Discussion Papers In Economics
And Business

Macroeconomic Dynamics with Limited
Commitment in Financial and Labor Contracts
Shingo Ishiguro
Discussion Paper 16-25

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMacroeconomic Dynamics with Limited
Commitment in Financial and Labor Contracts
Shingo Ishiguro
Discussion Paper 16-25

October 2016

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANMacroeconomic Dynamics with Limited
Commitment in Financial and Labor Contracts∗
Shingo Ishiguro†
Osaka University
October 7, 2016

Abstract
This paper presents a dynamic general equilibrium model to investigate the coevolution of employment and ﬁnancial systems in the process of economic development when ﬁrms’ commitment to ﬁnancial and labor contracts is limited. We showthat equilibrium modes of ﬁnancial and labor contracts endogenously change fromthe informal contracting phase in which both of them are implicitly self-enforced to
the formal contracting phase in which they are formally enforced and become moremarket-based as economies develop well. Furthermore, the formal contracting phase
is irreversible in the sense that, once the economy enters that regime, it never returns
back to the informal contracting phase.
Keywords: Dynamic General Equilibrium, Insider Lending, Implicit and Explicit Labor Contracts, Market Lending
JEL Classiﬁcation Numbers: D86, J41, J64∗

I am grateful to Takeo Hoshi, Hideshi Itoh, Michihiro Kandori and participants in Masahiko Aoki
memorial academic conference “Comparative institutional analysis and the future of economics” at University of Tokyo for useful comments and discussions. Of course all remaining errors are my own responsibility.
I would also like to thank a ﬁnancial support from JSPS KAKENHI Grant Number 20245031.
†
Correspondence: Graduate School of Economics, Osaka University. 1-7, Machikaneyama, Toyonaka,
Osaka 560 0043, Japan. e-mail: ishiguro@econ.osaka-u.ac.jp1

1

Introduction

In this paper we present a dynamic general equilibrium framework to investigate theco-evolution of employment and ﬁnancial systems over time in the endogenous process of
economic development when ﬁrms’ commitment to ﬁnancial and labor contracts is limited.
Among others both ﬁnancing and labor are the ones which play crucial factors in diﬀerent
phases of economic development. In particular, labor and ﬁnancial arrangements are
not governed by formally written contracts but they rely on non-contractual relations inemerging and growing stages of economic development. 1
Regarding ﬁnancial arrangements in the U.S. economic history, Lamoreaux (1994) reported that in the late 19th century the ﬁrms in New England relied on insider lendingthat they borrowed from the banks who had personal connections with them. As the U.S.
economy expanded in the early 20th century, they however shifted toward borrowing fromthose whom they did not know personally, i.e., they changed toward more market-based ﬁnancing. Allen (2001) also pointed out that major German banks such as Commerzbank,Dresdner and Deutsche grew rapidly in the 19th century by developing long-term relationships with industry enterprises. 2 Also, it has been often argued that the Japaneseﬁnancial system after the Second World War was characterized by the bank-oriented system in which so called the main banks engaged in not only lending to the client ﬁrms
but also helping their managements in several ways such as rescuing them in the caseof ﬁnancial distress and being represented as directors on the boards of these client ﬁrms
(Aoki (1994), Aoki and Patrick (1994)). These lending practices were not necessarily basedon formally written contracts but rather on informal and implicit agreements between theﬁrms and main banks. Although such bank-oriented system supported the rapid economic
growth in Japan during 1960s, it has been replacing by more market-based ﬁnancing suchas issuing bonds, convertibles and warrants since the late 1970s (see Hoshi and Kashyap(2004), Rajan and Zingales (2004)).
Employment systems have also evolved in the process of economic development (seefor example Gordon (1985) and Jacoby (1997)). Moriguchi (2003, 2005) argued that
the Japanese and U.S. ﬁrms had maintained the implicit labor contracts with employedworkers during 1920s. For example, they had provided employment security, several fringe
beneﬁts (company housing, health care, etc), and so on, to employed workers but the greatrecession in 1930s more seriously hit the U.S. ﬁrms than the Japanese ﬁrms so that the former was forced to breach the implicit labor contracts and adopted more formal and explicit

1

Greif (2006) investigates how informal contracting arrangements functioned in the Western economic
history when formal institutions such as courts were absent. Johnson, McMillan and Woodruﬀ (2002) andMcMillan and Woodruﬀ (1999a, 1999b) provide the evidence about the roles of non-contractual relations
in developing and emerging economies. See also Macaulay (1963) for a classical observation about how
non-contractual relations are widely used in practice.
2
These German banks oﬀered the ﬁrms low interest rates and were represented as directors on theboards of them. Such close and lasting relationships between large banks and ﬁrms contributed to therapid expansion of the German economy between the late 19th century and the First World War. Maurer
and Haber (2007) also provide related empirical evidence about Mexican banks during similar periods.

2

labor contracts. 3 On the other hand, the Japanese ﬁrms still maintained the implicit
labor contracts even after the Second World War. 4 Although the Japanese ﬁrms relied on
the implicit labor practices such as implicit employment security and seniority-based pay
during the rapid growth periods in 1960s, they have introduced more performance-based
pay system since 1990s (see for example Abe (2007) and Abe and Hoshi (2007)). 5
As the above historical facts show, both employment and ﬁnancial systems of an economy might change from the implicit-based contracts to the explicit-and market-based contracts over time as the economy grows and matures over time. In this paper we present a
dynamic general equilibrium model to account for how employment and ﬁnancial systems
do not separately evolve but they interact with each other in the process of economic
development. The previous literature has also emphasized the complementary roles ofﬁnancial and employment systems. For example, Aoki (1994) discussed that the Japanesemain bank system was institutionally complement with its employment system characterized by the imperfect labor market and long term employment relationships. Our paper
extends this view further to understand the complementary roles of employment and ﬁnancial systems in a dynamic perspective. To our best knowledge, our paper is the ﬁrst
attempt to investigate the dynamic macroeconomic implications about the co-evolution of
employment and ﬁnancial systems.
There are two key features of ﬁnancial and labor contracts in our model: ﬁrst, ﬁrms
can make only limited commitment to both ﬁnancial and labor contracts. Firms have the
incentive to renege on not only the implicit labor contracts with employed workers but also
the implicit ﬁnancial contracts with lenders (ﬁnanciers). Then the self-enforcing condition
must be satisﬁed such that ﬁrms voluntarily honor the implicit labor and ﬁnancial promises
at the same time, which we call dynamic enforcement constraint. Second, ﬁrms can rely on
both implicit and explicit contracts for ﬁnancing and labor. Workers are motivated to work
hard by the explicit wage contingent on a veriﬁable signal, for example their objective job
performances, as well as the implicit wage contingent on an unveriﬁable signal, for example,
the acquired ﬁrm-speciﬁc skill. Here, the veriﬁable signal is assumed to be less informative
than the unveriﬁable signal regarding worker’s eﬀort choice. For example, the ﬁrm-speciﬁcskill acquired by a worker may reﬂect more directly his or her eﬀort exerted for an assigned
job rather than his or her job performance which is determined by the worker’s ﬁrm-speciﬁcskill and further stochastic noises. Thus, the implicit wage is less costly to elicit workers’
eﬀorts than the explicit one unless enforcement problem arises. Then, ﬁrms try to use
the implicit wage rather than the explicit one but the former is limited by the presenceof the self-enforcing constraint. In addition, ﬁrms ﬁnance capital investment by not only
market lending in an anonymous competitive credit market but also insider lending with3

See also Jacoby (1997) about the formation of employment relationships in the early stages of economic
development in U.S.
4
Gordon (1985) describes the history of labor relations in Japan. See also Dore (1973) for a comparison
between the labor practices in the Japanese and U.K. ﬁrms.
5
Abe and Hoshi (2007) reported that the use of seniority wages is recently declining and is being replacedby more performance-based pay in the Japanese ﬁrms. Also the Japanese ﬁrms have focused more on theimportance of personnel management because they need to evaluate workers’ performances in the moreaccurate way which are based on their wages.

3

particular ﬁnanciers who lend to them and help their managements in the informal andunveriﬁable manners. Insider lending improves the managerial productivity of the client
ﬁrm more than market lending. Thus ﬁrms try to use insider lending rather than market
lending but, since ﬁrms cannot commit to make repayment to ﬁnanciers under insider
lending, its ﬁnancing capacity is limited by the self-enforcing constraint again.
The novel feature of our model is that the implicit wage and insider lending interact
with each other through the dynamic enforcement constraint. Each period young entrepreneurs purchase the ﬁrm ownerships from old entrepreneurs at its market price andbecomes new owners of the ﬁrm. When the old entrepreneurs owning a ﬁrm breaches theimplicit contracts with existing workers and ﬁnanciers, all young workers newly employedby the ﬁrm may quit that ﬁrm in which case the ﬁrm cannot produce at all in the next
period. This in turn triggers the punishment that the ﬁrm ownership will be not sold tothe next generation at a positive price. Then, the old entrepreneurs owning a ﬁrm havestronger incentive to self-enforce the implicit wage and insider lending when they expect
to sell the ﬁrm at a larger market value by honoring these contracts relative to the gain of
deviating from them. 6 When ﬁrms raise more capital by borrowing from ﬁnanciers via
insider lending, the dynamic enforcement constraint becomes tighter so that they ﬁnd itdiﬃcult to commit to pay larger implicit wages to workers. Thus, as ﬁrms invest more incapital as the economy grows, they shift to rely on market lending for ﬁnancing capital
investment and more on the explicit wage rather than the implicit wage.
We then show that there exist multiple equilibrium paths which change over time fromthe informal contracting phase in which ﬁrms rely on implicit wage and insider lending to
the formal (market) contracting phase in which they resort to explicit wage and market
lending. As ﬁrms depend more on explicit wage and market lending in such equilibrium
paths, they invest in more capital so that the economy develops well over time. We also
show that the change toward more formal and market-based contracts in both ﬁnancing
and labor is irreversible in the sense that the economy never returns back to the informal
contracting phase once it enters the formal contracting regime together with a large capital
accumulation. Thus our theoretical results conﬁrm the aforementioned historical evidence
that both ﬁnancial and labor contracts tend be more formal and market-based from theinformal and implicit ones as economies grow and mature over time.
These results are obtained due to the sequences of self-fulﬁlled expectations about theﬁrm market values as follows. 7
In the informal contracting phase, ﬁrms pay implicit wages to employed workers andﬁnance capital investment by insider lending. However, capital raised by insider lending isconstrained by the dynamic enforcement constraint to ensure that ﬁrms honor the agreed
upon implicit ﬁnancial contracts. When a ﬁrm expects a lower ﬁrm value in the next
period t + 1, it cannot commit to larger repayments to ﬁnanciers under insider lending,resulting in smaller capital investment in the current period t. Then, a lower demand for6

Our model is related to the literature emphasizing the role of the ﬁrm reputation as an asset. See
Kreps (1990) and Tadelis (1999, 2002)) for such issues.
7
More formally, we will deﬁne the ﬁrm value as the market price of a ﬁrm ownership divided by theaggregate outputs of the economy in the previous period.

4

capital investment in the current period allows more credit supply to use for purchasing
the ﬁrm ownerships (ﬁrm stocks), leading to a larger ﬁrm value in the current period t. In
this way, the expectation about a lower future ﬁrm value is associated with a larger current
ﬁrm value. Such expectation about a low ﬁrm value in period t + 1 is then self-fulﬁlled
by the expectation about a high ﬁrm value in further two periods ahead t + 2, and so on.
These sequences of self-fulﬁlled expectations that high and low ﬁrm values alternate overtime constitutes a part of equilibrium paths in the informal contracting phase.
On the other hand, as ﬁrms increase capital investment beyond the level constrainedby insider lending, they start to ﬁnance some of capital investment by market lending in
addition to insider lending. Then they enter the formal contracting phase. In that phase,when ﬁrms expect a lower ﬁrm value in the next period t + 1, the dynamic enforcement
contract is tighten in the current period t so that they must reduce capital ﬁnanced
by insider lending but rely more on market lending. The increase in the demand formarket lending reduces the credit supply in the whole economy which becomes available
to purchase the ﬁrm ownerships. Then the ﬁrm value in the current period t must decrease.
Moreover, as ﬁrms increase capital investment further, they ﬁnd it diﬃcult to commit toboth insider lending and implicit wage at the same time. Then, they shift to rely onmarket lending and explicit wage over time. However, explicit wage is more costly than
implicit wage because the former depends on more noisy signals regarding workers’ eﬀorts
than the latter. This reduces the ﬁrm’s ﬂow proﬁt, thus resulting in a further decline in
the current ﬁrm value when a lower future ﬁrm value is expected to tighten the dynamic
enforcement constraint. In this way, the expectation about a low ﬁrm value in the future
period t + 1 is associated with a lower ﬁrm value in the current period t in the formal
contracting phase. The expectation about a low ﬁrm value in period t + 1 is self-fulﬁlled
by the expectation about a low ﬁrm value in t + 2, and so on. Again, these sequences ofself-fulﬁlled expectations constitutes a part of equilibrium paths in the formal contracting
phase.
The above sequences of self-fulﬁlled expectations about the ﬁrm values cause multiple
equilibrium paths: some of them start with higher ﬁrm values in the informal contracting
phase and enter the formal contracting phase along with declining ﬁrm values over time.
Then the credit demand needed to purchase the ﬁrm ownerships falls over time as theﬁrm values decline. This implies that more credit supply becomes available to ﬁnancenew capital investment in the whole economy, leading to capital accumulation and economic development. Thus there exist equilibrium paths which exhibit the feature thatorganizational modes of labor and ﬁnance change from the informal contracting phase to
the formal contracting phase over time as capital proceeds to increase over time. This
theoretical insight implies that, as economies grow and mature over time, employment
and ﬁnancial systems co-evolve toward more explicit-and market-based ones, conﬁrming
the aforementioned historical facts.
Related Literature. The literature of relational contracts has attracted diﬀerent applications of economic models in which contracts are not formally enforced but self-enforced
via repeated agreements (Levin (2003)). Some of these papers have considered the impli5

cations about labor contracts (McLeod and Malcomson (1989, 1998), Malcomson (2013)).
Others have investigated the features of dynamic lending contracts (for example Albuquerque and Hopenhayn (2004), Clementi and Hopenhayn (2006)). Also there are somestudies which deal with the interactions between explicit and relational (implicit) contracts(Baker, Gibbons and Murphy (1994), Pearce and Stacchetti (1998), Schmidt and Schnitzer(1995)). However, most of these applications have focused on relational labor contracts
and relational ﬁnancial contracts separately, which is in contrast to our paper addressing
both of these contracts in an integrated framework. The recent paper by Barron and Li(2015) considers both relational labor and ﬁnancial contracts but they do not take into
account the interactions between relational and explicit contracts. Also our paper diﬀers
from them in that our main purpose is to investigate macroeconomic implications about
the interactions between ﬁnancial and labor contracts with limited commitment. In addition, our model is related to the papers which address the ﬁrm reputation as an asset(Kreps (1990), Tadelis (1999, 2002)). Our model builds the argument that the ﬁrm maintaining a good reputation can be sold at higher market price into the dynamic generalframework and consider its macroeconomic implications.
Our paper is also related to the studies which deal with macroeconomic implications
about self-enforcing debt (Hellwing and Lorenzoni (2009), Jeske (2006)) and long termlabor contracts (Francois and Roberts (2003)). Again, these existing papers have focusedon either self-enforcing ﬁnancial contract or self-enforcing labor contract, in contrast to
our paper
Furthermore, as we have already mentioned, our paper is inspired by the studies on
institutional complementarity (for example Aoki (1994)). These studies have identiﬁed an
economic institution as a set of complementary sub-systems. In particular Aoki (1994)
discussed that the Japanese employment system characterized by the imperfect labormarket and long term employment is institutionally complement with the bank-oriented
ﬁnancial system after the Second World War. Our paper extends this view to the dynamic
framework and consider the joint evolution process of employment and ﬁnancial systems
over time.
The rest of the paper is organized as follows: in section 2 we set up the basic model.
In section 3 we characterize the optimal ﬁnancial and labor contracts given the marketdetermined variables. Then in section 4 we embed the optimal contracts into the dynamic
general framework and in section 5 we derive full equilibrium dynamics. Section 6 concludes the paper. All formal proofs and some extensions are relegated to the appendix.

