Discussion Papers In Economics
And Business

Organizational Dynamics
Shingo Ishiguro
Discussion Paper 07-14

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOrganizational Dynamics
Shingo Ishiguro
Discussion Paper 07-14

May 2007

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANOrganizational Dynamics ∗Shingo Ishiguro†
Graduate School of Economics
Osaka University
May 2007

Abstract
This paper provides a dynamic general equilibrium framework toinvestigate how organizations change the modes to govern transactions
over time. We show that the agency problem becomes less serious whenthe economy is developed well so that large market size favors decentralized organizations having more specialization. We then show that
diﬀerent organizational modes endogenously emerge even in the same
economy and cause endogenous process of economic development.
Keywords: Economic Development, Incentive Contracts, Moral Hazard, Specialization
JEL Classiﬁcation Numbers: D82, G30, L22, O11∗
I am grateful to V Bhaskar, Eric Chou, Ryoji Ohdoi and seminar participants at
Contract Theory Workshop (CTW), Decentralization Conference, National Tsing Hua
University and Osaka University for useful comments and discussions. Of course all remaining errors are my own responsibility. This research was started during my visit to
University College London. I would like to thank for its great hospitality and support for
my research.
†
Correspondence: Shingo Ishiguro, Graduate School of Economics, Osaka University,
1-7 Machikaneyama, Toyonaka, Osaka 560–0043, Japan. Phone: +81-6-6850-5220, Fax:
+81-6-6850-5256, E-mail: ishiguro@econ.osaka-u.ac.jp1

1

Introduction

In his classical book of economics Wealth of Nations Adam Smith discussed
that specialization or division of labor is one of the most important engines toenhance the productivity of labor but the extent of specialization is limited
by market size (See Smith (1776, Chapter 1 and 3)). In this paper we will
elaborate these Adam Smith’s views further in the following three directions
and investigate how dynamical changes of organizational modes to manage
the division of labor cause the endogenous process of economic development.
First, the eﬀects of market size on specialization are not in one way.
Rather the market size aﬀects not only the degree of specialization but
also is aﬀected by how specialization proceeds in the economy. This isbecause specialization enhances the labor productivity and hence raises labordemand. This in turn results in higher wage income to make consumption
level higher and thus make market size actually larger. Thus the degree of
specialization and market size interact with each other through such generalequilibrium eﬀect.
Second, specialization (division of labor) is not only technological but
also organizational issue such as how total production process is divided
into specialized tasks, how these tasks are delegated to diﬀerent agents andhow they are motivated to work hard for their specialized tasks. More decentralized organizations involve more specialization so that diﬀerent tasks
are delegated to diﬀerent and specialized agents. Such specialization contributes to expanding the technological productivity of organizations. However, agents who are specialized on some tasks may obtain the powers anddiscretion to control their tasks in their own interests and hence may notwork in the interests of the principal of an organization. In particular theireﬀorts exerted for tasks may not be observable to the principal. This sortof asymmetric information between the principal (capitalist) and delegatedagents limits specialization and decentralization because the principal faces
the trade–oﬀ between incentives and rent extraction: Some rent must beleft to delegated agent over his reservation payoﬀ which could be otherwise
obtained by his outside opportunity in the economy.
Third, the general equilibrium interactions between the market size andspecialization are not static but rather dynamic through the change of organizational modes. When the market size changes over time, this may change
the trade–oﬀ between specialization and agency problem which then causes
the change of optimal organizational modes to balance such trade–oﬀ. Thischange of organizational modes has the dynamic feedback eﬀects on themarket size and hence the process of economic development.
We will integrate these issues in a uniﬁed framework and show that
optimal organizational mode (degree of decentralization) depends on thedevelopment stages of the economy: When the economy is developed wellso that the market wage is high enough, agents who work for the principal
2

of an organization face better outside opportunity which could be otherwise
obtained if they rejected the contracts oﬀered by the current principals andworked elsewhere in the economy. Then, although the agents have informational advantage over the principal, they may obtain smaller rent whenthe economy is developed well. This makes the trade–oﬀ between incentives
and rent extraction less serious so that it becomes more attractive for the
principals to choose more decentralized organizations in which more tasks
are specialized and delegated to agents.
However, the story does not end here because the general equilibrium
feedback comes in: When the principals choose more decentralized organizations, higher degree of specialization will contribute to expanding the
production level and hence labor demand in the economy wide, which thenpushes up wage income and actually makes the outside opportunity of agentsbetter.
This general equilibrium eﬀect through the choice of organizational modes
causes the diversion of the processes of economic development. If the principals choose decentralized organizations, they expect that specialization willproceed and hence result in higher economic development, which then actually makes decentralized organization attractive for them. On the other
hand, if the principals choose centralized organizations, they expect thatspecialization will be limited and hence the economy will be less developed,
which then actually makes centralized organization attractive for them.
Hence multiple equilibrium paths arise even in the same economy: Some ofthem take the advantage of specialization and involve decentralized organizations but others lose the gains from specialization and involve centralizedorganizations. These diﬀerent development paths emerge even when theeconomy starts from the same initial condition. 1
These results might explain why there are signiﬁcant diﬀerences in organizational modes to govern transactions across countries. Casual empiricism
suggests the diversity of organizational structures across countries. For example it has been argued that the Japanese automobile industry is lessvertically integrated than the U.S. counterpart.2 It was also reported that1
Some papers focus on the related issues such as how contract (or organizational)
designs aﬀect and are aﬀected by economic development. Esfahani and Mookherjee (1995)
consider the Shapiro–Stiglitz type shirking model and show that both high powered andlow powered contracts may emerge even in the same economy. Legros, Newman andProto (2006) consider the endogenous growth model with organizational design in which
a division of labor plays two diﬀerent roles; enhancing innovation and monitoring workers
by managers. Acemoglu, Aghion and Zilibotti (2002, 2006) investigate the endogenous
growth model in which investment technology choice is endogenous and related to contract
design and organizational choice such as vertical integration. See also Aghion, Griﬃth andHowitt (2006).
2
See Aoki (1988) for a comparative study on the Japanese ﬁrms. See also Abraham andTaylor (1996) and Helper (1991) for the evidence about supplier systems and outsourcing
decisions in U.S. industries.

3

in 1980s the Japanese electronics subcontractors had focused on fewer customers than the U.K. ﬁrms.3 Furthermore the U.S. electronics companies
like Apple and Dell recently have limited their main businesses to marketingand sales by outsourcing their production decisions to independent contract
manufacturers while Asian electronics companies like Sony, Toshiba andSamsung have maintained the strategy of in–house productions (See Berger(2005)).
Second, diversity of organizational modes is not only the phenomenon
observed across countries but also even in the same country and industry. 4
To account for this stylized fact, we show that there exist the mixed steadystates in which diﬀerent organizational modes, centralized and decentralizedorganizations, co–exist even in the same economy. We then show that there
exists equilibrium paths converging to such mixed steady states, and henceco–existence of diﬀerent organizations persists even in the long run. This
result will be also useful for understanding why diﬀerent patterns to organize
productions co–exist even in the same country and same industry.
Third, we show whether or not decentralized organization becomes dominant in the long run depends on the exogenous costs of starting up projects.
Such costs may be higher not only when the investment technology of entrepreneurs is ineﬃcient but also when severe regulations are imposed bygovernment, when contracting costs are large and when credit markets areimperfect. We then show that a long run equilibrium is uniquely characterized by decentralized organization when such staring up cost is suﬃciently
small. However, when the staring up cost is large, the economy is more likely
to be stuck in centralized organization equilibrium. Thus our result suggeststhat the severity and diﬃculty of staring up projects make organizationalmodes biased toward more centralization. This view is consistent with therecent empirical study by Acemoglu, Johnson and Mitton (2005, 2007) which
focus on the determinants of vertical integration across countries by using alarge data set. They found that countries with higher contracting costs andtighter regulation are more likely to be concentrated in vertically integrated
industries. 5
Furthermore our results shed lights on the dynamic relationship between
business practices and economic development. In fact we show that thesteady state with centralized organization corresponds to smaller economic
development level than the steady state with decentralized organization.
3

See Nishiguchi (Chapter 5 and 6, 1994) for this evidence.
For example Berger (2005) reported that diﬀerent companies in the U.S. microchip
industry have adopted diﬀerent strategies of organizing their productions.
5
Acemoglu, Johnson and Mittion (2005) also reported the result that countries with lessdeveloped ﬁnancial markets are more likely to be vertically integrated in the technology
intensive or human capital intensive industries. Acemoglu, Johnson and Mittion (2007)
emphasized the interactive eﬀects of ﬁnancial development and contracting costs on thepropensity of vertical integration.
4

4

This suggests the testable implication that ﬁrms are more likely to rely on
the business practices in the centralized and integrated manners when theeconomy is less developed. Khanna and Palepu (1997, 2000) provide therelated evidence that group aﬃliated ﬁrms, which are based on common
ownership and control, outperform focused, unaﬃliated ﬁrms in emerging
markets such as India. Moreover, as shown by Chandler (1962)’s famous
analysis about U.S. business history, in the late 19th century U.S. ﬁrms became bigger through vertical integration and re–organized their corporate
structures in the centralized way known as U–form (unitary form). However,these U.S. big ﬁrms shifted their corporate structures from U–from to more
decentralized form known as M–form (multidivisional form) in the early 20th
century as their focused businesses were expanded. 6 This U.S. business history might be consistent with our result that organizational modes become
more decentralized as the market size becomes bigger.
The remaining sections are organized as follows: In Section 2 we will set
up the basic model of the overlapping generations economy. In Section 3
we will characterize the equilibrium organizations and in Section 4 we will
derive the dynamical system to govern the change of organizational modes.
In Section 5 we will discuss the implications about contracting choices ofintermediate inputs such as outsourcing and vertical integration. Section 6
concludes the paper.

2

Model

2.1

Overlapping Generations Economy

We consider the overlapping generations economy in which a continuum of
new generation is born with a constant population size in each period. Timeis discrete and extends over inﬁnity (t = 0, 1, 2, ...). There is a single ﬁnal
good which is used both for consumption and investment. A newly bornindividual lives for two periods, say “young” and “old.” In each generation
there are two types of individuals, called principals and agents, with one
unit measure and θ unit measure respectively. Each agent is endowed with
one unit labor in each period of his life while each principal is endowed withno labor inputs. In the model there are three perfectly competitive markets,
i.e., credit market, labor market and good market. Each individual is risk
neutral and maximizes his or her utility when old (he or she can obtain
utility only from old consumption).
There are two types of workers, unskilled and skilled workers. All young
agents start as unskilled workers because they do not have any speciﬁc skills
when young. Thus in period t each young agent inelastically supplies one
6

In the early 20th century the U.S. market became geographically bigger via expanding
railway networks. Such expansion of the market size pushed the U.S. big ﬁrms toward thechange of organizational forms.

