e.g.
some Third World
[e.g.
many
is
therefore
accounts of economic
Mydral, 1968, North, 1990] and that high agency costs faced by
societies
North, 1990, p.
have been argued to prevent their economic development
59].
The change
in the factory
system at the time of the
British Industrial Revolution [see Mokjn:, 1991, for discussion], or the emergence
of hierarchical organizations
among the agency
relations that
process of development.
is
and professional management
[see
Chandler, 1977] are
seem to have played an important
Perhaps the most important example of agency relations
in the credit market. Entrepreneurs
be given the right incentives.
It is
borrowing funds
for their activities
need to
well-known that financial intermediation was
limited at the early stages of development,
and economic growth and the growth of
intermediated fimds went hand in hand over the past three centuries
1969, Kennedy, 1987,
role in the
King and Levine,
1994]. For instance.
[e.g.
Goldsmith,
Goldsmith (1987) shows
that in most pre-modern societies financial arrangements were extremely informal,
and the same pattern
arises
from Townsend's (1995) study of Indian
villages.
These
observations suggest that societies at the early stages of development were unable
to have wide-ranging agency relations
ties to raise
and
relied
predominantly on family or village
funds and ensure enforcement. This situation contrasts with the more
developed and complex credit relations that we observe today.
A number
of other features related to the evolution of incentive contracts
agency relations are also relevant to our investigation.
stages
and
First, wliile at the early
most firms were owner managed, the majority of large firms today have man-
agement separated from ownership. This suggests that the 'high powered' incentives
CEOs
of the owners have been replaced by the weaker incentives of current
day
Berle and Means, 1932, Jensen and Meckling, 1976]. Moreover, even
when
is
restricted to professional
Murphy
(1990)
managers
only, the
show that the pay-performance
[e.g.
attention
same pattern emerges. Jensen and
sensitivity for
CEOs
has decreased
substantially since the 1930s^ Second, while in less developed external financing re^ Unable to explain this pattern using any existing
theory, Jensen and Murphy suggest that this
due to political constraints. However, an impUcation of this explanation is that non-monetary
methods of control should be used more often and thus current day CEOs should be replaced more
frequently for poor performance. In contrast, Haddlock and Liuner (1994) find that the likeUhood
is
lies
almost exclusively on banks and other direct lending relations [Goldsmith, 1987;
Pry, 1995], stock
and bond markets play an increasingly important
role in
developed
economies. Since banks tjrpically screen and monitor the projects they finance [Dia-
mond,
1984], their diminishing role suggests that the informational requirements of
agency relations and thus the form of incentive contracts have been changing over
time.
This paper has three related objectives.
ple explanation for
Our main
costs
thesis
why agency
The first is
costs are high at the early stages of development^.
that the information structure of an
is
and formalize a sim-
to offer
economy determines agency
and that over the process of development the information structure changes
endogenously. According to North (1990, p. 57), the problem
nication
mechanism
is required'.
This 'communication mechanism'
economy develops, the
become cheaper to
A
weak
in
form a commu-
poor societies and as an
become more
flows of information
enforce.
is
'to
know when punishment
provide the information necessary to
to
is
efiicient
direct prediction of our theory
is
and incentives
the observed pat-
tern of evolution of incentive contracts: at the early stages of development, large
pimishments are necessary to induce the right incentives; but as the commimication
mechanism develops, better
risk-sharing (insurance) can be off^ered to entrepreneins
and other agents, and incentive contracts can become
second objective
is
to
show that the
less 'high-powered'.
analysis of agency costs
Our
and information has
important impUcations for the development of financial institutions. Our third objective
to assess the efficiency of the evolution of agency relations
is
and
financial
institutions.
As an example
to borrow
of the paper's
money for
main
idea, consider the case of
foreign trade. Given the
to the behavior of the entreprenevir
—
e.g.
and vmcertainty
of risk
has he picked a good trade,
effort, is
he steaUng part of the money, was
the ship?
—
,
amoimt
an agent who wants
it
is
relative
he putting
the weather or carelessness that sunk
high agency costs are to be expected. In
fact, in
pre-modern economies,
of dismissal has not changed since 1930s.
•^We should note that the main argmnent of this paper is about shadow rather than actual
agency costs. The actual agency costs in the villages studied by Townsend may be low because
no one engages in entrepreneurship nor invests in risky projects. What is very high is the cost
that an agent would have to incur if he decided to borrow money to become entrepreneiu, i.e.
the agency cost at the margin. Similarly, some aspects of incentives in the complex contemporary
organizations
may be quite distorted, but absent the relatively efficient information flows of modern
would be much more serious and perhaps such complex organizations would
society, distortions
not
exist.
while long-distance trade was an important activity, investors bore a large amount of
risk
and the risk-premium was very high
[see
a hypothetical situation in which there are
Braudel, 1979]. In contrast, consider
many
other entrepreneurs borrowing
funds for similar foreign trades. In this alternative scenario, investors can reduce
the agency costs by using the information they obtain from other entrepreneurs
regarding the unavoidable uncertainty of this trade,
performance evaluation
nmnber
in the jargon of the relative
they can filter out the
literature,
common
shock. This
is
the
At the early stages of development, hmited savings constrain
story of our paper.
the
i.e.
of projects (or entrepreneurs) as well as the information that can be
used to write incentive contracts. Limited information in turn leads to high agency
costs.
and
As the
their
increases,
more entrepreneurs
are active
performance reveals a substantial amount of information to the
which can be used
we
economy
capital stock of the
will also
society,
in devising the right incentives for each entrepreneur. Moreover,
argue that the relative scarcity of information at the early stages of
development favors banking over stock markets, and that economic development
can be associated with a
shift
from bank finance to stock and bond markets, as
observed in practice in the coiirse of financial development.
Our model has
three key featm'es:
subject to moral hazard;
amount
(ii)
(i)
Production requires entrepreneurial
Different projects have correlated returns;
of savings determines the
number
effort
(iii)
of projects that can be imdertaken.
The
As
a result, savings determine the amovmt of information which can be used in devising appropriate incentive schemes for entreprenein-s,
with accumulation.
Expressed
differently, in
an economy with moral hazard, the
compensation of agents depends on idiosyncratic and
ence their performance, and this lack of
As the economy becomes
richer
full
and agency costs decrease
common
shocks which influ-
insiuance introduces high agency costs.
and undertakes more
projects, the
compensation of
agents can be conditioned on the success of other projects, therefore the variability
introduced due to
common
diction of the model.
turnover of
shocks can be largely avoided.
Gibbons and Murphy (1990)
relation
find that the
compensation and
CEOs depend significantly on the performance of other firms in the same
industry and conclude that there
evaluations
In line with this pre-
among top
is
executives.
support for the presence of relative performance
Haddlock and Lumer (1994) find even a stronger
between these variables using data from the 1930s when the U.S. companies
were much
less diversified
to one industry.
than today, thus could more easily be
classified to
belong
Since each project's performance reveals information relevant for others, infor-
mation has pubUc good
features. It
is
then natural to question the extent to which
the market achieves an efficient allocation of resources.
To answer
this question
we
contrast the choice of a social plarmer subject to the relevant informational constraints to the decentralized equihbriiim.
We
show that the
social planner
would
always choose to produce more information than the decentralized equiUbriimi by
'experimenting'. Further, this constrained inefiiciency result
complex
to the formation of
In our
is
a pubhc good because stock prices of different
and reveal the performance of each project and thus
dertaken within the auspices of a bank
may be
two
all
is
not necessarily publicly observed, and as a
better equipped to deal with the free-rider problems at
the early stages of development.
to emphasize
to be robust
In contrast, detailed information regarding projects im-
the relevant information.
consequence banks
shown
financial coalitions.
economy information
firms are pubhcly observed
is
The comparison
of banks to stock markets leads us
distinct fimctions related to information: the first is information
aggregation; the aggregation of available information,
efiicient in this function.
and stock markets are more
The second is information production which will depend en-
dogenously on the financial incentives provided by different arrangements. Precisely
because the stock market
is
free-rider effects, therefore,
more
it is
disadvantage of stock markets
when
there
is
efficient at
aggregating information,
it
creates
not always good at producing information.
more dramatic
at the early stages of
This
development
information to be aggregated, and more need to produce addi-
is less
tional information.
This accords well with the emphasis of a number of economic
historians such as Cottrell (1992), Tilly (1992)
and Kennedy (1987) who emphasize
the role of banks in gathering information over the development process. Therefore,
at the early stages of development, as
it is
observed in practice [Goldsmith, 1987],
banks and other non-market institutions are the main channel of financial interme-
As development
diation.
proceeds, however,
more information can be aggregated
and stock markets emerge.
The
fact that
since the
more information reduces agency
work of Holmstrom
information structure
lation to the papers
is
a
costs has
been known at
least
(1979). However, to our knowledge, endogenizing the
new
step.
The main mechanism we propose has some
re-
on tournaments and yard-stick competition [Holmstrom, 1982,
Green and Stokey, 1983, Lazear and Rosen, 1981,
Shleifer, 1985]
which also argue
that conditioning on the performance of other agents improves incentives, but these
papers treat the niimber of projects and thus the information structure as given.
Prom a
large
different perspective,
number
Diamond
(1984) also discusses the advantages of a
Our paper
of projects in a moral hazard setting.
also related to
is
the literature on rational expectations equihbria, see inter aha Green (1977), Gross-
man
(1979) and Kyle (1989).
(1980),
where information
the information that
is
is
The
closest link
a public good and
relevant in their context
market traders, whereas the
perhaps to Grossman and
is
thus imderprovided.
is
is
However,
the private information of stock
role of information in our
of agency contracts. Ovir paper also shares a
Stiglitz
model
common ground
is
to reduce the costs
with the literature on
the financial development and growth. Here especially. Greenwood and Jovanovic
(1990), Bencivenga
how
and Smith (1991) and Greenwood and Smith (1993) discuss
financial intermediation interacts with growth.
in a related spirit discuss the interaction
between
Acemoglu and
Zilibotti (1995)
risk-diversification
through finan-
Newman (1993,1995) and Aghion and
may also be used to endogenize agency
arrangements and growth. Banerjee and
cial
Bolton (1993) propose a mechanism which
costs.