2

Model

2.1

Economic Environment

We consider an overlapping generations economy with a single good which is used bothfor consumption and investment. Time is discrete and extends over inﬁnity, denoted by
t = 0, 1, 2, ... Each period young capitalists and young workers with one unit mass each6are newly born. They live for two periods and are concerned with consumption only whenold. Among one unit mass of young capitalists, 1 − Nf are entrepreneurs who run theﬁrms and the remaining Nf are ﬁnanciers who help ﬁnancing the ﬁrms (we will explain
this in more details below.) We assume that there are N ﬁrms in the economy, which are
identiﬁed as N distinct projects or N physical assets, and that N is ﬁxed over time.
Each ﬁrm has one job to hire at most one old worker for completing its production.
The old worker employed by a ﬁrm exerts an unveriﬁable binary eﬀort at ∈ {0, 1}. Thena ﬁrm produces the output according to the following production function:yt = h(at )ktγ

(1)where yt ≥ 0 denotes the output, h(at ) the productivity of an employed old worker andkt ≥ 0 capital invested respectively. Here we assume that γ ∈ (0, 1) and h(1) ≡ 1 >s ≡ h(0). Also we assume that capital must be invested in one period advance before
production and it fully depreciates within one period.
In the economy there are three markets as follows:
Labor Market: We assume that hiring workers must be made in advance before they
become old, i.e., the ﬁrms need to hire young workers although they work when old. For
example, workers need to be trained when young before they become old and actuallywork at assigned jobs. Then we model the hiring process in the labor market in the simple
way as follows: a certain number of young workers arrives at each ﬁrm in every period.
Then each ﬁrm hires only one young worker among those who visited it. Since there are
N ﬁrms each of which employs one worker, N young workers are employed while 1 − Nyoung workers are unemployed where we assume N ≤ 1.
Firm Ownership (Stock) Market: There is a competitive market for ownerships of theﬁrms to be traded. Such market might be interpreted as a stock market where individuals
buy and sell the stocks of a ﬁrm at a given market price. Without loss of generality, we
normalize the number of ownership rights of each ﬁrm to unity. We denote by Vt themarket price of a ﬁrm at which each old individual having an ownership of the ﬁrm sells
it to young individuals in the end of period t − 1. 8
In what follows we will also conﬁne our attention to the case that only entrepreneurs
trade the ﬁrm ownerships but ﬁnanciers and workers do not. This assumption does not
lose any generality because, by the no-arbitrage condition (2) deﬁned below, every individual becomes indiﬀerent for purchasing the ownership of a ﬁrm and not purchasing it inequilibrium (see more details below).
Credit Market: There is an anonymous competitive credit market where everyone can8

As an alternative scenario, we can consider the bilateral bargaining in which old individuals are randomly matched with young individuals and make the take-it-or-leave-it oﬀer of the price at which thelatter buys the ﬁrm ownerships from the former. This situation can be interpreted as so called negotiatedblock trade (Au, Fong and Li (2015)). This alternative setting does not substantially change the following
results.

7

freely borrow and lend at a given market (gross) interest rate rt : one unit of good istraded today in exchange for rt units of good tomorrow. When the ﬁrms ﬁnance capital
investment by borrowing from the competitive credit market, we call this market lending. Market lending is based on formally written ﬁnancial contract and hence is legally
enforceable. The ﬁrms can commit to make the repayment of market interest rate rt .
There are two means for the ﬁrms to ﬁnance capital investments. One is market lending in the anonymous competitive credit market we have explained above and the otheris what we call insider lending as follows. 9
Insider Lending: The ﬁrms can ﬁnance capital investments from ﬁnanciers who have speciﬁc knowledge to help the managements of client ﬁrms, which cannot be provided bymarket lending. For example, ﬁnanciers not only lend to the client ﬁrms but also they
provide managerial advices to the client ﬁrms and are represented as directors on theboards of them, which help improving the management eﬃciency of the client ﬁrms. We
call these inputs provided by ﬁnanciers to the client ﬁrms relation-speciﬁc capital. However, such relation-speciﬁc capital is hard to be veriﬁed outside particular ﬁrm-ﬁnancier
relationships. Thus the ﬁrms cannot commit to formal ﬁnancial contracts contingent on
relation-speciﬁc capital provided by ﬁnanciers. In order for the ﬁrms to ﬁnance fromﬁnanciers, the ﬁrms simultaneously approach ﬁnanciers and oﬀer ﬁnancial contracts tothem where such contracts are enforced in the self-interest way of the ﬁrms (as we will see
below.)As one possible interpretation of insider lending, one might think the bank-ﬁrm relationships typically observed in Germany and Japan: The banks, called the main banks,
send directors to the client ﬁrms, own the shares of them, rescue them when they are in
ﬁnancially distress, and so on (see Aoki and Patrick (1994), Hoshi and Kashyap (2004)
for the case in Japan). These activities performed by the main banks are not formally
written agreements with the ﬁrms but implicitly enforced between them. The ﬁrms andmain banks usually engage in lasting relationships which allow them to enforce the implicit
ﬁnancial contracts in the self-interest manner. 10
The advantage of insider lending relative to market lending is that the former canlend to the ﬁrms in more productive manner than the latter by providing relation-speciﬁc
capital as we have mentioned above. We capture such gain of insider lending by assuming
that, in order to provide one unit of relation-speciﬁc capital, ﬁnanciers need only λ ∈ (0, 1)
units of good. 11 Under market lending one unit of good is used as one unit of capital
9

More precisely, the ﬁrms can ﬁnance capital investments by issuing equities and selling them at the ﬁrm
ownership (stock) market. However, such equity ﬁnancing is equivalent to borrowing from the competitive
credit market as shown in Appendix B. This is because the no-arbitrage condition (2), which we introduce
below, holds so that the market interest rate equals to the rate of stock return, implying that the ﬁrms are
indiﬀerent for borrowing in the credit market and issuing new equities. Thus we will not consider equity
ﬁnancing in what follows.
10
In that sense the ﬁrm-bank relationships are often called relational ﬁnancing (Aoki and Patrick (1994)).
11
This assumption is justiﬁed as follows: suppose that a ﬁnancier spends α > 1 units of good for creating
one unit of relation-speciﬁc capital and that one unit of relation-speciﬁc capital has its quality-adjusted8whereas only λ ∈ (0, 1) units of good is needed for one unit capital under insider lending.
On the other hand, the disadvantage of insider lending is that, since relation-speciﬁc capital
is not veriﬁable, the ﬁrms cannot formally commit to make repayments contingent on it.
We denote by xt relation-speciﬁc capital and by Rt repayment which the ﬁrm promise
to pay to a ﬁnancier under insider lending. We assume that, if the ﬁrm raises kt′ capitalfrom market lending and xt relation-speciﬁc capital from insider lending, its total capitalkt ≡ kt′ + xt matters for the production, i.e., yt = h(at )ktγ for kt = kt′ + xt . In other words
relation-speciﬁc capital raised by insider lending is perfect substitute to capital raised bymarket lending.
Note also that ﬁnanciers have no capacity limits for lending relation-speciﬁc capitalto the ﬁrms because they can access to the competitive credit market and borrow thenecessary fund at a given market interest rate rt in order to lend relation-speciﬁc capitalto the ﬁrms. Thus, even when multiple ﬁrms oﬀer insider lending contracts to a single
ﬁnancier, the latter can make ﬁnancial contracts and engage in insider lending with theseﬁrms at the same time.
We make the following assumptions on veriﬁability of capital and output: ﬁrst, as wehave mentioned, relation-speciﬁc capital xt provided by ﬁnanciers under insider lending
is not veriﬁable as well as it is not veriﬁed that ﬁnancier spent λxt for creating relationspeciﬁc capital xt . On the other hand, it is veriﬁable how much the ﬁrms borrowed
from the credit market. Second, the total capital kt and output yt are assumed to be
not veriﬁable as well. These restrictions on veriﬁability of capital and output imply thatthe ﬁrm cannot make repayments to ﬁnanciers under insider lending contingent on therelation-speciﬁc capital xt provided by the ﬁnanciers and the output yt produced by theﬁrm. 12

2.2

Endowment

To introduce the savings and close the model, we assume that young capitalists bornin each period can access to the private technology for producing wt units of good byvalue ρ > 1 in the production per job. For example, if z relation-speciﬁc capital is used per job, the ﬁrm
can produce the output y = (ρz)γ per. Thus, by spending αz units of good, the ﬁnancier can provide therelation-speciﬁc capital of z units which contributes to the production by its quality-adjusted value ρz. Letx ≡ ρz and assume that λ ≡ α/ρ < 1. Then, λx units of good are needed to create x units of the qualityadjusted relation-speciﬁc capital. To avoid complication, we will not use the term “quality-adjusted” but
we simply call xt ≡ ρzt the relation-speciﬁc capital in what follows.
12
In addition, it does not help by making a repayment Rt to the ﬁnancier contingent on his or her savings
denoted by St where wt−1 ≥ St . Let denote by R(St ) such repayment. Although the repayment can varywith the ﬁnancier’s savings, this does not ensure that the ﬁnancier provides a relation-speciﬁc capital xt :suppose that the ﬁrm wants to raise xt relation-speciﬁc capital from a ﬁnancier and induces him or her tosave Ŝt . Thus wt−1 − Ŝt ≥ λxt . If the ﬁnancier follows this, he or she earns rt Ŝt + R(Ŝt ). However, theﬁnancier can obtain at least this payoﬀ even if he or she does not invest λxt in relation-speciﬁc capital.
Furthermore, the ﬁnancier has the incentive to collude with the other party as follows: the ﬁnancier passes
amount of λxt to the third party who then lends it to the credit market in the behalf of the ﬁnancier. Suchsavings are not recorded on the account of the ﬁnancier but on the account of the colluding third party sothat the repayment to the ﬁnancier R(Ŝt ) is not aﬀected. Then the ﬁnancier and the third party can sharethe earned interest income rt λxt between them, which yields more than rt Ŝt + R(Ŝt ) to the ﬁnancier.

9themselves. Here wt depends on the aggregate productivity of the economy denoted by
At and we assume that wt = LAt where L > 0 represents the degree about how thecapitalist’s endowment reﬂects the aggregate productivity At . For simplicity, we assume
that the aggregate productivity depends on the social knowledge
embodied in the aggregate∫N
outputs produced in the current period, i,.e., At ≡ 0 yt (i)di where yt (i) denotes theoutput of ﬁrm i at period t, through the learning-by-doing eﬀect (Arrow (1962)). Then
young capitalists are endowed with wt units of good each in the beginning of period t.
This simple setting is made only for avoiding unnecessary complication of the model. We
can provide an alternative (and more reasonable) scenario which endogenously determines
the endowment of young capitalists (see Appendix B).

2.3

Production and Moral Hazard

Each old worker is endowed with one unit of labor skill and can access to the private
technology which produces ψAt units of good by using one unit of endowed skill whereψ > 0. Here, as in the case of capitalists, the worker’s private technology
also reﬂects∫N
the aggregate productivity embodied in the aggregate outputs At = 0 yt (i)di. After old
workers were employed by the ﬁrms, they decide whether to exert high or low eﬀort forthe productions, denoted by at ∈ {0, 1}, where at = 1 (at = 0) stands for high (low)eﬀort. When an old worker exerts high eﬀort at = 1, he must spend fully one unit oflabor skill endowed with him. When exerting low eﬀort at = 0, he does not needs tospend his endowed skill for the production but he can secretly spend it for his own private
technology to earn ψAt units of good.
The eﬀort choice by old worker is non-veriﬁable but the ﬁrm can observe two typesof the signal regarding it. One is the perfect but unveriﬁable signal st = at ∈ {0, 1}
about the worker’s eﬀort at . The ﬁrm can thus perfectly observe the worker’s eﬀort
choice at but it is not veriﬁed outside the employment relationship. The other is theimperfect but veriﬁable signal denoted by σt ∈ {σh , σl }. Here high signal σh is realizedwith probability qa ≡ Prob(σh |a) ∈ (0, 1) conditional on an eﬀort choice a ∈ {0, 1}. We
assume that 1 > q1 > q0 > 0, meaning that the veriﬁable signal σ is less informative
than the unveriﬁable one st regarding the worker’s eﬀort at ∈ {0, 1}. We use the notation∆q ≡ q1 − q0 > 0.
We denote by {bt (sh ), bt (sl )} an implicit labor contract which speciﬁes a wage bt (s)contingent on the worker’s unveriﬁable signal st . Here bt (s) ≥ 0 must hold for each s by
the worker’s limited liability. Since the signal (eﬀort) s = a is not veriﬁed, such contract
must be self-enforcing. We also denote by {vt (σl ), vt (σh )} an explicit labor contract which
speciﬁes a wage vt contingent on the veriﬁable signal σt . Here vt (σ) ≥ 0 for each σ. The
explicit wage vt (σ) is legally enforced.
More generally, we can think of worker’s wage as Wt (s, σ) which is contingent on bothof these signals s and σ in the interdependent way. However, we can show that generality is
not lost by restricting our attention to the additive separable form Wt (s, σ) = bt (s) + vt (σ)(see Appendix B in more details). We can also show that bt (0) = vt (σl ) = 0 and hence inwhat follows we will write bt ≡ bt (1) ≥ 0 and vt ≡ v(σh ) ≥ 0 and call them the implicit
10

bonus and the explicit bonus respectively.