5

unit labor and earns the competitive market wage wt . Since each agent
obtains no utilities from consumption when young, he simply saves all his
income wt . Let ρt+1 denote the (gross) interest rate paid in credit market inperiod t + 1. Then each young agent who was born in period t will earn theinterest income ρt+1 wt when old (period t + 1). There are two occupations
for each old agent in period t + 1: One is to supply one unit labor and earn
the competitive market wage wt+1 as an unskilled worker again. The otheroption is to work as a skilled worker for managing a project, which we will
explain more in details below. To be a skilled worker, each old agent needs
to spend his labor endowment in adulthood for learning and acquiring theskills which are necessary and speciﬁc for the implementation of the project.
Thus old agents consist of unskilled workers and skilled workers.
Each young principal can create projects by investing λ > 0 goods per
project in one period advance. Here λ captures the parameter value affecting the costs of starting up a project. These costs become higher not
only when the principal’s investment technology is ineﬃcient but also when
institutions underlying the economy are not well developed so that moreadministrative costs are needed to start up a project. The latter may be
due to severe regulations imposed by government and higher contracting
costs in the economy (We will give further justiﬁcation for λ later). Since
each young principal has no incomes, she needs to access to credit market
in order to ﬁnance these projects. If a young principal raises a project inperiod t, she will obtain stochastic returns from that project in the next
period (t + 1). To implement a project in period t + 1, each old principal
needs to hire one skilled worker for managing the project as well as employ
competitive unskilled workers for production at the market wage wt+1 .
In period t + 1 each project which was ﬁnanced in period t will yieldstochastic returns. If the outcome is failure, then the project yields nothing.
If the outcome is success, then the project yields the output Y by employing
unskilled workers L from the competitive labor market according to the
production function Y = F (L). We will make the standard assumptionthat F is continuously diﬀerentiable, monotone increasing, F ′ > 0, andstrictly concave, F ′′ < 0, with F (0) = 0 and Inada conditions F ′ (0) = ∞and F ′ (∞) = 0.
Since the labor market is perfectly competitive, the maximum proﬁt ofeach successful project becomes
π(w) ≡ max F (L) − wL.
L≥0

(1)

Here π is decreasing and convex, π ′ < 0 and π ′′ > 0. Let also g(w) denotethe corresponding labor demand, i.e., F ′ (g(w)) ≡ w.

6

2.2

Matching

At the beginning of each period the projects created in the previous period
and old agents (potential skilled workers) are randomly matched with eachother. Let Kt+1 denote the number of total projects created by young principals in period t and hence used for production in period t + 1. Then Kt+1
projects and θ old agents are randomly matched with each other according
to the matching function m(Kt+1 , θ). Here we assume that m is increasing
in its ﬁrst argument and satisﬁes the following condition: m(K, θ) ≤ γθfor all K ≥ 0 for some γ ∈ (0, 1). Also we assume that m(0, θ) = 0.
Then each project will succeed in matching an old agent with probabilityα(Kt+1 ) ≡ m(Kt+1 , θ)/Kt+1 while each old agent will succeed in matchinga project with probability β(Kt+1 ) ≡ m(Kt+1 , θ)/θ.
The old agent who succeeded in matching a project can be a potential
skilled worker who will handle the project he matched. Alternatively he canalways reject to work as a skilled worker even when he matched a project.
In that case such old agent will work as an unskilled worker to earn themarket wage in the competitive labor market. On the other hand, any old
agent who failed to match a project has nothing but to work as an unskilledworker.

2.3

Specialization (Division of Labor)

Each project has two tasks, called task 1 and task 2, and costly eﬀort isneeded for each task to be completed. We assume that by some technologicalreasons task 1 of each project can be handled only by old principal. However,task 2 of a project can be delegated to a matched old agent, called skilledworker. Each project’s outcomes, success and failure, depend on the eﬀorts
exerted for two tasks of the project. Eﬀort choice is assumed to be binary,high eﬀort (h) or low eﬀort (l), for any task. We will denote by a ∈ {h, l} theprincipal’s eﬀort exerted for task 1 of a project. Let also denote by ep ∈ {h, l}(resp. em ∈ {h, l}) the principal’s (resp. the skilled worker’s) eﬀort for task 2
of a project when the principal (resp. the skilled worker) handles that task.
Then we assume that each matched project will succeed with probability
P (a, ei ) ∈ (0, 1) but fail with probability 1 − P (a, ei ) respectively. We will
here assume that phh ≡ P (h, h) > phl = plh ≡ P (h, l) = P (l, h) > pll ≡P (l, l) > 0. Let also ∆p ≡ phh − phl and ∆q ≡ phl − pll . We will then assume
the diminishing returns: ∆q > ∆p.
When an old principal handles two tasks of a project together at a time,
she will personally incur her eﬀort cost as cp (a, ep ) where cp (h, h) = c̃ >cp (h, l) = cp (l, h) = c > cp (l, l) = 0. On the other hand, when tasks 2of a project is carried out by a matched old agent (skilled worker) (buttask 1 must be handled by the principal herself), the skilled worker will
personally incur his eﬀort cost as cm (em ) for task 2 of the project where7cm (h) = c > cm (l) = 0 while the principal will incur her eﬀort cost as cp (ep )for task 1 of the project where cp (h) = c > cp (l) = 0. 7 Here to simplify
notation we are assuming that skilled worker’s eﬀort cost c > 0 is same as
the principal’s one. 8 We also assume that each skilled worker has necessary
skills to implement one project and hence he is needed for implementationof one project even when he is not delegated task 2 of the project.

2.4

Organizational Modes

After the created projects matched old agents, each old principal decides
what organizational modes she should choose for implementing matchedprojects and what contracts should be oﬀered to the matched old agents whowill work as skilled workers for managing the projects. In other words eachold principal needs to decide whether or not task 2 of a matched project
should be delegated to a matched old agent who will work as an skilledworker to choose costly eﬀort em ∈ {h, l} for that task.
We will call the organizational mode in which task 2 of a project isdelegated to a skilled worker decentralized organization or simply D–mode.
In D–mode old principal and skilled worker choose eﬀorts a ∈ {h, l} for task
1 and em ∈ {h, l} for task 2 respectively. On the other hand we call theorganizational mode in which both task 1 and 2 of a project are handled
by the old principal herself centralized organization or simply C–mode. In
C–mode only old principal chooses both eﬀorts a ∈ {h, l} for task 1 andep ∈ {h, l} for task 2 of the project together.
We assume that the veriﬁable signals are only the realized outcomes ofeach project, i.e., success and failure. Then a contract oﬀered to skilledworker should specify the payments to him contingent on the outcomes ofthe project which he is working on. Let CtD = {RtD , rtD } denote the incentive
contract used for a project under D–mode in period t where RtD (resp. rtD )denotes the payment made from old principal to skilled worker in period t
when the project succeeds (resp. fails) in period t. Also let CtC = {RtC , rtC }denote the contract used for a project under C–mode in period t where RtC(resp. rtC ) is the payment made from old principal to skilled worker when
the project’s outcome is realized as success (resp. failure). Recall that weare assuming that one skilled worker is needed for implementing one projecteven when he is not delegated task 2. Thus even in C–mode the contract7

One interpretation for eﬀort cost ci (·) is as follows: Each task of a project gives player
i = m, p who handled it some private returns in terms of the good after the project iscompleted. For example, task 2 of a project yields B − ci (ei ) goods to the player who
handled it where B > 0. However, it is not veriﬁable whether or not each player obtained
the private returns.
8This assumption is not so essential: Even when we allow the principal’s eﬀort costcp (ep ) to diﬀer from the skilled worker’s one cm (em ), our results will not be substantiallychanged as long as these costs do not diﬀer too much.

8

CtC must induce the skilled worker to accept for working on the project.
The timing of events is summarized as follows:In period t:
1. Each young agent earns and saves the market wage wt .
2. Each young principal creates projects by investing λ goods per project.
In period t + 1:
1. The projects created in period t, denoted by Kt+1 , and θ old agents
randomly match with each other according to the matching functionm(Kt+1 , θ).
2. Each old principal chooses an organizational mode (D–mode or C–
C or C D ) to a matched old agent formode) and oﬀers a contract (Ct+1
t+1
each matched project.
3. Each old agent who matched a project decides whether or not to accept
the oﬀered contract. If he rejects it, he will work as an unskilled worker(thus he will supply one unit labor to labor market). If he accepts it, hewill spend one unit labor for acquiring necessary skills for the project
and will work as a skilled worker.
4. In D–mode the skilled workers who accepted the contracts and oldprincipals will simultaneously choose their eﬀorts em ∈ {h, l} for task
2 and a ∈ {h, l} for task 1 for matched projects. In C–mode oldprincipals will choose her eﬀorts a ∈ {h, l} and ep ∈ {h, l} for bothtasks of a matched project.
5. The projects’ returns are realized and payments are made according
to binding contracts.
In initial period (t = 0) there are a unit mass of old principals who
totally hold K0 projects, which are historically given in the model, and θ
old agents who have no initial wealth.

3Market Equilibrium and Organizational Choice

3.1

Optimal Contracts

We will ﬁrst characterize the optimal contract under each organizationalmode in period t ≥ 1, by taking as given the relevant market prices such as
the interest and wage rates. The optimal contracts oﬀered in initial periodt = 0 will be separately examined later.

9In each period each old principal chooses optimal organizational mode,
centralized or decentralized organization, as well as the associated optimal
contract oﬀered to skilled workers. In that stage the principals take themarket prices (wt and ρt ) as exogenously given because the markets are
perfectly competitive.
First we will start with the optimal contract under C–mode. Suppose
ﬁrst that old principal wants to implement high eﬀort only for one taskof a matched project. Since the probability of the project’s outcomes andeﬀort cost function are symmetric with respect to eﬀort levels, it does notmatter for which task the principal exerts high eﬀort (a = h or ep = h).
In either case the principal incurs eﬀort cost cp (h, l) = c > 0. Then the
success probability of the project is given by phl ∈ (0, 1). Given wt , ρt+1
C under C–mode is oﬀered to maximize
and wt+1 , period t + 1 contract Ct+1
the principal’s payoﬀ subject to the set of relevant constraints, i.e., it shouldsolve the following problem:
Problem CCC
max phl π(wt+1 ) − c − {phl Rt+1
+ (1 − phl )rt+1}

subject toCC(h, l) ∈ arg max pij π(wt+1 ) − cp (i, j) − {pij Rt+1
+ (1 − pij )rt+1} (ICPC)i,j=h,lCC
phl Rt+1+ (1 − phl )rt+1+ ρt+1 wt ≥ wt+1 + ρt+1 wt

(IRC)

C
C
Rt+1
≥ −ρt+1 wt , rt+1
≥ −ρt+1 wt(LLC)

Here (ICPC) says the incentive compatibility constraint for the principalthat she optimally chooses high eﬀort only for one task of the project,
C . Note here that, since the principal’s eﬀorts are
given the contract Ct+1
not veriﬁable, she cannot commit herself to her eﬀort levels when oﬀering
the contract. Thus (ICPC) must be satisﬁed.
(IRC) says the individual rationality constraint that the expected payoﬀ
C (the left hand side) must notof skilled worker by accepting the contract Ct+1
be less than the expected payoﬀ by rejecting it (the right hand side). By ourassumption one old agent (skilled worker) is needed for implementing one
project. Thus (IRC) must be satisﬁed for skilled worker to accept for working
on the project. Each old agent can obtain his interest income ρt+1 wt whichcomes from his saving, irrespective of accepting the contract. In addition tothis income each old agent can obtain the market wage wt+1 if he will work
as an unskilled worker but not skilled worker. Thus at least the total income
ρt+1 wt + wt+1 must be guaranteed for ensuring that matched old agent will
accept the contract and work as a skilled worker.