In these papers, the distribution of income
is
endogenous and
on the form of loan contracts; as the economy develops agents become
limited liabiUty constraints
a pattern opposite to what
contracts should
The plan
become
is
less serious.
it
impacts
richer, thus
However, this mechanism predicts
observed in practice; as an economy develops, agency
become more 'high-powered'.
of the paper
is
as follows.
The next
section lays out the basic
model and
characterizes the equilibriiun with stock markets. Section 3 characterizes the social
plarmer's choice and demonstrates that the decentralized equilibriiun
inefficient.
An Appendix
financial
development observed
contains the proofs of the
how our
in practice. Section 5
main Propositions and Lemmas.
The model
2
Set
2.1
2.1.1
We
constrained
Section 4 analyzes equilibrium with banking and demonstrates
model predicts the pattern of
concludes.
is
up of the model
Timing of Events and Preferences
consider a two-period
economy where a
large nxmiber
(M)
only derive utility from second-period consiunption. Each agent
of identical agents
is
risk-averse with
given by
utility
=
EoU{co,ci,ei)
where
second period consumption and
is
Ci
Agents who decide to become entrepreneurs
agents, ei
=
=
EoU{ci,ei)
Eo\ogc{ -v{ei),
will
have to exert
economy
side of the
consists of
We
entrepreneurial labor.
will refer to these as
'capital-intensive' (y) sectors, respectively.
two
ship spend the
end of
this period,
Each agent
who
hmnan
second period of their
consmne the proceedings
Output
is
is
/
The
in the labor-intensive sector dinring the
is
no consumption
the workers no longer work; they simply
of their investments
from the
in the labor-intensive sector of this
labor input and
rest
saved and invested in the capital-intensive
lives,
X
where
capital (at
rtm the capital-intensive sector firms. At the
become workers. They work
whole income
life
decide entrepreneur-
they receive their salaries and consume this amount.
in the first period, their
sector
in the first period of his
period of their fives and receive a wage income. Since there
sector. In the
first
the 'labor-intensive' (x) and the
period of their lives acquiring the necessary
first
in the second period, they
of the agents
first
'capital intensive' (y) sector.
economy
is
given by:
= Al,
A is aggregate labor productivity.
Since
all
factor markets
are assimied to be competitive, the entire output will accrue to the workers.
particular, since all first period
A{M —
letters
the
N), where
w
and
s
income
is
saved,
we have
of agents
who
w=
s
=A
and
In
W=S=
denote respectively wages and savings and capital case
denote aggregate variables. N, which will be
number
0.
for all other
The
sectors.
has a choice of whether to become an entrepreneur. Agents
and
and
>
and the second sector uses savings and
uses unskilled labor as the imique input,
cost)
eflFort
0, v'{.)
0.
The production
no
=
Also v{0)
ei is effort.
oiir
key endogenous variable,
decide to become entrepreneurs at time
is
0.
Production in the capital-intensive sector requires a project and an entrepreneur
to transform the savings of workers into output.
denoted by
'large'
that -^
C/
=
[0,A''];
nimiber (that
is
^More
constant).
presicely,
is
each project
is
The
set of available projects is
in our calculations
we
will let A^
—>
oo and
M
Each project can only be run by one entrepreneur
we assume
that
if
(M)
represented by an integer and A^
more than one entrepreneur run the same
—
>
^,
is
a
oo such
thus
N
is
project, they all
also the
the
number
amount
open
of
The
projects.
level of
production in project j depends on
of capital (kj), but there are decreasing returns at the project level [so
that in equilibrium not
productive only
[see discussion
all
the funds are invested in only one project]. Also, a firm
employs an amoimt of savings larger than some
if it
below on
The production
this feature].
ezk^
where ^ G
effort of the
as;
>D
<D
if kj
if kj
{0, 1, ^} is a stochastic variable
D
critical level,
function for an entrepreneur
with a project in the capital-intensive sector can be written
Vj
is
whose reaUzation depends on the
entrepreneur and on the state of nature. To simplify matters,
level of
we assume
that each entrepreneur decides between high and low effort, and the utility cost of
high effort
=
is v{.)
by no other agent
Whether the
e.
in this
entrepreneiir has exerted high effort
economy. The underlying state
is
can be Good with probability p and Bad with probabihty
2.1.2
We
observed
also unobservable
and
it
— p.
Uncertainty.
assume that the
set of projects
three subsets, such that
these subsets
is,
[/{o})
U =
respectively,
Each project has an
Pr(j e
1
is
-
1
U{oy
|
U
U
(where
|
\=
N) can be
partitioned into
U Urj^ where the cardinality of
— 7r)A'', [/{i} |= (tt — 6)N, Urj^
U{i}
U[oy \= (1
U
,
|
\
each of
|= 8N.
identical probability of belonging to each subset, thus Vj'
-
TT,
Pr(j G
t/{i})
=
TT-
6,
Common
captures idiosyncratic uncertainty in our model.
as the return of the projects in these subsets will
t/m) =
Piij e
6.
G
U,
This feature
shocks are also present
depend on the imderlying state of
nature. In particular,
-
Vj e
[/{o}
^
Oj
= 0.
^
r.
9i =0
-V;e[/,,}^Kj_^
{
.
6j
vi e U^e}
9j
6i
make
zero returns.
=
=6
=1
iff e^
=
low and state
is
Bad;
otherwise.
= low
= high
iff Cj
iff Cj
and
and
state
is
Bad;
state
is
Good;
otherwise.
This ensures that there are no property rights over the projects so that the
Even if there were property rights and the right to run
savers will always be paid the full returns.
a project could be traded, these rights would have a price of zero until the point when
therefore none of our key results would be aiFected.
N=
N,
Therefore, high effort increases the expected return of a projects in two ways;
it
reduces the probabiUty of a failure in bad times, and
we
assvune that this latter effect
implies that 8
is
=
is
Table
1
and
'small',
l/N, so the ex-ante probability
infinitesimal.
increases the probability
good times. For technical reasons which
of a very high return in
soon,
it
become
will
clear
f/r^ be a singleton. This
let
each project to belong to Urj^
for
simimarizes the conditional probabiUties of the different
realizations for each project.
Table
1.
Conditional probabiUties of realizations.
Production
Underlying
State
Effort
Good
High
(l-TT)
(prob.=p)
Low
(l-TT)
Bad
High
(l-TT)
Low
1
(prob.=l
With
and
- p)
ezk^
1
N
TT
stock markets, the realization of 9 for every firm will be publicly observed
this information
wiU be used to form the posterior public
improve the efficiency of contracts that induce high
costs. Intuitively,
is
Bad
was exerted).
effort
a stronger punishment of bad outcomes
(a failure signals that effort
than when the state
is
Good
The reason
signal extraction in
belief regarding the
A better inference of the miderlying state
underlying state of natin-e.
of nature
level
Zk^
is
of nature will
and thus reduce agency
called for
when the state
was not exerted with high probability)
(a failure is uninformative
about whether or not
to introduce the very high realization 6
a simple way — with 6 =
0,
when
all
is
effort
to enable
entreprenemrs exert
effort,
no information about the iinderlying state would be revealed. The presence of
very high return implies that
exert effort, in the
when
all
projects are open,
and when
all
this
entrepreneurs
Good state we would observe one project with return 6Z whereas
the very high return would not be observed in the
Bad
state.
Therefore,
when
all
projects are open and rim by high effort, the imderlying state would be revealed
and the common shock can be
infinitesimal
of retm-n
is
convenient as
it
filtered
out perfectly.
The assumption
that 6
ensures that the effect of 6 on the aggregate rate
and on the incentive compatibility condition of the entrepreneurs
negligible.
The important economic
rim, signal extraction will
is
point
is
be harder, thus, as
that
the
when only
8
be
a few projects are
number of projects
inference about the underlying state of nature will improve.
will
increases, the
It
Table
has to be noted that there are actually two important features embedded in
The
1.
first is
the one discussed in the above paragraph: as the nimiber of
projects increases, the information about the underlying state of nature improves.
The second
low
effort
feature which will play a crucial role in sections 3
and 4
that high and
is
have different consequences regarding information revelation. This second
aspect will be discussed in detail later.
The Stock Market
2.2
At the begirming of every period, a market which we
and functions without any
offer to
costs of transaction.
call
the stock market opens
Each entrepreneur makes a contract
the market which determines the pa3anent associated with one share (sale
price normalized to $1) of his business in each possible publicly observable state of
nature. Thus, the contract for entrepreneur j
is
a mapping from the space of publicly
observable events, denoted by E(n), into real numbers; Pj
of the world
a G 2(n)
is
:
reahzed, each consumer receives an
E(n) -^
5R.
After a state
amount (dividend)
Pj{cr)
per share. Since the savings used in the capital-intensive sector fully depreciate after
use, the capital value of each share after the dividend
is
zero, therefore Pj{(r)
the post-realization price of a $1 share inclusive of the dividend.
it is
We
also
is
assume that
possible to trade shares in the stock market after the realization of the state
of nature
and before the dividend payment, and
as a result, all agents observe the
share price of each 'firm' (entreprenevir) and, via the price, they infer the realization
of^.
The
set of observable events is conditioned
upon the
n
=
is
publicly observable. Contracts, P, are conditional
^, because as discussed above, n
will
fraction of
open
projects,
determine the amount of information that
upon the following
events:
(i)
what the information revealed about the
whether the project
is
imderlying state
Although the payment of each share could be conditioned upon
is.
successful or not;
(ii)
the performance of other specific projects, this will not have any information content
above and beyond the conditioning upon the public assessment of the underlying
state.
Therefore,
when
condition the payments and dividends on a
of two elements; the
successful
When
first
and the second
this will cause
we
convenient, instead of the overall state of nature a,
summary measure
aj
which
is
a vector
denotes whether the project in question, project
is
will
j, is
the pubhc behef about the imderlying state of nature.
no confusion we
will
drop the subscript
j.
The
ex-post price
of each share in each state
u
will be:
(1)
'
kj< D.
if
we
Finally,
will
assume that
all
contracts are signed at the
same point
in
time when
agents decide their profession, thus an entrepreneur offers a contract to the market
and the workers promise to
[this
invest a certain
amount once they
does not introduce any problems since there
is
receive their
wages
no vmcertainty regarding wages].
This assumption will make sure that entreprenemrs compete a la Bertrand and thus
obtain no rents above their reservation
assumed that entrepreneurs choose
The
2.3
utility.