2.4

Observability and Quitting Option

All the wages paid to old workers and repayments paid to ﬁnanciers are publicly observable.
In addition, we assume that a young worker employed by a ﬁrm in period t can observe
the unveriﬁable signal st = at ∈ {0, 1} of the old worker working at the same ﬁrm and therelation-speciﬁc capital xt provided by the ﬁnancier who engages in insider lending withthe ﬁrm.
Furthermore, we assume that it is publicly observable whether or not the young worker
employed by a ﬁrm in the previous period has been retained as the old worker by the sameﬁrm in the current period. Put diﬀerently, all other parties outside the ﬁrm can observe
whether or not the young worker employed by the ﬁrm in the previous period has left theﬁrm and has been hence not retained as old worker in the current period.
Finally, we assume that in the end of any period the ﬁrm and employed young worker
simultaneously decide whether or not to exercise the quitting option: by the quitting
option, it is meant that the young worker leaves the ﬁrm while the ﬁrm liquidates theﬁrm’s asset (project) in which case the ﬁrm cannot produce the outputs forever.

2.5

Market Price of Firm Ownership

The ownership rights of the ﬁrms are traded in the competitive stock market. Let Vt
denote the market price of one unit ownership of a ﬁrm determined in the end of periodt − 1. Then, if a young individual purchases one unit ownership of a ﬁrm at its market
price Vt in period t − 1, he or she expects that he or she will obtain a ﬂow proﬁt (dividend)
πt from the production of the ﬁrm and can sell the ownership to the young generation
at a market price Vt+1 in the next period t when old. On the other hand, if he or shelends one unit of good to the anonymous credit market, then he or she earns the market
interest rate rt . Thus, the following no-arbitrage condition must be satisﬁed for the ﬁrm
ownership (stock) market to clear: 13
rt =πt + Vt+1
Vt(2)

which means that the rate of gross return by holding one unit of the ﬁrm ownership must
be equal to the gross interest rate rt in the credit market. This can be re-written byVt =1{πt + Vt+1 }.
rt

13

(3)

If the left hand side of (2) is greater than its right hand side, then nobody buys the ﬁrm ownership,
which implies that its market price Vt goes down to zero. But then the right hand side increases beyond
rt > 0. Also, if the reverse inequality holds, then young individuals demand to buy the ﬁrm ownerships
as much as possible, resulting in an inﬁnite demand. This however implies that its demand exceeds the
supply and hence Vt must go up until the equality is preserved.

11

2.6

Timing

The events within each period t proceeds as follows:
1. The ﬁrm oﬀers a labor contract {b(st ), v(σt )} to an employed old worker. Then theold worker chooses an eﬀort at ∈ {0, 1}.
2. The ﬁrms hire young workers newly born in this period.
3. The signals of each employed old worker st = at ∈ {0, 1} and σt ∈ {σh , σl } arerealized and observed to the ﬁrm, the old worker himself and the young worker
newly employed by the ﬁrm. Then, the ﬁrm owned by old entrepreneurs produce
the output yt = h(at )ktγ by using old worker’s eﬀort at and capital kt invested in
the previous period t − 1. Then the ﬁrm decides whether or not to pay the implicit
wage b(st ) to the old worker and repayment Rt to the ﬁnancier who lent to the ﬁrm
via insider lending.
4. After the payment decision by the ﬁrm, the newly employed young worker and theﬁrm simultaneously decide whether or not to exercise the quitting option. If the ﬁrm
exercises the quitting option, it liquidates the ﬁrm’s asset and leaves the economy.
If the young worker quits, he is unemployed when old.
5. It is publicly observed whether or not the young worker employed by the ﬁrm haveleft the ﬁrm. The old entrepreneurs owning the ﬁrm earn a ﬂow proﬁt πt and sell
the ﬁrm ownership to the young entrepreneurs at a market price Vt+1 . Then theowners of the ﬁrms are changed from the old entrepreneurs to the young ones.
6. The ﬁrms owned by the young entrepreneurs simultaneously approach the young
ﬁnanciers and oﬀer insider lending contracts {xt+1 , Rt+1 } to each of them, which
speciﬁes a relation-speciﬁc capital xt+1 to be provided by a ﬁnancier and the cor′
responding repayment Rt+1 to her. At the same time, the ﬁrm raises capital kt+1
from market lending if necessary.
′
Then the economy moves to the next period t + 1: capital kt+1
and xt+1 invested in theprevious period t will be used for the production in period t + 1.

Remark. We are here assuming that the repayment Rt to ﬁnancier is not contingent on
the performance signals st and σt of employed old workers. This is a reasonable restriction
because it is the realistic case that the repayment to lenders do not vary with employedworkers’ task performances. However, we can extend the model to allow such case andshow that our main results still remain valid (see Appendix B).
Remark. Since ﬁnanciers are identical, we can suppose without loss of generality thateach ﬁrm approaches only one ﬁnancier and oﬀers an insider lending contract {xt , Rt } toher. Note that, since ﬁnanciers have no capacity limit to lend to the ﬁrms, it does notmatter how many ﬁrms they lend to via insider lending. In other words, each ﬁnancier
12can lend to multiple ﬁrms at the same time. Financiers are willing to lend to a ﬁrm aslong as they can earn at least what they obtain by lending to the credit market at theinterest rate rt .
In the following sections we will ﬁrst derive the optimal ﬁnancial and labor contracts
chosen by the ﬁrm in a given period t, by taking the market interest rate rt and the future
market price of the ﬁrm Vt+1 as exogenously given. Then we will embed the optimal
contract into the dynamic general equilibrium model, which endogenously determines theequilibrium paths of the interest rates and market prices of the ﬁrms.

3

Optimal Financial and Labor Contracts

3.1

Constraints

Consider a ﬁrm in period t which is hiring an old worker retained from the previous periodt − 1. Suppose that the old worker exerted high eﬀort at = 1 for the production. Thus theunveriﬁable signal (eﬀort) st = at is observed to the ﬁrm, in which case the ﬁrm must pay
the implicit bonus bt ≡ b(1) ≥ 0 to the old worker according to the implicit wage contract.
Also the ﬁrm must make the repayment Rt to the ﬁnancier from whom it borrowed byinsider lending.
When the ﬁrm reneges on the implicit bonus bt to the old worker and repayment Rt
to the ﬁnancier following the liquidation, it can totally save these payments bt + Rt . We
denote by Vt+1 the market price of the ﬁrm ownership (stock) when making the agreed
upon payments bt and Rt to the old worker and the ﬁnancier. Then, for the ﬁrm to honor
these payments, the following dynamic enforcement (DE) constraint must be satisﬁed
Vt+1 ≥ bt + Rt .

(DE)

Otherwise, the ﬁrm does not pay bt and Rt and then liquidates the ﬁrm’s asset, yielding
at least the gain of bt + Rt . 14
On the other hand, if DE is satisﬁed, then there exists an equilibrium in which theﬁrm honors making the agreed upon payments bt and Rt . To see this, suppose that theabove DE holds and consider the following strategies of the ﬁrm and the young worker
employed by the ﬁrm: The young worker stays in the ﬁrm in period t if the ﬁrm has paid
the implicit bonus bt to the old worker who worked at the same ﬁrm and chose high eﬀort
at = 1 as well as the ﬁrm has made the repayment Rt to the ﬁnancier. However, otherwise
the young worker quits the ﬁrm. Also, all the young workers who will be matched withthe ﬁrm in any future period will quit the ﬁrm if they observe that some young workers
14
One might consider an alternative timing of repayment to the ﬁnancier as follows: the ﬁrm pays Rt ﬁrst
before the ﬁnancier invests in relation-speciﬁc capital xt . In this scenario the ﬁrm has no rooms to breachthe ﬁnancial contract. However, the ﬁnancier will then breach the ﬁnancial contract by not delivering xt
after he or she has received the repayment Rt . Such breach by the ﬁnancier cannot be detected because
there are no ways to punish the deviating ﬁnancier: if the ﬁnancier collects Rt and does not provide xt
when young, he or she will consume all the earned income without any punishment when old.

13

who were employed by the ﬁrm in the past periods have quit and have been hence notretained as old workers in the ﬁrm. Otherwise, they stay in the ﬁrm. The ﬁrm continues
its production operation if it has paid the implicit bonus bt to the employed old worker
when the latter chose high eﬀort at = 1, it has paid Rt to the ﬁnancier in period t andit has retained the young worker employed in the previous period. Otherwise the ﬁrm
liquidates its asset. Then, we can show that the strategies speciﬁed above constitute anequilibrium supported by the continuation equilibrium in which the ﬁrm and the young
worker exercise the quitting option simultaneously after the ﬁrm reneged on bt or Rt in
period t. This triggers that all the young workers employed by the same ﬁrm in the future
will exercise the quitting option in any future period, making the market prices of theﬁrm down to zero in any future period (see Appendix A in more details). Thus, if DE issatisﬁed, it is not proﬁtable for the ﬁrm to renege on bt or Rt .
Next we turn to the incentive compatibility constraint for employed old workers. In
what follows we will pay our attention to the case that the ﬁrm wants to implement higheﬀorts from an employed old worker. 15 Note that bt (1) ≡ bt ≥ 0 and vt (σh ) ≡ vt ≥ 0
but bt (0) = 0 and vt (σl ) = 0. Then, given the ﬁrm honoring the implicit bonus bt , theemployed old worker chooses high eﬀort at = 1 if
bt + q1 vt ≥ q0 vt + ψAt
where the left hand side denotes the expected payoﬀ of the old worker when exerting higheﬀort at = 1 while the right hand side denotes the expected payoﬀ when exerting low
eﬀort at = 0 and spending his endowed labor skill secretly to the private technology whichyields ψAt outputs. When the old worker chooses high eﬀort, he earns the implicit bonus
bt plus the expected explicit bonus q1 vt while, when he exerts low eﬀort, he obtains noimplicit bonus but the expected explicit bonus q0 vt .
Thus the following old worker’s incentive compatibility (WIC) must be satisﬁed forhigh eﬀort a1 = 1 to be implemented:
bt + ∆qvt ≥ ψt ≡ ψAt(WIC)

where ψt denotes the opportunity cost of exerting high eﬀort. Also, since old workers have
no wealth, the following limited liability (LL) constraint must be satisﬁed: 16
bt ≥ 0,vt ≥ 0(LL)

In addition, each young ﬁnancier born in period t − 1 can always obtain at least what
he or she earns by saving all his or her endowment wt−1 to the credit market, i.e., rt wt−1 .
15Intuitively, when the old worker’s productivity s = h(0) when he exerts low eﬀort at = 0 is low enough,the ﬁrm has no incentive to implement low eﬀort. See Appendix B for more details.
16
In addition, old worker accepts the oﬀered contract {bt , vt } when the individual rationality constraint
bt +qh vt ≥ ψAt is satisﬁed where the old worker can obtain ψAt by rejecting the contract and spending theendowed labor skill to his own private technology. However, by WIC and LL, this individual rationality
constraint is satisﬁed so that we can ignore it.

14

Thus the following individual rationality (IR) constraint must be satisﬁed for each young
ﬁnancier to accept the insider lending contract {Rt , xt }:Rt + rt (wt−1 − λxt ) ≥ rt wt−1

(IR)

Here we are assuming that young ﬁnanciers do not purchase the ﬁrm ownerships. However,this is made without loss of generality because the no-arbitrage condition (2) means thatit becomes indiﬀerent for saving one unit of good at the interest rate rt and buying one
unit of the ﬁrm ownership at a market price Vt .17 IR states that the young ﬁnancier bornin period t − 1 spends λxt units of good for providing xt relation-speciﬁc capital underinsider lending and then receives the repayment Rt . In addition, the ﬁnancier saves (orborrows) the remaining amount wt−1 − λxt to (from) the credit market which yields theinterest earning rt (wt−1 − λxt ).
The ﬂow proﬁt of a ﬁrm πt is then given byπt ≡ ktγ − (bt + q1 vt ) − Rt − rt (kt − xt ).

(4)

Here ktγ is the output produced given high eﬀort at = 1 exerted by the old worker. The
second term is the expected total wage paid to an employed old worker who chooses at = 1.
Also Rt is the repayment to a ﬁnancier under insider lending, and rt (kt − xt ) is the costof market lending (kt ≥ xt ) respectively.
Finally, capital raised by market lending kt − xt must be non-negative:
kt ≥ xt ≥ 0.

(NE)

Before proceeding further, we explain the key feature of DE constraint which plays theimportant role in what follows. By combining DE with WIC and IR, we obtain
Vt+1 ≥ Rt + bt
≥ λrt xt + ψt − ∆qvt .

(5)

When the ﬁrm increases capital investment xt ﬁnanced by insider lending, the explicit
bonus vt must be increased as long as the above inequality (5) is binding. However, since
the explicit bonus depends on more noisy signal than the implicit one, the increase in
the explicit bonus vt costs the ﬁrm more when it relies more on insider lending to ﬁnancecapital. Then, as the ﬁrm invests more in capital, it shifts from insider lending to market
lending. In this way, the explicit bonus vt and relation-speciﬁc capital xt are related witheach other through DE, which causes the change from the regime of the implicit wage andinsider lending to the regime of the explicit wage and market lending as we will see later.

17
If a young ﬁnancier purchases ltf ﬁrm ownerships, then he or she obtains Rt + rt (wt−1 − λxt − ltf Vt ) +flt (πt + Vt+1 ) when he or she accepts the insider lending contract. When he or she rejects it, he or she
obtains rt (wt−1 − ˜
ltf Vt ) + ˜
ltf (πt + Vt+1 ) where ˜
ltf ̸= ltf may be the case. However, since rt Vt = πt + Vt+1 by

the no-arbitrage condition (2), we have the same condition as IR above.