10

(LLC) says the limited liability constraint that, since each old agent has
only the income ρt+1 wt , the principal cannot make lower payments than thisamount. 9
To solve Problem C, we will ﬁrst omit (ICPC) from the problem andCcheck this later. Then, (IRC) must be binding because otherwise Rt+1
C
can be slightly reduced and the principal’s payoﬀ can be inor/and rt+1
C
C =w
creased. In particular we take Rt+1
t+1 /phl and rt+1 = 0. Then (ICPC)
will be satisﬁed whenπ(wt+1 ) ≥ wt+1 /phl + (c/∆q)(ICPC′ )π(wt+1 ) ≤ wt+1 /phl + (c̃ − c)/∆p.

(ICPC′′ )andThe former says that the principal has no incentives to choose low eﬀorts
for both tasks and the latter ensures that she has no incentives to choose
high eﬀorts for both tasks.
We will make the following assumption to ensure that it is too costly forold principal to exert high eﬀorts for both tasks of any matched project at
a time:Assumption 1. (i) Minimum wage (v > 0): wt ≥ v and (ii) c̃ − c > π(v).
Assumption 1(i) says that some minimum wage requirement exists sothat the market wage wt must not be less than the minimum wage v > 0.
However, in what follows we will assume that v is so small and hence theminimum wage constraint is never binding in any equilibrium. The reason
why we introduced Assumption 1(i) is merely to state Assumption 1(ii) in thesimple manner. Assumption 1(ii) can then guarantee that c̃ − c > π(wt ) for
all wt in any equilibrium. Then, given Assumption 1, (ICPC′′ ) is satisﬁed inany equilibrium. In Appendix B we will also verify that (ICPC′ ) is satisﬁed
in any equilibrium.
Then the gross payoﬀ the principal can attain in Problem C (gross ofinvestment cost λρt+1 ) is given by
ΠC (wt+1 ) ≡ phl π(wt+1 ) − wt+1 − c.
9(2)

In addition to (LLC) the limited liability constraints on the side of the principal
must be also satisﬁed. The strongest version for these constraints is that the principal’s
CCmust not be greater than the realized revenues π(wt+1 ) and zeroand rt+1
payments Rt+1
C
andfor each project respectively. Then these constraints are given by π(wt+1 ) ≥ Rt+1
C
0 ≥ rt+1 . However, we can verify that the optimal contract to solve Problem C satisﬁes
these constraints. One might also think that since the principal handles many projects
which matched old agents, it is suﬃcient to consider weaker version for the principal’s
limited liability constraints such that total payments cannot be larger than the total
revenues for total matched projects but not per project. However, since the principal’s
limited liability constraints are satisﬁed even as the strongest version per project, they
are also satisﬁed by weaker version for total matched projects as well.

11

Second, suppose that under C–mode some old principal wants to implement low eﬀorts for both tasks of some project. Then, since the principal
must pay at least the market wage wt+1 to the skilled worker who accepts
C , the maximum gross payoﬀ the principal can obtain by
the contract Ct+1
implementing low eﬀorts for both tasks is given bypll π(wt+1 ) − wt+1 .
Thus each old principal never implements low eﬀorts for both tasks whenΠC (wt+1 ) ≥ pll π(wt+1 ) − wt+1 which can be written by∆qπ(wt+1 ) ≥ c.
We will assume that this inequality is satisﬁed in any equilibrium, although
we will give suﬃcient conditions for this in Appendix B.
Third, it is never optimal for any old principal to implement high eﬀorts
from both tasks of any matched project under Assumption 1.
We will also check later that each project under C–mode can cover thecredit repayment λρt+1 : ΠC (wt+1 ) ≥ λρt+1 (See (BEC–C) introduced below).
D under D–mode oﬀered
Next we will consider the optimal contract Ct+1
in period t + 1.
Suppose ﬁrst that each old principal wants to implement high eﬀorts
a = h and em = h from both tasks of a matched project, one high eﬀort
from the matched skilled worker and the other from herself. Then the successprobability of the project is given by phh ∈ (0, 1). Also the principal andskilled worker incur their eﬀort costs cp (h) = c > 0 and cm (h) = c > 0
D should solve the following problem:
respectively. Then Ct+1

Problem DDD
max phh π(wt+1 ) − c − {phh Rt+1
+ (1 − phh )rt+1}

subject toDDDD
phh {π(wt+1 ) − Rt+1
+ rt+1} − c ≥ phl {π(wt+1 ) − Rt+1
+ rt+1}(ICPD)

D
D
D
D
phh Rt+1+(1−phh )rt+1−c+ρt+1 wt ≥ phl Rt+1+(1−phl )rt+1+ρt+1 wt (ICM)D
D
phh Rt+1+ (1 − phh )rt+1− c + ρt+1 wt ≥ ρt+1 wt + wt+1
D
Rt+1
≥ −ρt+1 wt ,D
rt+1
≥ −ρt+1 wt

(IRD)
(LLD)

Here (IRD) and (LLD) mean the individual rationality and limited liability constraints of skilled worker again. 10 Here (ICM) and (ICPD) mean10

As we have discussed in footnote 9, the limited liability constraints on the side of theprincipal are ignored here because they are satisﬁed at the optimal solution.

12the incentive compatibility constraints on the side of skilled worker and theold principal respectively: Both of them can ensure that high eﬀort pair(a, em ) = (h, h) can be a Nash equilibrium in the game in which the skilledworker and old principal simultaneously choose their eﬀorts, given the conD .
tract Ct+1
As in the case of C–mode, we will ﬁrst omit the incentive compatibility
constraint for the principal (ICPD) and check this later. Then, if (IRD)
is not binding at the optimum, both (ICM) and the second (LLD) mustD
D − rDbe binding: Rt+1
t+1 = c/∆p and rt+1 = −ρt+1 wt where recall that∆p ≡ phh − phl > 0. This will be actually the case if phh (c/∆p) − c >ρt+1 wt + wt+1 , which means that skilled worker obtains some rent over the
payoﬀ of his outside opportunity due to limited liability. Otherwise, (IRD)
is binding at the optimum.
This argument leads to the following lemma:
Lemma 1. Suppose that (ICPD) is not binding in Problem D. Then the
D to solve Problem D is characterized as follows: (i)optimal contract Ct+1
D
D = c/∆p − ρ
Rt+1
t+1 wt and rt+1 = −ρt+1 wt if phh (c/∆p) − c > ρt+1 wt + wt+1
D =w
D
and (ii) phh Rt+1 + (1 − phh )rt+1
t+1 + c otherwise.
In what follows we will ignore (ICPD), although in Appendix B we will givethe conditions for (ICPD) to be actually satisﬁed.
Then, we derive the gross maximum payoﬀ the principal can attain in
Problem D (gross of investment cost λρt+1 ) as follows: phh π(wt+1 ) − c − phh (c/∆p) + ρt+1 wt if phh (c/∆p) − c > ρt+1 wt + wt+1
D
Π (wt , wt+1 , ρt+1 ) ≡
phh π(wt+1 ) − wt+1 − 2cotherwise.
(3)
Next suppose that some old principal wants to implement other eﬀort
pairs than the high eﬀort one for some matched project. Then, by similarargument to the case of C–mode, the maximum payoﬀ the principal canobtain by implementing other eﬀort pairs than (a, em ) = (h, h) is given byΠ∗ (wt+1 ) ≡ max{pll π(wt+1 ) − wt+1 , phl π(wt+1 ) − wt+1 − c}.
Thus it is optimal for each old principal to implement high eﬀort pair(a, em ) = (h, h) whenΠD (wt , wt+1 , ρt+1 ) ≥ Π∗ (wt+1 ).

(4)

Again we will assume throughout the main text that this inequality holds
in any equilibrium, although we will verify this in Appendix B.
We will also check later that each project under D–mode can cover thecredit repayment λρt+1 : ΠD (wt , wt+1 , ρt+1 ) ≥ λρt+1 (See condition (BEC–
D) introduced below).
13

Each old principal compares the payoﬀs (2) and (3) under diﬀerent organizational modes. Then each old principal will choose D–mode only if
ΠD (wt , wt+1 , ρt+1 ) ≥ ΠC (wt+1 ),(5)ΠC (wt+1 ) ≥ ΠD (wt , wt+1 , ρt+1 ),(6)

but C–mode only if

given the market prices wt , wt+1 and ρt+1 . Furthermore, when ΠD (wt , wt+1 , ρt+1 ) =ΠC (wt+1 ), old principals will be indiﬀerent for choosing D–mode and C–mode.
We will ﬁnally derive the optimal contracts in initial period t = 0. In initial period t = 0 old agents are assumed to be have ﬁxed wealth, normalized
to zero. Again we assume that each old principal wants to implement higheﬀorts for both tasks of any matched project. Thus the optimal contract
C0D = {R0D , r0D } under D–mode in period 0 should solve the following:
Problem D0
max phh [π(w0 ) − R0D ] + (1 − phh )[−r0D ] − csubject to (ICPD), (ICM) and
phh R0D + (1 − phh )r0D − c ≥ w0 ,(IRD0)

R0D ≥ 0, r0D ≥ 0.