Also, throughout the analysis
it is
their effort level after all contracts are signed^.
decentralized equilibrium
The Equilibrium Concept
2.3.1
The concept we
use for the decentraUzed equihbrium
will
is
an adaptation of the
notion of Perfect Bayesian Equihbrium to an economy with competing principals.
We require that
(i) all
be obtained by Bayes'
beliefs
rule,
(ii) all
entrepreneurs max-
imize their returns given their contracts and choose the best contract for themselves
and
(iii) all
workers behave optimally based on their beliefs taking
actions as given. In particular, this
projects
and investment
other agents'
means that a worker can take the
levels of other agents
have implicitly assumed that shareholders are Uable for the project's
project gives a zero return, the price turns negative
set of
his return.
losses.
When
the
and shareholders must pay to the entreprenetir
the agreed wage out of their personal income [note that
if
the consimiption of the entrepreneur in
the case of a failure were zero, nobody would choose to become entrepreneur since log(O)
An
open
and hence the information revealed
by these actions as given, and consider a deviation that maximizes
^We
all
= — oo].
model which would give exactly the same predictions with more notations is one
where entrepreneurs can buy insurances against personal failures. Denote the insurance payment
that the entrepreneur j receives in state a by ij {a) and the total compensation of the entreprenemby ujj{a). Since there are many projects, the insmrance provision can function without any residual
alternative
risk,
thus
we have ^^g£(n)
to simi to zero, thus
V
entreprenem- {since this
tj(a)
is
= 0. Also in each state, the total insurance transactions have
— 0, Vct e S(n). Savers observe all insurance contracts of the
ij {o^)
crucial for incentives).
r
Pj{&)
=
\
The ex post
price of each share
ej(a)Zfcf-(wJ'(a)-i,(
^i^
'f ^i
if kj
It is
would then
be:
we
not
^^
< D.
easy to check that the two models give identical results. In the rest of the paper
introduce the insurance market in order to keep the notation simpler.
10
will
Analysis: Preliminary Results
2.3.2
We first characterize an equilibrium in which a number N < N projects are open and
all entreprenevirs choose to exert high effort. We will then determine the equilibrium
number
N, and,
of projects
finally,
prove that imder some parameter restrictions,
the unique equilibriiun will entail aU entrepreneurs exerting high effort.
Since
all
use the convention that
if
j
>
that aggregate risk induced by sampling randomness
studying the model for the range in which
N of
We
f, then project j will open after /.
N
suppose throughout the analysis that the number of open projects
size D].
we
projects are ex ante symmetric, without loss of any generality,
A
is
is
will
will also
so
is 'large',
neghgible [that
is
we
are
minimiun project
large relative to
This assumption implies that by a law of large number argvmient, the subset
the
N projects which are open will consist
and a proportion
of projects belonging to U{oy
(approximately) of a proportion
—
(1
tt)
belonging to U{iy.
Now
tt
the
maximization problem of a representative saver can be written as
max J2 Vi^j) log Yl Pji^j)^j
^^>>
j,aj
where
Sj is
\
j
=^
^J
J
the amovmt that savings invested in project (entrepreneur)
Now, consider a
situation in which
P(a-), to the market,
and
this induces
Given high
all
effort in
a;
e
will
K,
was Good (the high rate of return 6Z
is
high rate of return
and
6Z
let ujq
is
in each of the
be conditioned.
{cDq, (jJi,u!o,
it is
(lui);
and
ri{aj)
N
contract,
effort,
projects
-
and
[it
is
details are
aU projects, there are two pieces of information
wages paid to the entrepreneiu when
and successful
same
each entreprenevir to exert high
whether or not the high return
reward to an entrepreneur by
offer the
equiUbria must have this property
upon which each entrepreneur's reward
project. Second,
j,
a-j.
aU entrepreneurs
investors decide to invest an equal amount,
straightforward to show that
(2)
j=l
denotes the probability associated with the state
omitted].
S
s-t-
9Z
is
First, the return of his
observed. Let us denote the
^i}- In particular, let
publicly
known
observed) and
Coq
and
a)i
be the
that the underlying state
when he
is
luisuccessful
and ui denote the corresponding wages
(tJo)
in case the
not observed. Once entreprenemrial rewards are defined,
the reaUzations of the ex-post price (or dividends) of each share are determined
according to equation
(1).
11
Prom
(1)
V(n,K)
and
=
where we have
(2),
the utility of each saver can be written in a simple form
+
+pn log
^"l
f—
\m — nj
-
(ttZK^
-
wui
(1
-
ttui
case, the entrepreneurial
6Z
1"^
reward
is a;
€
and contribution to
and
N
is
Bad, probability
u 6
1
—
p, or
and
Finally, there are
M — N savers who
open
pn.
is
In this
is effectively
leave the determination of this
may
not be observed because the imderlying
Good, but the
is
was not open, probability p{\
Since
all
nN
—
n).
entrepreneurs exert high
In this
effort, in
projects that are successful
and pay
— 7r)iV which are not successful and pay negative dividends.
positive dividends
N
we
because the underlying state
{ujq,uji} is paid.
(1
open, probability
savers' utility out of the analysis [formally,
both underlying states of nature there are
of the
+
^^ oo that makes this the exact solution to our model].
project that would be very successful
case a reward
is
Note that since there
{Coq^Cji}.
Alternatively, the high rate of return 6
state
Tr)u;o]
(3)
publicly observed
is
only one project with the high rate of return dZ,
we have ^ -^
-
^, and n = ^ had already been defined above. Let us
The high retiirn 6Z is only observed when the tmderlying state
Therefore, the overall probabihty that
project's contract
{I
7r)i:;o)
Good, probability p, and the project with the high return 6
n.
-
m=
set
explain this equation.
is
(l-p)]log (-^^)[ttZK'^ -
[p(l-n)
:
projects, thus
have invested an equal amount in each
will
we need
to divide the revenue
by
logarithms and dividing the numerator and the denominator by
M—
iV.
Taking
N gives the utility
of the representative saver (worker) as (3).
We next write the participation and incentive compatibility constraints that need
to hold for a representative entrepreneur to
be
willing to choose this profession
and
to exert high effort.
(1
— pn)[7rloga;i +
(1
— 7r)loga;o]
(1
The
first
constraint
is
+pn[7rloga;i
- p)7r(logu;i -
for participation.
+
(1
log Wo)
The
- 7r)loga;o] >
e
> V{n,K)
e
left-hand side
(4)
(5)
is
the entrepreneur's
expected utiUty and since the entrepreneur can decide to become a worker this
has to be no smaller than V{n, K). Note that
therefore (4) requires the utiUty of
value of a saver.
all
agents are free to become savers,
an entrepreneur to be greater than the maximized
This makes the problem somehow non-standard in that
12
it is
no
longer a simple constrained maximization but a fixed-point problem.
constraint
expected
is
The second
for incentive compatibility. It requires that the entrepreneur
utility
from high
effort
than from low
In other words,
effort.
has higher
it
requires
the loss of utiUty from lower rewards in the case of low effort to be less than the
cost of high effort.
and u{a) are determined by maximizing
In equiUbrium, P(o-)
(3)
subject to
the participation constraint (4) and the incentive constraint (5) with respect to
^1,
cjo,
(^0)
and uJi. Proposition
together with
all
other proofs
1
summarizes the results
in the Appendix].
is
Proposition 1 In an equilibrium where
1.
The constraints
2.
Each entrepreneur
and
(4)
G = exp
entrepreneurs choose high effort:
all
(5) hold with equality.
receives the following rewards:
UJl
=
=
Cjq
u>i
where
Tj-f-c;:
>
1,
B=
ttG
+
GuJq
= Buq
Savers obtain the safe return -^^^^{ttZK^
4.
The expected
utility
obtained by
all
—
(1
3.
—
agents
tt)
>
with partial derivatives Vn{n,
K) >
for the safe return
1,
and
E = exp{e} >
1.
Bojn).
is
equal to
-+^
As the expression
[the proof of this Proposition
Q and Vxin,
(^)
K) >
-
(7)
1
0.
shows, Bujq can be interpreted as the aver-
age (per project) cost of entrepreneurship borne by the savers. Furthermore, from
equation
(7)
we can observe
the entrepreneur
is
that
not getting
receives a salary that
{^j
full
is
the cost incurred due to the fact that
insm:ance; instead, with probability (1
depends on the outcome of the project and hence he
13
—pn), he
is
bearing
some
As n
risk.
increases this probability
Finally, notice that savers are fully insinred
by
effort
all
feature as
costs
it
entrepreneurs, there
is
and the associated
due to the
costs will go down.
fact that conditional
no aggregate uncertainty. This
is
on high
a very useful
enables us to completely isolate the impaxit of information on agency
by removing the interactions between aggregate uncertainty and agency
The Number
2.3.3
we
Next,
of Projects:
characterize the choice of
entrepreneurs exert high
Assumption
1
costs.
K
and n
still
assuming that in equilibrivmi
all
effort.
^-^m >
E (^) —
1
This condition implies that decreasing returns to capital in each project are
sufficiently strong (low
that,
if
possible, to
/3),
compared to the
effort cost.
therefore guarantees
It
open a new project and pay the compensation to one more
entrepreneur wiU always be more profitable than to expand the scale of production
in existing firms. In the absence of this
would be
affected,
assumption none of our qualitative results
but restricting attention to this set of parameter values simplifies
the exposition.
Recall
now
that aggregate savings, S,
is
equal to the wage income of the previous
period. Then:
Proposition 2 Suppose Assumption
1 holds.
Then, in an equilibrium with high
effort:
(ii)If^>§=^S = A{M-N),
Therefore, the
maximimi mmiber
kj^j^,
of projects which
andN = N.
is
consistent with the tech-
nological constraints will be opened in a high effort equiUbrium,
of savings increases
more
projects will be imdertaken.
Once
all
and
as the stock
the projects are
open, expansion wiU follow in the form of higher investment in each project.
consequence the
determines
is
level of savings
how much
As a
determines the nrnnber of projects which in turn
information becomes pubficly available and thus
to induce the right incentives. Since the level of savings
14
is
how
costly
it
a one-to-one function
of the level of labor productivity, A,
respect to
A
(or interchangeably
Proposition 3 Let u denote
we
will
do
oiir
comparative static analysis with
with respect to S)^.
the entrepreneurial
(^^)
fine the agency costs by the ratio,
[recall
wage with
equation
contractible effort.