15

3.2

Optimal Contracts

We next move to derive the contracts the ﬁrms oﬀer to old workers and ﬁnanciers. The
ﬁrm oﬀers a labor contract {bt , vt } to an employed old worker as well as it approaches
a ﬁnancier and oﬀers an insider lending contract {Rt , xt } in period t. We focus on the
optimal contract that maximizes the current ﬁrm market price Vt or equivalently the ﬂow
proﬁt πt subject to WIC, DE, IR, LL and NE, given the market interest rate rt and the
ﬁrm’s belief about its future market price Vt+1 .
The optimal contracts depend on what beliefs the ﬁrm forms about the future market
price Vt+1 . To be consistent with the continuation equilibrium strategy speciﬁed in theprevious subsection, we suppose that, when considering a contract oﬀer {bt , vt , xt , Rt } forthe production in period t, the ﬁrm has the belief about the market price Vt+1 in the next
period t + 1 as follows: (i) Vt+1 does not vary with what contracts the ﬁrm oﬀers to oldworker and ﬁnancier in the current period t as long as the ﬁrm honors paying according
to the oﬀered contract, 18 and (ii) Vt+1 will be zero when the ﬁrm reneges on the implicit
bonus bt or the repayment Rt because then the young worker employed in period t willquit the ﬁrm. These beliefs are consistent with the strategy of young workers speciﬁedin the previous subsection: the newly hired young worker will quit the ﬁrm if the latterreneged on the payments to the old worker and ﬁnancier under the oﬀered contract or
the young worker employed in the previous period was not retained in the current period.
When the participants in the stock market observe that the young worker hired by a ﬁrm
has left the ﬁrm, they believe that such ﬁrm has lost the trust from employed workers,
which in turn triggers the separation of all young workers in the future periods. Then,
nobody purchases the ownership (stock) of such trust-losing ﬁrm in any future, resulting
in zero market prices of the ﬁrm in the future.
There are two remarks on the above restrictions on the beliefs about the ﬁrm market
prices as follows. First, young workers trust the ﬁrm as long as the latter does not breachthe contracts oﬀered to the employed old workers. Thus, even when the ﬁrm oﬀers adiﬀerent contract from the equilibrium one, it is not the object to be punished as long as
the ﬁrm has honored the agreed upon implicit contracts. Second, the ﬁrm believes thatits market prices will go down to zero in any future period once it reneged on the implicit
bonus or repayment to ﬁnanciers. According to the workers’ strategy, all young workers
will quit the trust-losing ﬁrm in any future period, resulting in zero production in the
future. When the outputs of a ﬁrm will be zero forever, it is reasonable to suppose thatits market price will be zero as well. In fact, a sequence {Vs }∞s=t such that Vs = 0 for alls ≥ t satisﬁes the no-arbitrage condition Vt = (1/rt )Vt+1 when πs = 0 for all s ≥ t.
Given the above belief about the future market price of the ﬁrm Vt+1 , the optimal
contract solves the following problem for t ≥ 1: 19
18

In fact, the contract used in period t does not aﬀect the production in the next period t + 1 so that itis reasonable to suppose that the market price Vt+1 of the ﬁrm in the next period t + 1 does not changewith the contract in the current period t.
19
In the initial period t = 0 old entrepreneurs are endowed an initial capital stock k0 each which is used

16

Problem (P):max Vt =1{πt + Vt+1 }rtsubject to DE, WIC, IR, LL and NE, given rt and Vt+1 .
The immediate features of the optimal solution to Problem (P) are as follows: First,
since a decrease in bt can raise the ﬁrm’s ﬂow proﬁt and weaken DE as long as WIC is
satisﬁed, WIC must be binding, i.e., bt = ψt −∆qvt . By substituting this into the expected
wage bt + q1 vt , we obtainbt + q1 vt = ψt + q0 vt .
Second, since the reduction of Rt can increase the ﬁrm’s proﬁt and weaken DE as long as
IR is satisﬁed, IR must be binding as well, i.e., Rt = λrt xt . Then DE can be written by
Vt+1 ≥ Rt + bt = λrt xt + ψt − ∆qvt ,(6)

from which we havevt ≥ (1/∆q)(λrt xt + ψt − Vt+1 ).
Then, the explicit bonus vt should be set as}{
1(λrt xt + ψt − Vt+1 ), 0 .
vt = max∆q(7)

When the future market price of the ﬁrm Vt+1 is larger, DE becomes weaker so that the
optimal wage involves only the implicit bonus, i.e., vt = 0 and bt = ψt .
We substitute the above expression of vt into the ﬁrm’s ﬂow proﬁt (4) in order to
re-write Vt as
{
{
}
}
1
q0
Vt =
ktγ − λrt xt − (kt − xt )rt − ψt − max(λrt xt + ψt − Vt+1 ), 0 + Vt+1 . (8)rt∆qThe optimal contract is characterized depending on whether or not DE becomes binding and whether or not the explicit bonus vt is used. If DE can be ignored, the ﬁrm uses
only insider lending to ﬁnance capital investment because it is more eﬃcient than market
lending.
For a further analysis, we deﬁne three key capital levels: ﬁrst, we deﬁne the capital
investment, denoted by kt∗ , which maximizes the ﬁrm’s proﬁt without DE (kt = xt ),yielding the ﬁrst order condition:γ(kt∗ )γ−1 = λrt .

(9)

to produce the outputs y0 = h(a0 )k0γ . When implementing high eﬀort a0 = 1, they oﬀer the contract{b0 , v0 } to the initial old worker to maximize the proﬁt π0 = k0γ − (b0 + q1 v0 ) subject to WIC, LL and DE:
V1 ≥ b0 . When implementing low eﬀort a0 = 0, the proﬁt becomes sk0γ . The diﬀerence from subsequent
periods is that DE in the initial period does not include capital raised by insider lending.

17

Second, when the ﬁrm ﬁnances capital investment only by market lending (xt = 0), theoptimal capital, denoted kt∗∗ , is given by the following ﬁrst order condition:γ(kt∗∗ )γ−1 = rt .

(10)

Here kt∗ > kt∗∗ holds due to λ < 1. Third, we deﬁne the capital x̂t which makes DE binding
as follows
1(Vt+1 − ψt )
(11)
x̂t =
λrt
whenever Vt+1 ≥ ψt .
Throughout the paper, we will maintain the following assumption:
Assumption 1. λ(1 + ql /∆q) > 1.
Assumption 1 says that the veriﬁable signal σ of each old worker’s eﬀort is not so
informative, i.e., ∆q is small, relative to the eﬃciency of insider lending captured by theparameter value 1 − λ. One unit of capital can be raised by one unit of good via market
lending but by only λ ∈ (0, 1) units of good via insider lending. Thus, for ﬁnancing
one unit capital, (1 − λ)rt can be saved under insider lending relative to market lending,which is the beneﬁt of using the former. On the other hand, when the explicit bonus
vt is paid together with insider lending, one unit increase in relation-speciﬁc capital xt
must be accompanied with the rise of the explicit bonus by the amount of (ql /∆q)λrt in
expected term due to (7). Assumption 1 then ensures that such capital cost accompaniedwith the explicit bonus is greater than the beneﬁt of insider lending (1 − λ)rt . When
Assumption 1 does not hold, the beneﬁt of insider lending always dominates the capital
cost accompanied with the explicit bonus, implying that market lending is never used forﬁnancing capital investment (see Appendix B for more details.) In order to make market
lending viable, we will maintain Assumption 1 in the following analysis.
By solving Problem (P), we show the following result.
Lemma 1. Suppose that Assumption 1 holds. The optimal contract solving Problem (P)is characterized as follows.
(i) Suppose that Vt+1 ≥ ψt and rtγ ≥ (γ/λγ )(Vt+1 − ψt )γ−1 . Then capital investment
is ﬁnanced by only insider lending and attains the unconstrained optimum kt = kt∗
without DE. The optimal wage involves only the implicit bonus, bt = ψt and vt = 0.
(ii) Suppose that Vt+1 ≥ ψt and (γ/λγ )(Vt+1 −ψt )γ−1 > rtγ ≥ γλ1−γ (Vt+1 −ψt )γ−1 . Then
capital investment is ﬁnanced by only insider lending and given by kt = x̂t where DE
is binding. The optimal wage involves only the implicit bonus, bt = ψt and vt = 0.
(iii) Suppose that Vt+1 ≥ ψt and γλ1−γ (Vt+1 − ψt )γ−1 > rtγ . Then total capital kt is given
by kt = kt∗∗ where capital x̂t is ﬁnanced by insider lending and the remaining capital18

kt∗∗ − x̂t is ﬁnanced by market lending. The optimal wage involves only the implicit
bonus, bt = ψt and vt = 0.
(iv) Suppose that ψt > Vt+1 ≥ 0. Then capital is ﬁnanced by only market lending andgiven by kt = kt∗∗ (thus xt = 0). The optimal wage involves both the implicit andexplicit bonus, vt = (1/∆q)(ψt − Vt+1 ) > 0 and bt = Vt+1 ≥ 0.
Since the implicit bonus bt is less costly than the explicit bonus vt to motivate old workers,
the ﬁrms want to use the former ﬁrst and then the latter if necessary. Also the ﬁrms want
to ﬁnance capital by insider lending which is more eﬃcient than market lending.
The optimal contract has the diﬀerent features depending on the future market priceof the ﬁrm Vt+1 and the current interest rate rt . When the future market price Vt+1 is
expected to be high enough relative to the current interest rate rt (case (i) of Lemma 1),
DE is not binding so that the optimal capital coincides with the unconstrained optimum
kt∗ and only the implicit bonus is used to motivate employed old workers. When Vt+1 falls
to the range in case (ii) of Lemma 1, DE becomes binding so that capital investment isconstrained by DE and smaller than the unconstrained one kt = x̂t < kt∗ although the
optimal wage still uses only the implicit bonus. When Vt+1 falls further to the range incase (iii) of Lemma 1, DE becomes tight so that the ﬁrms must ﬁnance capital by both
insider and market lending (kt = kt∗∗ > xt = x̂t ) while still keeping the use of the implicit
bonus. When Vt+1 is so small that ψt > Vt+1 (case (iv) of Lemma 1), the ﬁrms neverﬁnance capital by insider lending but they resort to market lending fully. In addition, theﬁrms combine the explicit bonus vt = (1/∆q)(ψt − Vt+1 ) with the implicit bonus bt = Vt+1
because DE is so severe that the implicit bonus is not suﬃcient to motivate workers andhence the explicit bonus must be introduced.
Remark. The optimal contract never involves both insider lending (xt > 0) and the
explicit bonus (vt > 0). The reason for this is as follows. The explicit bonus vt depends onthe veriﬁable but noisy signal σ while insider lending can save the cost of raising capitaland its gain is captured by the parameter value 1 − λ. By (7) the explicit bonus is given by
vt = (1/∆q)(λrt xt + ψt − Vt+1 ) whenever the ﬁrm ﬁnances capital investment via insider
lending (xt > 0). However, when the veriﬁable signal σ is not so informative (∆q is small)as made in Assumption 1, the cost of raising relation-speciﬁc capital xt becomes larger
through increasing the explicit bonus by the term (1/∆q)λrt . Assumption 1 then implies
that the eﬀectiveness of insider lending captured by 1 − λ is so small relative to its cost(ql /∆q)λ arising from the use of the explicit bonus at the same time. Thus the ﬁrms never
use the explicit bonus when they ﬁnance via insider lending. 20

20
In contrary to this, if we drop Assumption 1, the ﬁrms always choose insider lending but market
lending never emerges (see Appendix B).

19

4

Full Equilibrium

We embed the optimal contract derived in the previous section into the dynamic generalequilibrium framework which endogenously determines the market interest rate rt , capital
kt , and the market price of the ﬁrm ownership Vt .

4.1

Firm Ownership (Stock) Market

As we have mentioned, all young individuals become indiﬀerent for purchasing the ﬁrm
ownerships and not purchasing them because the no-arbitrage condition (3) implies thatthe net return of purchasing one unit of the ﬁrm ownership becomes zero, i.e., −rt Vt + πt +Vt+1 = 0. Then, without loss of generality we can suppose that only young entrepreneurs
purchase the ﬁrm ownerships.

4.2

Credit Market Equilibrium

In the credit market in period t there is the excess credit demand by Nf young ﬁnanciersas
Nf (wt−1 − mλxt )(12)where each of them lends λxt to each of m ﬁrms via insider lending. Since ﬁnanciers are
indiﬀerent regarding how many ﬁrms they lend to, m is indeterminate but it is irrelevantin the following analysis. Here, note that Nf m = N holds. On the other hand, the excess
credit demand by 1 − Nf young entrepreneurs is given as follows(1 − Nf )[wt−1 − lt ((kt − xt ) + Vt )](13)

where each of them purchases lt ownerships (stocks) of the ﬁrms at the price Vt andborrows capital kt − xt from market lending per ﬁrm they own.
Then the total excess credit demand must be zero in equilibrium:
Nf (wt−1 − mλxt ) + (1 − Nf )[wt−1 − lt ((kt − xt ) + Vt )] = 0.

(14)Here, by noting that the total demand for the ﬁrm ownerships by young entrepreneurslt (1 − Nf ) must be equal to the total ownerships of the ﬁrms, N , we have lt (1 − Nf ) = N .
21 Also, N young ﬁnanciers lend to m ﬁrms each so that N m = N holds. Then, byff
re-arranging the above condition, we obtain the credit market equilibrium (CME):
wt−1 = N (kt − (1 − λ)xt + Vt )(CME)This states that total resources available in period t−1 for the young generation to invest in
new capital and purchase the ﬁrm ownerships from the old generation is given by the total
21

Recall that there are one unit mass of ownerships per ﬁrm. Thus the total number of ownership rightsof the ﬁrms should be equal to N . Note that ﬁrms have no reasons to issue new equities for ﬁnancing
capital investment as we have mentioned. They ﬁnance capital investment kt − xt which is not covered byinsider lending via borrowing from the credit market but not issuing new equities.

20

endowment of young capitalists, wt−1 . Insider lending needs λxt units of good per ﬁrm forﬁnancing xt capital whereas market lending requires xt units of good per ﬁrm for ﬁnancing
the same amount of xt capital, which implies that the former can save (1 − λ)xt units ofgood relative to the latter. Thus the net capital demand kt − (1 − λ)xt after subtracting
such saving by insider lending must be ﬁnanced via the credit market. Finally, young
entrepreneurs purchase N ﬁrms from the old generation at the market price Vt , resulting
in N Vt credit demand in the economy. The sum of these credit demand must be equal to
the total endowment of the economy wt−1 , which is the credit supply of the economy.
The endowment of young capitalists wt−1 depends on the aggregate productivity At−1
embodied in the aggregate output N yt−1 , i.e., it is given by wt−1 = LAt−1 = LN yt−1 .
An equilibrium path of the economy is described as a sequence {Vt , kt−1 , xt , bt−1 , vt−1 , rt }∞t=1
which solves the optimal contract and satisﬁes both CME and the ﬁrm market price (8),given the initial capital stock k0 .
In what follows we will assume N = 1 without loss of generality. Thus the aggregate
output N yt is simply given by yt = ktγ and the worker’s opportunity cost of choosing higheﬀort at = 1 is given by ψt ≡ ψAt = ψyt = ψktγ respectively.