(LLD0)

This problem diﬀers from those in other periods only in that old agents have
no wealth initial period and thus they can obtain only the market wage w0
when they reject the contract. By this reason, the individual rationality
constraint (IRD0) and the limited liability constraint (LLD0) were changed.
Then the optimal contract to solve Problem D0 should be given as follows: R0 = c/∆p and r0 = 0 if phh (c/∆p)−c > w0 and phh R0 +(1−phh )r0 =w0 +c otherwise. Thus, in initial period (t = 0) the gross (per project) payoﬀ
of old principal under D–mode becomes phh π(w0 ) − c − phh (c/∆p) if phh (c/∆p) − c > w0D
Π0 (w0 ) ≡(7)phh π(w0 ) − w0 − 2cotherwise.
Next we consider the optimal C–mode contract in initial period t = 0.
By Assumption 1, any old principal never exerts high eﬀorts for both tasksof any matched project. Also we assume that it is optimal for each oldprincipal to implement high eﬀort only for one task of any matched project14

as in the case of other periods (t ≥ 1). Then, the gross payoﬀ each oldprincipal can obtain per project under C–mode is given by
ΠC
0 (w0 ) ≡ phl π(w0 ) − w0 − c.

(8)Thus each old principal will choose D–mode in period 0 only if ΠD
0 (w0 ) ≥In contrast to other periods only the market wage w0 determines
the equilibrium organizational mode in initial period (t = 0).
ΠC
0 (w0 ).

3.2

Labor Market Equilibrium

The labor demand in period t is totally given by g(wt )phh m(Kt , θ) when all
old principals choose D–mode in period t. This is because m(Kt , θ) projects
are matched with θ old agents and phh fraction of them succeeds, each ofwhich requires labor demand g(wt ). When all old principals choose C–mode,
total labor demand is given by g(wt )phl m(Kt , θ) because each project ownedby the principals will succeed with probability phl ∈ (0, 1). Of course, when
D–mode yields strictly higher (resp. lower) payoﬀ ΠD (wt , wt+1 , ρt+1 ) thanC–mode ΠC (wt+1 ) does, all old principals will choose D–mode (resp. C–mode) with certainty. When old principals obtain the same payoﬀ underboth C–mode and D–mode, they will be indiﬀerent between these modes,
and hence in this case we suppose that xt ∈ [0, 1] fraction of them will choose
D–mode.
On the other hand, labor supply comes from all θ young agents andthe old agents who failed to match the projects, θ(1 − β(Kt )). All of them,
θ+θ(1−β(K)), supply their labor inputs to work as unskilled workers. Thus,
recalling that we denoted by xt ∈ [0, 1] the fraction of the old principals
choosing D–mode in period t, the labor market will clear in period t when{xt phh + (1 − xt )phl }g(wt )m(Kt , θ) = θ + θ(1 − β(Kt )).

(LMC)

Since g ′ < 0, we can derive a unique market clearing wage to solve the above
equation. In particular we will denote by wt = wD (Kt ) the market clearing
wage for xt = 1 (when all old principals choose D–mode) and by wt =wC (Kt ) the market clearing wage for xt = 0 (when all old principals choose
C–mode) respectively. Since β is increasing in Kt , the market wages wD
and wC are both increasing in Kt . Since phh > phl , we also have wD (K) >wC (K) for all K.
Note here that, since m(K, θ) ≤ θ for all K ≥ 0, and phh > phl , (LMC)
implies that g(wt ) ≥ 1/phh which then shows wt ≤ g −1 (1/phh ) where g −1 isthe inverse function of g. This shows that the market wage must be bounded
above by g −1 (1/phh ) in any equilibrium.

15

3.3

Credit Market Equilibrium

Suppose that all old principals in period t + 1 will choose D–mode. Then inperiod t the following break even condition for each young principal mustbe satisﬁed:α(Kt+1 )ΠD (wt , wt+1 , ρt+1 ) = λρt+1 .
(BEC–D)
Here the left hand side means the expected payoﬀ of a young principal per
project in period t because she can match an old agent with probabilityα(Kt+1 ) and obtain the gross payoﬀ ΠD (wt , wt+1 , ρt+1 ) in the next periodt + 1. The right hand side means the cost of raising one project (recall that
λ goods are needed to create one project). If the left (resp. right) handside of the above inequality is greater than the right (resp. left) hand side,
each young principal will require inﬁnite (resp. zero) credit demand, which
will not make the credit market clear. Thus the above equality must holdin credit market equilibrium.
Similarly, when all old principals choose C–mode, the following breakeven condition must hold:α(Kt+1 )ΠC (wt+1 ) = λρt+1 .

(BEC–C)When each old principal is indiﬀerent for choosing between D–mode andC–mode, she must obtain the same expected payoﬀ ΠD (wt , wt+1 , ρt+1 ) =ΠC (wt+1 ) under both modes. This case can be thus included in the above
break even condition (BEC–D) or (BEC–C).
In credit market equilibrium the total credit demand (total investment)
must be also equal to the total credit supply (total saving):
θwt = λKt+1 .

(9)Here the total saving comes from θ young agents who will save all theirwage incomes wt . The total investment comes from one unit mass of young
principals who need λ goods per project to be ﬁnanced.
Note that (BEC–D) and α(Kt+1 ) < 1 imply ΠD (wt , wt+1 , ρt+1 ) > λρt+1 .
This shows that each project under D–mode can cover the credit repayment
λρt+1 after matching but before projects’ outcomes are realized.11 The same
argument can be applied to C–mode.

3.4

Equilibrium Organizations

Given the market wage wt determined in the previous period, say t, wedeﬁne the static equilibrium in the current period (t+1) which consists of the11

Since each old principal handles many projects, she can perfectly diversify the risksof the projects’ failure due to the law of large numbers. Thus her expected payoﬀ per
project can be viewed as her average one and hence the actual revenues of all her projects
can cover her total credit repayments.

16

number of total projects Kt+1 , the market wage rate wt+1 , the organizationalmode, and the interest rate ρt+1 . In the static equilibrium D–mode emerges
in period t + 1 when (5), (LMC) with xt+1 = 1, (BEC–D) and (9) are allsatisﬁed. We will call this D–mode equilibrium. On the other hand, C–mode
emerges in period t + 1 when (6), (LMC) with xt+1 = 0, (BEC–C) and (9)are all satisﬁed. We will call this C–mode equilibrium. Furthermore, whenΠD (wt , wt+1 , ρt+1 ) = ΠC (wt+1 ), (LMC) with some xt+1 ∈ (0, 1), (BEC–D)(or (BEC–C)) and (9) will give an equilibrium in which both C–mode andD–mode co–exist. We will call this mixed mode equilibrium.
Then, in order to see which type of the static equilibrium arises, wedeﬁne the following functions:ρ̂D (wt+1 , Kt+1 ) ≡α(Kt+1 ){phh π(wt+1 ) − c − phh (c/∆p)},λ(1 − β(Kt+1 ))(10)α(Kt+1 ){phh π(wt+1 ) − wt+1 − 2c},(11)λα(Kt+1 ){phl π(wt+1 ) − wt+1 − c}.
(12)ρC (wt+1 , Kt+1 ) ≡λBy using (9) and α(Kt+1 )Kt+1 /θ = β(Kt+1 ), we can readily verify thatρ̂D means the interest rate to ensure the break even condition (BEC–D) of
each young principal in period t, provided D–mode will be chosen and (IRD)
will not be binding in period t+1. ρ̃D is the interest rate to ensure the breakeven condition (BEC–D), provided D–mode will be chosen and (IRD) will
be binding in period t + 1. ρ̃C means the interest rate which ensures thebreak even condition (BEC–C), provided C–mode will be chosen in periodt + 1. Then, in order to ensure that these interest rates are all positive,
we must have phh π(wt+1 ) − phh (c/∆p) − c > 0, phl π(wt+1 ) − wt+1 − c > 0
and phh π(wt+1 ) − wt+1 − 2c > 0. To this end we will make the following
assumption:ρ̃D (wt+1 , Kt+1 ) ≡Assumption 2.
c/phh }.

π(g −1 (1/phh )) > max{(g −1 (1/phh ) + 2c)/phh , c/∆p +Since the market wage wt+1 is bounded above by g −1 (1/phh ) in any equilibrium as we have noted before, Assumption 2 can guarantee the following:First, π(w) > c/∆p and hence ρ̃D (w, K) > ρC (w, K) in any equilibrium.
Second, all ρC (w, K), ρ̃D (w, K) and ρ̂D (w, K) are positive in any equilibrium.
By deﬁning the following functions:Γ(w, K) ≡ ρ̂D (w, K) − ρC (w, K),(13)

Φ(w, K) ≡ ρ̂D (w, K) − ρ̃D (w, K),

(14)

and

17

we can show the following lemma:
Lemma 2. Given K, if Γ(w, K) = 0 has solutions for w, its number isgenerically two. Also, given K, if Φ(w, K) = 0 has solutions for w, its
number is generically two.
Proof. See Appendix A.
Since Γ(w, K) > Φ(w, K) for all w and all K by Assumption 2, Φ(w, K)
has solutions as long as Γ(w, K) does so. Let w(K) and w(K) denote thesesolutions to Γ(w, K) = 0 for w, given K, where w(·) > w(·). Then, we canverify that w (resp. w) is decreasing (resp. increasing) in K. Let also denoteby w∗ (K) and w∗∗ (K) the two solutions of w to the equation Φ(w, K) = 0
for a given K, where w∗ (K) < w∗∗ (K) for all K. Again, w∗ is increasing
but w∗∗ is decreasing. Since Γ(w, K) > Φ(w, K) for all w, we also havew∗ (K) < w(K) < w(K) < w∗∗ (K) for all K ≥ 0.
Now we can show the following result:
Proposition 1. Suppose that wt is given (t ≥ 0). Suppose also thatΓ(wt+1 , Kt+1 ) = 0 has the solutions of wt+1 for a given Kt+1 = (θ/λ)wt .
Then the static equilibrium in period t + 1 is characterized as follows:(i) C–mode equilibrium emerges if and only if wt+1 and Kt+1 satisfy
Kt+1 = (θ/λ)wt ,(15)

wt+1 ∈ [w(Kt+1 ), w(Kt+1 )],(16)

wt+1 = wC (Kt+1 )(17)(ii) D–mode equilibrium emerges if and only if wt+1 and Kt+1 satisfy
Kt+1 = (θ/λ)wt ,(18)

wt+1 ∈ (0, w(Kt+1 )] ∪ [w(Kt+1 ), ∞)(19)

wt+1 = wD (Kt+1 )(20)(iii) Mixed mode equilibrium emerges if and only if
Kt+1 = (θ/λ)wt ,(21)

wt+1 ∈ {w(Kt+1 ), w(Kt+1 )}(22)

respectively.
Proof. See Appendix A.