(4)]-
De-
Then, agency costs
are a decreasing function of the aggregate stock of savings.
At the
early stages
when S
is
n
low,
also low
is
and thus agency
In richer economies, the nvunber of projects which can be financed
better information
our analysis.
and thus agency costs are
lower. This is
larger, there is
one of the key results of
demonstrates that, as often claimed informally,
It
is
costs are high.
e.g.
North (1990),
development goes hand in hand with the reduction of incentive (agency) costs and
the reduction in these costs at the later stages
is
due to an improvement
in the
information structure of the economy. Moreover, as the economy develops, n, the
probability that the underlying state
underlying state
Good,
is
we have
is
cjq
discovered increases and because
=
(Di,
when the
and together with development, the
agent will bear less risk and the incentive contracts wiU become less and less 'highpowered'. This
High
2.4
is
in
all
-
as long as the cost of effort
entrepreneurs choose high
Assumption 2
We
in the introduction.
Effort as Equilibrium
We now prove that
equilibrium
Une with the evidence discussed
E ( J^) < f ^^ +
also introduce
is
not too high, in the decentralized
effort.
1.
some additional
notation:
p denotes the probabiUty of discovering ex post that the underlying state has
been Good conditional on the actual state being Good.
^Note that while the model
forward.
is static,
to extend these results to a growing
In Acemoglu and Zilibotti (1996a),
assume that production
we
economy
is
straight-
and
on the labor
consider an overlapping generation model,
in the capital-intensive sector exerts a positive externaUty
productivity of the labor-intensive sector, such that At — BY^, and At is the state variable of
the model. In this case, the model exhibits standard neoclassical dynamics with convergence to a
N grows and agency
and the stock of savings increase. Therefore, all results can be easily re-interpreted
the context of a growing economy (an earUer version with the details is also available on request
stationary steady-state. Along the transitional path towards the steady-state,
costs fall as At
in
from the authors).
15
-
V{n, K\p) denotes the indirect
entrepreneurs exerting high
-
b{n,
K\p) and
b'^{n,
utility of
each worker conditional on
p,
with
all
effort.
K\p) respectively denote the average
retiurn of
one project
net of the salary paid to the entrepreneur conditional on high effort and low
effort.
Let us
now estabhsh
that the return to a representative saver in the case of high
effort for all projects is increasing in
Lemma
1
Then^p' >
Lemma
Consider an allocation in which
V{n,K\p')
p,
all
entrepreneurs choose high
> V{n,K\p) andb{n,K\p') >
2 Suppose Assumption 2 holds, then h{n,K\p)
With low
The
the amount of available information.
effort,
effort.
b{n,K\p).
>
U^{n,K\p), Vp.
the cost of effort and the related agency costs are not incurred.
rate of return of the project
informational issues
the
-
is
lower.
first effect is
Lemma
2 estabUshes that
-
leaving aside
outweighed by the second, and savers receive
a higher retvirn from a high effort project than from a low effort project.
Proposition 4 Suppose Assumptions
ized in Propositions 1
Given Assumption
and 2
2,
is
1
and 2
hold.
Then, the allocation character-
the unique equilibrium.
high effort has higher return and this induces each en-
treprenem: to offer a contract that promises high
effort.
To
see the intuition, sup-
pose that an entrepreneiu offered a contract that would make him choose low
effort,
because the rate of return from this project would be lower than the alternatives,
each saver wovild prefer to invest in other projects. Therefore, the entrepreneur with
low
effort
would not be able to
raise
enough funds.
We conclude this section with two remarks about the robustness of omx specification. First, in
our formulation agency costs decrease as the nimiber of firms grows,
because the probability of observing the
nvraiber of firms.
The same mechanism,
fully
reveahng
signal, 6, is increasing in the
that the inference of the imderlying state
improves with the number of firms, would work more generally as long as the number
of possible realizations of the underlying state were of the
ber of projects.
number
Our
specification with just
of projects captures this
as the
num-
two underlying states and a very large
mechanism
16
same order
in a parsimonious way.
Second, our
results
depend on some form of technological non-convexity.
be opened
at all stages of development, then there
However, the assimiption of minimvmi
costs.
function (that
arising
is,
from the
given exactly the
ioi
k
< D,
y
= 0)
is
same retmrn
would be no dynamics
inessential as there
already a non-convexity
is
one entrepreneur who needs to be
it
D
would
would complicate the analysis because the
equilibrium size of projects in this case would depend on the
discontinuity at
agency
as the savers. Thus, removing this assumption
not alter our qualitative results, but
The
in
requirement in the production
size
fact that each project requires
the projects can
If all
number
of projects.
enables us to keep the project size constant irrespective of
the nvunber of projects are open.
Constrained Inefficiency
3
Constrained Efficiency and Experimentation
3.1
We
have now established that decentrahzed equiUbrium induces
may
to choose high effort. However, this
different information
1; if
than high
effort.
a project that has exerted low
information revelation
This
is
the second feature
we
Although the exact
a special featine of our model,
is
which
is
embedded
link
it is
between
than high
amount
instance, consider a
is
but
it
more complex matrix
return but reveal more information.
it
We
can
has a lower
has the potential of revealing socially useful
considerably
actions would require high effort,
and
of socially
revealed by different production techniques.
effort
information. This featmre
effort
an example of a more
think of choosing low effort in this setting as experimentation because
direct return
in Table
discover that the state
general issue; the trade-off between the private return and the
useful information
entrepreneurs
not be optimal since low effort produces
effort is successful,
of nature has definitely been Good.
all
more general than our formahzation:
of actions
and a subset
The
and payoffs than Table
of these
1:
for
some
would have lower private
actions that reveal
more information would
play the same role as low effort in our example.
We now
analyze the choice of a social plarmer
representative agent subject to the
who maximizes
same informational
the welfare of a
constraints as the decentral-
ized economy. Namely, the social planner will directly observe neither the underlying
state nor the effort choices of the entrepreneurs.
Under
this informational constraint,
the planner will choose and announce the set of contracts that will be offered to the
17
who
agents
decide to become entrepreneurs^.
DN
Proposition 5 3 Sh <
(i)
S <
If
effort in the
N—
-^
Sh, the social planner would induce no effort in r projects
> S >
open projects
(Hi) If
N — r projects,
remaining
DN
(a) If
such that;
S > DN,
where
<
<
r
N
and
N=
and high
-^.
Sh, then the social planner would induce high effort in all
(r
= 0).
then
N
all
and r
projects are open
=
0.
This proposition states that the social planner would choose to experiment at
earlier stages of development
Low
effort yields
if
low
effort is
that the underlying state of nature
is
chosen and the project
it
0,
the decentralized equilibrium
However note that even
in this range.
feature that the
amount
we
discover
increases p, the probability that
the society discovers the state of nature. Since for
>
successful,
is
Good. Therefore, investing in low effort projects
is
eqmvalent to social experimentation because
allocation has r
(or labor productivity) is low.
a lower return, thus everything else being equal, choosing low effort
However,
is costly.
when the stock of savings
S<
is
Sh, the constrained efficient
constrained Pareto inefficient
in the social planner's choice
we have the
of information available to the society on which incentive
contracts can be conditioned increases along the process of development, thus agency
costs are
decreasing in
A
and
S.
Impossibility of Experimentation in Equilibrium
3.2
The
still
previous subsection demonstrated that the constrained efficient allocation in-
volves
some degree
of experimentation. However,
the stock market such an allocation
fact that the stock
is
we know from
not sustainable. Yet this
section 2 that with
is
market set-up forces each entrepreneur to act alone and thus
provides no means for internaUzing the informational externality.
ition
is
partly due to the
A
possible intu-
that financial coalitions can form in order to internalize this externality [see
Boyd and
Prescott, 1987
who
suggest financial coahtions as a solution for a different
type of externahty]; for instance, a group of entrepreneurs can get together and
offer shares as
an investment fund. Then,
^Note also that we axe
finitesimal.
This
is
still
is
may be
conjectured that some positive
maintaining the assumption that iV
equivalent to assuming that the level of savings
small, then the social planner
there
it
is
S
is
large or that
would again prefer not to experiment. For instance,
obviously no point in experimenting.
18
n
is
not in-
N were
when N =
not too small.
If
\,
amount of experimentation can be sustained
an equilibrium.
as
We
will
show that
this conjecture is not correct.
We model
the formation of financial coalitions
any number of projects and
these projects. There exist a finite
to
maximize
make
aries will
zero
a set of contracts
j
G
Ji]
and compete a
<
profit'''.
>
u^
More
number I
:
a
if
thus in equilibrium
we assume
specifically,
to each of the entrepreneurs
—
>
^'^
and P*
:
E(n) -^
a and
will get in state
u^jicr)
3?,
i
deals with [that
actually
With
do
this set-up
some amount
no
all
any generaUty that
who
gets in state
is
all
In the
risks.
intermediaries
risk.
socially desirable.
stated and proved in Proposition
6;
here
we
can be
far, it
equilibriimi in the range of saving levels {S
is
If
each
— P*(cr) — Z)jeJi '^]{'^)
the idiosyncratic
of experimentation
tion impossible.
for
is
an intermediary rims a large number
loss of
they bear no
to the existing literature.
offers
again a mapping
and the equilibrium concept we have used so
exists
i
each $1 invested. Thus we have
Z)jej. 9j{a)Z'Kkj
can diversify
we suppose without
this, therefore
shown that there
it
is
amoimt that entreprenein j
as the
in state a. If
of projects, then in equilibrimn,
rest of this section,
level for
intermedi-
with P*(cr) as the amount the investor
he accepts the contract. The amount
the profit of the intermediary
it
it
all
assumed
that intermediary
and an investment package (fund) to the market which
Ti{n)
put $1
of financial intermediaries
la Bertrand,
from the observable states to a pajnnent
u^j
fi-
a dividend stream that combines the returns of
offer
all
profits
entrepreneurs through
A financial intermediary can costlessly
nancial intermediaries (or investment funds).
'run'
among
<
Sh) where
This result will be formally
briefly discuss the intuition
First, the presence of free-entry
and
relate
makes experimenta-
an intermediary engages in experimentation,
it
also needs to
run some other projects with higher returns to cover the costs of experimentation.