5

Equilibrium Dynamics

5.1

Static Equilibrium

We deﬁne a new state variable Qt ≡ Vt /yt−1 by dividing the market price of a ﬁrm Vt inγ
the current period t by the aggregate output yt−1 = kt−1
in the previous period t − 1. We
call this the ﬁrm value simply.
Before moving to the analysis of full equilibrium dynamics, we deﬁne a static equilibrium in period t as {Qt , kt , xt , rt , bt , vt } given Qt+1 and kt−1 such that all these variables
satisfy Lemma 1 and CME given the capital in the previous period kt−1 and the expectation about the ﬁrm value Qt+1 in the next period t + 1. Then, we can show that there arethree diﬀerent regimes of static equilibrium as depicted in Figure 1.
When the future ﬁrm value Qt+1 is expected to be so high that Qt+1 ≥ λγ + ψ, theﬁrms have enough incentive to honor the promised implicit bonus bt = ψt and repayment
Rt to ﬁnanciers. Thus, insider lending and implicit wage become a static equilibrium inthis regime. When Qt+1 is in the middle range (λγ + ψ > Qt+1 ≥ ψ), DE becomes tight sothat the ﬁrms ﬁnd it diﬃcult to ﬁnance capital via only insider lending. In that case theﬁrms complement market lending with insider lending while they still maintain only the
implicit bonus. When Qt+1 is so low that ψ > Qt+1 , the ﬁrms must rely on the explicit
bonus to motivate workers but then it becomes too costly to use insider lending. As aresult, the ﬁrms shift to using only market lending together with the explicit bonus.
In the following we examine these three regimes in more details.
Implicit Contract Regime. First we consider the implicit contract regime in whichlabor contract is based only on the implicit bonus bt = ψt (i.e., vt = 0) and ﬁnancing is21

based only on insider lending (kt = xt ). There are further two cases: one is that DE isnot binding and the other is that it becomes binding. When DE is slack, the ﬁrm chooses
capital kt to maximize its proﬁt ktγ −λrt kt which yields the unconstrained optimum kt = kt∗satisfying γktγ−1 = λrt . By using this and the fact that vt = 0 and bt = ψt , the ﬁrm market
price Vt (see (8)) is given by
Vt = (kt1−γ /λγ){(1 − γ)ktγ − ψt + Vt+1 }.
By using ψt = ψyt = ψktγ and the deﬁnition of Qt = Vt /kt−1 , this can be written byγ
Qt = (kt /kt−1){(1 − γ) − ψ + Qt+1 }.

(15)γ
In this regime CME becomes wt−1 = Lkt−1= λkt + Vt because kt = xt holds. Then weγ
can write CME as λkt /kt−1= L − Qt which we substitute into (15) to obtain

Qt= (1 − γ) − ψ + Qt+1 .
L − Qt(16)

This can be a static equilibrium when DE is not binding in the optimal contract (case(i) of Lemma 1). By substituting rt = (1/λ)ktγ−1 into the non-binding DE constraint,
Vt+1 ≥ λrt kt + ψt , and using the deﬁnition of Qt+1 and ψt ≡ ψktγ , we obtain
Qt+1 ≥ γ + ψ.
Then we show the following.
Lemma 2. Suppose that Assumption 1 holds. Suppose that Qt+1 ≥ γ + ψ. Then thestatic equilibrium in period t has the following features: (i) DE is not binding, (ii) theﬁrm value Qt satisﬁes (16), (iii) the market interest rate is given by rt = (1/λγ)ktγ−1 , andγ(iv) capital kt is determined by (L − Qt )kt−1= λkt .
Suppose next that DE is binding in the implicit contract regime. Then, case (ii)of Lemma 1 becomes a static equilibrium. In that case the ﬁrms still ﬁnance capital
investment by insider lending but DE becomes binding (kt = xt = x̂t ). By using kt =x̂t ≡ (1/λrt )(Vt+1 − ψt ) and the deﬁnitions of Qt+1 and ψt , the condition in case (ii) of
Lemma 1 is shown to be equivalent to γ + ψ > Qt+1 ≥ λγ + ψ. Also, CME becomesγ
Lkt−1= λkt + Vt again because of kt = xt , implying thatγkt = (L − Qt )kt−1/λ.

Given such capital kt , the interest rate rt which makes DEt binding becomesrt =Qt+1 − ψ γ−1kt .
λWhen we substitute this into (8), we can show the following result.

22

(17)

Lemma 3. Suppose that Assumption 1 holds. Suppose that γ + ψ > Qt+1 ≥ λγ + ψ. Thenthe static equilibrium in period t has the following features: (i) the ﬁrms choose capital kt =γγ
(L−Qt )kt−1/λ, (ii) the market interest rate satisﬁes rt = (Qt+1 −ψ)((L−Qt )kt−1)γ−1 /λγ ,(iii) DEt becomes binding, (iv) the ﬁrms pay only the implicit bonus bt = ψt but not the
explicit bonus (i.e., vt = 0), and the ﬁrm value Qt is given byQt
1=.
L − QtQt+1 − ψ(18)

Mixed Contract Regime. Next we consider the regime in which the ﬁrms use both
insider and market lending (kt > xt > 0) but still maintain only the implicit bonus(bt = ψt and vt = 0), which we call mixed contract regime. This occurs in case (iii) of
Lemma 1, which is equivalent to λγ + ψ > Qt+1 ≥ ψ because rt = γktγ−1 holds in thisregime. Also CME is written byγ
Lkt−1= kt − (1 − λ)xt + Vt

and DEt becomes binding, Vt+1 = λrt xt + ψt which yields rt = (Vt+1 − ψt )/λxt . The ﬁrm
chooses total capital kt satisfying γktγ−1 = rt . By combining this with the binding DE, weobtainrt = γktγ−1 = (Vt+1 − ψt )/λxt .
(19)Here the second line implies that xt = (Qt+1 − ψ)kt /λγ. Thus CME shows that
)
]
[
(
1−λ
γ
(Qt+1 − ψ) kt .
(L − Qt )kt−1 = 1 −
λγ

(20)

Then we can show the following:
Lemma 4. Suppose that Assumption 1 holds. Suppose that λγ +ψ > Qt+1 ≥ ψ. Then, thestatic equilibrium in period t has the following features: (i) the ﬁrms choose total capital
kt satisfying (20) and the relation-speciﬁc capital xt satisfying xt = (Qt+1 − ψ)kt /λγ, (ii)the market interest rate rt is given by rt = γktγ−1 , (iii) DE becomes binding, (iv) the ﬁrms
pay only the implicit bonus, i.e., bt = ψt and vt = 0, and (v) the ﬁrm value Qt satisﬁes
Qt
1 − γ + (1/λ)(Qt+1 − ψ)
(
)
].
= [
L − Qtγ 1 − 1−λ (Q
− ψ)
λγ

(21)

t+1

Explicit Contract Regime. Finally we consider the case (iv) of Lemma 1, ψt > Vt+1 >0. This is equivalent to ψ > Qt+1 > 0. In this regime the ﬁrms choose total capital kt
satisfying γktγ−1 = rt and ﬁnance all capital kt by market lending. The credit market
equilibrium (CME) in period t − 1 is then given byγ
Lkt−1= kt + Vt

23

(22)

γ
because xt = 0 in this regime. This can be written by kt /kt−1= L − Qt , which we
γ−1
substitute into rt = γkt
in the expression of the ﬁrm price Vt given by (8) in order toshow the following:

Lemma 5. Suppose that Assumption 1 holds. Suppose that ψ > Qt+1 > 0. Thenthe static equilibrium in period t has the following features: (i) the ﬁrms ﬁnance capitalγ
investment kt = (L − Qt )kt−1
via only market lending but not insider lending (xt = 0),(ii) the market interest rate rt is given by rt = γktγ−1 , (iii) the ﬁrms pay the explicit bonus
vt = (1/∆q)(ψ − Qt+1 )ktγ > 0 and the implicit bonus bt = Qt+1 ktγ > 0, and (iv) the ﬁrm
value Qt satisﬁes
[
(
)
(
)
]
1
q0
q0
Qt
=
1−γ− 1+
ψ+ 1+
Qt+1 .
(23)
L − Qt
γ
∆q
∆q

5.2

Endogenous Changes in Employment and Financial Systems

Now we combine all Lemma 2-5 to obtain the full equilibrium paths of the economy. Once∞a path of the ﬁrm values {Qt }∞t=1 is determined, we can recover a path of capital {kt }t=0from CME starting with the initial capital stock k0 . The path of relation-speciﬁc capital∞∞{xt }∞
t=1 is then obtained as well by using the paths of {kt }t=0 and {Qt }t=1 . We can also∞
recover the path of the interest rates {rt }t=1 from using these paths. Thus it suﬃces todetermine the path of the ﬁrm values {Qt }∞t=1 which evolves from Lemma 2-5 as follows
(1/γ)[(1 − γ) − ψ + Qt+1 ]if Qt+1 > γ + ψ




1


if γ + ψ ≥ Qt+1 > λγ + ψQt+1 − ψQt=
1 − γ + (1/λ)(Qt+1 − ψ)[()]
L − Qt 
if λγ + ψ > Qt+1 ≥ ψ


1−λ

γ
1
−
(Q
−
ψ)

t+1
λγ



(1/γ){1 − γ − (1 + (q0 /∆q))ψ + (1 + (q0 /∆q))Qt+1 } if ψ > Qt+1 > 0
given Qt ∈ [0, L) for all t ≥ 1. We deﬁne the above relation between Qt and Qt+1 as afunction Qt = Φ(Qt+1 ) where Φ is increasing except the interval [λγ + ψ, γ + ψ] which is
the implicit contract regime with binding DE (see Figure 2).
Note that DE constraint becomes binding in all the regimes except the implicit contract
regime with high enough ﬁrm value Qt+1 > γ + ψ. In all these regimes the future ﬁrm
value Qt+1 inﬂuences capital investment kt and worker’s wages bt and vt in the current
period t, which is in contrast to the case that DE is non-binding.
For a comparison, we will consider the benchmark case in which DE can be ignoredby supposing that relation-speciﬁc capital xt and worker’s eﬀort at (thus the unveriﬁable
signal st = at ) are veriﬁable. To ensure that the ﬂow proﬁt of the ﬁrm in the benchmark
case becomes positive, we make the following assumption.

24

Assumption 2. 1 > γ + ψ.
The equilibrium path of the economy in the benchmark is governed by {Qt }∞t=1 satisfying (16), which is same as in the implicit contract regime without binding DE. We thendeﬁne a unique steady state satisfying (16) at Qt = Qt+1 , denoted by Q∗ , as follows
γQ∗= (1 − γ) − ψ + Q∗ .
L − Q∗
Under Assumption 2 the equilibrium path {Qt } in the benchmark case is unique andgiven by Qt = Q∗ for all t ≥ 1. 22 Thus the ﬁrm value Qt does not change over time in
the benchmark equilibrium. We also denote by k ∗ the corresponding steady state capitalwhich satisﬁes CME:
(L − Q∗ )(k ∗ )γ = λk ∗ .
γ
Since Qt = Q∗ holds for all t, CME implies that kt−1
(L − Q∗ ) = kt and hence the path of∞capitals {kt }t=0 is uniquely determined as well. As a result, kt goes to the steady state k ∗as t → ∞.

We now return back to the case that the relation-speciﬁc capital xt and worker’s eﬀort
at are not veriﬁable. We deﬁne the three cut oﬀ ﬁrm values as follows (see Figure 2):Q̂ ≡ L(1 − γ),
Q′′ ≡

L
,
γ+1

Q′ ≡

L
λγ + 1

andwhere ψ = Φ(Q̂), λγ + ψ = Φ(Q′ ) and γ + ψ = Φ(Q′′ ) are satisﬁed respectively. Here wecan verify that Q̂ < Q′′ < Q′ hold. Then, the implicit contract regime with binding DE
prevails when Qt ∈ (Q′′ , Q′ ) and Qt+1 ∈ (λγ + ψ, γ + ψ) while the implicit contract regime
without DE arises when Qt > Q′′ and Qt+1 > γ + ψ respectively. Also, equilibrium mode
becomes the mixed contract regime when Qt ∈ (Q̂, Q′ ) and Qt+1 ∈ (ψ, λγ + ψ) while it
becomes the explicit contract regime when Qt ∈ (0, Q̂) and Qt+1 ∈ (0, ψ) respectively.
Although equilibrium dynamics becomes complicated, we can show that the economy
never moves from the mixed or explicit contract regime to the implicit contract regime inany equilibrium path.
Proposition 1. Suppose that Assumption 1 and 2 are satisﬁed. Then, in any equilibrium
path Qt < λγ + ψ holds for all t ≥ T once QT < λγ + ψ in some period T .
22
If Q1 ̸= Q∗ , we can show from (16) that Qt goes to either a negative value or positive inﬁnity under
Assumption 2, either of which cannot be an equilibrium path.

25

Proposition 1 states that, if the economy enters the mixed or explicit contract regime insome period T −1 which implies that QT < λγ+ψ, then it never enters the implicit contract
regime from period T onwards, i.e., Qt < λγ + ψ for all t ≥ T . In this sense organizationalchanges in ﬁnancial and labor contracts are irreversible such that, once market lending or
explicit wage is adopted in some period, such market-and-explicit contract based featureof employment and ﬁnancial systems persists over time.
From Proposition 1, there are only two possible candidates for equilibrium paths. First,
the economy moves from the implicit contract regime to the mixed or explicit contract
regime over time. Second, the economy stays in the implicit contract regime forever. In
the latter case the equilibrium path either ﬂuctuates over time in the implicit contract
regime or stays in the steady state in the implicit contract regime from the initial period
forever. The steady state ﬁrm value, denoted by Qi , in the implicit contract regime isdeﬁned as follows{
Qi
1 − γ − ψ + Qi if Qi > γ + ψ=
1/(Qi − ψ)
if λγ + ψ < Qi < γ + ψ
L − Qi
Note that Qi coincides with the ﬁrm value in the benchmark case, Qi = Q∗ , when Q∗ >γ + ψ.
We then show that the ﬁrm value must be constant at the steady state value Qi if
the economy stays in the implicit contract regime forever. Thus there are no equilibrium
paths which ﬂuctuate forever in the implicit contract regime.
Proposition 2. Suppose that Assumption 1 and 2 are satisﬁed. Then, in any equilibrium
path in which Qs ̸= Qi for some period s, it must be that Qt < λγ + ψ for all t ≥ T fromsome period T onward.
Proposition 2 together with Proposition 1 implies that in any equilibrium path theeconomy eventually enters the mixed or explicit contract regime unless it stays in theimplicit contract steady state Qi from the initial period forever. Thus any equilibrium
path must be either (i) that the economy stays in the implicit contract regime with theﬁrm value being constant at Qi in all periods or (ii) that the economy enters the mixed
or explicit contract regime from some period onward.
The ﬁrst type (i) of equilibrium path exists if the implicit contract steady state Qi
exists, which is shown to be equivalent to the condition that Q′ > λγ + ψ. 23
Our next task is to provide the conditions under which the second type (ii) of equilibrium path actually exists. In particular we show that there exists an equilibrium path
which starts from the implicit contract regime and then moves to the mixed contract
regime and explicit contract regime over time. To this end, we make the following additional assumption.
23
When this condition is satisﬁed, Φ(λγ + ψ) = Q′ > λγ + ψ holds. Then, since Φ is increasing andconvex with Φ(∞) = L when Q > γ + ψ and Φ is decreasing when λγ + ψ < Q < γ + ψ, there exists a
unique Qi such that Φ(Qi ) = Qi and λγ + ψ < Qi .