18

Although the formal proof for Proposition 1 is relegated to Appendix
A, the intuition behind this result can be understood as follows: Roughly
speaking, the reservation payoﬀ of skilled worker ρt+1 wt + wt+1 , which hecould obtain if he rejected the contract oﬀered by matched principal andworked as an unskilled worker, is U–shaped with respect to the market wage
wt+1 in period t+1, given the previous period wage wt (See Figure 1). When
wt+1 is large, this directly raises the opportunity cost of working as a skilledworker instead of an unskilled worker. When wt+1 is small, such opportunity
cost also rises because the proﬁt of successful project π(wt+1 ) becomes larger
and hence the equilibrium interest rate ρt+1 must go up in order to satisfy
the credit market equilibrium. Hence, by recalling that (IRD) is binding
under D–mode when phh (c/∆p) − c ≤ ρt+1 wt + wt+1 , (IRD) is more likely
to be binding when the market wage wt+1 becomes larg or small. Since thetrade–oﬀ between incentive and rent extraction becomes less serious when(IRD) is more likely to be binding, D–mode becomes more proﬁtable foreach old principal than C–mode when wt+1 is either large or small. Thismeans that D–mode emerges as an equilibrium when the market wage wt+1
in period t + 1 is large or small, given the previous period wage wt .
In initial period t = 0 the argument should be modiﬁed as follows: Recall that the maximum gross payoﬀs of initial old principals under D–mode
and C–mode are given by ΠD (w0 ) and ΠC (w0 ) respectively. The only difference from any other period t ≥ 1 is that the maximum payoﬀ of old
principal under D–mode is changed from phh π(wt ) − c − phh (c/∆p) + ρt+1 wt
to phh π(w0 ) − c − phh (c/∆p) when (IRD0) is not binding in Problem D0.
Thus, when (IRD0) is not binding, the payoﬀ diﬀerence ΠD (w0 ) − ΠC (w0 )between D–mode and C–mode becomes(phh − phl )π(w0 ) − phh (c/∆p) + w0 ,which is the same as the function Γ we have deﬁned for other periods (t ≥ 1)
once we set β(Kt ) ≡ 0 or equivalently Kt ≡ 0. Thus we can show that for
all w0 ∈ (w(0), w(0)) we have ΠC (w0 ) > ΠD (w0 ) but ΠD (w0 ) ≥ ΠC (w0 )for all w0 ∈ (0, w(0)] ∪ [w(0), ∞). Then D–mode equilibrium emerges ininitial period if w0 ∈ (0, w(0)] ∪ [w(0), ∞) and the labor market clears,
i.e., phh g(w0 )m(K0 , θ) = θ + (1 − β(K0 ))θ. Here the initial value K0 is
historically given. On the other hand, C–mode equilibrium emerges in initial period if w0 ∈ (w(0), w(0)) holds and the labor market clears, i.e.,phl g(w0 )m(K0 , θ) = θ + (1 − β(K0 ))θ. Also, mixed mode equilibrium arises
in initial period when w0 = w(0) or w0 = w(0).

194Equilibrium Dynamics

As we have shown, if they exist, the two cutoﬀ market wages w(K) andw(K), which determine the equilibrium organizational mode, are decreasing
and increasing in K respectively. In Figure 2 we depicted these two curves
and the diﬀerent regions, labeled by D1, C and D2. The shaded area, labeled
by C, means the set of pairs (w, K) such that C–mode equilibrium emerges
while in all other areas, labeled by D1 and D2, D–mode equilibrium emerges.
Furthermore, on the boundaries of the curves w(K) and w(K), mixed mode
equilibria arise where old principals are indiﬀerent between C–mode andD–mode as we have shown in Proposition 1.
A can be seen in Figure 2, both D–mode and C–mode static equilibria
may arise for the same number of projects K because the market wage
wD (Kt ) under D–mode belongs to region D1 as well as the wage wC (Kt )under C–mode belongs to region C even when the economy has the same
number of projects Kt . Also, mixed mode equilibrium can co–exist together
with C–mode and D–mode static equilibria: Given the same Kt , the market
wage w(Kt ) satisﬁes the labor market clearing condition (LMC) and thus
some xt ∈ (0, 1) fraction of old principals choose D–mode while 1−xt fractionof them choose C–mode. This fraction xt can be determined by (LMC) as
well. Such mixed mode equilibrium is depicted as point M in Figure 2. Thus
for the same value of Kt there may exist three possible static equilibria, i.e.,
C–mode, D–mode and mixed mode equilibria.
Note that the market wage wC (K) (resp. wD (K)) must belong to thecorresponding region, C (resp. D1 or D2) when C–mode (resp. D–mode)
equilibrium arises. Note also that in a mixed mode equilibrium the market
wage must be w(K) or w(K). Then we can derive the entire equilibrium
market wage which is a function of the number of total projects Kt as follows:wC (Kt ) if wC (Kt ) ∈ (w(Kt ), w(Kt )),wD (Kt ) if wD (Kt ) ∈ [0, w(Kt )) ∪ (w(Kt ), ∞)w∗ (Kt ) ≡(23)w(Kt )
if w(Kt ) ∈ [wC (Kt ), wD (Kt )],w(Kt )
if w(Kt ) ∈ [wC (Kt ), wD (Kt )].
Given this wage function w∗ (K), we can characterize the equilibrium dynamics which governs the path of the number of created projects {Kt }∞t=0
which satisﬁes the credit market clearing condition (9) as well as (23):Kt+1 = (θ/λ)w∗ (Kt ).

(24)

We will here make the following assumption:
Assumption 3. wi′ (0) = +∞, i = C, D.
Assumption 3 will be satisﬁed in the following example: F (L) = Lη , where20η ∈ (0, 1), and β ′ (0) > 0. In this example we can show that dwi /dK|K=0 =η(1 − η)2η−1 z 1−η β ′ (0)β(0)−η = +∞ for z = phh when i = D (resp. z = phlwhen i = C).
Then we can deﬁne the steady state value of K in D–mode equilibrium,
called the D–mode steady state and denoted K D , asθwD (K D ) = λK D
and the steady state value of K in C–mode equilibrium, called C–mode
steady state and denoted K C , asθwC (K C ) = λK C .
By Assumption 3 and the fact that wi (K) ≤ g −1 (1/phh ) for all K, there exist
the steady state values K D and K C for all λ > 0. To avoid complication,
we will assume that the steady state K i uniquely exists for each i = D, C.
Note here that K D > K C holds because wD (K) > wC (K) for all K > 0.
The properties of equilibrium dynamics (24) depend on the parametervalue λ which captures the costs of investing and starting up a project.
As we have mentioned, λ may measure both technological and institutional
ineﬃciencies underlying the economy when starting up a project. We canalso give further interpretation of λ as the measure of contracting costsof ﬁnance in credit market by extending the basic model to allow default
possibility (See Appendix C for more details).
We will ﬁrst consider the case that the investment cost λ is so small.
Then the economy eventually converges to the D–mode steady state K D .
Proposition 2. Suppose that Assumption 1–3 hold. Suppose also thatthe investment cost λ is suﬃciently small. Then the economy eventuallyconverges to the D–mode steady state, i.e., Kt → K D as t → ∞.
Proof. See Appendix A.
When the investment cost λ is small enough, the number of createdprojects Kt becomes much larger. Then the market equilibrium wage wi (Kt )is pushed up under both D–mode and C–mode (i = D, C). Thus, as the
economy grows, it must enter the region in which the market wage is large
enough and hence the rent of skilled workers becomes small enough as well.
This means that D–mode becomes optimal choice forever from some period
onward. Thus, in this case there are no diversities of organizational modes
to govern transactions in the long run.
One implication of this result is as follows: When the institutions underlying the economy are developed well so that it becomes cheaper for young
principals to raise the fund and start up projects, λ becomes smaller. Then
Proposition 2 suggests that decentralization is more likely to be dominant
21

form and the economy is more likely to enjoy the beneﬁts from specialization
when well developed institutions emerge to reduce the costs of starting up
the projects.
Next we will turn to more interesting case that the investment cost λis moderately large. To make the analysis interesting, we will impose thefollowing assumption:
Assumption 4. (i) minw≥0 Γ(w, ∞) < 0, and (ii) g −1 (σ/phl ) > w(∞)
where σ ≡ (2 − γ)/γ.
Assumption 4(i) ensures that the two cutoﬀ wages w(K) and w(K) actually exist for all K ≥ 0. This is because Γ(w, K) is increasing in K and hence
minw Γ(w, ∞) < 0 implies minw Γ(w, K) < 0 for all K ≥ 0. Then, by deﬁnition of Γ(w, K), we have two solutions of w to the equation Γ(w, K) = 0 for agiven K, due to Lemma 2. Thus there exists the region in which C–mode becomes a static equilibrium (labeled by C in Figure 2). Assumption 4(i) also
implies that w(K) and w(K) exist for all K ≥ 0 and hence w(∞) = w(∞)
exists. Assumption 4(ii) then guarantees that these cutoﬀ curves w(K) andw(K) intersect with the market wage curves wC (K) and wC (K) corresponding to D–mode and C–mode respectively (as depicted in Figure 2). This isbecause wD (K) > wC (K) for all K and limK→∞ wC (K) = g −1 (σ/phl ) by
limK→∞ m(K, θ) = γθ. In Appendix B we will give the parametric conditions for Assumption 4 to be satisﬁed and consistent with other assumptions
we have made.
Given Assumption 4, we will consider the range of the investment cost
parameter values λ such that wC (K C ) ∈ (w(K C ), w(K C )) and wD (K D ) >w(K D ) are satisﬁed. Then, as depicted in Figure 4, there exist multiple
equilibrium paths some of which converge to the C–mode steady state K C
but others of which converge to the D–mode steady state K D even whenthe economy starts from the same initial value K0 . To see that this isactually the case, we deﬁne the values of K, denoted K ′ and K ′′ , such thatwC (K ′ ) = w(K ′ ) and K ′′ ≡ max{K > 0 | wC (K ′′ ) = w(K ′′ )}. 12
Then we can show the following result:
Proposition 3. Suppose that Assumption 1–4 hold. Then there exists somerange of the investment cost parameter λ, denoted (λ, λ), such that for allλ ∈ (λ, λ) the following statements hold: (i) If K0 ∈ (K ′′ , K ′ ), then multiple
equilibrium paths arise, some of which converge to the D–mode steady state
K D and others of which converge to the C–mode steady state K C . (ii) If
K0 ≥ K ′ , then there exist no equilibria which converge to the C–mode steadystate K C .
Since w(0) > wC (0) = 0 and Assumption 4, such K ′′ is well–deﬁned. In Figure 2 and
3 we assume that such K ′′ is unique to simplify argument.
1222