However, another financial intermediary can then bid away the profitable projects
who are cross-subsidizing the low effort
entrepreneurs and leave only the loss-making
experimentation project(s) to the
intermediary. Therefore, even with complex
financial coalitions, competition
first
and public
availability of information
make
sure
that any intermediary engaging in costly information creation will not be able to
enjoy the monopoly power required to cover
allocation without experimentation cannot
^For ofF-the equilibrium path behavior,
owned by
all
its
among
the
we assume that
M agents.
19
on the other projects. Next, an
be an equilibriiun either since a
the agents in the Scime generation and
distributed equally
losses
if
all financial
they
make
financial
intermediaries are jointly
profits or losses, these
wiU be
intermediary can do better than this allocation (through some degree of experimen-
and attract a
tation)
non-existence result
Stiglitz (1976).
is
In
large portion of the funds.
is
Grossman and
related to
Grossman and
arises
when
Grossman and
no information
Stightz, in our
equilibrium in the original
game
[see
is
stealing high
eff'ort
effort projects in
and
Stiglitz,
and
and
revealed and no equilibrium
is
in a sense natural
exist in general. Also, in Rothschild
poohng equilibrium by
same consequences
free-riding
our model. However,
it
stealing
as a financial intermediary
has to be noted that again in Rothschild
the non-existence problem arises without coahtions and also that nonis
much more common than
moral hazard models as ours.
We wiU now show that
as in Rothschild
and
Stiglitz 's
insm-ance model a natural
refinement of omi original equilibrium concept. Reactive Equilibrium,
restore a unique equilibrium
tions 1
and
2,
make
is
sufficient to
which coincides with equilibrium outcome of Proposi-
where aU entrepreneurs exert high
effort. Intuitively,
previous equilibriimi concept, to disturb equilibrium
to
and
on the information created by the low
existence problems in adverse selection models as theirs
in
market traders
Bernheim, Peleg and Whinston, 1987 for a def-
this hcis the
projects
and Rothschild and
equivalent to looking for a coahtion-proof
and such an equiUbrimn does not
profitable types
This
exists.
economy the non-existence prob-
Stightz's (1976) paper, entry can always destroy a
away
is
financial coaUtions are introduced. This
since introducing financial coalitions
inition]
Stiglitz (1980)
StigUtz, the information of stock
costs of gathering information, thus
lem only
no equilibrium
This implies that no trader wants to incur the
revealed by the market price.
exists. In contrast to
So,
it
was
according to oiu
sufficient for
positive profits given the set of existing financial contracts.
an entrant
Whereas the
Reactive Equifibrium imposes the additional requirement that the entrant should
not be subject to yet another round of entry which would
make her
incur negative
profits^.
Before providing the formal definition,
we introduce the
following notation.
denotes a set of contracts offered to savers by a financial coafition.
A
game
U.{C\C
U
C
C")
is given in Acemoglu and Zihwhich financial intermediaries are sequentiaUy off'ering
contracts or are withdrawing the contracts that they have previously oflFered. An equiUbriimi is
reached when no intermediary wants to withdraw or make a fiu-ther offer. We show that for any
cost of withdrawing contracts e > 0, there is a imique equifibrium which is the same as the Reactive EqmUbriiun. It is significant to note that Wilson's (1979) equilibriimi concept, which is in
spirit very different than Reactive Eqmfibrium, was motivated by intermediaries withdrawing their
contracts from the market but, in om- game, this equiUbrivun only exists when e = 0.
®
theoretic justification for this equifibrium concept
botti (1996b). Briefly, consider a
game
in
20
denotes the profits of the intermediary that offers
C
when
also
C
is
offered in the
market. Then:
A
Definition 1
(i)
{Ci}
vector of contracts {Cj}
is feasible; all
is
a Reactive Equihbrium
agents are optimizing conditional on {Cj},
>
derived by Bayes' rule, andVj, Tl{Cj\{Ci})
(ii)
VC
:
n{C'\{Ci}liC')
C U C") <
The
>
0,
3C"
:
all beliefs are
0.
U{C"\{Ci}UC'[JC") >
0,
andU{C'\{Ci}U
0.
part of the definition
first
iff
is
common with
our previous definition of equihb-
rium; aU agents need to optimize and based on this, no intermediary makes negative
profits.
We also need to impose that these contracts are feasible,
intermediary promises to undertake project
second part
j, it
contracts are optimal only against
all
is if
a financial
enough funds to do
raises
the 'reactive' equilibriiun restriction.
is
that
It
so.
The
requires that the existing
deviations which themselves would not turn
unprofitable.
Proposition 6
(i)
V5 <
Sh no (Perfect Bayesian) Equilibrium
exists.
and 2 where
(a) V5, the allocation characterized by Propositions 1
all
entrepreneurs
exert effort is the unique Reactive Equilibrium.
The second
that there
is
part of this proposition
is
and fimctioning
costlessly.
always constrained inefficient because
produces too
4
important
for
two reasons.
First, it
shows
a well-defined and unique equilibrium in our economy even with complex
coalitions forming
have
is
little
Second, the unique equilibrium
it
we
leads to no experimentation, thus
information.
Financial Institutions and Information.
We have so far established that
(i)
the amoimt of information increases with develop-
ment, hence agency costs are lower in developed economies and
(ii)
the equihbrium
of our economy, especially at the early stages of development, produces too little
information. This second feature brings the question of
may
arise to deal
strates that
when
with this
inefficiency.
access to information
The
is
what
financial institutions
result in the previous subsection
demon-
not restricted, coalition formation will not
21
prevent the inefficiency. In this context
we
argue that banks can be thought of
will
on access to information and/or producing information that
as placing restrictions
cannot be easily transmitted.
Before
we
start
is
it
we have
useful to recall that
also suggested that stock
markets have good information revelation and aggregation properties. That
success of a project
is
observed publicly through the performance of
its
is,
the
share on the
stock market and this impUes that the information contained in this performance
regarding the underlying state
this informational
transmitted to
is
all
the agents in the economy. Yet,
advantage of stock markets creates a free-rider problem.
wants to bear the cost of experimenting [by choosing the low private return
to reveal information that
useful to the whole society.
is
of stock markets in information aggregation
mation production. In the
rest of the
As a
result,
will
agent
activity]
the advantage
makes them disadvantageous
paper we
No
show that the market
at infor-
failure just
discussed can explain the emergence of a speciahzed financial institution, which
wiU
We
refer to as 'banks'.
reasons
we wiU
emphasize two roles of banks [and
treat each separately
banks are not as
make them
will
efficient at
though they are in no way
for expositional
excliisive].
First,
information aggregation as stock markets but this will
well-suited to information production at the early stages
by avoiding the
Second, banks have alternative ways of obtaining information,
free-rider problem.
in particiilar as
we
emphasized by Diamond (1984), they have the capacity to monitor
individual effort choices.
The comparison
empirically
of stock markets
we observe
and banks
is
of considerable interest because
that banks and other direct lending institutions played an
important role in developing economies and stock and bond markets only emerged
much
later in the
development process.
For instance, in an important historical
study, Goldsmith (1987) analyzes the financial structures often pre-modern societies
and
finds that
institutions
banking was developed in a number of them while stock market type
were not observed at
all
except in Holland and there only due to special
circumstances. However, since the nineteenth century stock markets have played an
increasingly important role in the financing of
Western economies. Despite
to date has offered
this
new and
existing ventures in
many
well-known historical sequence, economic theory
no explanation^.
^An exception is Greenwood and Smith (1993) who discuss debt versus equity and conclude
that equity mcirkets will never arise in equihbrium without government intervention. Another
important contribution
is
Greenwood and Jovanovic (1990) who
22
also motivate the existence of
Banks
4.1
as 'Experimenters'.
Equilibrium With Beuiks Only
4.1.1
we
In this subsection,
drawn
in this section
outline the
is
first
role of banks.
The important
distinction
between institutions that lend to a group of borrowers
through a bilateral relation and institutional arrangements whereby each borrower
comes into contact with the whole market. Banking
That
sive bilateral relation.
entrepreneurs and provides
tion that
the bank.
the funds to these entrepreneurs, and the informa-
produced by these entrepreneurs
is
Intuitively,
would observe
its
if
be modelled as an exclu-
each bank enters into a relation with a number of
is,
all
will
is
not observed by anyone other than
a company enters the stock market, the whole economy
performance. In contrast,
if it
obtains
all its
finances from a bank,
We
the economy will only have limited information on this company.
that the fimctioning of banks
Ce{Nb) that
is
is costly; if
a bank runs
iVj,
projects, there
incurred in terms of final output in every state.
increasing and weakly convex.
One
also
Ce(.)
possible justification for these costs
is
assume
is
a cost
positive,
comes from
the fact that banks often gather information by making most of their investment
in
a particular industry and thus are not well-diversified. Also, banks in practice
incur administrative costs which would be part of Ce(.).
economy when the only
equilibrixun of this
We now
characterize the
financial intermediation possibility
is
banking.
Proposition 7 Let Ce{Nb)
(i)3
bank
is
Se{Ce) such that
active
choosing low
f
=
S<
then
V Ce < nZV^ such
Se{Ce), there
is
that:
a unique equilibrium in which one
carries out experimentation in the
form of f >
1
entrepreneurs
effort.
S >
(a) If
and
if
= CeN b,
Se{Ce), then
we have a unique equilibrium without experimentation,
0.
Corollary 1 There
Ce(.) such that in
number
I
>
1
(i)
exists
and
an open set of
(ii)
strictly increasing
in Proposition
7,
and convex functions
instead of a unique bank,
we have a
of banks.
by arguing that they run small scale experiments to extract information
about the state of natiu-e. However, since each intermediary finances a continuum of projects
but only experiments on a countable subset of them, experimentation is a costless activity in
their model, and there is no comparison of the information production roles of diflFerent financial
financial intermediaries
institutions.