26

Assumption 3. Γ ≡ (1 + q0 /∆q)ψ − (1 − γ) > 0.
Assumption 3 says that the veriﬁable signal σ is not so informative (∆q is small) again
as assumed in Assumption 2. Assumption 3 ensures that Φ(0) > 0 (Figure 2).
We deﬁne the two cut oﬀ values of the parameter L measuring the size of savings (thecredit supply) of the economy as follows
L̂ ≡

ψ
1−γ

where Q̂ > ψ holds when L > L̂, andL′ ≡ (1 + λγ)(λγ + ψ)
where Q′ > λγ + ψ holds when L > L′ , respectively.
We also deﬁne (Qe , k e ) as the steady state in the explicit contract regime, i.e., itsatisﬁes (23) at Qt = Qt+1 :
[
(
)
(
) ]
Qe
q0
q0
1
1−γ− 1+
ψ+ 1+
Qe .
(24)
=
L − Qeγ∆q∆qThen, from CME at kt = kt−1 and Qt = Qe , we obtain the steady state capital k e in the
explicit contract regime:k e = (L − Qe )(k e )γ .
(25)
Then we can verify the following claim.
Lemma 6. Suppose that L > L̂ and Assumption 3 hold. Then the steady state in the
explicit contract regime (Qe , k e ) exists. Furthermore, Qe is decreasing in L and Qe = ψat L = L̂.
Given Lemma 6, we can show the following result.
Proposition 3. Suppose that Assumption 1-3 and L̂ > L′ hold. 24 Then there exists
some L > L̂ such that for all L ∈ (L̂, L) the following multiple equilibrium paths exist with
diﬀerent switching periods Ti and Te such that the economy stays(i) in the implicit contract regime for t ≤ Ti ,(ii) in the mixed contract regime for Ti < t < Te ,(iii) in the explicit contract regime for t > Te

24

When λ is small, L̂ > L′ is satisﬁed.

27

where the switching periods Te and Ti are not unique.
The condition L > L̂ > L′ stated in Proposition 3 ensures that Q̂ > ψ and Q′ > λγ +ψ(Figure 3). 25 By combining this with Assumption 3, Q̂ > ψ implies that there exists a
steady state in the explicit contract regime (depicted by Qe in Figure 3). Also L > L′ canguarantee that the implicit contract steady state (Qi , k i ) exists (Figure 3). In addition,
L < L is used to show that the ﬁrm value in the explicit contract steady state Qe is solarge that the ﬂow proﬁt of the ﬁrm becomes positive in the explicit contract steady state.
26

As we have already mentioned, equilibrium path is unique in the benchmark casein which DE can be ignored. In contrast, equilibrium paths become multiple when DE
constraint becomes binding. It can be seen in Figure 3 that multiple equilibrium paths
are possible with diﬀerent initial ﬁrm values Q1 . The main intuition behind multiple
equilibrium paths is due to the existence of the sequences of self-fulﬁlled expectations overtime, although such mechanism works diﬀerently in the implicit contract regime and otherregimes.
We ﬁrst consider the implicit contract regime with binding DE and see that the expectation about a high (low) future ﬁrm value Qt+1 is associated with a low (high) current
value Qt . Thus high and low ﬁrm values alternate overt time and constitute an equilibrium
path in this regime. To see this, suppose that the ﬁrm value Qt+1 is expected to be lower
in the next period t + 1. Then, since DE becomes binding and all capital is ﬁnanced byinsider lending in this regime, capital investment kt = xt in the current period t decreases(note that (DE) Vt+1 = λrt kt + ψt implies that Qt+1 = λ(rt /yt )kt + ψ.) This reduces theexcess credit demand in the credit market so that more credit supply is left for ﬁnancing the purchase of the ﬁrm ownership (note that (CME) Lyt−1 = λkt + Vt implies thatL = λkt /yt−1 + Qt .) This increases the ﬁrm value Qt in the current period t. In this way
the expectation about a low future ﬁrm value Qt+1 results in a high current ﬁrm value
Qt . The expectation about a low ﬁrm value Qt+1 is self-fulﬁlled by the expectation abouta high ﬁrm value Qt+2 in period t + 2, which will be self-fulﬁlled by the expectation abouta low future ﬁrm value Qt+3 in period t + 3, and so on. Thus, the path of alternating highand low ﬁrm values becomes an equilibrium in the implicit contract regime with bindingDE.
On the other hand, in the mixed contract regime, the expectation about a low ﬁrm
value in the future period Qt+1 results in a low ﬁrm value Qt in the current period t. To
see this, suppose that the future ﬁrm value Qt+1 is expected to be lower. Then, since
DE is binding, capital investment ﬁnanced by insider lending xt in the current period t
decreases. However, in contrast to the implicit contract regime, since both insider andmarket lending are used in this regime, the decrease in capital ﬁnanced by insider lending
implies that more credit is needed to ﬁnance the remaining capital by market lending.
25

In Figure 3 we depict the case that γ + ψ > L so that DE is always binding in the implicit contract
regime.
26
The ﬂow proﬁts in the implicit contract and mixed contract regimes can be non-negative due toAssumption 2.

28

This increases the excess credit demand kt − (1 − λ)xt in the credit market in the current
period t, implying that less credit supply is left for ﬁnancing the purchase of the ﬁrm
ownerships in the current period t. This then decreases the ﬁrm value Qt in the current
period t because of CME: L = (kt − (1 − λ)xt )/yt−1 + Qt . In this way, a low future ﬁrm
value Qt+1 is associated with a low current ﬁrm value Qt .
In the explicit contract regime, the expectation about a low future ﬁrm value Qt+1
makes the explicit bonus vt larger in the current period t because DE is binding so thatvt /yt = (1/∆q)(ψ − Qt+1 ). Then, since the explicit bonus is more costly than the implicit
bonus and the ﬁrm must rely on the former in this regime, the ﬂow proﬁt πt in the current
period t becomes lower, resulting in a decrease in the ﬁrm value Qt in the current periodt. Again, the expectation about a low future ﬁrm value is associated with a low current
ﬁrm value Qt .
Thus, in either mixed or explicit contract regime the expectation about a low ﬁrm
value Qt+1 in period t + 1 will be self-fulﬁlled by the expectation about a low ﬁrm value
Qt+2 in period t + 2, which will be self-fulﬁlled by the expectation about a low ﬁrm value
Qt+3 in period t + 3, and so on.
As a result, these sequences of self-fulﬁlled expectations about the ﬁrm values Qt
cause multiple equilibrium paths. Some of them have the feature that ﬁnancial and labor
contracts change between the diﬀerent regimes toward more explicit-and market-based
systems as shown in Proposition 3. Such equilibrium paths can be constructed by having
an initial value of the market value Q1 in the implicit contract regime and having thesubsequent values of Qt in the mixed contract regime. As we have discussed above, once
Qt enters the mixed contract regime, a decreasing sequence of Qt can be a self-fulﬁlled
expectation equilibrium which eventually enters the explicit contract regime. Such pathcan be actually an equilibrium as long as Qt does not diverge to negative values, i.e.,
Qt ∈ (0, L) holds in any period t.
To see this last point, consider the explicit contract regime in which the ﬁrm must
rely on the explicit bonus vt /yt = (1/∆q)(ψ − Qt+1 ) measured in terms of the current
output yt . When the future ﬁrm value Qt+1 decreases by one unit amount, this raises
the current explicit bonus by (1/∆q) units. Such loss tends to be larger as the veriﬁable
signal σ becomes more noisy (i.e., ∆q is smaller), which implies that one unit increasein the future value Qt+1 results in more than one unit decline in the current value Qt .
This ensures that ∆Qt /∆Qt+1 > 1, although ∆Qt /∆Qt+1 < 1 holds at the steady statein the benchmark case without DE constraint. Thus, by reversing the above inequality,∆Qt+1 /∆Qt < 1 holds in the explicit contract regime when the veriﬁable signal is not so
informative as assumed in Assumption 1 and Assumption 3. Then, as Qt becomes lower,
a further decrease in the current value of Qt does not reduce the next period value Qt+1
too much, guaranteeing that Qt does not diverge to negative values even when it decreases
over time.

29

5.3

Organizational Changes in Economic Development

Next we investigate how the changes in ﬁnancial and labor contracts aﬀect and are aﬀectedby capital accumulation and economic development.
As we have shown in Proposition 1 and 2, equilibrium paths are characterized as either
one of the following: (i) Qt = Qi for all t ≥ 1 or (ii) Qs ̸= Qi for some s and Qt < λγ + ψ
for all t ≥ T from some period T . In the former case (i) equilibrium capital follows the
dynamic equationγ(L − Qi )kt−1= λkt
due to CME in the implicit contract regime. Then it is veriﬁed that kt monotonicallyincreases over time when (L − Qi ) > k01−γ and kt eventually reaches the steady state in
the implicit contract regime, denoted by k i , satisfying(L − Qi )(k i )γ = k i .
In the latter case (ii) capital dynamics becomes more complicated because the ﬁrm
value Qt may ﬂuctuate for some periods (but not permanently) in the implicit contract
regime and then moves to the mixed contract or explicit contract regime as depicted in
Figure 4. However, we know from Proposition 2 that the economy eventually enters themixed or explicit contract regime in either which the dynamic equation Φ governing thepaths of the ﬁrm values Qt is monotone decreasing. Thus, capital dynamics becomes easier
to be characterized in these regimes than in the implicit contract regime.
We denote by gt ≡ kt /kt−1 − 1 the net growth rate of capital. Then we show thefollowing result.
Proposition 4. Suppose that Assumption 1-3 hold. Suppose also that L > max{L̂, L′ }.
Then equilibrium path must be either one of the following features:(i) Qt = Qi for all t ≥ 1 and capital kt converges to the steady state k i in the implicit
contract regime. Furthermore, the net growth rate of capital gt is positive if L − Qi >k01−γ .
(ii) Qs ̸= Qi for some s and Qt → Qe as t → ∞ where capital kt converges to the steadystate k e in the explicit contract regime. Furthermore, the net growth rate of capitalgt+1 > gt holds if gt < 0 and gt+1 > 0 holds if gt > 0 in the mixed and explicit
contract regimes.
Thus, when the size of economy savings L is large so that L > max{L′ , L̂}, the economy
eventually reaches either the implicit contract steady state (Qi , k i ) or the explicit contract
steady state (Qe , k e ) in the long run.
In the second case in Proposition 4, as capital kt increases over time, the equilibrium
modes of ﬁnancing and labor contracts change from the implicit contract-based to the explicit and market-based together. The key feature of this result is that dynamical changes
30

in ﬁnancial system are accompanied with the changes in employment system: as ﬁnancial
system moves from insider lending to market lending, employment system also moves fromthe implicit-based contract to the explicit-based one. Such co-evolution of ﬁnancial andemployment systems is caused by the interactions between the dynamic enforcement (DE)constraint and market equilibrium conditions. As we have already seen in Proposition 3,
the fall of the ﬁrm values Qt is supported by self-fulﬁlled expectation. Then, the credit
demand for purchasing the ﬁrm ownerships falls over time as well, resulting in more credit
supply used to ﬁnance larger capital investment over time. Thus capital kt increases if
its growth rate gt is positive or the growth rate gt increases if gt is negative in the mixed
contract and explicit contract regimes. 27
Our theoretical insights help understand how ﬁnancial and employment systems evolve
over time in the process of economic development. As we have discussed, the rapid economic growth of the Japanese economy during 1960s was associated with the bank oriented
ﬁnancial system and implicit employment relationships. However, as the Japanese economy matured and slowed down after 1970s, they have shifted toward ﬁnancing in anonymous ﬁnancial markets from the bank-oriented ﬁnancing (Hoshi and Kashyap (2004), Rajan and Zingales (2004)). At the same time, the implicit labor practices of the Japaneseﬁrms based on implicit employment security and seniority-based pay system have beenchanging to be replaced by more performance-based explicit contracts in 1990s (Abe andHoshi (2007)). These changes in employment and ﬁnancial systems might suggest thatthe implicit contracts in both ﬁnancing and labor contexts change together toward moremarket-and explicit-based systems. In that sense employment and ﬁnancial systems are
dynamically complement with each other, which might support the aforementioned historical evidence (Abe and Hoshi (2007), Aoki (1994), and Aoki and Patrick (1994)).

5.4

Long Run Welfare

As we have shown, the economy admit multiple equilibrium paths converging to diﬀerent
steady states under certain conditions. In this section we investigate the comparison
among the long run welfare in these diﬀerent steady states. To allow multiple steadystates, we maintain Assumption 1-3 and L > L̂ which ensure that the explicit contract
steady state (Qe , k e ) always exists.
When λ is so small, L̂ > L′ holds so that L > L̂ > L′ and hence there exists theimplicit contract steady state (Qi , k i ) in addition to the explicit contract one (Qe , k e ) (seeProposition 4). When λ is close to 1, we may have L′ > L > L̂ so that the there exists a
steady state in the mixed contract regime but the implicit contract steady state (Qi , k i )disappears. We denote by (Qm , k m ) such mixed contract steady state if it exists. Notethat Qe < Qm and Qe < Qi hold.
Thus, under Assumption 1-3 and L > L̂ the long run equilibrium is characterized by
two steady states: (i) (Qe , k e ) and (ii) (Qi , k i ) or (Qm , k m ). In the latter case (ii) the longrun steady state becomes (Qi , k i ) when λ is small whereas it becomes (Qm , k m ) when λis close to 1.
27

However, capital may temporarily decrease when the economy moves from one regime to the other.