Proof. See Appendix A.
Proposition 3 states that the economy may be diverted into diﬀerent
development phases, which are characterized by diﬀerent organizationalmodes, even when it starts from the same economic condition. If all theprincipals expect higher economic growth (and hence higher market wage),
D–mode becomes more proﬁtable for them than C–mode as we have seenin Proposition 1. Since D–mode is more productive than C–mode in thesense that it can exploit the beneﬁts from specialization of tasks (note thatphh > phl ), production and hence employment levels are expanded when all
the principals choose D–mode. This in turn pushes up the labor demand
and thus the market equilibrium wage, which actually fulﬁlls the principals’
original expectation. Thus D–mode can be an equilibrium. On the other
hand, if all the principals expect lower economic growth (and hence lower
market wage), C–mode becomes more proﬁtable for them than D–mode,
which actually makes the market wage lower. Thus the principals’ original
expectation is fulﬁlled again and thus C–mode becomes an equilibrium as
well. Which development path, the path converging to the D–mode steadystate or the C–mode steady state, the economy will eventually reach depends
on the expectation held by the principals.
The equilibrium paths we have shown above eventually converge to either the D–mode steady state or the C–mode steady state in the long run.
Thus, although equilibrium paths are not unique, each long run steady statecorresponds to a unique organizational mode, one for D–mode and the other
for C–mode. This result will be useful for understanding why diﬀerent countries are sometimes characterized by diﬀerent organizational modes: Diﬀerent countries might be on diﬀerent equilibrium paths converging to diﬀerent
steady states. However, diversity of organizational modes to govern transactions is observed not only across countries but also even in the same country
and same industry (See Berger (2005)). To address this issue, we show that
there exists the mixed steady state in which both C–mode and D–mode
co–exist together. These steady states are deﬁned as
θw(K M ) = λK M ,andθw(K m ) = λK m .
Here K M > K m holds (See Figure 5). Since w(0) > 0 and w(K) is bounded
above, the steady state K m exists. Also, since w is decreasing in K, there
also exists the steady state K M as well.
First, when λ is large, there exists a low mixed steady state K m wherem
K < K C as well as some equilibrium path converging to such steady state(See Figure 5). In this equilibrium path the number of total projects is23

low and the resulting market wage is low as well. Second, when λ is in
the moderate range, a high mixed steady state K M exists where K M >K C (See point M in Figure 5). Since the equilibrium wage curve w(Kt ) is
decreasing and the curve of credit demand λKt is increasing, there may exist
equilibrium path which ﬂuctuates around high mixed steady state K M andeventually converges to it in the long run ( as shown in Figure 6). Thenthe mixed steady state can be attained and hence co–existence of diﬀerent
organizations persists even in the long run.
Proposition 4. There exists some range of the investment cost parameter
λ for which the mixed steady states exist in which both C–mode and D–mode
co–exist. Furthermore, there may exist an equilibrium path which ﬂuctuatesaround the high mixed steady state K M and converges to it in the long run.

Proposition 4 shows the possibility that the economy exhibits endogenous ﬂuctuation due to the endogenous non–monotonic change of the composition of organizational modes: The fractions of D–mode in the economy
ﬂuctuate over time. Furthermore, depending on the costs of starting up
project λ, there may exist multiple mixed steady states, K m and K M , which
diﬀer each other in terms of the fractions of the principals choosing D–mode.
In Figure 7 we depicted such situation where there also exist multiple equilibrium paths some of which converge to the low mixed steady state K m butothers of which converge to the high mixed steady state K M even from thesame initial condition.
We will ﬁnally comment on the welfare property of diﬀerent steadystates. We deﬁne the social welfare of this economy in a steady state as
the sum of all individuals’ payoﬀs in each generation (the Benthamian welfare). All individuals obtain utilities from their old period consumption. The
total consumption of all old principals and old agents in each generation is
given by the total output F (g(w))[xphh + (1 − x)phl ]m(K, θ) they generatefrom matched projects minus total wage paid to young workers in the next
generation θw. Since θw = λK, this payment is equal to the investment
cost incurred by the principals in each generation. Here x ∈ [0, 1] denotes
the fraction of old principals choosing D–mode in a steady state. Also, total eﬀort costs in a generation under C–mode are given by c because only
old principal exerts high eﬀort only for one task of each matched project.
Under D–mode both old principal and skilled worker exert high eﬀorts forboth tasks of each matched project, and hence the total eﬀort costs in a
generation is given by 2c.
Let denote by wC ≡ wC (K C ) and wD ≡ wD (K D ) the equilibrium wages
in the C–mode and D–mode steady states. Then the sum of the utilities of24

all individuals in each generation is given byV D ≡ F (g(wD ))phh m(K D , θ) − m(K D , θ)2c − θwD(25)in the D–mode steady state andV C ≡ F (g(wC ))phl m(K C , θ) − m(K C , θ)c − θwC(26)

in the C–mode steady state respectively.
Which V D or V C is larger depends on the two opposite eﬀects: Since
D
K > K C , more projects are ﬁnanced in the D–mode steady state thanthe C–mode steady state. This gives larger number of successful projects
in the D–mode steady state than the C–mode steady state. On the other
hand, since wD > wC , the cost–rising eﬀect arises under D–mode. First,
the output per successful project becomes lower in the D–mode steady statethan the C–mode steady state. Second, the investment cost θwD in the
D–mode steady state becomes larger than the C–mode steady state, θwC .
When the former project expansion eﬀect dominates the latter cost–rising
eﬀect, the D–mode steady state attains higher welfare than the C–mode
steady state. However, when the latter dominates the former, the rankingof the long run welfare in these steady states is reversed.
Although it is diﬃcult to derive general conclusions on the welfare comparison, we can give some parametric example to show that more decentralization yields larger steady state welfare than less decentralization:
Example. Suppose that F (L) = ALη where η ∈ (0, 1) and A > 0. Let
K = K(m) denote the inverse function of m = m(K, θ). Then we haveK(0) = K ′ (0) = 0 and limm→γ K(m) = ∞ under our assumptions. Under
this speciﬁcation we can derive the steady state welfare under D–mode as
follows:1−η
V D (m) ≡ Aphh
(2θ − m)η m1−η − 2cm − λK(m)for m = mD ≡ m(K D ) and the welfare under C–mode as follows:1−η
V C (m) ≡ Aphl
(2θ − m)η m1−η − cm − λK(m)for m = mC ≡ m(K C ) respectively. Then we obtain that wD = Aη(2θ −mD /ph mD )η−1 and θwD = λK(mD ). Here note that mD > mC by K D >K C . Under our assumptions, the equilibrium value mD tends to be zero
when η goes to zero. Similarly, mC tends to be zero as η goes to zero.
Also, by letting f (m) ≡ V D (m) − V C (m), we can show that f (0) = 0 andf ′ (0) = +∞. 13 Then V D (m) > V C (m) holds for all small m, and hence wehave V D = V D (mD ) > V C (mD ) > V C (mC ) = V C when η is small enough
1−η
η 1−η
In fact we can derive f (m) = A[p1−η−cm and hence f ′ (0) = +∞hh −phl ](2θ−m) m
due to phh > phl .
1325

because both mC and mD are small and mD > mC when η is small. Thisshows that the steady state welfare V C under D–mode become higher thanthat under C–mode V C when η is small enough.
Furthermore the social welfare in the high mixed steady state (K = K M )is given by
V M (m) ≡ A[xphh + (1 − x)phl ]1−η (2θ − m)m1−η − [x2c + (1 − x)c]m − λK(m)for m = mM where mM is deﬁned by K(mM ) = K M . V M (m) lies between
V D (m) and V C (m) for all m. Thus, since mC < mM < mD , we also have
V D (mD ) > V M (mM ) > V C (mC ) when η is small enough. This shows thatthe steady states with more decentralization attains higher social welfare
than those with less decentralization.

5

Implications

5.1

Outsourcing vs. Vertical Integration

In this subsection we discuss the implications of the model about contracting
choices of intermediate inputs, i.e., outsourcing and vertical integration. In
contrast to the basic model, we assume that there are two types of goods,
one ﬁnal good and one intermediate input. The principals are the ﬁnal good
manufactures and one unit of the ﬁnal good can be produced by using one
unit of the intermediate input. The ﬁnal good manufacturers can create theprojects when they are young. These projects will yield stochastic returns
when they are old by using the intermediate inputs. The output of theintermediate input Y is produced by the production function Y = F (L)by using labor L. Each old agent is the intermediate input producer who
initially owns the technology or knowledge to develop the intermediate input.
The projects created by the ﬁnal good manufacturers are randomly
matched with the intermediate input producers. The intermediate input
is not traded in the market but its trading and payment decisions are madeby bilateral bargaining between the ﬁnal good manufacturer and the intermediate input producer who matched the project owned by the ﬁnal good
manufacturer.
For each matched project there are two tasks to be completed, one for“input development” and the other for “output development:” To enter the
production process of ﬁnal good and intermediate input, both the input andthe ﬁnal good must be developed by ex ante eﬀorts. These two tasks correspond to task 1 and task 2 in the basic model. We assume that the output
development can be carried out only by the ﬁnal good manufacturers. However the input development can be handled by either the input producer orthe ﬁnal good manufacturer. We here assume that the party who developed

26

the input can access to the production technology of the input. 14
Then each ﬁnal good manufacturer has two options for handling the
input development: One is to keep the arm’s length transaction with the
input producer who owns the technology to develop the input and hence
produces the input. This is called outsourcing. The other option is tobuy the technology of the input development from the input producer andproduce the intermediate input by herself. This is called vertical integration.
Extending the basic model in this way, decentralized organization (resp.
centralized organization) which we have deﬁned in the basic model corresponds to the mode of contracting choice as outsourcing (resp. vertical
integration). Then Proposition 1 gives a prediction that outsourcing becomes dominant in well developed countries with smaller costs of setting
up the projects such as smaller contracting costs and less regulations. Thisprediction might be consistent with the recent ﬁnding by Acemoglu, Johnson and Mitton (2005, 2007). Proposition 3 also suggests that both vertical
integration and outsourcing emerge as organizational modes to govern the
productions of intermediate inputs even in the similar economies. This might
explain the stylized fact that contracting modes of intermediate inputs are
diﬀerent across countries. 15

5.2

Decentralization and Economic Development

Our result also shows that the steady state with centralized organization corresponds to smaller economic development in terms of the number of total
projects K C than the steady state with decentralized organization K D . Thismight suggest the testable implication that organizational modes to governtransactions become more centralized and integrated in less developed countries. Kahnna and Palepu (1997, 2000) provide supportable evidence for thisand found that group aﬃliated ﬁrms which are based on common ownershipand control outperform focused, unaﬃliated ﬁrms in emerging markets such
as India.
Also, as Chandler (1962) has extensively analyzed in his famous book,in the late 19th century U.S. ﬁrms became bigger through vertical integration and changed their corporate structures in the centralized way known as
U–form (unitary form). However, as the U.S. market was expanded throughincreasing connections of railway networks, in the early 20th century the
U.S. big ﬁrms shifted their corporate structures from U–form to more decentralized form known as M–form (multidivisional form). Since top management had incurred large overload costs under U–form when the market
was expanded, the U.S. big ﬁrms were forced to be restructured toward more
decentralization such as M–form. This U.S. business history might be con14

This may be because the party who developed the input can own some knowledge andskills which are necessary for the production of the input.
15
See for example Berger (2005) for recent evidence.

