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SETTING STANDARDS: INFORMA TION A CCUMULA TION IN
DEVELOPMENT
Daron Acemoglu
Fabrizio Zilibotti
No.
97-6
March, 1997
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
WORKING PAPER
DEPARTMENT
OF ECONOMICS
SETTING STANDARDS: INFORMA TION A CCUMULA TION IN
DEVELOPMENT
Daron Acemoglu
Fabrizio Zilibotti
No.
97-6
March, 1997
MASSACHUSEHS
INSTITUTE OF
TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASS. 021329
•TiTUTE
MUl.
Setting Standards:
Information Accumulation in
Development*
Daron Acemoglu
Massachusetts Institute of Technology and
CEPR
and
Fabrizio Zilibotti
Universitat
Pompeu Fabra and CEPR
March, 1997
Abstract
We
propose a model in which economic relations and institutions in ad-
vanced and less developed economies differ as these societies have access to
different amounts of information. This lack of information makes it hard to
give the right incentives to managers
and entrepreneurs.
We argue
that differ-
ences in the amount of information arise because of the differences in the scale
of activities in rich and poor economies; namely, there
is
too
little
repetition
of similar activities in poor economies, thus insufficient information to set the
Our model predicts a number
transformations as the economy accumulates
appropriate standards for firm performance.
and structural
and information.
of institutional
capital
Keywords:
Information, Development, Agency Costs, Incentives, Relative
Performance Evaluation, Risk Sharing, Sectorial Transformations.
JEL
Classification: D82, M13, 013, 014, O40.
*We thank Piero Gottardi, Dilip Mookherjee, Jaume Ventura and seminar participants at
Boston University, IGIER-Bocconi for useful comments. Zilibotti acknowledges the kind hospitality and financial support of the European Forum of the EUI.
Introduction
1
The
traditional approach views economic development as a set of interrelated changes
ranging from the structure of production to social institutions.
In this paper
we
propose a theory of development based on the evolution of principal-agent relations
which emphasizes some of the same themes. Even though we are
far
from providing
a unifying theory that can do justice to such a complex phenomenon, our
model
generate a host of structural changes in the process of growth. In particular, in
will
our economy, efficiency and productivity
will increase, the extent of risk
sharing will
change, agents will use more sophisticated production techniques and more division
of labor, the sectorial composition of output
economy
will evolve,
and some new
and the degree of specialization
financial institutions will
in the
emerge while others
disappear. Naturally, our analysis and therefore these results will be extremely stylized,
but we
will argue,
they give some of the flavor of the complex process that
is
development.
Our main argument can be broken
1.
into the following steps:
Delegation of tasks, employment relations and entrepreneurial activities are
important
for
wealth creation. All of these give a first-order role to principal-
agent relations in the organization of production (see for instance Mokyr, 1991,
North, 1990, Pollard, 1965,
2.
Principal-agent relations
formation
it is
is
fail
when information
is
scarce
and thrive when
in-
abundant. In particular, in the absence of adequate information,
excessively costly to give the right incentives to workers,
entrepreneurs
3.
Stiglitz, 1987).
Holmstrom, 1979,
(e.g.
managers and
for a formal analysis).
Societies accumulate information about a certain activity
by repeating
it,
in
other words, by allocating some of their scarce resources to this line of business.
When
a certain activity
developed a standard iov
it.
is
repeated
many
times, the society will have
This standard can be used by the entrepreneurs to
increase their efficiency, or by principals to control the entrepreneurs/managers
by comparing
their
performance to
historical study of Pollard (1965)
ficiency were very
in mining, textiles
this standard.
who
is
echoed by the
explains that embezzlement
common among managers
and other new
This view
at
first,
and
inef-
led to high failure rates
industries during the seventeenth century.
and were eliminated only slowly over
ment became part
time. In the end, professional
of the efficient organization of production. Regarding the
importance of setting standards, Pollard writes:
'^as
ever there was consid-
erable difference between the pioneers, ploughing a lone furrow,
groups large enough
will use as
a body of knowledge...''
to develop
case of developing a standard
which we
manage-
is
(p.
and those in
A
122).
special
the use of relative performance' evaluation
our main example to illustrate
how information improves
principal-agent relations.
4.
scarce,
and
there-
in great quantity to every single activity.
The
result
In a less developed economy, capital (broadly construed)
fore cannot
be allocated
lack of repetition in
is
economy accumulates
of activities,
many
capital,
is
areas and thus informational scarcity.
it
will start
As the
devoting more resources to a range
and the accumulation of information
improve
in each activity will
the principal-agent relations and productivity.
These steps bring us to our
formation which implies greater
first
main conclusion: prosperity implies more
efficiency,
and, in turn again, prosperity.
in-
This
is
a process of growth that would be called a "virtuous circle" by Singer (1949) or
"circular cumulative causation"
The mechanism through the
by Myrdal (1956).
accumulation of information and change of incentives
an important, and to date
is
unexplored, alternative to the existing formalization of these ideas
Shleifer
and Vishny, 1989,
or
(e.g.
Murphy,
Matsuyama, 1994). What distinguishes our model
is
not only that we are offering an alternative microfoundation for "sectorial externalities",
but the implications. The improvements principal- agent relations due to
information accumulation lead to a number of the structural transformations which
are not predicted
by
existing models.
Before discussing the implications of our approach, we outline our formal model.
Our economy has many
sectors (islands).
Production in each sector takes place
within firms run by entrepreneurs (managers) using capital and labor.
of each firm
shock.
The
depends on managerial
effort choice of the
effort,
an idiosyncratic shock and a sector-specific
entrepreneur
is
her private information, introducing
the principal-agent problem in our economy.
if
She can be induced to exert
her compensation depends on her performance. Since
this
is
costly.
As the
capital stock increases,
different activities, enabling the parallel
The output
more
employment
all
effort
agents are risk-averse,
capital will be allocated to
of
more entrepreneurs
in
each
With many entrepreneurs
sector.
as
an adequate standard and
is
be increasing
effort
earlier stages of
and the
level of productivity
economy. Since productivity
in the aggregate capital stock of the
endogenously lower at
performance can be used
out sector-specific shocks ensuring better
will filter
As a result, the level of entrepreneurial
incentives.
will
in a sector, average
development, growth
will
be slower relative
to a pure neoclassical version of our model.
Now
a)
We
the implications:
economy develops and more information
find that as the
is
accumulated,
there are two opposing forces impacting on the extent of risk sharing.
one hand, more information enables greater risk sharing at a given
effort.
On
to exert
more
effort
which reduces
As a
risk sharing.
explain the interesting findings of
Asian villages he finds the degree of
if
the
level of
the other hand, with more information entrepreneurs are induced
result, risk sharing
may
U
shape pattern. This result
Townsend
(1995a,b). In his studies of
decline with development or follow an inverse
may
On
risk sharing
is
lower in richer villages; "as
consumption insurance, whether indigenous or otherwise, deteriorates with
growth" (Townsend, 1995b,
p. 95).
We know
of no other explanation for this
interesting finding.
b) Our model predicts that a
less
developed economy
may
be highly specialized in
The
order to economize on agency costs by investing only in a few sectors.
structure of production will then
ops. In particular,
vary across
when
is
economy
devel-
production and productivity in the sectors with
will increase
with development.
pattern that the share of agriculture
income, and this
as the
the importance of effort and the nature of uncertainty
activities, relative
more agency problems
become more balanced
is
often explained by
It is
a well-known
strongly correlated with per capita
some
sectorial "increasing returns" in
manufacturing (Matsuyama, 1991, see Backus, Kehoe and Kehoe, 1992,
some
evidence).
Our model
offers
for
a microfoundation for this sectorial increas-
ing returns in manufacturing whereby, even though technologically unbiased,
growth
c) Since
If
will favor
Adam
some
sectors at the expense of others.
Smith, division of labor
is
seen as an important engine of growth.
the process of production can be separated into smaller or more special-
ized tasks
and delegated
to different agents, efficiency can be improved.
And
yet,
with division of labor there
a need to provide the right incentives to
is
Our model
the agents working in different parts of the production process.
shows that as information
is
accumulated and agency costs decline, produc-
become more
tion with division of labor (delegation) will
general prediction
is
that a developed society should use
production techniques than
d) Economic development
is
The more
attractive.
more "sophisticated"
developed economies.
less
typically
accompanied by urbanization. Bairoch (1988,
pp. 246-248), for instance, describes
how England (and many
other countries)
got transformed from an essentially rural society to an urban one.
enables us to think of another dimension of this transformation.
Our model
The
close-knit
structure of villages provides "efficient" direct monitoring of agents, which
certainly related to Townsend's
and other
is
researchers' findings that simple vil-
lage institutions achieve a high degree of consumption risk sharing. In contrast
to villages, monitoring agents in cities
tion that cities provide
more privacy
is
much
harder, hence the
to individuals.
common
Our model then
predicts
that at the early stages of development, the close monitoring of villages
be very valuable, but
mation
is
this
advantage diminishes as more decentralized
"
may
infor-
accumulated, and the right incentives can be given to entrepreneurs
in cities without excessive costs.
common
no-
More
generally, this reasoning explains a
perception, expressed succinctly by Rosenberg and Bridzell (1985):
The move
to
wealth
is
a
move towards
greater possibilities of privacy
and
individual choice'' (p. 4).
e)
A
similar reasoning to (d) also gives us a
way
of understanding the transforma-
tion of financial arrangements over the process of development.
historical study,
Goldsmith (1987) shows that premodern
societies
tions to intermediate funds, but these were quite different
today. Intermediation
was
local
and
relied
In a classic
had
institu-
from what we have
on heavy monitoring. In contrast,
today a large fraction of funds are intermediated by stock and bond markets
and even the banks which
still
play an important role do
relative to village "usurers" of older times.
cross-section
when
The same pattern
is
monitoring
observed in a
the financial institutions of low income economies are com-
pared to their western counterparts
is little
little
(e.g.
Besley, 1995).
information in the economy regarding
be conducted, close monitoring
is
beneficial.
how
As a
Again,
when
there
a certain business should
larger scale of activity in-
creases the
amount
more
relatively
of decentralized information, different institutions
efficient.
As the discussion
suggests, our paper
the literature, and without doing
growth aspect of our model
structure
is
full justice to
rather standard.
evaluation using the linear structure
now on H-M,
first
many
(1981),
The key
Holmstrom
More important
We
model
differences
the information
is
is
the
performance
relative
H-M,
1991).
Models of
for instance,
Lazear and
Green and Stokey (1983) and
between our paper and these contributions
endogenous
our paper
for
The
introduced by Holmstrom and Milgrom
see also the apphcations in
(1982),
different strands of
the insights of some of these.
performance evaluation date back even further,
relative
Rosen
related to
is
we borrow from information economics.