31

Now we investigate which long run equilibrium attains a higher welfare. We deﬁne thesocial welfare of the economy as the sum of the utilities of all individuals in a steady state.
Since individuals consume only when old and their utilities are linear with respect to theirconsumption equal to incomes they earn, we can derive the social welfare by summing all
the incomes of old capitalists and old workers in a steady state. In a steady state, 1 − Nfold entrepreneurs earn
(1 − Nf ){r(w − lE (k − x + V )) + lE (π + r(k − x) + V )}(26)

where lE denotes the number of ﬁrms owned by each old entrepreneur and satisﬁes lE (1 −Nf ) = N = 1. Also, Nf old ﬁnanciers earn
Nf {r(w − mλx) + mR}(27)

where each ﬁnancier lends to m ﬁrms. All old workers earn
N EW + (1 − N )ψN y(28)

where N y = N k γ is the aggregate output, EW ≡ b + q1 v denotes the expected (average)
wage per employed old worker and (1 − N )ψN y is the income unemployed old workers
earn by accessing to their private production technology.
By summing all these payoﬀs and using lE (1 − Nf ) = N = 1 and Nf m = N = 1, we
obtain the social welfare as
SW (Q, k) ≡ r{w − N (k − (1 − λ)x + V )} + N {π + r(k − x) + R + EW + V } + (1 − N )ψN y= N {k γ − R − EW + R + EW + V } + (1 − N )ψN y= N {k γ + V } + (1 − N )ψN y= N k γ (1 + Q + (1 − N )ψ)

(29)

γ

= k (1 + Q)

(30)

where we used w = N (k − (1 − λ)x) + V ) due to CME and N = 1.
In the implicit contract steady state (Qi , k i ), CME implies that (L − Qi )k γ = λk andhence k = k i ≡ ((L − Qi )/λ)1/(1−γ) . Thus the social welfare in the implicit contract steadystate is given as follows) γ(
L − Qi 1−γi
i i(1 + Qi ).
(31)
SW ≡ SW (Q , k ) =λOn the other hand, in the explicit contract steady state (Qe , k e ), the social welfare is given
as follows
γ
SW e ≡ SW (Qe , k e ) = (L − Qe ) 1−γ (1 + Qe )(32)
due to CME: (L − Qe )k = k γ . Note that both Qi and Qe are independent of λ. Finally,in the mixed steady state (Qm , k m ), the social welfare is given by
(
) γ
1−γ
m
L
−
Q
SW m ≡ SW (Qm , k m ) =(1 + Qm )(33)
m − ψ)
1 − 1−λ(Qλγ
32

by using CME in the mixed contract regime: (L − Qm )(k m )γ = (1 − ((1 − λ)/λγ))(Qm −ψ))k m . Qm depends on λ.
Then we can show the following result:
Proposition 5. Suppose that Assumption 1-3 and L > L̂ hold. Then, the long run welfare
in the implicit contract steady state (Qi , k i ), SW i , is the highest among those in all thesteady states when λ is small enough. However, the social welfare SW e in the explicit
contract steady state is the highest among those in all the steady states when λ is close to1.
When insider lending is highly eﬃcient so that λ is small, the implicit contract steadystate which involves fully insider lending achieves the highest social welfare among all thesteady states. However, when the eﬃciency of insider lending is not so large, the implicit
contract steady state or the mixed contract steady state does not attain larger welfare
than the explicit contract steady state. The reason for this is as follows: in the implicit
contract steady state the ﬁrm value Qi is larger than those in other steady states, which
implies that more resources must be used for ﬁnancing the purchase of such high ﬁrm value
in the implicit steady state. This reduces capital investment and aggregate production of
the economy. On the other hand, in the explicit contract steady state the ﬁrm value Qe is
the lowest among all the steady states so that more credit supply can be used to ﬁnancelarger capital investment. This raises the aggregate output of the economy. When insider
lending is not so eﬃcient, its advantage of saving the lending costs is dominated by theeﬀect of lower aggregate outputs, showing that the explicit contract steady state achieves
the largest social welfare.

6

Conclusion

In this paper we have investigated the dynamic macroeconomic implications of ﬁnance andlabor contracts with limited commitment of the ﬁrms. Employed workers are motivatedto work hard by not only the implicit wage but also the explicit wage. The ﬁrms alsoﬁnance capital investment by not only insider lending but also market lending. For the
implicit labor contract and insider lending to be self-enforcing, the dynamic enforcement(DE) constraint must be satisﬁed such that the ﬁrms have no incentive to renege on theimplicit wage and repayment to ﬁnanciers under insider lending. Then we have shownthat the dynamic interactions between DE constraint and market equilibrium conditions
endogenously lead to the dynamic changes in organizational modes of ﬁnancial and labor
contracts: in equilibrium paths ﬁnance and labor contracts move toward more explicit andmarket-based system over time. Furthermore, we have also shown that the market-basedand explicit-based feature of employment and ﬁnancial systems persists in the long run
even though equilibrium paths become non-monotonic and complex.
We conclude this paper by discussing some extensions of the model left for the future
research. First, in our model workers and ﬁnanciers contract with ﬁrms only once so33

that they do not form long term relationships. Even so, the ﬁrms have the incentive
not to renege on the implicit ﬁnancial and labor contracts because they expect that theirmarket prices will go down when they lose reputation by not honoring contracts with thecurrent workers and ﬁnanciers. When the ﬁrms can engage in long term relationships
with workers and ﬁnanciers, it becomes easier for them to enforce the implicit labor andﬁnancial contracts. Then, the implicit contract regime may be more likely to be sustained,
which may aﬀect the regime switching patterns of equilibrium paths. Second, there areother ways to model the roles of ﬁnanciers than the one presented in this paper. For
example, one important role by banks is to monitor the client ﬁrms at lower costs thanmarket lending. The bank ﬁnancing is then interpreted as monitored loans while market
lending is the non-monitored loans by which investors do not directly monitor the ﬁrms’
activities (see Baliga and Polak (2004), Tirole (2005)). Then it will be interesting to see
how the bank monitoring is combined with diﬀerent forms of employment contracts andhow such determination of the corporate governance is linked to the process of economic
development.

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36

Qt+1
Implicit contract regime with non-binding DE:
Insider lending and implicit wage
(kt = xt < x̂t and vt = 0)γ+ψImplicit contract regime with binding DE:
Insider lending and implicit wage
(kt = xt = x̂t and vt = 0)γλ + ψ
Mixed contract regime:
Insider and market lending, and implicit wage
(kt > xt > 0 and vt = 0)ψExplicit contract regime:
Market lending and explicit wage
(kt > xt = 0 and vt > 0)Qt

0

Figure 1: Diﬀerent regimes of static equilibrium
Qt+1
Qt = Φ(Qt+1 )γ+ψ

γλ + ψ

ψ

0

Q̂

Q′′

Q′

Qt

Figure 2: Equilibrium Relations between Qt and Qt+1

37

Qt+1
45o
Qt = Φ(Qt+1 )L

γλ + ψ

ψ

Qe

0

Q1

Qt

L

Figure 3: Endogenous Organizational Changes
Qt+1
45o

Qt = Φ(Qt+1 )γ+ψ

γλ + ψ

ψ

0

Q′′

Q′ L

Q1 Qi

Qt

Figure 4: Irreversible Change to Mixed or Explicit Contract Regime

38

Appendix for Online Publication
7

Appendix A: Proofs

In this section we will provide the formal proofs for the results presented in the main text.

7.1

Suﬃciency of DE

In the main text we have claimed that, if DE is satisﬁed, there exists an equilibrium inwhich the ﬁrm honors for paying the implicit bonus bt to the old worker and repayment
Rt to the ﬁnancier. We will show that the strategies of the ﬁrm and the workers deﬁned
in the main text leads to such equilibrium when DE is satisﬁed.
Suppose that DE is satisﬁed. Suppose also that the ﬁrm breached to pay bt or Rt insome period t. Then we can show that it becomes a continuation equilibrium for the ﬁrm
and the young workers employed by the ﬁrm to exercise the quitting options simultaneously
as follows: given the ﬁrm’s liquidation, the young workers are indiﬀerent for quitting and
not quitting. Also, one optimal response of the ﬁrm to the young worker’s quitting choice
is to liquidate the ﬁrm’s asset because it expects that the young workers who will be
employed by the ﬁrm in the future periods will quit the ﬁrm according to their strategies,
implying that the ﬁrm cannot produce in any future period. Then, one possible path of
the ﬁrm’s market prices which is consistent with zero productions in the future is thatVs = 0 for all s ≥ t which actually satisﬁes the no-arbitrage condition, Vs = (1/rt )Vs+1 .
Thus its ownership rights will be never sold at positive prices. Here note that the young
workers born in period t + 1 can observe that the young workers employed in period t
have quit the ﬁrm and have been hence not retained in the ﬁrm in period t + 1. According
to the speciﬁed strategy, the newly employed young worker will quit the ﬁrm in periodt + 1 whenever the young workers employed by the ﬁrm in period t have quit. Similarly
the young workers born in period t + 2 will observe such quitting behavior of the young
workers in period t + 1 and then will quit the ﬁrm as well, and so on. Thus, all the young
workers in any period s > t will quit once the ﬁrm reneges on the payments and then theyoung worker quit the ﬁrm.
The above continuation equilibrium ensures that the market price of the deviating ﬁrm
goes down to zero in any future period s > t, i.e., Vs = 0 for all s > t. Thus, although thedeviating ﬁrm can save the payment bt or Rt in the current period t, it will be sold at zero
market price after the deviation. Then, if DE is satisﬁed, such deviation is not proﬁtable
for the ﬁrm.

7.2

Proofs for Lemmas and Propositions

Proof of Lemma 1. We deﬁne the ﬁrm’s ﬂow proﬁt in period t as}{q0γ(λrt xt + ψt − Vt+1 ), 0 .
Π(kt , xt ) ≡ kt − λrt xt − (kt − xt )rt − ψt − max∆q39Note that Π(kt , xt ) is increasing in xt when xt < x̂t and decreasing in xt when xt ≥ x̂tunder Assumption 1 where recall that x̂t ≡ (1/λrt )(Vt+1 − ψt ) when Vt+1 ≥ ψt . Thus the
optimal choice of xt which maximizes Π(kt , xt ) subject to (NE) 0 ≤ xt ≤ kt is given by{
kt if kt ≤ x̂txt = x(kt ) ≡x̂t if kt > x̂t
When ψt > Vt+1 , the optimal choice of xt must be zero, x(kt ) = 0. By substituting theseresults into Π(kt , xt ), we re-deﬁne the ﬁrm’s ﬂow proﬁt as
 γ
if kt ≤ x̂t
 kt − λrt kt − ψtγ
1−λπ(kt ) ≡ Π(kt , x(kt )) =k − rt kt − ψt + λ (Vt+1 − ψt )if kt > x̂t ≥ 0
 tγ
kt − rt kt − (1 + q0 /∆q)ψt + (q0 /∆q)Vt+1 if x̂t < 0First we suppose that Vt+1 > ψt (that is, x̂t > 0).
(i) Suppose that γ x̂γ−1≤ λrt which is equivalent to the condition stated in (i) of Lemmat
1. Then π(kt ) has a unique optimum kt∗ less than x̂t and hence vt = 0 holds.
(ii) Suppose that rt ≥ γ x̂γ−1> λrt which is equivalent to the condition stated in (ii) oftLemma 1. Then, π(kt ) is increasing in kt for kt < x̂t but decreasing in kt for kt > x̂t .
Thus π(kt ) has the maximum at kt = x̂t . Then DE is binding and vt = 0 still holds.
(iii) Suppose that γ x̂γ−1 > rt which is equivalent to the condition stated in (iii) of Lemma
1. Then, π(kt ) is increasing in kt < x̂t and has a unique maximum at kt = kt∗∗ > x̂t . Thus
the ﬁrm invests in capital kt = kt∗∗ totally and xt = x̂t is ﬁnanced by insider lending whilekt∗∗ − x̂t is ﬁnanced by market lending. Also vt = (1/∆q)(λrt x̂t + ψt − Vt+1 ) = 0 holds.
Next, suppose that ψt > Vt+1 (that is, x̂t < 0). Then the optimal choice of xt must
be xt = 0 under Assumption 1. Thus we must have vt = (1/∆q)(ψt − Vt+1 ) > 0, showingthat the explicit bonus must be used. Then the ﬁrm’s ﬂow proﬁt is given byπ(kt ) = ktγ − rt kt − (1 + q0 /∆q)ψt + (q0 /∆q)Vt+1
which is maximized at kt = kt∗∗ . Q.E.D.
Proof of Lemma 2. The implicit contract regime with non-binding DE occurs whenγ x̂γ−1≤ λrt due to Lemma 1 (i). In this case kt = kt∗ , i.e., γktγ−1 = λrt . Thus, the abovet
condition γ x̂γ−1≤ λrt can be written by
t
(
γ

)γ−1
kt1−γ
(Vt+1 − ψt )≤ λrt = γktγ−1γ

by using the deﬁnition of x̂t . This is further written by Vt+1 ≥ γktγ + ψt so that Qt+1 ≥40γ + ψ. Then, by using the deﬁnition of Vt and CMEt−1 , we obtain
Vt =
=
=
=
=

1 γ
{k − λrt kt − ψt + Vt+1 }rt t
λkt1−γ{(1 − γ)ktγ − ψktγ + Vt+1 }γ
λkt{1 − γ − ψ + Qt+1 }
γ
γ
γ
kt−1
(λkt /kt−1){1 − γ − ψ + Qt+1 }γγ
kt−1(L − Qt ){1 − γ − ψ + Qt+1 }γwhich yields
γQt= 1 − γ − ψ + Qt+1 .
L − Qt
The implicit contract regime with binding DE occurs when (γ/λ)x̂tγ−1 > rt ≥ γ x̂tγ−1(case (ii) of Lemma 1). In this case we know kt = xt = x̂t from Lemma 1 (ii), implyingthat rt = (1/λkt )(Vt+1 − ψt ). Then the above inequalities (γ/λ)x̂tγ−1 > rt ≥ γ x̂tγ−1 can be
written by γktγ + ψt > Vt+1 ≥ λγktγ + ψt , which shows that γ + ψ > Qt+1 ≥ λγ + ψ. Then,γ
by using the binding DE, ψt = ψktγ and CME (λkt /kt−1= L − Qt ) together with π(kt ),we obtain
Vt = (1/rt ){ktγ − ψt − λrt kt + Vt+1 }λktkγ=Vt+1 − ψt t
λkt=
Qt+1 − ψwhich yields
γ
Qt = Vt /kt−1=

implying thatγ
λkt /kt−1
Qt+1 − ψ

1 + (1 − p1 )ψ
Qt=.
L − QtQt+1 − ψ

Q.E.D.
Proof of Lemma 3. The mixed contract regime occurs when γ x̂tγ−1 > rt and Vt+1 ≥ ψt(case (iii) of Lemma 1). In this case we know kt = kt∗∗ , i.e., γktγ−1 = rt holds, and DE
becomes binding at xt = x̂t . Thus γ x̂γ−1> rt can be written by
t
)γ−1
(
kt1−γ
(Vt+1 − ψt )γ
> rt = γktγ−1
λγ
41which is further arranged as λγktγ + ψt > Vt+1 , implying that λγ + ψ > Qt+1 . Since
Vt+1 ≥ ψt is equivalent to Qt+1 ≥ ψ, we must have λγ + ψ > Qt+1 ≥ ψ.
By using π(kt ) in the proof of Lemma 1 and the binding DE, we have
Vt = (kt1−γ /γ) {(1 − γ)ktγ + (1/λ)(Vt+1 − ψt )} .
γ
By dividing both sides of this expression by kt−1
and using CME ((20) in the main text)
and the deﬁnition of Qt and Qt+1 , we can verify that

1 − γ + (1/λ)(Qt+1 − ψ)
Qt
)
].
(
= [
L − Qtγ 1 − 1−λ (Q
− ψ)
λγ

t+1

Q.E.D.
Proof of Lemma 4. The explicit contract regime occurs when ψt > Vt+1 , i.e., ψ > Qt+1
(Lemma 1 (vi)). Then, by using π(kt ), the ﬁrm’s market price is given by
Vt = (1/rt ){ktγ − rt kt − (1 + q0 /∆q)ψt + (1 + q0 /∆q)Vt+1 }γ
where rt = γktγ−1 . By using CME (kt = (L − Qt )kt−1) and rt = γktγ−1 , we obtainγ
Qt = Vt /kt−1
1=[(1 − γ)ktγ − (1 + (q0 /∆q)ψt + (1 + (q0 /∆q))Vt+1 ]γ
rt kt−1==γkt /kt−1[1 − γ − (1 + (q0 /∆q))ψ + (1 + (q0 /∆q))Qt+1 ]γ
L − Qt[1 − γ − (1 + (q0 /∆q))ψ + (1 + (q0 /∆q))Qt+1 ]γwhich yields the equilibrium value of Qt . Q.E.D.