23
Since banks do not automatically reveal the information that
is
produced by the
projects they are financing, they are subject to less severe free-rider problems. In
bank
particular, a
economy
in this
receive outside information
the information that
Note since
and there
on an
effectively
is
model there
derlying state of nature,
if
is
isolated island;
no possibiUty of other banks
produces since this information
it
in our
is
it
does not
free-riding
on
not pubhcly observed.
is
no source of micertainty other than the un-
the aggregate performance of the bank were observed,
the relevant information woiild again be perfectly revealed, thus the restrictions
on information transmission introduced by banking could not work. However,
transmission would not occur easily
if,
as
it
seems plausible, other unobservable
we have
ables affect the performance of the bank. Further,
left
this
vari-
out of the analysis
the possibility that banks trade in information. Although interesting, this extension
would not
alter the quaUtative result as long as the costs of banking, Ce{.), are
positive.
Financial Development: Banks versus Stock Mcirkets
4.1.2
When
will intermediation
be carried out through banks rather than stock markets?
In answering this question,
The
linear cost functions.
marginal cost
is
we
restrict the analysis to the case in
generalization of this result to the case of increasing
straightforward.
At
this stage, a
most convenient. To write the utiUty of a saver
aggregate saving level
S < ND, we
Then we have the
(7).
but with C
=
Cn
which banks have
diagrammatic analysis
in the stock
K=D
substitute
will
be
market economy with
and n
= -^
in equation
utihty of a representative saver given by V{S) = log (('S')
tZD"C can be thought as the average rate of
M+%
return on aggregate savings.
Instead of the decentrahzed equilibrimn,
where the underlying
for the
The
1
now
state is publicly observable;
consider a hypothetical
we then have the same
average rate of return on aggregate savings with
difference
is
due to the
fact that the true state
(^
=
(^gg
=
we would have C
= Cce =
expression
TvZD^-'^
M+^
becomes known with probabiUty
rather than n. Finally, in another hypothetical world in which effort
contractible,
economy
^^"^^^ ^^'^
is
perfectly
entreprenemrs will only be
m+£\e-i\
paid their reservation return in the form of a constant salary.
Now
let
us turn to banking.
When
all
24
projects are financed through banks.
the tmderlying state
perimentation.
On
is
inferred
more
precisely since the banks will carry out ex-
the other hand, banking comes at the cost of Cg per project.
=
Therefore, the average rate of retm:n on savings would be given by C
<zZD^-'^-Ce%
where
\{i-pp')
(1-PP-,
M+%
p' is
the probabiUty that the Good state
depends on the optimal extent of experimentation and n
<
p'
<
=
revealed, p'
when
than
less
projects can be opened. Figure la plots the inverse of these four rates of return
A'^
(plotting the inverse simplifies the diagrams).
As
given [but of course no experimentation
Ss{Ce)
SliCe)
is
ND
SUCm)
intuitive reason
Cos^
why
starts at the
these curves start at the
not matter relative to
imderlying state
At
Figure lb.
same point but
are zero, the utihty of consimiption
is
minus
lies
Bank
not observed,
it is
above
infinity
(^'^^
ND
as 'monitors'.
retiurn
with contractible
everywhere above C~^ [the
same point
^~^ also starts at the
this].
this point, all projects are
is
other savers as
Ss{Cm)
below the other curves, implying that the rate of
effort is highest.
all
possible with stock markets]
Figure la. Banks as 'experim^enters'.
(^~^ lies
max-
before, the equiUbriimi will
imize the utility of a representative saver taking the decision of
is
I
is
(be
is
that
when
and the cost of
same
savings
effort
does
point, but because the
imtil aggregate savings reach
ND.
nm with high effort and the imderlying state of natture
revealed almost surely, thus the two curves meet again. Next to understand the
shape of
Q^
(thicker curve), suppose that
then the (inverse of the) rate of return
Constrained Pareto
Optimum
is
banks fimction at no cost
(i.e.
given by the dashed curve which
of this economy.
25
is
Cg
=
0),
also the
This line starts at the same point
as the other ciirves, then
case can use
it
remains below
more information than the decentralized economy], and then
the point where the bank finds
optimal to stop experimentation,
it
the cost of banking Ce increases,
Figure
is
up
(^"^ shifts
it
finally, at
meets
(^~^
As
.
Inspection of this
multiplicatively.
suS&cient to establish the following proposition [except for the multiplicity
aspect which
is
discussed below].
Proposition 8 3 Ce,SL{Ce), Ss{Ce) such
S G
If
(i)
[because the agency contracts in this
(^~^
{SL{Ce),Ss{Ce)), then
that for Ce
>
(^be
<
Ce;
Cnn o,nd in equilibrium all funds are
intermediated by banks.
(a) If
S >
Ss{Ce), then
<
(he
Cnn, o,nd in the absence of
the stock market, there exist multiple equilibria;
investment funds in
one with banking and one with stock
market intermediation. With investment funds, the only (Reactive) equilibrium has
all
savings intermediated through the stock market.
Returning to Figure
levels
such that banking
is less efficient
is
(say,
no information
when S > Ss{Ce) banking
coaUtions among entrepreneurs can be
If
through financial intermediaries), stock market
an equilibrium with banks.
if
we do not
in the
predictions of our
state of nature
observed in the development experience of
many
countries.
on banks and
Goldsmith, 1987]. The intuition for
two types of
institutions
duce information but are
is
why
there
is
higher)
less
less devel-
is
this historical switch
between the
worth discussing. As noted above, banks actively pro-
inefficient at
and since there
experimentation diminishes.
becomes
typically
bilateral direct lending relations [see
information aggregation. In more developed
economies the amoimt of information that needs to be aggregated
N
is
Developed economies
use institutions such as stock and bond markets quite widely whereas
rely
and
a project that uses the stock market.
model are consistent with the pattern which
oped economies exclusively
be the
no firm enters the stock market, then
economy regarding the underlying
for
will
allow costless coahtions, there also exists
Intuitively, if
agency costs would be very high
The
not too large, there will be a range of saving
the only eqmlibrium. Instead,
tinique equilibrimn. However,
is
is
than the stock market.
formed costlessly
there
Ce
la, if
is
is
larger (that
is,
information available at no cost, the need for active
As an economy grows
rich,
information production
important and information aggregation, the activity at which stock
markets have a comparative advantage, gains importance, hence the switch from
26
banks to stock markets^". In support of
this thesis,
stock market of Holland 'Hhe exchange's price
we can quote Goldsmith on the
list...
was one of the most important
sources of information for the period's international trade."
and Thomas, 1973
for
statements to the same
argue in this paper, even the
first
effect].
[p.
217, see also North
This suggests that, as we
proper stock market of economic history appears
to have acted as an important source of information.
Therefore, our model explains the late emergence of stock markets by the different comparative advantages of various financial institutions in producing infor-
mation. Naturally, the prediction that once the stock market
disappear
is
unrealistic,
but
it is
is
introduced banks
due to the simplicity of om- framework
banks only perform 'experimentation'
activity, as well as
in
which
our assumption that
all
projects are homogenous.
Banks
4.2
The
as
Monitors
other role of banks
is
to reduce transaction
and agency
costs via monitoring.
Assimie that each project can be interim monitored at the some constant cost, which
we denote
as
Cm- Monitoring enables the bank to observe whether the entrepreneur
exerts effort or not. Therefore, monitoring forces high effort. For simplicity,
assimie that these banks necessarily monitor
tions in
projects.
all
More
we
also
realistic specifica-
which banks can decide to randomly monitor some projects, or to ex-post
monitor only unsuccessful projects
[as in
Diamond,
1984]
would give the same
re-
sults.
Since each bank deals with a large
number
insurance to the entrepreneur and pay him a
3]
so as to
meet
his participation constraint.
of projects,
flat
wage
it
can provide perfect
[O as defined in Proposition
Thus we can write the return
^°Note that the model also predicts that at the very primitive stages of development
fi-om
(for
S <
SL{Ce)), the stock market performs better than banking. This is because stock markets are
supposed to fxmction at no admistrative cost which is an unrealistic assmnption but makes the
rest of oiu"
mechanism
easier to appreciate.
Goldsmith (1987)'s analysis of pre- modern financial
systems shows that at the very early stages, there was no intermediation, and it was only after a
number of necessary economic conditions were met that banks started to play an important role
in financial intermediation. Therefore, the historical pattern suggests a sequence of a period of
almost no intermediation, then banking and then finally a period of stock market intermediation.
In Acemoglu and Zihbotti (1996a), we show that when we allow for an alternative technology
without division of labor and no need of financial contracts this technology will be chosen at the
very early stages of development. Therefore combining the results of these two papers, we have
predictions which are in line with the styhzed fact about financial development.
27
=
banking in this case as (
C^
is
the same as
(^~J-
(~^ over
some range. But
preferred, thus
=
"^gijiT^r^f
and
in Figure lb,
Cm
Therefore, as long as
Cbm
than a
also
monitoring
if
critical
is
This time for
Cm >
for
is less
•
0, it shifts
Cm =
up
value Cm, the curve
the curve
0,
multipUcatively.
(^
will
be below
very cheap, stock markets will never be
Cm needs to be greater than another critical value Cm and in this case
Proposition 8 will apply to monitoring banks exactly as to banks as experimenters.
The sequence
of financial institutions that arise in equihbriimi will again be in line
with the historical pattern.
and
is
only necessary
do not work.
when
Intuitively,
heavy reUance on direct monitoring
more information
is
revealed,
and
it
to rely on the stock market to aggregate this information rather than
monitoring should play a
societies is in line
It is also
less
is
optimal
make heavy
interesting to observe that the conclusion that
important role in modern society than in more ancient
with the view that information has become more decentralized
and privacy has acquired a value
5
costly
other methods of providing incentives to entrepreneurs
In rich economies
use of direct monitoring.
is
it
did not have before.
Concluding Comments
Agency
costs feature importantly in
many
accoimts of the process of development.
High risk-premia, distorted incentives, corruption, limitations on the division of
labor can
all
be related to the high agency costs faced by
less
developed economies.
Why are agency costs high? We suggest an answer to this question based on the idea
that the hmited range of projects xmdertaken in less developed economies leads to
relatively less information for the society to
for agents.
be used in devising the right incentives
This argument explains a number of styUzed facts about development of
financial institutions
and evolution of incentive contracts.
The mechanism proposed
in this paper opens
a nimaber of avenues
search. If indeed information reduces agency costs
and
is
for future re-
increasing in the
amoimt
of savings, similar arguments can be applied to imderstand the emergence of or-
ganizations that rely on
Fm-thermore,
on
financial
more complex
we have omitted
divisions of labor
and deeper
in this paper the importance of
arrangements and agency
costs.