(1987) (from
become
in our setting;
and
Shleifer (1985).
are: (a)
the fact that
(b) the application of
endogenous
information accumulation to the process of development.
Other papers closely related to our work are Greenwood and Jovanovic (1990),
Greenwood and Smith
man
(1993,96).
Acemoglu and
(1993),
Acemoglu and
Greenwood and Jovanovic
Zilibotti (1996,97),
(1990),
Greenwood and Smith (1993) and
Zilibotti (1996) discuss issues of financial
two compare the
roles of
Banerjee and New-
development and the
banks and stock markets. Acemoglu and
last
Zilibotti (1997)
explain some patterns of the development process by exploiting the endogenous
risk-return trade-off but does not feature agency problems. Banerjee
and Newman
(1993) account for the patterns of occupational choice over the process of develop-
ment using a model
shift of
of agency.
In Banerjee
population from villages to
are less severe in villages than
cities,
and Newman (1996) they explain the
based on the assumption that agency costs
The main
cities.
difference
between their approach,
which also brings back some of the important insights of the older
velopment economics, and ours
is
literature
on
de-
that they concentrate on the general equilibrium
implications of wealth effects on the interest rate.
In particular,
when agents
are
poor, they have no collateral, and cannot borrow, but with development, these bor-
rowing constraints are relaxed. Since
of information
mentary to
is
like all
previous contributions, accumulation
absent in their papers, their mechanism
is
different
but comple-
ours.
The plan
of our paper
is
as follows:
section 2 describes the environment
characterizes the equilibrium in the absence of imperfect information.
is
the core of the paper.
equilibria
It
and
Section 3
introduces imperfect information, characterizes the
and determines the impact
of capital accumulation
on the organization of
production and productivity. Section 4 discusses a number of implications of our
model. Section 5 concludes and the Appendix contains
all
the proofs.
The Model Without Informational Imperfec-
2
tions
Technology and Preferences
2.1
We
consider an
Each generation
generations.
Throughout
economy populated by a sequence
his
t is
is
of agents where A^
>>
1.
The
utility of
an agent of dynasty h born
as:
^(c^6^e,^)
where e^
A'^
each agent inherits and invests his (her) parent's savings, earns
life,
given
continuum
consists of a
labor income and makes savings decisions.
in period
of one-period lived altruistic
effort, c^
= -exp
(1)
'--^{i'-ri^y^-M'^^^
denotes consumption and
denotes the funds
b^
left for
bequest.
These funds are invested at the market rate of return, and the returns accrue to
the offspring as bequest.
the
amount
Two
features are worth noting.
First, agents care
about
of bequest they leave rather than about their offspring's utility (An-
dreoni, 1989). Second, preferences over total wealth exhibit Constant Absolute Risk
Aversion (CARA).
Each agent has a career
He can become
choice.
either a worker
wage w, or a manager (entrepreneur) and earn the managerial
income
will
and earn the
salary z. His total
then be given as follows:
'rtb'l_i+Wt
^t
=
ftbt~i
where x denotes income, and
if
worker
<
r^
+ Zt
if
manager
denotes the gross rate of return on savings.
further assume that capital depreciates
upon
use, thus rj will also
We
will
be the net rate of
return on savings. Given the separability between the saving and effort decisions,
utiUty maximization impHes that
c^
=
(1
—
s)xi
and
6j'
—
sx'l,
thus the warm-glove
bequests and Cobb-Douglas preferences ensure a constant savings rate s which will
simplify the dynamics in our model.
The
indirect utility of agent
/i
can then be
written
as:
U{xle1)
-pU-yM
= -ex^
(2)
2/5
=
where we have defined p
s^(l
- s^'^p and P =
cause no confusion, we will drop superscript
There
j G
[0, 1].
a continuum
is
1
of islands.
Every period there are
N
All islands produce exactly the
this
in island j at time
i
=
Vijt
lijt
is
(/^
Cijt
t, yijt
+ eijt + ajt)
total labor hired
a constant,
is
output
Within the
island,
,
min
The amount
of final
role of
[l, ltjtk]^t'^\
aff^ects all
+
(3)
^ijt
kijt is capital,
cr^)
and
ajt '^
firms on island j,
We
and
eiji
is
an
is
an
think of e^t as capturing the
managerial ability ex-ante unknown to the
All productivity shocks are independent
N{0,
good
given by:
is
the effort exerted by the manager of this firm, ajt
is
importance of luck as well as the
e^jt '^
final
some others become workers.
capital.
idiosyncratic shock which only affects this firm.
and we have
economy.
by the firm inclusive of the manager,
island specific productivity shock which
agents.
N agents on each island
same good.
Production requires one manager, labor and
produced by firm
this will
agents in island j will have to work there, but they
of the agents will choose to be managers while
where
When
s^'^p.
h.
can invest their capital in any of the islands of
some
-
Labor cannot be transferred between islands but capital and
can. Therefore, each of the
/x
5^(1
N{0,
u"^).
from each other and over time,
The Law
of Large
Numbers imphes
(ignoring technical details related to the continuum) that in each period /q ajtdj
and
/o J2i ^ijtdj
The form
=
0.
of the production function captures the idea that a
sary for production
there
is
(
"division of labor"
)
and has
to exert
some
manager
effort
is
neces-
but also that
only a limited amount of capital and labor that the manager can produc-
tively use (see also footnote 6 below).
is
=
no need
for
workers to exert positive
workers also have to exert
Capital
is
effort
Though managers have
effort (or equivalently,
imply that wealth
we could assume that
but they are perfectly monitored).
owned and supplied competitively by the
constant savings rate and
to exert effort, there
agents.
As we will
CARA preferences over the uncertain
effects are absent, thus
see shortly,
income process
will
income distribution among agents does
not matter for occupational choices and the dynamics of the aggregate capital stock.
Also because capital
is
perfectly mobile, the distribution of wealth across islands
is
of
no importance. Therefore, the
be the unique state
total stock of capital, Kt, will
variable of this economy.
we assume
Finally,
we
that there exists a large set of potential intermediaries, which
They can
refer to as firms.
owned
economy and
are
capital, labor
and one manager
at time
by generation
t
all
distributed
among
the owners, and
the owners. Since there
owns an equal share
if it
makes a
no aggregate
is
of all the firms bears
makes
a firm
If
no
risk.
positive profits, these are
economy, an agent who
Therefore,
situation in which
some agents owned a small subset
depending on the
each agent owns an equal
risk in this
profits.
risks
active firm rents
the losses are also distributed
loss,
maximize expected
It is
Each
agents.
We assume that
of the firms in the economy.
among
t
this
Both workers and managers have to
for production.
be hired from a competitive spot market.
share of
any of the islands of
freely decide to enter into
all
firms will simply
we
started from a
of the firms
and thus faced
straightforward to see that
if
then there would be Pareto improving
profits of these firms,
trades in shares.
The Equilibrium Concept
2.2
Throughout the paper we
will use a
concept of equilibrium very close to the standard
notion of competitive equilibrium, and we will model
stage
game
in the presence of unfettered competition
In the
1983).
first
be active.
will
denote this by
If
i
stage, potential firms
firm
€
first
announce
announces at time
i
M.jt-
announced that they
take the
it
We
also let
Mjt
=
t
it
among
in
j.
island,
be active
number
Mj =
and
We
'We
also
manager,
will
time
i is
it
and labor and
assume that
if
enter.
UJt
e.
first
G M.jt
{^jt)j0Qi\ ^nd the
We restrict each
to denote the publicly observed
stage announcements
Mt, Kt) and
capital
demands
Zj (ut,
for all
M(, Kt)
i
summarized
6
[0, 1],
G M.jt and j G
[0,1],
for all j
it will be active in island j and does not hire a
This assumption ensures that in equilibrium only firms which
a firm announces that
incurs a small cost
be active
also use
a set of
factor return functions Wj {ut,
r {u)t\yi.t)Kt),
all i
t.
static equilibrium at
by Mt,
we
economy, Kt, as given and compete to hire workers and
firm to hire at most one manager."'
A
any, they
of firms that have
managers from island j and capital from the economy-wide market.
state of nature at time
if
in island j,
In the second stage
stage announcements as summarized by
total capital stock of the
firms (see Townsend,
which
will
H^M.jt be the
be active in island
will
that
as the equilibrium of a two-
^
lij
(Mt, Kt) and
1.
No
by
firm
kij
(Mt, Kt)
€ A4jt
i
,
any j G
for
No
firm can change
expected
3.
[0, 1]
offering a different function
amounts of labor and
2.
such that:^
The
its
can
than Wj,
capital than
lij
strictly increase its
and
Zj,
and
r,
or by
expected profits
demanding
different
kij.
entry decision in the
first
stage
and
strictly increase its
profits.
resulting allocation
is
feasible, in the sense that:
kjdi<N,yje
/
[0,1]
(4)
JieMji
f
f
kijdidj< Kt.
(5)
Jo JieMjt
4.
E
In every island j
[0, 1],
given Wj
[ut]
Mj, Kt) and
Zj
{cut]
M(, Kt), Mjt agents
choose to become managers.
5.
All
Note
managers choose
in particular that
concept that
all
maximize
all
price for capital. This
is
their utility.
we have already imposed
firms in island j will pay the
and managers, and
Finally, a
effort to
firms in the
same
economy
will
as part of the equilibrium
(state-contingent) price for labor
pay the same (state-contingent)
a straightforward implication of competition
djmamic equilibrium
is
among
firms.
simply a sequence of static equilibria linked through
bequest decisions.
We
1.
will
now
analyze the equilibrium of the economy under two scenarios:
Effort choices of
managers are publicly observed (the case of perfect informa-
tion).
An equivalent definition of equilibrium can be given where all agents would be pure price-takers.
This does not affect our results as long as the space of commodities is chosen appropriately, in
particular, insurance contracts need to be allowed, and some incentive compatibility constraint
would have to be imposed on these when there is imperfect information. Yet another equivalent
model would involve managers hiring workers and capital, and writing co-insurance contracts with
^
each other while also respecting incentive compatibility.
^It is however possible in theory that two firms in island j offer different state-contingent
functions, say Wjiloj) and w'J^{uJ) but workers are indifferent between these two functions and firms
make the same expected profits. Leaving this case out is without loss of generality, since it does
not happen in equilibrium, and enables us to keep the definition of equilibrium relatively simple.
.
2.
managers are not observed by any other agent
Effort choices of
economy
in the
(the case of imperfect information). In this case firms will have to obey the
incentive compatibility constraints of their managers.