27

sistent with our result that organizational modes become more decentralized
to pursue the gains from specialization as the market size is expanded.

6

Conclusion

In this paper we have developed the dynamic general equilibrium framework to investigate how organizations change the modes to govern transactions and how such organizational change leads to endogenous processof economic development. Then we have argued that specialization or division of labor makes decentralized organization advantageous while it islimited by the agency problem caused by asymmetric information in organizations. We have shown how centralized and decentralized organizations
emerge in the economy and how they dynamically interact with the market equilibrium conditions. Depending on the ﬁnancing cost underlying the
economy, equilibrium organizational mode becomes centralized and decentralized. Furthermore, there exist multiple equilibrium paths some of whichconverge to the steady state with decentralized organization and others ofwhich converge to the steady state with centralized organization. We havealso shown that such endogenous choice of organizational modes causes endogenous ﬂuctuation of the economy such as business cycles.

7

Appendix A

7.1

Proof of Lemma 2

Γ(w, K) = 0 can be written by
]
[
)
]
(
[
1
phh
phh
− phl π(w) + w − c+ 1 − 1 = 0.
1 − β(K)1 − β(K) ∆p
Since π is convex function with π(0) = ∞ and π(∞) = 0, the number of thesolutions for w to the above equation is generically two, given K.
Similarly Φ(w, K) = 0 generically has two solutions of w, given K.
Q.E.D.

7.2

Proof of Proposition 1

To save notation, we will denote K ≡ Kt+1 , ρ ≡ ρt+1 and w ≡ wt+1 . We
will also use the fact wt = (λ/θ)Kt+1 .
Given K, both ρ̂D (w, K) and ρ̃D (w, K) coincide with each other at apoint w if and only if
phh (c/∆p) − c = ρ̂D (w, K)(λ/θ)K + w.
As we have seen in Lemma 2, there are two such points w, denoted w∗ (K)and w∗∗ (K), and we obtain ρ̃D (w, K) > ρ̂D (w, K) if and only if w ∈28

(w∗ (K), w∗∗ (K)). Furthermore, since both ρ̃D and ρ̂D are decreasing andconvex functions of w, we havephh (c/∆p) − c ≥ ρ(λ/θ)K + w
for any (w, ρ) such that ρ = max{ρ̂D (w, K), ρ̃D (w, K)} and w ∈ (w∗ (K), w∗∗ (K))
whilephh (c/∆p) − c < ρ(λ/θ)K + w
for any (w, ρ) such that ρ = min{ρ̃D (w, K), ρ̂D (w, K)} and w ∈ (0, w∗ (K))∪(w∗∗ (K), ∞). Thus, if D–mode becomes an equilibrium and (IRD) is notbinding, then w ∈ (w∗ (K), w∗∗ (K)) and ρ = ρ̂D (w, K) must be satisﬁed.
Also, if D–mode becomes an equilibrium and (IRD) is binding, then w ∈(0, w∗ (K)) ∪ (w∗∗ (K), ∞) and ρ = ρ̃D (w, K) must be satisﬁed.
Furthermore, since ρC (w, K) < ρ̃D (w, K) for all w and K due to Assumption 2, we always have the relation that w∗ (K) < w(K) < w(K) <w∗∗ (K) for any K.
Note also that, since λ > α(K)(λ/θ)K = λβ(K), we have ρ > (<)
Dρ̂ (w, K) if and only ifα(K){phh π(w) − c − phh (c/∆p) + ρ(λ/θ)K} < (>)λρ.

(*)(i) Suﬃciency Part: Now take an interest rate ρ = ρC (w, K) and the market
wage w = wC (K) such that w ∈ [w(K), w(K)], given K = (θ/λ)wt . Then,since w ∈ [w(K), w(K)] implies w ∈ (w∗ (K), w∗∗ (K)), we havephh (c/∆p) − c > ρC (w, K)(λ/θ)K + w
which implies that (IRD) is not binding under D–mode, given ρC (w, K) andw. Then, by using inequality (*) and the fact that w ∈ [w(K), w(K)] impliesρ̂D (w, K) ≤ ρC (w, K) and (IRD) is not binding for ρC (w, K), we obtainΠD (wt , w, ρC (w, K)) = phh π(w) − c − phh (c/∆p) + ρC (w, K)wt
= phh π(w) − c − phh (c/∆p) + ρC (w, K)(λ/θ)K≤ ρC (w, K)(λ/α(K))= ΠC (w).
Hence each old principal optimally chooses C–mode, given w and ρ =ρC (w, K). Also, by deﬁnition, all ρ = ρC (w, K), wt = (λ/θ)K and w =wC (K) satisfy the credit market equilibrium and labor market equilibrium
conditions as well. Thus these constitute an equilibrium in period t + 1.
Next we take the interest rate ρD ≡ min{ρ̂D (w, K), ρ̃D (w, K)} and themarket wage w = wD (K) such that w ∈ (0, w(K)] ∪ [w(K), ∞). Then, sincew ∈ (0, w(K)] ∪ [w(K), ∞), w(K) > w∗ (K) and w(K) < w∗∗ (K), we have29ρD ≡ min{ρ̂D (w, K), ρ̃D (w, K)} ≥ ρC (w, K) for such w and K. Then, we
have(λ/α(K))ρD = ΠD (wt , w, ρD )≥ (λ/α(K))ρC (w, K)= ΠC (w).
Hence all old principals will choose D–mode, given w and ρ = ρD ≡ min{ρ̂D (w, K), ρ̃D (w, K)}.
Finally suppose that w = w(K). Then we can take ρ = ρC (w(K), K) =Dρ̂ (w(K), K) due to the deﬁnition of w(K). Given these market prices, wehaveΠD (wt , w, ρ) = (λ/α(K))ρ̂D (w, K)= (λ/α(K))ρC (w, K)= ΠC (w),which shows that each old principal is indiﬀerent for choosing C–mdoe andD–mode given such w and ρ.
(ii) Necessity Part: Suppose that C–mode becomes an equilibrium in periodt + 1. Then w = wC (K) must hold (labor market equilibrium). Also w ∈[w(K), w(K)] must be satisﬁed. Suppose not. Then we have min{ρ̂D (w, K), ρ̃D (w, K)} >ρC (w, K). Thus ρ̂D (w, K) > ρC (w, K) or ρ̃D (w, K) > ρC (w, K). Then,ρ = ρC (w, K) must hold for C–mode to be an equilibrium in period t + 1((BEC–C): α(K)ΠC (w) = λρC (w, K)). First, when (IRD) is not bindingfor ρ = ρC (w, K), by using inequality (*), we obtainΠD (wt , w, ρC (w, K)) = phh π(w) − c − phh (c/∆p) + ρC (w, K)(λ/θ)K> (λ/α(K))ρC (w, K)= ΠC (w),which shows that each old principal should choose D–mode, given w andρ = ρC (w, K). This is a contradiction. Second, when (IRD) is binding forρ = ρC (w, K), we haveΠD (wt , w, ρC (w, K)) = phh π(w) − w − 2c
= (λ/α(K))ρ̃D (w, K)> (λ/α(K))ρC (w, K)= ΠC (w)which shows that each old principal should choose D–mode again. This is acontradiction again.
Next suppose that D–mode becomes an equilibrium in period t+1. Thenw = wD (K) must hold (labor market equilibrium). Also w ∈ (0, w(K)] ∪30

[w(K), ∞) must be satisﬁed. Suppose not. Then we have ρC (w, K) >ρ̂D (w, K) and (IRD) is not binding under D–mode (since w ∈ (w∗ (K), w∗∗ (K))).
Thus ρ = ρ̂D (w, K) must hold in such D–mode equilibrium. However, thensince ρC (w, K) > ρ̂D (w, K), we obtainΠC (w) = (λ/α(K))ρC (w, K)> (λ/α(K))ρ̂D (w, K)= phh π(w) − c − phh (c/∆p) + ρ̂D (w, K)(λ/θ)K= ΠD (wt , w, ρ̂D (w, K)),
which shows that each old principal should choose C–mode, given ρ =ρ̂D (w, K) and w. This is a contradiction.
Finally suppose that there exists a mixed mode equilibrium with theequilibrium prices w and ρ. Then ρ = ρ̂D (w, K) = ρC (w, K) must be
satisﬁed because ΠD (wt , w, ρ) = ΠC (w) = (λ/α(K))ρ must hold in themixed mode equilibrium. Solving ρC (w, K) = ρ̂D (w, K) for w gives w ∈{w(K), w(K)}. Q.E.D.

7.3

Proof of Proposition 2

For suﬃciently small λ > 0, we have θw∗ (K) > λK for all K ∈ (0, K D ) because w∗ (K) ≥ wC (K) > 0 for all K > 0. Then, since Kt+1 = (θ/λ)w∗ (Kt ) >Kt for all t ≥ 0 and for all Kt ∈ (0, K D ), when the initial condition is
given by K0 ∈ (0, K D ) Kt increases over time at least until it reaches K D .
Also, when λ is small enough, K D becomes large enough and wD (K D ) approaches to g −1 (σ/phh ), where recall that σ ≡ (2γ)/γ, because (LMC) andlimK→∞ m(K, θ) = γθ by our assumption. Thus when λ is suﬃciently small,
we have K D → ∞ and wD (K D ) → g −1 (σ/phh ).
Note that, due to Proposition 1, the region in which C–mode becomesa static equilibrium is given by QC ≡ {(w, K) ∈ ℜ2+ | w(K) ≥ w ≥ w(K)}.
Note also that w is increasing and w is decreasing. Thus, if w(K̃) = w(K̃)
exists for some ﬁnite K̃, we have (wD (K D ), K D ) ∈/ QC for small enoughλ. If limK→∞ w(K) = limK→∞ w(K) exists, we have (wD (K D ), K D ) ∈/ QCagain except for the knife–edge case that limK→∞ w(K) = limK→∞ w(K) =g −1 (σ/phh ). Hence, generically the economy eventually converges to theD–mode steady state K D in the long run when λ is suﬃciently small.
When the initial condition K0 satisﬁes K0 > K D , we have λKt >θwD (Kt ) for all Kt > K D . Since (wD (K D ), K D ) ∈/ QC generically holds
when λ is small enough, we have (w(Kt ), Kt ) ∈/ QC for all Kt > KD as
well when λ is small enough. Thus Kt eventually converges to the D–mode
steady state K D in the long run. Q.E.D.