The
hierarchies.
income distribution
interaction between these
two
as-
pects has not yet been investigated but potentially important in imderstanding both
the development of financial institutions and the incentive problems faced by
societies today.
28
many
APPENDIX:
Proofs
Proof of Proposition
1.
1.
Suppose that
holds with strict inequaUty for the entrepreneur running project v.
function could be increased by reducing the entrepreneur's salary in all
states. Next consider the case where (5) holds with strict inequality. Then a meanpreserving contraction in cjj and Uq would still satisfy the incentive compatibility
constraint but would increase the utility of the entrepreneur, thus cOi and a>Q can
be reduced without violating the participation constraint and hence the objective
function would be increased [note that due to log utility, we can always reduce these
(4)
The objective
salaries
2.
Let
B
D
without hitting a boundary].
and
G
be as
in the Proposition, uji
Then by
this Proposition.
problem can be written
=
Gloq follows from (5) and Part
1
of
substituting from (5) into (3) and (4), the maximization
as:
V{n, K)
=
m_ax_
- pn) logiir ZK'^ -
(1
Bloq)
+
UlQ ,i'l jliJO
pn log [(ttZX^ -
-
TTWi
(1
- 7r)tDo)l + log
L
J
"
(
(A.l)
)
\m — n)
subject to:
(1
The
— pn)[7rlogG-|-loga;]
conditions that
(Do
+pn[7rlog(Di
= cji =
the F.O.C's with respect to
cjo,
-7r)loga;o]
wo and
= max
log
wo
Cj\
(we omit the
IttZK^
J
The
tives
part follows from the previous point.
straightforward.
Proof of Proposition
can be opened.
We will
first
(A. 2)
2. First part.
maximize
Then, the objective
—
(A.3)
= V{n,K) + e
(A.4)
— nj
\m ^1
f
for
V{n, K) and
u}q
which
is
given
D
3.
first
is
= V{n^K)
as:
(A.3) and (A.4) uniquely determine the solution
by expressions (6) and (7) in the text.
Substituting wq into (A.3) proves
details).
- Bujq] + log
L
log(Ba;o)-(l-pn)[logB-7rlogG]
4.
e
as:
and the participation constraint (A.2)
3.
-
-Bwq (part 1 of the Proposition) are immediate from
function can be written (eq. (A.l))
y(n, K)
+ (1
(7)
The
calculation of the partial deriva-
D
Suppose A{M — N) < ND, so not all projects
with respect to n and
subject to the resource
K
29
{nK < ^) and
constraint
the equilibrium value of
the
minimum
n and
S. Since y„
By
constraint must be binding.
{K > D). Then, we
size constraint
>
0, V^-
>
will
determine
(see Proposition 1) the resource
using the resource constraint to eliminate
K, we can then
rewrite (7) as:
(l-pn)
V{n)
=
Lz (-|^
log
This expression
constraint that
j
to be
is
+
g
first
/x
>
condition for
is
/i
order condition
\(l-pn)
m + n E (*)
the Lagrangean multiplier of the
to be strictly positive
is
size)
-/i
=
(A.6)
-1
minimum
size constraint.
A
sufficient
that:
\(l-pn)
Q
is
(A.5)
is:
\(l-pn)
^
1-/3
>
n
which
-1
(J^^
E (#)"--' plog(j.)
^(*)
n
where
m + n fE
- /?) log(n) - log
maximized with respect to n subject to the (minimum
n < j^. The
1-0
(1
^(*)
-1
\(l-3m)
Q
m + n E (*)
(A.7)
-1
equivalent to:
1-/3
m> n
/3
m"
pn)
-1
(A.8)
< E
(A.9)
Finally,
\ (i-pn)
Vn €
(0, 1)
Therefore, Assumption 1
(resource constraint) and
5=
-j^DM
and iV
=
,
n
H^
is sufficient
to ensure that
A{M — N) = S
-^,
and the
first
K
(savings equal
ND
= D. Next, since
= S
wage income), we obtain that
part of the Proposition
is
proved.
A{M
— N) > ND, the first part implies that the economy will open as
part. If
projects as possible, thus n
1. Then, from the fact that VK{'i-,K) > 0. it follows
= S. Finally, a maximum of agents
that the resource constraint will be binding, so
—
become
entrepreneurs,
can
hence
N)
S. This completes the proof.
Second
=
many
N
NK
A{M
=
W=
Proof of Proposition 3. First, consider the case in which effort were contractible.
Then the salary of the entrepreneur would be independent of n and thus of the amount of
savings, which implies
V{n, K)
= loginZRl^ -Q)+ log
n
m—n
(A.10)
and
logu)
= V(n,K) + e
30
(A.ll)
where (A.ll)
is
the relevant participation constraint. Hence:
l
From equation
(6)
we know
(A.12)
+ E-2Tn—n
that:
n
bA(^-p")
B^, =
Then agency
^^^-"^
, „-" ^ZK^
B\(l-P") n
l +
^(^)
costs are given by:
BuQ
Tin— 71
is
decreasing with n, as
Proof of Lemma
it
1. Since the
(A.14)
m
Q
UJ
which
(A.13)
can be
N(l-pn)
verified
by
D
differentiation.
probabihty of discovering the Good state
is p,
the program
to be solved becomes:
max y(n,ii:)
==
[p(l
- p) +
+pplog
(1
-p)] log
{-kZK^ f—
— nj
\m ^^
{-kZK^ f—
— nj
\m ^^
-
TTcDi
(1
-
TxuJi
-
(1
- tx)ujq) +
7r)wo)
(A.15)
subject to the constraints
(1
-p)[7rloga;i
-
-7r)logwo] +pp[7rloga;i
(1
(1 -p)7r(loga;i
The
first
u = Buq
-
-
(1
-7r)loga;o]
log wo)
>
-e > V{n,K)
(A.16)
(A. 17)
e
wi = Guq and
whatever the probability of discovering the underlying state, p. Following the
order conditions of the problem, as in Proposition
steps above,
we can
(1), yield
solve:
V[n,K\p)
=
log(7rZir^)
b{n,K\p)
=
- log
\^ + E
Q
\i^-PP)
[^
(A.18)
{kZKP)
{
pn)
i(^+^(*r
(A.19)
-1
Straightforward differentiation shows that both expressions are increasing in p everywhere.
D
Proof of Lemima
2,
We
entrepreneurs choose low
characterize the
effort.
To
minimum
cost of entrepreneurship when some
when some projects are with
start with, note that
31
effort, the economy can be subject to aggregate uncertainty since in the Bad state,
of the low effort projects fail. In this proof, we will ignore the cost induced by the
introduction of uninsurable risk, since this risk would make low effort less desirable and
thus make our argument true a fortiori. Ignoring the uninsurable risk implies that the
entrepreneurs who are induced to exert no effort are offered a flat wage, which we denote
low
all
by
a;'^
The
On
participation constraint for the no effort entrepreneur implies log(a;'^)
V{n,K).
the other hand, the participation and incentive constraints for the rest of the en-
trepreneurs (4) and (5) together with the
log
=
(
^
+ e + V{n,K).
The two
first
part of Proposition
give log{Bu)o)
1
=
conditions together yield:
Buo
J' =
E (*)
(A.20)
(i-pp)
Then:
{nZK^ -
h{n,K\p)-b''{n,K\p)
Bujo)
- pnZK^ -
BujQ
B
\
-kZK^
(1-p)
The RHS of (A.23)
N, i.e. ^^xH ^^ ^*
i
Proposition
1].
T^ZK^
/
minimum at N = N [since the only term which depends on
maximum at A'^ = iV
see the expression for Bujq in the proof of
reaches a
^^^
BuQ
(A.21)
\(i-P^)
B
E (*)
\(i-PP)
—
At the minimum, the
RHS
of (A.21) becomes:
-nZK^
(1-pp)
1 + -^
^^(Jr)
^
M-N~^\
^
M-N
(A.22)
\G^,
and some simple algebra using the definition of E, G and B shows that the term within
D
is always positive as long as Assumption 2 is satisfied.
brackets
Proof of Proposition 4. Consider, first, an allocation such that Nd < % agents are
entrepreneurs. Then there is an opportimity for a Nd+V*^ agent to become an entrepreneur
and increase his expected utility since there are enough funds. Therefore, any equilibrium
= -^ entrepreneurs. Consider now an allocation where r > of
allocation must have
N
N
entrepreneurs raise funds with a fiat wage contract at uie. Savers will naturally
anticipate that they will exert no effort. By Lemma 2, the contribution of each low effort
project to the savers' portfolio is lower than that guaranteed by high effort projects. So,
each low effort project will raise less funds than
and will not be feasible. Therefore in
the
D
N
equilibrium we cannot have r > 0. Finally, consider the case in which
entrepreneurs
offer contracts that satisfy the incentive compatibility constraint and thus savers anticipate
high
Now, if a iV + P* agent decides to become entrepreneur, he will not be able
enough funds with either a high effort or low effort contract. Further none of the
effort.
to raise
32
entrepreneurs can improve on this allocation, therefore the allocation with
D
contract is an equilibrium and there cannot be any other.
Proof of Proposition
induced for
all
5.
=
projects (r
N high effort
Consider as a benchmark the allocation in which effort is
Now suppose that for a small number of entrepreneurs r
0).
replace the incentive compatible contract with a flat wage w'^. Let p, p' be respectively
the probabilities of discovering the Good state before and after the contract replacements.
we
To be
we have p
precise,
= §
and
=
p'
^+
(l
-
^)
[1
-
(1
-
^
So, p'
Tr)^.
p
+
[l-(l-7rr](l-|) + ^(l-7rr.
This increase in probability of discovering the underlying state from p to
a change in the expected return equal
(A^
where the
{N — r)
first
- r)
term
-r
[b{n,
K\p')
-
the
eflFect
of the increase in the information which the remaining
is
b{n, K\p)]
b{n,
incentive contracts can be conditioned upon,
from inducing no
effort in r firms. (A.23)
N [b{n,K\p') - b{n,K\p)]
where the
first
term
is
cause
p' will
to:
positive
r \h{n,K\p')
and the second
is
U^{n, K\p')
and the second term
can be written
-
-
K\p)
is
(A.23)
the loss return
as:
-
negative by
b'^{n,K\p')
Lemmas
1
and
(A.24)
2,
respectively.