Equilibrium With Perfect Information
2.3
Since output in this economy will be non-random and markets are complete, riskneutral firms will pay non-random wages, managerial salaries and interest rates.
Therefore, Wj
(cvt;
Mt, Kt)
=
Wj (Mt, Kt), and similarly
for Zjt
and
r^.
Moreover,
agents in the same occupation in island j will receive the same payment, and
in island j will adopt the
same technology and
hire the
same amount
within an island must be indifferent between these two occupations.
island j,
—
WjCM-t, Kt)
+
and Zj(M.t,Kt)
is
ZjCM-t, Kt)
agreed level of
where
jg^^t
Cjt is
firms
of capital
Further because agents are free to become managers or workers,
labor.
all
the effort exerted by each
all
all
and
agents
Therefore,
manager
in
the salary of the manager conditional on exerting the
effort, Cjt.
Before characterizing the equilibrium occupational choice in each island,
we
also
assume that
Kt>N~^,
holds at time
at time
t
t.
(6)
This condition ensures that there
so that
if
this capital
is
enough capital
is
allocated equally across
one firm in each island can be run with productive
technology such that
Ij'^kjt"
=
productive efficiency and fully
state the following
Lemma
1, \/j.
When
(6)
is
utilize all of their
all
efficiency,
economy
in the
the islands, at least
namely adopting a
satisfied, all firms are
managerial input.
run with
We
can
now
Lemma.^
1 Suppose (6) holds at time
t
and there
is
perfect information.
Then, in
equilibrium:
—
e^^
=P
in all islands;
1.
Cjt
2.
the equilibrium
is
islands (for all j
symmetric in the sense that
E
[0,1],
Kjt
=
Kt).
capital is equally allocated across
The number affirms (managers) which
''Throughout the paper we ignore integer problems and use differential calculus with
10
Mj
are active in each island
is
Mt = M{Kt)
same amount of
All firms hire the
kiKt)
3.
= (fy
sothatl?kl-°'
=
the economy-wide interest rate,
€
tion in every island j
r {u,-
=
N'^K]-''.
and
labor
1 establishes
and
{-^i
l-
and the wage rate and managerial compensa-
are given by:
[0, 1]
M„ K,) = r{K,) =
a
=
l{Kt)
capital:
{I
-
a)
(^)"
zM\ Mt, K,) = z{K,) = «(§)'
Lemma
(7)
number
^
(m
+
^
f)
+
l^i
+
of results. First, since there
(8)
;
f
(10)
•
no informational
is
=
imperfections, managers will agree to exert the first-best level of effort, e^^
which equates marginal cost to the marginal benefit of higher return (part
ond, decreasing returns to capital ensures that the equilibrium
sense that
all
same amount
islands receive the
Mt is increasing in Kt, thus
the number of managers (firms)
(7),
as the
amount
of capital in the
result,
agerial occupation.
Finally, although
economy expands,
=
f^ (^
+ Cjt)
min
[l, i^^/cj""]
Mjt dj
is
output in any particular island
=
{fx
+ /3) N'^K}'''.
associated
who choose
thanks to the large number of islands, total output in this economy
Y^
Note that from
development
with capital deepening and a growing proportion of the agents
Sec-
1).
symmetric in the
of capital (part 2).
As a
increases.
is
P,
is
is
the
man-
random,
non-random:
Together with the lack
of informational imperfection, this ensures that risk-neutral firms can offer full in-
surance to the factors of production that they hire (part
(9) state
In particular, (8)
and
that in equihbrium the expected revenue of each firm net of the additional
—
a) and a.
+
to capital and labor with
f ) is distributed
(10) ensures that managers get exactly the same expected
cost of managerial compensation,
shares (1
3).
(a*
,
return as workers.
Given
all
Lemma
1,
equilibrium dynamics are straightforward. Since a fraction s of
earnings are saved, the equilibrium capital stock follows the recursion:
11
= sYt^s{fi + P)M{Kt)
= s{f, + /3)N-K',-^.
Kt+,
We
(11)
(12)
also assume:
+ P)>
s{tx
This condition - which
is
always satisfied
the steady state level of capital
is
large
N-^
when
'
(13)
A'^ is
sufficiently large
- ensures that
enough so that more than one firm per island
can be opened, and guarantees that the capital stock of the economy does not
below a certain lower bound. Then
Proposition
is
1
Assume
perfect information.
:
that (6) holds att
Then, there
fall
is
=
Q, that
(13)
is satisfied,
and
that there
a unique equilibrium sequence of allocations
where within every period, the number of firms, wages and managerial salaries in
each island, and the interest rate in the economy are given by
aggregate capital stock, Kt, follows (11).
steady-state capital stock,
The
sical:
K^^
—
[s (/x
+ /?)]
Lemma
Kt uniformly converges
to
1,
and the
the unique
'"A''.
equilibrium dynamics of the economy under perfect information are neoclas-
there
is
accumulation until a steady state
capital decreases monotonically in the process.
is
reached and the rate of return on
Furthermore, given the absence of
informational asymmetries, neither the variability of rewards nor the power of incentives
change over time. As a
result, the
behavior of managers and the organization
of production are independent of the stage of development.
3
We
The Economy With Imperfect Information
will
now assume
that the effort choice of a manager
is
his private information.
This introduces standard moral hazard considerations and implies that the manager should be rewarded conditional upon his performance, and thus will have to
receive a
random
return.
We
also
assume that while the ex-post performance of
(a^)
nor
the rate of return to capital,
and
each individual firm can be costlessly observed, neither the island-specific
firm-specific (ejj) productivity shocks are publicly observed.
We will first characterize the equiHbrium wages,
the form of equilibrium managerial contracts conditional upon the allocation of the
12
We
capital stock to different islands.
will
next show that under certain conditions,
only a unique symmetric equilibrium will exist, and we will fully characterize the
dynamic equilibrium
in this case.
and a discussion
static results
Static Equilibrium
Let us
first
earnings.
manager
the case of perfect information,
Lemma
1.
2 Suppose
for
is
all
j e
[0, 1],
the
given by Mj{Kjt)
for the effort cost
we had
(6) holds at
time
Q=
t.
=
Mjt such
=
N'^K]-'' and
k{I<,t)
=
interest rate
=
j' Kjtdj
{^y
economy wide
he takes
(in
for the risks
are active in island j
that:
(^)'"" and
the
his
Then, in equilibrium:
same amount
=
compen-
|).
All firms in island j hire the
kjt
and
as the
wage component of
number of firms (managers) which
Mjt
2.
with some comparative
this section
in island j receives additional to the
compensate him
will
end
Q {ut, Mt, Kt) = Zj {ut] Mt, Kt) — Wj {ut] Mj, Kt)
define
Q
will
of constrained efficiency.
3.1
sation that a
We
Kt.
and
of labor
so that If.k]^
-
(14)
capital:
=
Ijt
l{Kjt)
=
1;
and wages in every island j G
[0, 1]
are given
by:
r
(a;,;
wj
{ut;
where Et
of time
Part
1
of
M^, K^)
is
= r (M„ IQ =
-Q
^f^ft bjt
M„ Kt) = wj {Mt, Kt) =
jj^.Et
[yjt
the expectations operator conditional
(^t;
-Q
M„ K^)]
{cof,
(15)
,
M„ Kt)]
(16)
on the public information
set
t.
Lemma
2
is
identical to part 2 of
Lemma
1,
except that under imperfect
information the equilibrium does not necessarily have capital equally invested in
islands. Part 2 of
Lemma 2,
issue of risk-taking
the analogue of part 3 of Lemma
by labor and capital (hence
r
,
states that there
is
no
and Wj do not depend on the state of
nature Ut). The large number of islands ensures that there
13
1
all
is
no aggregate
risk,
thus
risk-neutral firms can once
more
offer full insurance to the factors for
no incentive problem. Therefore,
which there
is
as in the case of perfect information, the expected
revenue of firms net of managerial premium will be distributed between capital and
labor, again with shares
a and
managerial compensation
—
1
be random because individual managers
will
and
In contrast to returns to capital
a.
will
labor,
have to
bear some risk to provide them with the right incentives. The rest of .this section
will characterize the contract
which determines the managerial compensation.
We start the analysis with two observations.
structure, normally distributed
to the results of
adjustment of
CARA
H-M
(1987)
random
effort levels, the
CARA
and
variables,
who prove
economy has a linear
First, since the
utility,
we can appeal
that with this structure and continuous
optimal contract
is
Moreover, thanks to
linear.
we can simply work with the
preferences and normally distributed returns,
H-M,
certainty equivalent of the income process faced by the agents (see also
an application). Even though we have so
for
and
our structure would carry over unchanged
for all,
to be a segment of continuous time,
chosen once
we considered each period
if
their effort
Using the result of Holmstrom and Milgrom
this slightly modified interpretation of
attention to linear contracts.
effort as
and managers continuously adjusted
after observing previous performance.
and
thought of
far
1991,
Second, we
our timing structure, we will restrict
know from standard agency theory
any variable which contains information about the
effort level will
that
be useful in giving
incentives to the agent (Holmstrom, 1979). In our economy, average output in the
manager
island of the
upon.
is
in question will
Average output of island j
is
be a useful variable to condition contracts
correlated with
beneficial because the variability generated
incentive contracts.
very poorly.
If all
To
by
ajt,
this
and conditioning on
shock reduces the power of
see the intuition, imagine that firm
i
other firms in the island did well, this suggests that the island
have been due to low
same shock
also
Let us
effort.
performed badly,
an adverse island
specific shock,
now drop time
it is
In contrast,
if all
likely that the
not to low
Zij
=
^oij
+
of the
manager
other firms affected by the
effort.
subscripts whenever this will cause no confusion.
(puj (Vij
- A*) +
</'2ij
is
poor performance was due to
optimal compensation contract for the manager of firm
the form:^
performed
in island j
must have received a favorable shock and the bad performance
likely to
ajt
{vij
-
V'ji-i))
i
in island j will
where
is
7/J(_ )
The
then take
the average
^See Theorem 7 of H-M (1987). Note that with this contract, the manager will sometimes
have to receive a negative payment. However, for fi large enough this will happen very seldom,
14
productivity of
all jf-th
island firms except firm
i.e.
i,
y"(_i)
\
=
lit
-
~^^^ri~^-
Rewriting
in
terms of the compensation of the manager additional to the wage, Qj
Zij
=
+
Wj
C,ijJ.
(ij
where
=
(poij
~
0oij
=
'^j-
hij
+
(puj ivij
-
f)
+ ha
of the firm (the term
formance compared to the average output of
term
—
yij
y'^i^-^).