31

7.4

Proof of Proposition 3

Note ﬁrst that λ aﬀects only for determining the steady state values K D
and K C . Then we can take the values of λ > 0 such that K ′′ < K C < K ′and wD (K C ) > w(K C ) are satisﬁed. This is done by taking the values of
λ to ensure that K C < K ′ but K C is close to K ′ (See Figure 4). Since all
the functions w, w and wi (i = D, C) are independent of λ, we can actuallyﬁnd the range of such parameter values of λ.
Then, given such λ, if the economy starts from K0 ∈ (K ′′ , K ′ ), multiple
equilibrium paths arise where some of them converge to the D–mode steadystate K D but others converge to the C–mode steady state K C (See Figure 4).
To see this, note that wC (K C ) ∈ (w(K C ), w(K C )) and wD (K C ) > w(K C ).
Also we have wC (K0 ) ∈ (w(0), w(0)) for K0 ∈ (K ′′ , K ′ ). Thus C–mode
becomes an equilibrium at initial period (t = 0). Then, since wC (K C ) ∈(w(K C ), w(K C )), the economy can converge to the C–mode steady state
K C . On the other hand, since wD (K C ) > w(K C ), there exists some period,
say T ≥ 1, such that D–mode can be an equilibrium from period T onward
and the economy converges to the D–mode steady state K D as well.
However, if the economy starts at K0 ≥ K ′ , then we have wC (Kt ) >w(Kt ) for all t. Thus there are no equilibria which converge to the C–mode
steady state K C . Q.E.D.

8

Appendix B

In the main text we have made the following conditions:• Assumption 2. π(g −1 (1/phh )) > max{(g −1 (1/phh )+2c)/phh , c/∆p+c/phh }.
• Assumption 4. (i) minw≥0 Γ(w, ∞) < 0, and (ii) g −1 (σ/phl ) >w(∞).
• It is optimal for any old principal to implement high eﬀort pair (a, em ) =(h, h) from any matched project under D–mode:Π(wt , wt+1 , ρt+1 ) ≥ Π∗ (wt+1 ) ≡ max{pll π(wt+1 )−wt+1 , phl π(wt+1 )−wt+1 −c}.
• (ICPD) is satisﬁed at the optimal contract in Problem D.
• It is optimal for any old principal to implement eﬀort pair (a, ep ) =(h, l) from any matched project under C–mode.
• (ICPC′ ) is satisﬁed at the optimal contract in Problem C.
In this Appendix we will give some parametric conditions for all theserestrictions to be satisﬁed simultaneously. To this end we will assume that32

the production function Y = F (L) takes the following form: Y = ALη whereA > 0 and η ∈ (0, 1).
Then we can show the following:π(g −1 (1/phh )) = (1 − η)Ap−η
hh ,andThus we haveg −1 (1/phh ) = ηAp1−ηhh .
π(g −1 (1/phh )) > g −1 (1/phh )/phl

as long as(1 − η)/η > phh /phl .

(A1)(i) Given (A1), Assumption 2 will be satisﬁed when c > 0 is small enough.
In what follows we will assume (A1). We also deﬁne the following function:
G(A) ≡ π(g −1 (1/phh )) − c/∆p
= (1 − η)Apηhh − c/∆p.
Next, letting Γ(w) ≡ Γ(w, ∞), we can show that Γ(w) attains its minimum at w = wmin where)1−η(1wmin ≡ ηA
phh − phl
1−βwhere β ≡ limK→∞ β(K) < γ ∈ (0, 1) by our assumption. Then we canshow that
[
)]
(
1
phh
1+η
wmin + c 1 −+1 .
min Γ(w) = H(A) ≡w≥0η
1 − β ∆p
Now, by taking small β (so small γ) and small ∆p (letting phl close to phh ),then we can verify that H ′ (A) > 0 takes a small value but G′ (A) = (1−η)pηhh
is not changed. Thus, in such case there exist some A and A such that for
all A ∈ (A, A) we have G(A) > 0 > H(A) where H(A) = G(A) = 0. Thisshows that when β and ∆p are small we obtainπ(g −1 (1/phh )) > c/∆p,andmin Γ(w) < 0.
w≥0Given such A ∈ (A, A), we can ﬁnd some small c > 0 such that Assumption
2 is satisﬁed.16 In what follows we will assume A ∈ (A, A).
16

In fact we can take small c/∆p by keeping ∆p small as well.

33

(ii) Next we will verify Assumption 4(ii). Taking A close to A deﬁned above,
limK→∞ w(K) = limK→∞ w(K) = wmin because H(A) = minw Γ(w) = 0.
Recalling that wmin = ηA((1/(1 − β))phh − phl ), when A → A we have
Assumption 4(ii):g −1 (σ/phl ) > w(∞) ≅ wmin
as long as((σ/phl )1−η

>1phh − phl
1−β

)
(A3)

which we will assume.
(iii) Next we will show that it becomes optimal for any old principal to
implement high eﬀort pair (a, em ) = (h, h) from both tasks of any matchedproject under D–mode. First note that, due to Assumption 2, we haveΠ∗ (wt+1 ) = phl π(wt+1 ) − wt+1 − c.
Thus it is suﬃcient to show thatphh π(wt+1 )−c−max{ph h(c/∆p)−ρt+1 wt , wt+1 +c} ≥ phl π(wt+1 )−wt+1 −c,which will be satisﬁed if
π(wt+1 ) > phh (c/∆p2 )because ρt+1 wt ≥ 0, wt+1 ≥ 0 and Assumption 2. However, since wt+1 ≤g −1 (1/phh ) in any equilibrium, this inequality will be satisﬁed if
π(g −1 (1/phh )) > phh (c/∆p2 )which can be written by2(1 − η)Ap−η
hh > (c/∆p ).

(A4)

Now let k ≡ c/∆p2 and keep k constant and small. Then, we can takesmall ∆p and c/∆p while keeping k constant and small. Hence (A4) will besatisﬁed for small k, given A ∈ (A, A).
(iv) Next we will check that (ICPD) in Problem D will be satisﬁed at the
optimal D–mode contract without (ICPD). To see this, ﬁrst consider the casethat (IRD) is binding at the optimal D–mode contract without (ICPD). In
D = 0 and RD = (wparticular let rt+1
t+1 + c)/phh which satisfy (IRD) and
t+1(LLD). Then (ICPD) can be written byπ(wt+1 ) ≥ c/∆p + (wt+1 + c)/phh ,34which will be satisﬁed if
π(g −1 (1/phh )) > c/∆p + (g −1 (1/phh ) + c)/phh .

(A6)

However, given (A1), when c/∆p and c are small, (A6) will hold as well.
Second, suppose that (IRD) is not binding at the optimal contract without
D
D = −ρ
(ICPD). Then we have rt+1
t+1 wt and Rt+1 = phh (c/∆p) − ρt+1 wt by
Lemma 1. Then (ICPD) can be written byπ(wt+1 ) ≥ c/∆p + phh (c/∆p)which will be satisﬁed if
π(g −1 (1/phh )) > (1 + phh )(c/∆p).

(A7)

However, when (c/∆p) is small, (A7) will hold as well.
(v) Next we will show that it is optimal for any old principal to implement
eﬀort pair (a, ep ) = (h, l) from any matched project under C–mode. As
we have seen, it is never optimal to implement high eﬀort pair (a, ep ) =(h, h) due to Assumption 1. Thus the remaining case is that the principal
implements low eﬀort pair (l, l): In that case the principal can obtain at
most pll π(wt+1 ) − wt+1 . Thus it is not optimal to implement low eﬀort pairif ∆qπ(wt+1 ) ≥ c which will be satisﬁed by (A7) due to ∆q > ∆p.
(vi) Finally (ICPC′ ) will be satisﬁed when π(g −1 (1/phh )) > g −1 (1/phh )/phl +c/∆q. This will hold when c > 0 is small, given (A1).

8.1

Appendix C

We will modify the basic model by introducing the possibility of default by
young principals in credit market and show that the parameter λ can beinterpreted as the measure of contracting costs in credit market.
Instead of assuming perfectly competitive credit market, we consider the
Bertrand competition among young principals who compete each other toattract the fund for project ﬁnancing by oﬀering interest rates to creditors(young agents). Suppose that each young principal may renege on making
repayment and be in default after she borrows from creditors. Suppose alsothat each young agent (creditor) can access to the storage technology whichconverts one unit good in the current period into ε > 0 goods in the next
period. Thus, each young agent never lends to a young principal unless
she can commit herself not to go on default (otherwise the agent will use
the storage technology instead of lending to credit market). However, eachyoung principal can develop the veriﬁcation technology by which she canperfectly reveal the hard evidence about her repayment decision and persuade creditors not to be in default. To develop such veriﬁcation technology,
each young principal needs to raise λ goods per project.
35

Then we consider the following two–stage game in credit market in eachperiod (say period t): In the ﬁrst stage young principals simultaneously oﬀer
the loan contracts which specify the interest rate to be paid to creditors inperiod t + 1 and commitment to the development of the veriﬁcation technologies. Let denote by ρit+1 the interest rate oﬀered by young principal i inperiod t (thus it will be paid in period t + 1). In the second stage, given theloan contracts oﬀered by young principals, young agents (creditors) decide
to which young principals they should lend their wage income wt . Then
only the young principals who developed the veriﬁcation technologies canattract credit from young agents (because otherwise young agents can use
the storage technology to obtain ε unit goods when they are old). Also, by
the Bertrand competition among young principals, they will bid up the interest rates to make them break even. Thus they will oﬀer the same interest
rate ρt+1 ≡ ρit+1 in equilibrium, and the equilibrium interest rate ρt+1 isdetermined by the same break even conditions (BEC–C) and (BEC–D) as
we have already derived in the main text. Finally we assume that ε is smallenough so that young agents never use the storage technology in equilibrium.
By modifying the model in the above way, we can interpret λ as the parameter value to measure how it is costly to detect default in credit market.
Put diﬀerently λ captures the contracting costs incurred by the principals(borrowers) to commit themselves not to go on default in credit market.

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36

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37

wt+1 + ρt+1 wt

ph (c/∆p) − c0wt+1(IRD) is not binding
Figure 1: Optimal D–mode Contract38

w
wD (K)D2
wD (Kt )wC (K)
M

w(Kt )w(K)C
wC (Kt )

w(K)

D1

0

Kt

Figure 2
Patterns of Equilibrium Organziations
D1, D2: D–mode. C: C–mode

39

K

w∗ (K)

wC (K)wD (K)
w(K)

w(K)

0

K

Figure 3: Market Wage Function40

λKt+1Dθw∗ (Kt )C

0

K ′′ K0KC

K′

Figure 4: Multiple Equilibrium Paths

41

KD

Kt

λKt+1θw∗ (Kt )

M

m

0

K0

KM

Km

Figure 5: Mixed Steady States

42Kt

λKt+1θw∗ (Kt )

0

K0

KM

Figure 6: Endogenous Fluctuation

43

Kt

λKt+1θw∗ (Kt )

0

Km

K0

KM

Figure 7: Multiple Mixed Equilibrium Paths

44

Kt