We now proceed in two steps. First, we show that when a non-negligible fraction of projects
are not open, i.e. when h = N — N has the same order of magnitude as A'' [that is 4- > 0],
is always desirable. Second, we analyze the case in which h is small
with respect to A'' so that n —> 1~ [that is ^ !^ 0]. In the first case, since
then experimentation
(infinitesimal)
N
is by assmnption 'large', then p' "^^ p -\- 7r(l — n) [note that since (1 - n) = j^, the
assumption that h is not infinitesimal with respect to
is crucial for this approximation]
N
Then, since b(n^K\p')
(A.24)
is
—
b{ji,K\p)
>
0,
there exists r
>
0,
small relative to TV, such that
strictly positive.
namely the limiting case in which n —+ 1~. Now,
the previous argument does not apply because 6(n, K\p') — b{n, K\p) becomes infinitesimal.
Next, consider the case of 'small'
In this case,
we need
/i,
to write the explicit expression for the benefit of experimentation,
namely:
N [6(n,ir|p') - b(n,K\p)\
= ^-kZK^N
1
<
1
+
B
\(1-PP')
^^(*)
1
+ "^^[w]
[\-VP)
(A.25)
which, after rearranging, gives:
-zk^^^e[-8^)
1
(*)"""')
+ ;=fes
(i
(i-p)
-(#)
+ ;#.^ (*)""'")
"
1/iV
(A.26)
33
can now take the limit of this expression as A'' —> oo and n —> 1~. Note that to
calculate the limit of the second term (which is of the type ^) we use L'Hopital Rule. The
We
resulting expression
is:
lim
N—oo.n—*1-
E(^)
(l-P)
1
+ E (^)
(i-p)
-
b{n, K\p)]
=
N
M-N
ttZK^-
A^ [6(n, K\p')
,p\og
N
(^)
[/I
-
- i^Yih + r)] = Q[h -
(1
(1
- nYih + r)]
M-N
(A.27)
which
is
a positive function of h [where the expression
Next, consider the cost.
We
Q
is
defined for future use]
have:
r \h{n, K\p')
lim
N->oo
x),n—!"
- b^\n, K\p') =
1-
1rn ZK^
rH
,1-p
1
The
^<M
+ lih^^
(A.28)
( J^j
net value of experimentation - which depends on r as well as on
/i
-
is
Q[h-{l-'Ky{h + r)]-rH
and, for given h, experimentation
positive. For r
=
0, this
(A.29)
is
is
beneficial
iff
(A.29)
3r € N'^ such that this expression
,
equal to zero and for r
7^
it
can be written
fc>r(r).r-'^/«+'l-'')'
1
Equation (A. 23)
-
(1
is
as:
(A.30)
- -kY
gives:
Sh^N
min Tir)
reN+
(A.31)
the critical savings level such that such that V5 < Sh experimenting is optimal.
Sh < ND, we immediately obtain that S > Sh experimenting has higher
cost than benefit, therefore, in this range r =
as claimed. This completes the proof. D
where Sh
is
Finally, since
Proof of Proposition 6. Part (i). First, we will show that there will exist a contract
that makes positive profits if the allocation has positive amoimt of experimentation [Result
(1)]. Then, we will prove that for S < Sh, there exists a contract that will make positive
profit when the allocation in question has no experimentation [Result 2] These two results
will prove that there exists no equilibrium with our previous equilibrium concept. Then
we will move to prove the existence of a tmique Reactive Equilibrium.
Result 1. Experimentation means that a financial intermediary that markets a set
of projects U* is inducing a subset of this u** C U* to exert no effort - that is, r > 1
.
entrepreneurs are offered a flat wage a;'®. Let us also denote the probability of discovering
that the underlying state was Good by p as before. By Lemma 2, the return on these
— r projects b{n,K\p).
r projects, b'-^{n,K\p), is lower than the return on the other
N
34
Now, another intermediary can attract all entrepreneurs in the set U*\u** by offering
them a slightly higher reward in all states, and market this security to savers at rate
of return b{n,K\p) — e, who will prefer this new security for e small enough; as long as
u** ^ 0, this deviant intermediary would make positive profits. Thus no allocation with
D
experimentation can be an equilibrium.
Result 2. Consider an allocation where a subset of projects U* <ZU = [0, N] is open
Now another intermediary can enter, attract all the
all entrepreneurs exert effort.
entrepreneurs in the set U* by offering all but one of them a slightly higher reward in all
and
and the remaining one a
states,
the
first
flat
wage
(cj'^
+ e),
where
lj^^ is
part of Proposition 5 this portfolio gives higher utility to
coalition can
make
defined by (A.22). By
edl savers and thus the
D
positive profits.
Part (ii). We will first show that the high effort allocation is an equilibrium [Result
and then that no others exist [Result 4]- We focus on S < Sh, which is the region where
an equilibrium did not exist. For the other cases an equilibrium exists with our previous
equilibrium concept and is the high effort equilibrium. Let p = n he the information
available in the absence of experimentation, and p' > p the information available when
some experimentation is carried out (see the proof of Proposition 5 for details). Also,
remember from Proposition 1 that savers bear no risk, thus we can work with average
rates of returns without worrying about variability of returns.
Result 3 Consider the candidate equilibrium allocation where a number L of intermediaries offer identical incentive compatible contracts of the type described by Proposition 1.
Now, if we call
this set of contracts, [Result 2] above shows that 3C'\U{C'\{Ci}L)C') > 0.
However, as shown by [Result 1], for such
there also exists C" such that a second round
of entry is profitable, i.e. U{C"\{Ci} U
U C") > 0. Then, to show that the candidate is
a Reactive Equilibrium we only have to show that U{C'[{Ci} U C" U C") < 0.
As it is characterized in the proof of [Result 1], C" steals all projects that had high
effort, and leaves projects that have no effort with the incumbent intermediary that offered
3]
d
C
C
is
2,
.
The average
h{n,K\p')
Lemma
2,
>
when there is experimentation to a small nimiber of projects
provides savers with a return up to 6(n, K\p'). But, by Lemma
rate of return
6(n, K\p'). Therefore,
must have
C
C"
h'-^{n^K\p').
Furthermore, in order to attract savers away from {Ci\,C'
from high effort that is b{n,K\p), where, again by
offered at least the return
b{n^K[p)
>
b''^{n,K[p).
Now we
claim (postponing by few lines the proof) that
b^^{n,K\p') < b^^{n,K\p). This implies that the intermediary which had entered at the
first stage offering
is left only with the no effort projects and therefore making a loss,
because it is offering savers a return b{n,K\p). This establishes that all entrepreneurs
exerting effort is an equilibrium.
C
To
finish the
b'^^{n,K\p).
J' =
To
proof of Result
5,
we need
to
show
see this, observe that: b^^{n,K[p')
that, as claimed above, 6'^(n, K\p')
=
{pirZK^ — cv'-^)
is
<
decreasing with p
/Xpn) = i+^(g^j
-VJi-^n) ^ is increasing with p.
;;r=;r
Result 4- We now show that no other equilibrium exists. Consider an allocation in
which some projects choose low effort (set ui /g = 0). If we can show that no such
since:
,
E[-^)
allocation can be an equilibrium,
we
will
have established the uniqueness of the high
by the above argument (i.e. that b{n,K\p) > b''^{n,K\p')),
the low effort projects must be run by an intermediary who also runs a number of high
effort projects - let us call the set of these projects with high effort Uh- But as long as
Uh 7^ 01 there exists another intermediary who can offer C" and attract all these projects
in Ufi- And this intermediary can offer a rate of return as high as b{n,K\p') because
effort equilibrium.
First,
the experimentation of the
first
intermediary
35
who
offered
C
still
reveals the underlying
Thus this new intermediary with C" can make positive profits.
Therefore, to show that this candidate allocation is not an equilibrium it is sufficient to
would no^ make negative profits, that is n(C|{Ci}U
show that for any further entry C",
does not do cross-subsidization
C" U C") > 0. However, this is true by definition since
of projects; the worst that can happen is that it loses u^ because C" offers a better
return (say closer to the maximum rate fe(n,K|/j')), but C" never makes negative profits.
state with probability p'
.
C
C
D
< txZD^
ensures that the return
to each project remains positive after paying the banking cost.
Second, free entry im-
Proof of Proposition
7. First,
the condition that Ce
implement the Constrained Social Optimum
subject to payment of the 'administrative' cost of Ce per project. Note that no entrant
can free-ride on the experimentation carried out by the incumbent bank because the replies that
if
a unique bank
is
sulting information does not
adopted
if
and only
if
active,
become
it
will
public. Finally, the proof that experimentation will
is below some threshold (5e) follows the proof
which
needs to be subtracted from the return of
Ce
the stock of savings
of Proposition 5 except for the cost
each project. The corresponding expression
Se{Ce)
=
is:
mm = r
(A.32)
^_^^_^y
where
,(i-p)
rr { _B_Y'-P>
_N_
1-P
This completes the proof of the Proposition.
G'^
JV
D
liFirst note that as long as Ce(l) < nZK^, there exists an equilibrium with banking. This follows immediately from the fact that the allocation in which
each project is run by one bank dominates no production. Second, note that in all equilibria, banks have to make zero profits; otherwise, another bank could enter and run exactly
the same projects and pay e more to the savers in all states. Now we will construct an
example where Ce(.) is increasing and convex and where there are a number of banks
Proof of Corollary
larger
than
1
carrjdng out active experimentation. Let the savings level he
S<
Sh, then
consider the following cost function for banks:
^
where Nb < -^ and
> 0. Also Ce < nZD^ and Ce > Ce- Let Ce —> oo, then we will have
at least two banks in this economy. Next, since at least two banks will have Nb projects
and -j^ > 0, experimentation is profitable by the argument of the proof of Proposition
5. This establishes that for Cf{.) increasing and convex, we have an equilibrium with
many banks and each bank (with the possible exception of one that may be too small)
runs active experimentation. Next, suppose a sequence of increasing convex functions,
{Cg(.)}fc — C|°(.), and a sequence of economies {£''^}fc such that economy k has the
>
36
banking cost function C^. Then it immediately follows that 3k* such that \/k > k*, the
equilibrium of economy E'^ has a number Z*^ > 1 of banks where each bank (but possibly
one) runs experimentation with r*^ > 0.
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