(vij
-
Expressed
all
—
yij
(17)
yj(-i))
Note that the compensation of the manager
upon the performance
ditional
(recall
fi),
and the
is
made
con-
relative per-
other firms in the same island (the
differently, (17) will
be a type of relative performance
evaluation contract which sets the average performance of other agents as the stan-
dard relative to which the manager
firms in island
j,
Mj, increases,
is
Z/j(_i)
judged. Quite importantly, as the
Next,
it
be more strongly correlated with
will
the standard that the society can set will
number
become more
Ujt,
and
accurate.
can be shown that productive inefficiency
is
never compatible with
equilibrium as long as there are at least two active firms in each island, and thus
will limit
we
our attention to equihbria with productive efficiency (as defined in section
Then, the problem of firm
2.3).
of
i
on island j
is
equivalent to solving the following
program:
^^\ ^ [y^i ~
^
<P0ij,<Pliii'P2ij
subject
^
=
arg
is
yij
The
that the
given by (3) and Qj
first
eflFort
condition, (19),
I
(20)
-
Sij
rf^^J
I
—e% -
4) - ^^«KOi) =
is
is
as in (17). (18)
=
e*)
(18)
'
is
*^Var{Qj)
(19)
^</
(20)
clearly the expected profit of the
the incentive compatibility constraint.
It
requires
choice of the agent should be the one that maximizes his payoff given
the managerial contract. Note that
will
"^i^o
max E{Cij) -
£^(Cv
firm.
~
to:
e*^
where
^^i
we have made use
of the fact that the
manager
simply maximize the certainty equivalent of his income minus the cost of
is
effort.
the participation constraint which requires that the certainty equivalent of
the additional return (over and above the wage Wj) that the manager receives should
and moreover, since there
follows,
we
is
accumulated wealth, negative payments are not problematic. In what
ignore the constraint that wealth should be non-negative.
15
.
exactly compensate
him
for the cost of effort.
The
participation constraint (20)
necessary and sufficient to characterize the equihbrium occupational choices.
see why, recall that there are
no wealth
manager rather than a worker,
become managers
effects,
thus
if
is
To
one agent prefers to be a
agents in the same island would also prefer to
all
irrespective of their wealth level.
Therefore, in equilibrium
all
agents must be indifferent between the two occupations and (20) ensures .this. Also,
note that in the objective function
and the
price of capital, r,
we have imposed that each firm takes the
(18),
price of labor in island
j,
Wj, as given.
This
is
understood as each firm taking the capital stock of the economy, Kt, and the
stage announcements of
price of capital
all
to be
first-
other firms, M^, as given and anticipating the equilibrium
and the wage
rate in island j in the second stage of the entry
game
(see section 2.2).
CARA preferences
together with linear contracts simplify the problem, allowing
us to proceed in two steps.
maximize the sum
first
and
(j)2ij
affect the solution, this
max
subject to (19). Next,
The
Lemma
1.
and the manager's
of the firm's
is
utility
transferable,
we can
with respect to
(jjnj
subject to the incentive compatibility of the manager, (19). Ignoring terms
which do not
(20).
In particular, because utility
following
(poij
E{y,j
|
maximization problem can be written
e*,)
-
^e^ -
as:
^Var{Q,)
(21)
can be determined by solving the participation constraint,
Lemma
establishes three important intermediate results.
3 Under imperfect information:
effort choice of
manager
i
in island j
e*j
=
is
given as:
Picf>uj
+ cl>2ij).
2.
the average productivity of firm
3.
the variance of managerial compensation for
i
in island j
is:
(22)
E{yij
e*j)
\
manager
i is
=
(fj,
+ e*j)
given by:
0-2
Var{zij)
Lemma
=
Var{Qj)
{<Puj
+
02ii)
0-2
+
</-2..j,2
_^
(/»2,.___
3 enables us to fully characterize the set of equilibrium contracts.
16
(23)
Proposition 2 Suppose
have contracts:
Q
=
Then
(6) holds.
+
(p*oj
(pij {Vij
-
l^)
managers
in equilibrium all
+4'*2j (vij
-
T^^^'^
Vji-i))'
4
in island j
"=
^i
+
Qj)
where:
0-2
^^^-
=
'^^^''^•^
0;,
=
'^^(/^i)
<i>h
and Var{Q)
is
=
=
(M,.^
=
...... fr2)£|^+a2
.„. + J../, J,+ (Mj
=
given by (23) with
+
f (01,
(fjuj = (p^j
This proposition establishes that
managerial contract and
function of
as the
e*j
=
J3
Kj from
number
f^jj
by the
(f)2ij
=
managerial
^27-
of both
(j)\j
and
productivity.
effort
- depends on the number
there are
Combining the
Symmetric
We
more
firms, the society
results of Proposition 2
vs.
equilibrium whereby
all
>
0.
Intuitively,
can set more accurate
and an increase
with those of
in effort
Lemma
2,
we
on the allocation
i.e.
Asymmetric Equilibria
we
established that there could only be a symmetric
islands receive the
With imperfect
-j^
next discuss this allocation.
In the perfect information case,
Mt).
a
on Mj implies that
de'
of capital across islands.
=
itself is
of firms in the island,
can easily obtain equilibrium factor returns conditional on Mj,
Mjt
(which
same
the organization of production - here captured solely
standards; this enables a change in managerial contracts
3.2
Mj
(p^j
statics in section 3.4 will establish that
commented above, when
and
(26)
of firms in island j increases incentive contracts change. Also, since
Mj. The comparative
as
and
+ ^^«KC;)-
firms in island j will choose exactly the
The dependence
+ 02j j (Lemma 3),
level of
all
-^^i)'
(25)
^
.
l)iy2
uniquely determined for given
this is
(14))"
....
(M,i/2
'^o(/<i)
(24)
+ ,2)^ + ,2 + (M,-i).2'
same amount of
capital, Kjt
=
Kt (and
information, relative performance evaluation and the en-
dogeneity of information introduce an island-specific externality, and as a result, the
^
As we
will see in
more
in island j information
detail later, this feature implies that
problems are
profitable to increase the
number
less severe.
of firms
by
This
may
when the number
suggest that
it
of firms
is
larger
could sometimes be
sacrificing productive efficiency.
However, the form
of our production function (3) precludes this possibility. What matters is not the number of other
firms producing in the same island, but total production. Mj appears in our expressions because
when
all
firms are run with productive efficiency, total output
explains the particular form of the production function chosen.
17
is
proportional to Mj.
This also
equilibrium
may
the islands.
More
involve an asymmetric distribution of the total capital stock across
specifically,
information and incentives improve with the scale of
production within an island and this counteracts decreasing returns to capital in the
island.
In an asymmetric equilibrium, islands which receive a higher (lower) than
average amount of investments have higher (lower) capital to labor ratios, and this
same time, the
depresses (increases) the rate of return to capital. But at the
(smaller)
number
of firms improves (reduces) information
larger
and productivity, and
this
increases (reduces) the rate of return to capital.
Lemma
4 below establishes the uniqueness of a symmetric equilibrium under
simple restrictions.
Lemma
4
3/2
such
equilibrium which
We
for
that,
all
Kt
satisfying (6),
can see the intuition
for this
lemma
which maximizes the number of firms
across islands. Thus,
of reducing the
large, the
high,
3.3
>
there exists a unique
/2,
symmetric.
is
each firm produces irrespective of the
(i.e.
"iji
as follows,
fi is
manager. The allocation
effort level of the
in the
the amount of output
economy has
capital allocated equally
the opportunity cost of allocating capital asymmetrically
fx is
number
As a consequence, when
of firms in the economy).
opportunity cost of an asymmetric distribution of capital
and there only
exists a
Dynamics
of
is
fi is
prohibitively
unique symmetric equilibrium.
Symmetric Equilibria
Since asymmetric equilibria are complicated and of only indirect interest for the
focus of this paper, in this section
we
will fully characterize
cumulation and development with sjonmetric equilibria.
only the dynamics of ac-
We will discuss asymmetric
equilibria in section 4.6.
In the case of a symmetric equilibrium, managers in
the same contract,
and
(^S^-^
4>2{Kt) are given
equilibrium Mjt
=
=
(f)*o{Kt),(pljt
=
(l^li^t)
,
<Pht
=
all
islands will receive exactly
<P*2{^t),
where
(p*o{Kt),
^KKt)
by Proposition 2 and simply depend on Kt since in a symmetric
N°'Ki~°'.
economy adopt the same
Furthermore, given Proposition
2,
all
firms in the
technology, and workers in different islands receive the
same wage.
Since a fraction s of
all
income
is
saved, the law of motion of capital with sym-
18
)
metric equilibria can be written
/^,+i
We
=5[/i
as:
+ y9(0*(X,) +
0;(/^,))]iV"/^^".
can summarize our findings in (proof in the
Proposition 3 Assume
that the conditions of
(27)
text):
Lemma
4 cind condition (6) are sat-
isfied.
Then, given Kt, there exists a unique symmetric equilibrium allocation at
time
In this equilibrium Mjt
t.
=
contract (17) with (^^{Kt) 4>]{Kt)
,
=
and
(j)2U'Ct)
all
j
6
[0, 1],
and
all
managers sign
as given by Proposition 2,
and choose
Factor prices are:
effort level as in (22).
r{K,)
N°'Kt~°' for
-
(1
a)
ill
+ 13 miQ +
r2{Kt))
- W<t))
[0
(28)
=
w{K,)
The evolution of
a{ii
+ P {cPlilQ +
<p;{Kt))
-
l-Q
4^1{K,))
(f
the physical capital stock is given by (27).
This proposition estabhshes that in the dynamic equilibrium of our economy,
capital accumulation
more
is
accompanied by an increase
The information
repetition.
that
is
in the
accumulated as a
number
of firms
and
result of this process
enables the society to set better standards, and improves the effort level of managers
and
total factor productivity. Thus, the interaction of
incentives creates a form of "scale externality".
may be
endogenous information and
Because total factor productivity
increasing in the capital stock over a certain range, dynamics are no longer
purely neoclassical, and multiple interior stable steady states cannot be ruled out in
general (though
is
can be established that
it
for
/i
sufficiently large, the
steady state
unique).
Remark We have
so far talked of islands. For
consisting of different sectors
results
some
may be more
of our applications
appropriate.
would apply exactly to an economy where there
sectors, each agent has a strong
is
start with,
our
a continuum of
comparative advantage for one sector and the
output of different sectors are perfect substitutes.
would involve
To
an economy
A more realistic formulation
producing imperfect substitutes. In this case,
different sectors
aggregate consumption could be defined as a composite of different sectors'
output,
e.g.
Ct
=
exp
which sector to work
of this paper
-
/q Cjtdj
in.
,
and agents could be homogeneous and decide
This setup - which was analyzed in a previous version
gives similar results, but the analysis
"Jensen's inequality" terms in aggregation.
19
is
more involved due
to
Some Comparative
3.4
Statics
Equations (24) and (25) lead to a number of interesting comparative static
results.
Straightforward differentiation establishes:
First,
when
cr^
To understand
S<0,
^<0;
S<0>
S>0,
^>0;
^<o
^>o
^>0
both
increases,
(f)^
and ^2 decrease, and
this result, recall that in this
which makes
variability
S^<0,
it
and productivity
effort
economy
costly to induce effort.
(29)
precisely idiosyncratic
it is
If cr^
=
fall.
0,
managerial contracts
and 0| = 1, and provided that there are at least two firms in
managers would bear no risk, and the first-best would be implemented.
would specify
the island,
(fi^
—
Idiosyncratic variability introduces noise to the signal coming from the individ-
ual performance
and makes managerial compensation random. Since managers are
risk-averse, this lack of full insurance
insurance and incentives.
and there
is
costly,
and managerial contracts trade
increases, lack of insurance
lower effort in equilibrium. Second,
increases because with
more
As a^
is
more
(j)^
dominates and the net
level of effort
and productivity increase with the
Third, as the
number
is
more
evaluation (higher ^2 ^^'^ lower 0j).
managerial
effort
0,
costly,
and
(j)^
become
effect is that
the
volatility of island-specific shocks.
of firms grows (as a result of capital accumulation), the stan-
dards improve, and again there
^>
v^ increases, ^j falls
island specific variability, relative standards
informative. Overall, the change in
and we have
when
becomes more
off
of relative
and
Once more the
less of
effect
absolute performance
through
(j)^
dominates,
thus accumulation of information (and capital) leads to higher
and
productivity.
This contrasts with the economy with perfect
information where there was no interaction between incentives and development.
3.5
Agency Costs and Development
Agency
costs are the costs incurred
in principal-agent relations. In our
exert less effort in the
and
(ii)
by the society due to the imperfect information
model these have two components:
economy with asymmetric information than
(i)
managers
in the first best;
they require a risk-premium to be compensated for the variability in their
20
income.
We
capture both components with our concept of
TAC{K),
total
agency
costs that the society incurs per firm:
TAC{K) = (/5-e*)-^(/3^-(e*f) + ^Var{C)
(30)
2P
where
and
e*
and Var{C) depend on M{K),
(23),
and
this
TAG
makes
component and the second
(jj\{K)
and ^1{K) via equations
depend on K. The
term of
first
(30)
is
(14), (22)
the effort
the loss of utihty in certainty equivalent terms due to
is
the risk borne by the managers.
Another useful concept
is
SAC(e, K) {shadow agency
certainty equivalent of income that
is
cost)
which
is
given by the
foregone in order for managers to be induced
to exert the effort level e (as different from the optimal level of effort, e*). Formally;
SAC{e, K)
=
min
{4'i,4'2}
where Var{Q
is
^
'^'
+Var (C)
s.t.
(jjj+cp^^^
(31)
given by (23).
Proposition 4 Both
TAC{K)
and SAC{e,K) are weakly decreasing functions of
the capital stock of the economy.
This proposition establishes that more information
as
it
always useful in our setting
enables better incentives and risk sharing. Therefore, the cost of more effort
at the
margin and the
in the
amount
due to incentive problems are decreasing
total loss of utility
of information,
and thus
in the total capital stock of the
economy.
Constrained Efficiency
3.6
We
is
have established that as the economy develops,
and productivity. Can a
achieves higher levels of effort
social planner subject to the
formational constraints ensure a more
we analyze
it
efficient
same technological and
outcome? To answer
the static problem of a planner maximizing the
agents in the economy without any distributional concern.^
sum
this question,
of the utility of all
We start with three sim-
ple observations. First, as in the decentralized economy, as long as there
capital to
open
at least
two firms with productive
in-
is
enough
efficiency in every island, the
planner would never choose productive inefficiency. Second, given our assumptions.
^We
ignore saving decisions which will depend on the "discount rate" of the planner.
21
the planner will also choose linear contracts. Finally, the planner will offer the same
contracts to
written
managers
all
in the
same
island.
Then, the planning problem can be
as:
/ ^^
s.
."'f/^ X
{<plj,<p2jMj<f^j,ej}Jo
^^
+
^^^
^i^«KO)^i
"^^-ol
Jo
2,
-^
Zp [
Mj e]d3
(32)
'•
Jo
subject to the incentive compatibility of the managers, (19), and the resource constraint (14), with
Var{Q) given by
(23).
Proposition 5 Conditional on Mf,
induces e^
=
for
e^
all
The proposition
j
6
[0, 1]
the planner chooses
=
(plj
(p\j,
(p^-
=
^2,-
o-nd
as given by equations (22), (24) and (25).
on the allocation of capital across
states that conditional
islands,
the planner would choose the same allocation as the decentralized economy, or in
other words, she would choose exactly the same contracts and induce the same level
of effort.
Although there are many
from decentralized competition are
result, first
externalities at work, the contracts that result
efficient.
To understand the
intuition of this
note that the effort level of a manager does not create an externality
on other managers
in the
same
as he exerts the effort level
he
island.
is
Given the additive structure of
(3),
as long
expected to (a requirement in any pure strategy
equilibrium), the signal extraction problem faced by
all
other firms
is
unaffected.
This can also be seen in equations (24) and (25) where the power of incentives given
to the
the
manager only depends on the
number
of firms) in the
same
total
amount
More
if II is
it is
the allocation of capital across
straightforward to show, along the lines of
sufficiently large (say greater
equilibrium. Therefore,
we
equilibrium that
However,
it is
tuitively,
when a
j' will
efficient,
by the planner does not necessarily coincide with the equilibrium.
specifically,
externalities
on
island.
Despite the fact that contract choices are
islands chosen
of production (or equivalently,
when
/^ is
than
/i^),
larger than
characterized above
is
be worse
both
/2
and
/2'^,
that
the unique symmetric
also the constrained efficient allocation.
how
Jjp
compares to
firm decides to locate in island j rather than /,
creating: workers in island j will
off
4,
the planner will choose a symmetric
not possible to establish unambiguously
it is
Lemma
be better
because labor demand and wages
off
it
/2.
In-
ignores two
and those
in island
will increase in island j
and
decrease in island /. These two effects do not always cancel out in the equilibrium,
22
hence the distribution of capital across islands
useful to note at this stage that
it is
many
is
not necessarily
of the applications
Moreover,
efficient.
we
will consider in the
next section features technologies or sectors with different degrees of agency costs,
and
4
in these situations, the decentralized equilibrium
more
is
likely to
be
inefficient.
Applications
Risk Sharing
4.1
Since workers and capital owners bear no
risk,
the extent of risk sharing in our econ-
omy is capturedby the variance of managerial returns Var
This
is
{(^{M[K),(j)l[K), (fj2{K))).
a component of total agency cost (see (30)) analyzed in the previous section.
TAC{K)
However, although
true for the variance.
That
decreases with accumulation, this
is,
the degree of risk sharing
is
not necessarily
may be non-monotonic
or
even decreasing with development.
=
Proposition 6 LetV{K)
1.
ifp{a^
2.
if
+
ifpa^
>
how
then V'{K)
p{o^^
+
<
J^^),
M
(i.e.
<
for
K
then 3
all /<;
s.t.
if
K <K
then V'{K)
>
0,
and
K)
<
for
K.
all
increases,
shadow agency
cost,
SAC{e, K),
falls
but in
the equilibrium level of effort e* also increases, and this requires
more
risk.
if
0;
p, then V'{K)
mean time,
agers to bear
13,
then V'{K)
Intuitively, as
the
<
< P <
pa^
K >k
3.
u"^)
Var{C{M{K),(j)l{K),(Pl{K))). Then:
man-
This interaction between two opposing forces determines
the variability of managerial returns will change over the development process.
Proposition 6 shows that the link between risk sharing and growth depends on the
degree of risk-aversion and on the amount of noise that contractual arrangements
are subject to. For instance,
or the variance of the shocks
risk sharing will improve.
if
is
agents have a low degree of risk-aversion (small p)
small, then, as
more information becomes
The opposite occurs when
the idiosyncratic variability,
o"^) is
high.
the degree of risk aversion (or
In this case, because incentives are very
low powered, the variability of managerial returns
and increases over time. In intermediate
available,
is
very low in poor economies,
cases, the variance
23
is
non-monotonic, and
risk sharing increases first,
and decreases afterwards. Even though only managers
bear risks in our economy, the opposing forces impacting on risk sharing will apply
more generally
The
to
all
agents bearing risks due informational problems.
possibility that risk sharing
U-shaped
inverse
(case 2), provides
empirical evidence.
is
decreasing with accumulation (case 3), or
an interesting interpretation to some recent
often argued that less developed economies §uffer from
It is
serious agency problems (see for instance. North, 1990). However,
Townsend
(1994,
1995a,b) and other recent studies (as reviewed by Morduch, 1995) have shown that
in
Asian villages there
is
relatively
low variance of consumption, thus quite good
risk sharing arrangements. Moreover,
pears to be lower in n'c/ier villages.
Townsend (1995b)
It is
finds that risk sharing ap-
tempting to interpret these recent findings
as evidence that less developed economies do not in fact suffer serious incentive
problems, and that growth and modernization
"efficient"
the organization of production
prohibitively high.
available,
As the
is
decline.
agerial
whereby
Our model provides an
in less developed economies,
highly inefficient because shadow agency costs are
scale of
economic activity grows, more information be-
and more sophisticated contractual arrangements can be devised.
Because these contracts induce higher
may
the factors destroying the
organization of small communities and villages.
alternative interpretation to these findings
comes
may be
effort,
the extent of observed risk sharing
Possibly at even later stages of development, the variability of
and entrepreneurial returns may again
start falling as relative
man-
performance
evaluation becomes more powerful.
A
related feature worth noting
is
that our model also has implications about
the distribution of (labor) income. Income distribution
of effort
by managers and the
3 of Proposition
it
6,
growth
is
is
determined by the choice
variability of managerial returns.
In cases 2
and
"unequalizing" in less developed economies because
leads to an increase in managerial effort
and higher
variability of managerial in-
comes, and therefore to a greater difference between the average income of managers
and workers. In case
managerial returns
among
2,
however, as capital accumulates further, the variability of
will decrease,
and
this will
reduce both the observed dispersion
entrepreneurs and the risk-premium that managers are paid over workers.
the decline in risk-premia dominates the increased compensation for higher
growth
will bring
about a more equal distribution of income
in
Therefore, our model with intermediate levels of risk aversion
Kuznets curve.
24
If
effort,
advanced economies.
is
consistent with the
Sectorial Transformations
4.2
Sectors typically differ in terms of the structure of uncertainty they face as well as
be more serious
their technology. This will imply that agency problems will
sectors than others.
When
the scale of production
is
in certain
be very
limited, there will
little
information to be used in agency relations, and sectors where agency matters more
will
accompanied by
result, capital
Even though
applications, in this section
we
this insight
may
accumulation will be
where agency problems
sectorial transformation towards activities
more important.
are
As a
have relatively lower productivity.
have a number of potential
will consider the implications of
our model for
how
the share of agriculture in total production evolves over the process of development.
As discussed above, the variance
of
common
of idiosyncratic shocks relative to the variance
Now
shocks are crucial for the extent of agency costs.
implausibly, that agriculture
is
subject to large
common
suppose, not
shocks due to weather,
while the variation due to idiosyncratic uncertainty (or managerial talent) are more
important in industry (manufacturing). This assumption
provided by Pollard (1965)
pirical evidence
(1995b)
more
most
who
volatile
Let us
accord with the em-
the very high failure
and eighteenth century
of this to managerial mistakes or negligence,
Britain,
and
and with Townsend
reports that individuals involved in entrepreneurial activities suffer
consumption than farmers.
now
develop a very simple version of this sectorial transformation story
by making three strong assumptions:
as
in
who documents
rates of managerial firms during the seventeenth
attributes
is
There are two goods.
(i)
We
think of the
first
an agricultural and the second as a manufacturing (industrial) product. Half of
the islands can only produce agricultural goods and the other half can only produce
industrial goods.
On each
agriculture, while
on each island j E
industrial goods,
(iii)
The
(ii)
island j
E
[0, 1]
[1, 2]
there are
A'^
there are A^ agents which can only produce
Agricultural and industrial products are perfect substitutes.
variance of idiosyncratic shocks in agriculture
for the idiosyncratic shocks in
manufacturing
is
a]
>
absent in "agriculture".^ However, note that since u^
^ All
agents which can only work in
0.
>
is o"^
=
and the variance
Thus, agency problems are
0,
agricultural output
may
three assumptions can be relaxed. For instance, islands can be allowed to choose whether to
two goods may be imperfect substitutes
with elasticity of substitution greater than one, and this would also enable us to match the relative
price movements over the development process, but again is not crucial for our argument. Also,
specialize in agriculture or industry. Instead of perfect, the
2
^>
2
;^ rather than
ct^
=
would be
sufficient in general.
25
be subject to more variability than manufacturing.
The technology
is
essentially the
have a quasi-Leontieff technology
same
as in the one-sector economy.
Workers cannot move across islands but
like (3).
can invest their wealth in a balanced portfolio of
will
we
bear no
Since in equilibrium
risk.
All firms
all
the firms in the economy, thus
firms are run under productive efficiency,
all
have:
2/4
=
+ e<;, + a^)
Z(//
Vie
[0,1],
(33)
yijt^f^
Z measures
where
+ eijt + ^ijt +
will also
aj~iV(0,i.J),
and
e(,
~ iV(0, a^).
assume, in analogy with the result of
are such that within each sector there
M^
[1,2],
the productivity of agriculture relative to industry. Furthermore:
af<^N{0,,^l),
We
Vie
c^jt
M/ = M'
for all j
e
is
[0,1],
Lemma
4,
(34)
that parameters
only a symmetric equilibrium,
though in general
managerial contracts in agriculture z^ (or
C/^^ will differ
M^ ^ MK
Mj^
i.e.
=
Moreover,
from managerial contracts
in industry z^ (or C^) because of the differences in the structure of uncertainty.
Consequently, managerial effort in agriculture, e
,
will differ
from managerial
effort
in industry, e^. In particular, since the return to agricultural firms within each island
are perfectly correlated, the first-best effort level can be implemented in agriculture,
that
=
e^
is
number
=
+
—
0.
Instead, industrial contracts will induce the
with 0J and (f)^ given by (24) and (25). Note that the
the industrial sector will be conditional on the information - thus, the
effort level e'
contract in
ZjS and Var{z'^)
of firms
^{(pl
-
cf)^),
in the industrial islands.
now write the rate of return on capital in the two sectors, r and
make zero profits and labor is paid its marginal product. As in the
Let us
firms
r
,
when
previous
sections, these are given as:
-^
(1
r
-
a)
1^//
+ e^ - IfO!,
y- -
=
=
^
^
kl
f l/ar(CO
j
(i
_
^^
(35)
(/x
+
f
- TAC\M'))
(36)
k'
k'
26
where the second equaHty of
from defining
(36) follows
costs in industry analogously to (30).
to this factor will be equalized across
Since capital
all islands,
is
TAC^
as the total
agency
perfectly mobile, the return
therefore,
we must have
=
r^
then
^
r^
which implies:
^_
First, observe that if there
and
-^
(/x
+ f-TAC^(MO)
were no agency costs
in industry [a'j
=
0),
would be constant irrespective of the stock of capital of the economy. This
implies that the perfect information version of this model would have sector-balanced
growth. Next, consider the case with
M^
>
a"]
0.
In this case, as capital accumulates
grows, and from (37), jj increases. Therefore, there
than in agriculture.
in industry
As a
is
faster capital
result, the shares of industrial
deepening
production
over total production and the share of expenditure in industrial goods over total
expenditure also grow with development. Additionally, productivity and wages also
increase in industry but remain constant in agriculture.
in the presence of imperfect information
fact that technical progress
is
is
As a result, economic growth
endogenously sector-biased,
neutral across the two sectors.
very salient pattern in the development process of almost
This
is
despite the
consistent a
countries: at the early
all
and
stages of development, a large fraction of resources are allocated to agriculture,
as the
economy grows, more
development
growth
is
is
much
resources are transferred to industry. This pattern of
usually explained by assuming that the potential for productivity
higher in manufacturing than in agriculture due to some "sectorial
externalities" (e.g.
Matsuyama,
1991).
Our mechanism can
therefore be viewed as
suggesting a microfoundation for these externalities.
4.3
From
Villages to Cities
In the model discussed in the previous subsection, by construction, the share of
total
employment
and the model has
in agriculture
remained constant. This feature
is
easy to change,
interesting implications about migration from "rural villages" to
"industrial cities", another salient pattern of
economic development. To do
introduce an additional factor of production, say land, which
to previous sections,
we now assume
that labor can
agents can choose in which island to work.
27
The
move
is
this,
we
immobile. In contrast
freely
islands,
and
model can
also
between
interpretation of the
)
be modified to
example. Agricultural islands have low agency costs not only
this
fit
because of the different structure of uncertainty
(as in the previous subsection),
but also because of the way villages are organized: the close-knit communities lead
to tight peer group monitoring.
do not allow easy
In contrast, the privacy
and anonymity
in cities
hence principal-agent relations have to use
direct monitoring,
decentralized information and incentive contracts to induce effort. This point
emphasized by Banerjee and
cities as
Newman
who
(1996)
less severe.
these issues, let us modify the Leontieff part of the technology in both
agricultural
and industrial have
(where q
land). In this case,
is
also
obtain migration from villages to
a result of borrowing constraints becoming
To model
is
to:
min
l,q°''>l°''k^^~°'>~°'''>
instead of min
l,l°'k^^'°'^
both the rate of return to capital and wages
will
be
equalized across islands, but the rental rate on land will not be.
The formal argument
Now, with labor
is
very similar to that used to analyze the two factor case.
perfectly mobile, wages
and the rate of return on capital
will
be
equalized across islands, and this implies:
aiZ
(Ai
V
- =
a,Z
=
^
where
l^ {k^)
and V
+
(/i
+
f
)
f
_
-
dustrial firms, respectively.
'
a,
number
total
villages to cities.
mechanism:
(^9)
From
Then, from
{f^
+ l-TAC\M^))
Z
(40),
in agriculture
The reason
in-
(38), it follows that:
jx
+
(/.
f
number
and
^
^
of firms
is
declining
-^,
^
and decreasing agency
have to grow.
of firms in industry relative to agriculture,
employment
(3«)
denote labor (capital) employed by agricultural and
Capital accumulation implies an increasing
the
f
k^
V__k^_
l^-k^costs in industry.
- TAC^M'))
'
V
- TAC^M^))
(/. + f
+
(/i
=—
)
\^
[k^)
ai
is
This implies that
growing. Therefore,
and agents must be migrating from
for this transformation
is
again related to our main
villages provide direct peer-monitoring thus reduce
agency costs
(i.e.
without informational problems, there would be no migration). At the early stages
of development, this
is
a very important resource and a large fraction of the popu-
lation lives in villages, even
because there
is
though productivity
excessive capital
and labor
28
is
lower there (partly endogenously
in these islands).
As more
capital
and
information are accumulated,
cities
take advantage of reduced agency costs and ex-
pand. Finally, note that during the process of industrial expansion and migration,
the productivity of capital and labor in agriculture
also increasing because fewer
is
workers and capital are working with the same amount of land.
4.4
Direct Monitoring and Financial Development
At
stages of development, specialized financial institutions intermediate funds
all
from savers to
However, there are very important differences between the
firms.
institutions in poor
economy and those
in a
more developed
society. In his historical
study Goldsmith (1987) finds that in most pre-modern societies funds are provided
by
direct lending institutions such as usurers, or local intermediaries, or at best,
local banks. This contrasts with the larger banks, stock
A
developed economies.
by
carried out
and bond markets of more
crucial difference concerns the degree of direct monitoring
different financial institutions (see
Diamond, 1984, on the monitoring
role of financial intermediaries).
In this section,
with
we assume that
each type.
free entry into
tutions which
we
there are two types of financial intermediaries,
First, in
each island there exist local credit
will call village intermediaries {VI).
These intermediaries
funds from savers in the whole economy at some market rate
firms located in their
managers that they
services
and
is
is
c
own
island.^
finance.
The
A
r,
insti-
collect
but can only lend to
VI can perfectly monitor the
effort of the local
cost of providing these intermediation/monitoring
per firm. For simplicity, we assume that monitoring takes place interim
publicly observed. This implies that a V7 monitors
before their final performance
stochastic monitoring
is
all
the managers
it
lends to
revealed (more realistic assumptions, for instance,
would not change the
results).
Second, there are some global
intermediaries (GI) which offer their service at some lower cost, but cannot monitor
(for instance,
generality,
they lack the local expertise). For simplicity, and without loss of any
we assume that
of these GIs as of banks
we can think
and
which operate
at the
economy-wide
is
zero.
We
intermediaries
more
may have
can think
scale, or, alternatively,
that firms can raise their funds through a stock market (see
Zilibotti, 1996, for a
^Many
the cost of providing these services
Acemoglu
detailed analysis of these different institutions in a
local "expertise"
which would justify
this.
The assumption
that Vis borrow from savers in the whole economy may not be very realistic. In practice, they can
do so by borrowing from other financial institutions. Witli Vis only using funds from their own
islands,
our analysis would be more involved, but the main result would not be affected.
29
.
model with moral hazard problems).
The important assumption here
is
the absence of informational imperfections, intermediation through GI
ficient
than intermediation through
firms which receive funds via
Vh
We
VI.
also
more
ef-
assume that the performance of
by firms run through
are observed
is
that, in
GI,
and
this
assumption rules out potential coordination problem.
Clearly, the comparative advantage of local intermediation
clines as the scale of
economic activity expands and more information
We
the activity of firms in the economy.
Proposition 7 Let
ation
is
is
and mor\itoring de-
K
be such that
is
revealed by
can prove the following proposition:
TAC{K) —
provided by "village intermediaries"
,
c.
Then, for
while for all
Ki <
all
Kt >
f^
K
intermedi-
intermediation
provided by "global intermediaries"
This result
tions are
is
consistent with the empirical evidence that local financial institu-
predominant
in
poor economies and decline as development proceeds
(see
Besley, 1995, Fry, 1995, Goldsmith, 1987). Intuitively, Vis do not need decentralized
information since they carry out direct monitoring. Therefore, as more information
is
accumulated, global intermediaries become relatively more attractive. Note also
that even though our simple model predicts an abrupt switch from local to global
intermediation, the analysis could be easily extended to yield a smoother transition.
4.5
Division of Labor
Division of labor
on
is
is
a complex phenomenon, and different approaches concentrate
different aspects of this process.
An
important aspect of the division of labor
the delegation of tasks to agents
who
are not residual claimants of the returns
they generate. In our economy, the most important form of delegation
managers running
information
is
firms.
In this extension
we show
will
be limited.
This view
is
to have
that in poor economies where
scarce, production techniques that do not rely
lationships will have a comparative advantage
is
on principal-agent
and the extent
re-
of division of labor
in line with a simple reading of historical patterns.
During the First Industrial Revolution firms were predominantly family-managed,
and
this
managerial structure
recently, especially starting
chical organizations have
is still
dominant
in
many
developing economies.
More
with the Second Industrial Revolution, complex hierar-
emerged and played an important
distribution (see Pollard, 1965, Chandler, 1977).
30
role in production
and
We
analyze the issue of division of labor with a very simple extension model.
We refer to our benchmark technology of production
The
alternative
is
primitive production (PP) which entails no delegation of tasks to
Output
a manager.
of unit
Vijt
Since with
is
as the factory production (FP).
PP there
=
no
is
in island j using
i
+ +
{f^
ajt)
min
PP is
given as:
+
l,klfi"'lijt°'
"fixed cost" of production, the
(41)
eijt
number
of firms using
PP
indeterminate. However, to facilitate the discussion and without loss of generality,
we impose that
for all i,j, lijtkljt^
=
so that
1,
we can
"number of
talk of the
still
firms".
We now
assume that
€ (|
— TACocf — TACi)
where
agency cost incurred with factory production when there are
TACm
m firms in the island.
PP
This assumption ensures that when there are very few firms,
FP. In contrast, when there are
sufficiently
many
is
because,
FP generates
when
there are
m
less
than some
critical level
K
to division of labor, there
fi
+ etofx +
an increase
is
e*{k) (where e*{K)
It is useful
intensive but that
is
would be preferred
division of labor
The
If
more
in this case
if
economy
capital,
becomes
given by
economy switches from
^^ - ^^Var{C{K)) =
PP
from
0).
economy the switch from primitive produc-
PP was
is
more
capital
intensive. In other words, in the absence
preferred to
FP at
is
some
capital level Kq,
and thus more information, agency
relatively
more
costs decline
,
the
and
attractive.
more general
"sophisticated" products or production techniques are at the
intensive"
when
limited by the extent of information:
analysis of the last two subsections also suggests a
more "information
is
at all other capital levels too. Hence, loosely speaking, the
division of labor in our
economy has more
the
K
in the level of productivity per firm
more information
it is
FP.
of division of labor,
not due to the fact that division of labor
of informational imperfections,
it
-
to emphasize that in our
tion to division of labor
for all capital stocks
K, the economy does not make use
> K, all production takes place with
TAC{K) = ^ — 6. Note also that at the point when
and when
/Li
an analogue of Proposition 7 whereby
state
preferred to
is
+ | — TACm whereas with
income equal to /x + 6. It is now
a certainty equivalent of income equal to
we can
preferred to
active firms in the island, each firm with
PP, each firm produces a certainty equivalent of
clear that
is
firms so that the society can set
accurate standards, agency costs are low and division of labor {FP)
PP. This
the total
is
because monitoring
31
is
principal.
same time
harder or because they involve
Figure
1:
Asymmetric Equilibria
m.ore delegation, then as a society develops, the range of products will expand, the
production methods
With
tasks.
all
will
be become more refined and there
will
be more delegation of
of these changes, as in our division of labor example, the productivity
will often increase.
Development and Specialization
4.6
Less developed economies are typically highly specialized and invest a large share
of their resources in only a few narrow sectors. This
when we view
is
consistent with our
islands as sectors. In particular, the analysis of section 3
model
showed that
asymmetric equihbria are possible with imperfect information, even though with
perfect information, only symmetric equilibria can exist.
Asymmetric
equilibria
correspond to the economy being specialized in a few activities at the expense of
the
rest.
some
The reason
is
to economize
on agency costs by generating information in
selected sectors.
To
get
more
insights into
development and specialization, we can consider a
simple diagrammatic exposition of asymmetric equilibria.
the rate of return to capital in island j
if
32
Figure
1
an amount of capital Kj
draws rj{Kj),
is
invested in
that island (thus with
schedule
downward
is
and our condition
and rj{Kj)
is
in
M{Kj)
active firms).
straightforward to see that
It is
sloping everywhere, there can be no asymmetric equilibria,
Lemma 4,
that
/^
>
/I,
ensures
Suppose instead that
this.
now
increasing over a certain range. Let us
<
K2-
It is
K2, and
K
€
no island
increasing around K'
make
fix
,
thus
will
if
all
K2 must be
capital
the remaining
as
drawn
in Figure
an additional firm opened in
is
this island,
it
is
determined
equal the aggregate stock of capital.
It is
now
would
in order to
have the
clear that conditional
specialized. For instance, thinking of the islands as sectors, this
an economy which invests a large fraction of
As the economy becomes
economy becomes
societies
In
characterized as follows:
on Ki and K2, a smaller aggregate stock of capital impHes that the economy
the
1.
as in Figure 1, then compute;-'"
demand
sectors.
—A
1
have a capital stock equal to K' because rj[Kj)
the fraction of islands with more capital
is
/2
islands can have capital equal to
positive profits. Then, an asymmetric equilibrium
Ki and K2
That
K2) so that not
straightforward to see that Ki and
particular, note that
is
(A'l,
<
/i
concentrate on asymmetric
K2 and
equilibrium in which a fraction A of islands have capital
have Ki
this
if
richer
less specialized.
and
its
is
very
would correspond to
resources in a small fraction of the
K increases,
X(K)
increases too,
and
Therefore, our model predicts that poorer
tend to be more specialized as a way of economizing on agency costs.
Conclusion
5
This paper
crucial role.
offers a
theory of development where principal-agent relations play a
Wealth
is
generated by delegating tasks to agents
residual claimants of the returns they generate.
costly, productivity is low.
the society generates
is
We
When
who
are not the
the control of these agents
is
argue that the amount of decentralized information
a crucial determinant of
how
easy
it is
to control the agents.
In turn, the structure of information depends on the scale of production.
When more
agents are engaged in the same activity, the society will develop a better standard
for this activity,
10,
and
As the argument
this will enable relative
suggests,
performance evaluation. Therefore, as
when one asymmetric equiUbrium
others.
33
exists, there will also exist
many
a society accumulates more capital,
will lead to higher
As
managerial
well as explain
number
why
it
will also
accumulate more information. This
and productivity.
effort
the process of development
may be
slow, our theory has a
of important implications, reminding us of the older theories of development
We
with their emphasis on structural transformation.
find that the extent of risk
sharing will change with development, the sectorial composition of output will
there will be
more
division of labor, there will be less reason for close-knit village
communities to survive, and financial institutions
some of the same problems
as our
will
be transformed. Approaching
paper with a somewhat different emphasis, Besley
(1995, p. 121) writes that "(local institutions
do disappear as capital markets develop.
and enforcement)... do seem
cause, the decline of this type of
in general
This reflects the fact that monitoring and
other technologies improve in the development process.
...
Whether a symptom or a
non-market institution in the development process
vividly illustrates the idea that they use certain information structures
ment
shift,
technologies that are eroded by the transformation to a
and enforce-
modern economy." In
terms of Besley's quotation, we are arguing that the relative decline of a host of
institutions
and
sectors
is
a consequence of information accumulation and develop-
ment, but also that such structural transformations have important implications
regarding the range of products, organization of firms and productivity.
Our
list is
not even close to being exhaustive of the transformations on the way
from poverty to prosperity.
It is also
not exhaustive of the development implications
of changing principal-agent relations.
that
more
results
specification.
can be done.
Our model
is
sufficiently simple
and tractable
can be obtained by modifying certain aspects of the baseline
However, we hope the implications we draw give the fiavor of what
We
also
hope that our model suggests other approaches to the same
problem: more information will improve agency relations not only via better relative
performance evaluation but also through alternative uses of information. Some of the
implications will be similar while others will
are derived
differ.
As
different testable implications
from these approaches and are confronted with data, our understanding
of the development process will improve.
34
Appendix
Lemma
Proof of
Part
1.
Since markets are complete, firms and managers will
1.
agree on the first-best level of effort which, from (2) and
Part
We
2.
start
by showing that
we prove that
all
islands have the
First, since firms
game
=
^
that
=
the capital per firm in island
kj,
tonically increasing function of Mj, the
(3), is eijt
number
same number
compete to maximize
/3.
j, is
Given
of firms in this island.
of firms using proof
profits,
we must have
a mono-
by contradiction.
in the second-stage
Y (where the superscript — indicates that these are
'^
.
this,
partial
derivatives "from below"). Then, from the unitary elasticity of substitution between
=
labor and capital, 7^
Since
when
holds
(6)
case from (4) that
island,
*)
we have
(all
firms in island j adopt the
firms are productively efficient {l^kj~°'
all
lij
= ^^^^
7^
=
Ij
^
=
=
^ind kij
kj
=
same technology).
=
1), it
must be the
-^. Then, aggregating within each
that:
N^KJ-" = Mj
which can be rearranged to
give:
^^A"
Now, suppose that there
>
that Mji
-f-
>
Mjii.
l-Q
_fN
„.,
are two islands, /, /' such that Kji
This implies from (43) that
and because
-f—
(42)
Iji
<
Ijn
both islands managers exert e
in
/
capital in firms of island
> Kju. (42) imphes
and kji > kjn. Therefore,
=
the rate of return to
jS,
lower than in j". Since the rate of return to capital
is
has to be equalized across islands, this gives a contradiction. Therefore, we must
have Kj,
=
Part
3.
Kjn
=
Kt and Mji
From
plies that all firms
a)E
\yj
—
[zj
—
part
2,
=
;j^
Mj,,
= Mf
= ^^.
Hence,
and
Wjlj
= aE
[y^-
—
{zj
—
=
(8), (9)
and
fi
+
Cj, Cj
(10).
=
/?
"
f-^)
profits.
=
and
first
kj
stage
= {jff
game im-
Therefore, rkj
Wj)]. Since, agents
ent between becoming managers or workers, hence Zj
that yj
=
Free-entry in the
must be making zero expected
Wj)]
Ij
Wj + -^-
must be
=
(1
—
indiffer-
Now using the facts
and that the equilibrium must be symmetric, we obtain
This concludes the proof of
35
Lemma
1.
QED
Proof of Proposition
that
if
Kt
^t
^
+ P)N.
s(/i
Since Kt+i
is
By assumption Kq >
,
strictly
Then
N'i^^.
neighborhood of N^"^ Kt+i
in the right
is
an increasing and
is
1:
>
condition (6) ensures
Kt- Next, given (7), Kt+i
concave function of Kt and since
Mt < N, we have
Therefore, there exists a unique steady state level of Kt, K^^.
a strictly concave function, this unique steady state
also globally
is
stable.
To
=
characterize the steady state value K^^, note that K^^
Then using
K^^
(7) gives the expression of
+
P)M{K^^).
The
rest of the
s{fj,
in the proposition.
QED.
Proposition follows immediately from the analysis discussed in the text.
Proof of
2,
Lemma
Part
2:
1.
The proof
Lemma
identical to the proof of
is
part
1
except for the symmetry argument.
Part
Since firms are risk-neutral, by a standard argument, they will offer a
2.
non-random return
compatibility constraints.
Therefore, the rates of return to labor and capital are
non-random. The exact expressions
by the same argument
Lemma
Proof of
Ui{eij,
.)
=
where Qj
E{Cij
is
3.
eij)
I
Lemma
as part 3 of
Part
The
1.
+ wj - ^e^ -
given by (17), and
straightforward. Part
3.
1.
(16), follow
QED.
manager
utihty of
^Var{Qj)
=9+
i
{(puj
in island j
+
(j)2ij)
(m
is
+
we have
= E [Qj -
e^j
—E
0oij]
=
^{(puj
+
(pujio-j
+
given by
eij)
terms that do not depend on
collects
Var{C,ij)
and
for these rates of return, (15)
the manager chooses effort to maximize Ui,
{cf>nj
which there are no incentive
to the factors of production for
- ^4'
Cij.
t/'aij)-
Since
Part 2
+ <j^2ij{^ij -
<^ij)
is
l2
Es^i
+ 4>2ijf a^ + <Pliji^' + ^hj$^] QED.
Proof of Proposition
max
[/i
2.
Using
Lemma
3,
we
+ p{(j)uj + 02ii)]-x(<^iii+<?^2ij)^-^
L
<Plij,<P2ij
write the maximization of (21) as:
ihij
Z
\2
2
<p2ij) (j'
+
v2
,,2
+ 0H,i^' +
,
/2
'^
2
I
r2iojf^z\
(44)
Solving the two first-order conditions gives
(25).
e*.
=
To
find 0oij
13 (0*^.
+
=
05j)
0^,.)(from
we use the
Lemma
3)
(^uj
=
4>\j
and
(f)2ij
=
4>*2j
as in (24)
and
participation constraint, (20), and the facts that
and E{C
36
\
e^)
=
(/-S^
(from (17)).
QED.
^sj)
Proof of Lemma 4. Let us define the rate of return to capital in island j when a
total amount of capital Kj is invested there, the labor market clears, firms choose
the optimal contracts and make zero profits as:
=
rj{Kj)
The
+ /? mKj) +
(!-«)(/.
same
fact that capital should receive the
that for
all
€
j
rj{Kj)
[0, 1]:
A necessary condition for
-
4^;{Kj))
<Pl{K,))
^^^
rate of return in
islands implies
all
= r (Mf, IQ.
asymmetric equilibrium
capital, Kji, Kj", such that rji(Kji)
=
is
that there exist two levels of
Therefore, a sufficient condition for
rjii{Kjii).
<
We
the equilibrium to be unique and symmetric
is
now prove
always the case. To see this note that:
(i)
the
that for
first
not depend on
0,
^^^
and there
sufficiently large this
term on the
second term of
Kj >
/J,
RHS
fi;
(iii)
from
< S" <
Then,
and
3/i
exists a unique equilibrium
QED.
Proof of Proposition
Recall that
ject to (19).
(7^)
Given the definition of
(14)
it
such that V/i
and
(pi
TAG
<f)2
from
,
is
(l)*2{Kj))
>
/I
[0, 1],
(5{(P\+(l)l)
Then, we can write
rem imphes
that
and Var{C)
=
-
(30), this
[((</-!
(f>o{Kj)]
35" such
we have that
Kj
=
+
is
will
(ii)
(;^)"
,
the
does
that for any
r'j{Kj)
K (capital
are chosen to
strained maximization of fi+^—TAC{K) with respect to
Lemma 3 that e* =
for all Kj.
decreasing in Kj]
follows that
whereby Vj G
distributed across islands).
4.
a)fj,
- a)p [(</'J(/<j) +
{I
(24), (25)
00.
—
of (45), (1
=
(45), <i/{Kj)
is
that r'^[Kj)
is
maximize
<
0,
equally
(21) sub-
equivalent to the uncon-
(pi
^a)'^'
and
+
^2-
Now recall from
(-/-l)'^'
+ i^l?-^)
TAC{K) = T AG {M {K) (t>\{K) (j)l{K)) The envelope theo=
= 0. Therefore TAC'{K) = ^,M'{K) =
^gf
,
,
^^
.
-f('/'2)'(^AF(l-^)(f^<0•
For the second part, from the definition of
2
SAC{e, K)
=
min
02
(a'
K
-
+ |-*2
+ i'') I
Differentiating this with respect to
/
SAG we
have that:
\ 2
2
•''+
,,Z,
J
l
and once more using the envelope theorem,
we have SAC2{e, K) <0. QED.
37
Proof of Proposition
Conditional upon Mt,
5.
first-order conditions of (32) once Cj
and
ferentiation leads to (24)
Proof of Proposition
and ^2
by the
are given
substituted from (19). Straightforward
is
(23), (24)
and
it
(25),
dif-
QED
(25) exactly as in the decentralized equilibrium.
From
6.
(pij
follows that:
+ (M-l)i/2)(cr2 + Mi/2)
+ (M - l)z/2) + (a2 + Mz/2) 2£i
((72
(a2
To
Let Ttv denote the numerator and
the denominator of the right hand-side ex-
pression. Then:
=
V'iK)
^
{[{a^
'
D
+ (M - ly) + (a2 + Miy^)] TD-2(l +
[((72 + Mjy2) E^-(^a'' + {M- l)i/2)l
where some straightforward algebra
line.
If
This expression establishes that, sign
/90-2
p(o-2
+
M(A:)
1)
>
0,
>
/?,
1/2)
>
then
<
then V'{K
i3,
K)
^<
all
M(/<:)
=
pa^
<
1,
if
V'{K)
and
all
for
where w^^ and
r"^^ are
r^^k{Kt)
The
all
.
u"^),
0,
try in the first-stage
game
^
we have V'{K
(part 2).
+ Mv^)
= 1. If
(cr2
j
M
K
for all
\
such that
M{K) =
Moreover, for
M (or
K such that
Therefore, there exists
QED.
K as defined in the proposition. We
Kt <
through Vis
profit,
since
an equilibrium.
intermediation through GIs
to the second
Next, set
3.
M (thus K).
K, V'{K) <
i.e.
profit of this firm will
is
+
p{a^
—
fl
and V'{K) <
0,
is
the unique equilibrium. Suppose
(fx+^ —
cj
the equilibrium factor returns
= c - TAC{Kt) <
tion through V7s
f3
—
—
w^\Kt)l{Kt)
when
there
Now, consider a deviation from a firm which
diation through VI.
instead of VI.
sign v^
first
In this case, free entry ensures that active firms
intermediation through V7s.
which are using V7s make zero
<
<
1)
7. First, consider
is
=
which proves part
V'{K) <
K>
intermediation
[1^'(/'C)]
decreasing in
is
sufficiently large
V'{K) >
show that
K
Finally,
Proof of Proposition
will
for all
I
but since
K <K,
V\K) >
(part 1).
1
equivalently
for
necessary to go from the
is
'f) T^v}
is
be
(//
+
f
only interme-
decides to use
GI
- TAG{Kt)) - w^^{Kt)l{Kt) -
TAC{Kt) > TAC{k) =
We
is
+ r'^^k^Kt),
c.
Hence, intermedia-
then show that in the same case {Kt
not an equilibrium.
Assume
it is,
< K)
then free en-
ensures that active firms which are using GIs
make
zero
- TAC{Kt)) = w^'{Kt)l{I<t) - r^'k{Kt). Now, consider a devif
ation from a firm which decides to use VI instead of GI. The profit of this firm will
profit,
i.e.
(a*
+
38
be (//+ f
-
c)
- w^\Kt)l{Kt) -
r^'k{Kt)
= TAC{Kt) - c>
0.
This establishes
that there exists a profitable deviation, therefore intermediation through GIs
an equilibrium.
firms use GIs
through V7s
A
similar
argument would show that no equilibrium
and some others use
is
and
not
which some
Kt < K, intermediation
the unique equilibrium.
Next consider Kt > K. In
exactly,
V7s can exist. Thus, with
in
is
this case, the reverse of the previous
this establishes that only intermediation
QED
39
through GIs
argument applies
is
an equilibrium.
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