We provide some preliminary results to characterize the equilibrium paths.
Lemma A1. In any equilibrium path the ﬂow proﬁt of the ﬁrms must be non-negative
πt ≥ 0 in each period t ≥ 0.
Proof. Suppose that πt < 0 holds in some period t. Then, the ﬁrms cannot pay out someparts of bt , vt and Rt from the total revenue ktγ in that period t. Thus the equilibrium
contract {bt , vt , Rt , kt , xt } in period t cannot be enforced. Instead, the ﬁrms must adjust ex
post to make diﬀerent payments b̃t , ṽt and R̃t the sum of which can be covered by the total
revenue ktγ , ensuring that the revenue is not less than the total payment ex post. Then,the ﬁrms should have oﬀered such contract which attains a non-negative proﬁt without
loss of generality. Q.E.D.
42

Lemma A2. In any equilibrium path the ﬁrms never shut down the productions in eachperiod t ≥ 0.
Proof. Suppose that the ﬁrms shut down in some period t. Thus the ﬁrms make nocapital investment kt = 0 and earn zero proﬁt πt = 0 in period t. Then, the ﬁrm market
price in period t satisﬁes Vt = (1/rt )Vt+1 by the no-arbitrage condition. Also the credit
market equilibrium in period t implies that 0 = Lktγ = kt+1 − (1 − λ)xt+1 + Vt+1 and hence′′
kt+1 = Vt+1 = 0 (note that kt+1 = kt+1
+ xt+1 ≥ 0, kt+1
≥ 0 and xt+1 ≥ 0). This showsthat Vt = 0 by the above no-arbitrage condition. Then the credit market equilibrium inγ
period t−1 in turn implies that Lkt−1= kt −(1−λ)xt +Vt = 0 and hence kt−1 = 0 because
kt = 0 implies xt = 0 as well. Repeating this process, we have ks = 0 and Vs = 0 for all
1 ≤ s ≤ t. However, in the initial period 0 the credit market equilibrium is then given
by Lk0γ = k1 − (1 − λ)x1 + V1 = 0 so that k0 = 0, contradicting to the initial condition
k0 > 0. Thus, kt > 0 holds in any period t ≥ 1. Also, in the initial period t = 0 theﬁrms can alway earn a positive proﬁt sk0γ > 0 by using the endowed initial capital k0 and
implementing low eﬀort from old workers. Q.E.D.
By Lemma A1 and A2 equilibrium paths must be that the ﬁrms produce and earn nonnegative proﬁts in each period. In addition, we can show that the ﬁrms must induce
high eﬀort from employed old workers in each period in any equilibrium path when the
productivity s = h(0) of shirking old worker is suﬃciently small (see Proposition B1 inthe appendix B below), which we assume in what follows.
Proof of Proposition 1. Suppose contrary to this claim that there exists some period
T such that the equilibrium path is in the mixed or explicit contract regime at T − 1:
QT ≤ λγ + ψ
but it is in the implicit contract regime at T :QT +1 ≥ λγ + ψ.
This implies that L > λγ + ψ: otherwise, QT +1 ≥ λγ + ψ ≥ L and then CME in period
T must imply that 0 ≥ (L − QT +1 )kTγ = λkT +1 and hence kT +1 ≤ 0. Then kT +1 = 0
must hold so that ks = 0 and Qs = 0 for all s > T + 1 due to CME from periodt onward. However, then QT +1 = 0 must be also satisﬁed because we have VT +1 =(1/rT +1 )(πT +1 + VT +2 ) = 0 by πT +1 = 0 (due to kT +1 = 0) and VT +2 = 0, a contradiction
to QT +1 > 0. Thus L > λγ + ψ.
Case 1: DE is binding in period T . In this case we have
QT
1=
L − QT
QT +1 − ψ43

Since QT ≤ λγ + ψ, this implies thatλγ + ψ
1≥.
L − (λγ + ψ)
QT +1 − ψWe deﬁne Q̃ such thatλγ + ψ
1.
=
L − (λγ + ψ)Q̃ − ψ
The above two (in)equalities imply that Q̃ ≤ QT +1 .
L
Next we show that Q̃ > Q′ ≡ λγ+1
if Q′ > λγ + ψ. In fact Q̃ > Q′ is equivalent to
λγ + ψ1< ′L − (λγ + ψ)Q −ψwhich can be in turn written by
L > L′ ≡ (λγ + 1)(λγ + ψ)
which holds if Q′ > λγ + ψ.
Now we consider three sub-cases as follows:
Case 1-1: Q′′ > γ + ψ. Then Q′′ > γ + ψ ≥ QT so that Q′′ > QT . However, then there
are no QT +1 such that QT = Φ(QT +1 ) and QT +1 ≥ λγ + ψ by deﬁnition of Q′′ (Figure
A1). Thus the equilibrium path never moves to the implicit contract regime in period T ,a contradiction.
Qt+1
Qt = Φ(Qt+1 )

45o

γ+ψ

λγ + ψ

Qt
λγ + ψ

γ + ψ Q′′

Qi

Figure A1: Q′′ > γ + ψCase 1-2: Q′ < λγ + ψ. Then Q′ < λγ + ψ ≤ QT +1 so that Q′ < QT +1 . Since in this case
we have Φ(Q) < Q for all Q > λγ + ψ, Qt increases over time and Qt → ∞ as t → ∞(Figure A2).
44

Qt+1

Qt = Φ(Qt+1 )
45o

γ+ψ

λγ + ψQ′

Qt
λγ + ψ

γ+ψ

Figure A2: Q′ < λγ + ψ
Case1-3: Q′′ < γ + ψ and Q′ > λγ + ψ (Figure A3). In this case we know that Q̃ > Q′ .
Thus Q′ < Q̃ ≤ QT +1 . However, when Q′ < QT +1 , the next period value QT +2 which
satisﬁes QT +1 = Φ(QT +2 ) must be above γ + ψ. Then, since Φ(Q) < Q for all Q > γ + ψ
if Q′′ < γ + ψ, Qt increases over time from period T + 1 and Qt → ∞ as t → ∞, acontradiction.
Qt+1
45o
Qt = Φ(Qt+1 )γ+ψ

λγ + ψQt
Q′′ Qi Q′γ+ψ

Figure A3: Q′′ < γ + ψ and Q′ > λγ + ψCase 2: DE is not binding in period T . In this case we have QT +1 > γ + ψ.

45

Case 2-1: Q′′ > λγ + ψ. Then Q′′ > λγ + ψ ≥ QT so that Q′′ > QT . Thus there exist no
QT +1 such that QT = Φ(QT +1 ) and QT +1 lies in the implicit contract regime in period
T + 1.
Case 2-2: Q′′ < λγ + ψ < Q′ . This must imply that Q′ < γ + ψ. If this is not the case, wehave Q′′ < λγ + ψ and Q′ ≥ γ + ψ. However, we then obtain(λγ + 1)(γ + ψ) < L < (γ + 1)(λγ + ψ)
due to the deﬁnitions of Q′ and Q′′ , which in turn implies that 1 < ψ. This contradicts
to Assumption 2. Thus Q′ < γ + ψ. Then, QT +1 > γ + ψ > Q′ must imply the following: QT +2 deﬁned as QT +1 = Φ(QT +2 ) has the property that QT +2 > γ + ψ. Also
QT +2 > QT +1 . Thus Qt → ∞ as t → ∞, a contradiction.
Case 2-3: Q′ < λγ + ψ. Since Qt increases over time when Q′ < λγ + ψ and QT +1 > γ + ψ,
we have Qt → ∞ as t → ∞, a contradiction. Q.E.D.
Proof of Proposition 2. Suppose that Qt ̸= Qi holds in some period t in some equilibrium path {Qt }∞t=1 which stays in the implicit contract regime forever from the initialperiod.
Case 1: Q′′ > γ + ψ (Figure A1). In this case Qi must satisfy Qi > γ + ψ so that DE is notbinding at Qi . Also, we must have Φ(Q) > Q for all Q ∈ (λγ + ψ, Qi ) and Φ(Q) < Q for
all Q > Qi . Thus, if Qt ̸= Qi , the equilibrium path {Qs }∞s=t from period t either diverges
to a positive inﬁnity or eventually becomes below λγ + ψ. In the former the equilibrium
path cannot be an equilibrium and in the latter the equilibrium path cannot stay in theimplicit contract regime anymore.
Case 2: Q′ < λγ + ψ (Figure A2). In this case we must have Φ(Q) < Q for all Q > λγ + ψ:
since Q′′ < Q′ < λγ + ψ < γ + ψ, it must be that Q′′ < γ + ψ and hence Φ(Q) < Q for all
Q > γ + ψ under Assumption 2. Also Φ(Q) < Q must hold for all Q ∈ (λγ + ψ, γ + ψ)
because Φ is decreasing in Q ∈ (λγ + ψ, γ + ψ) and Q′ < λγ + ψ. Then, the equilibrium
path {Qs }∞s=t diverges to a positive inﬁnity as long as it stays in the implicit contract
regime. This cannot be an equilibrium.
Case 3: Q′′ < γ + ψ and Q′ > λγ + ψ (Figure A3). In this case Qi satisﬁes λγ + ψ <Qi < γ + ψ. Also, Qτ < γ + ψ must hold for some period τ because otherwise Qs diverges
to inﬁnity, which cannot be an equilibrium. To see this, note that Φ(Q) < Q for all
Q ≥ γ + ψ in Case 3, implying that Qs → ∞ as s → ∞ if Qt ≥ γ + ψ for all t ≥ 1. Thus
Qτ < γ + ψ must hold. In what follows we suppose τ = t + 1 without loss of generality sothat Qt+1 < γ + ψ. Then, since Qt must be in the implicit contract regime and DE must
be binding in period t, we obtainQt
1=.
L − QtQt+1 − ψ46(A1)

First, suppose that Qt < Qi (Figure A4). Then there must exist some integer n ≥ 1such that Qt+n+1 < γ + ψ again: otherwise, Qs > γ + ψ must hold for all s > t sothat Qs diverges to inﬁnity as s → ∞, which cannot be an equilibrium. Furthermore,since Qt < Qi , it must be that Qt < Qi < Qt+n . Also, since Φ is decreasing in Q whenQ ∈ (λγ + ψ, γ + ψ), by using Qt < Qi and λγ + ψ < Qt+1 < γ + ψ, we must haveQt+1 > Qt . Given Qt+n+1 < γ + ψ, for the equilibrium path {Qs }∞s=t+n to stay in theimplicit contract regime, Qt+n+1 ≥ λγ + ψ must hold. Thus Qt+n+1 ∈ (λγ + ψ, γ + ψ).
Then, since the equilibrium path lies in the implicit contract regime, it must be that DE
is binding in period t + n:
1
Qt+n
=
.
(A2)
L − Qt+n
Qt+n+1 − ψQt+145o
Qt = Φ(Qt+1 )γ+ψ

Qt+n+1λγ + ψ

Qt
Qt+n+1

Qt

Qi

Qt+n

Figure A4: Q′′ < γ + ψ and Q′ > λγ + ψ
Furthermore, since Qt+n > Qi and Φ is decreasing in Q ∈ (λγ + ψ, γ + ψ), Qt+n+1 < Qimust be satisﬁed. We then show that Qt+n+1 < Qt . Suppose contrary to this claim that
Qt+n+1 ≥ Qt . Then, by using (A2) together with Qt+n > Qt+1 , we can show that
1
Qt+n
Qt+1
1
≥
=
≥
Qt − ψ
Qt+n+1 − ψ
L − Qt+n
L − Qt+1
which implies that
1
Qt+1
≥
.
Qt − ψ
L − Qt+1

47(A3)

By combining (A1) with (A3), we then obtain
Qt+1
L − QtQt+1 − ψ×≤
L − Qt+1
Qt
Qt − ψ
which is in turn rewritten by
ψL(Qt+1 − Qt ) ≥ Qt Qt+1 (Qt+1 − Qt ).
Since Qt+1 > Qt , the above inequality shows that ψL ≥ Qt Qt+1 . Thus Qt ≤ ψL/Qt+1 .
By using this, (A1) implies that
ψL/Qt+1Qt
1≥=
L − ψL/Qt+1
L − QtQt+1 − ψwhich in turn shows that ψ ≥ 1. This however contradicts to Assumption 2. Thus
Qt+n+1 < Qt must be satisﬁed. The above result has established that, if Qt < Qi ,Qt+n+1 < Qt must hold for some positive integer n. Then, Qt+n+1 < Qt < Qi so that
Qt+n+1 < Qi . By a similar argument, we can show that there exists some integer m > nsuch that Qt+m+1 < Qt+n+1 < Qi . By repeating this process, we obtain a decreasing
sequence Qt+k(j+1) < Qt+k(j) < · · · < Qt < Qi where k(j) is a positive integer andk(j + 1) > k(j) for j = 0, 1, 2, ... and k(0) ≡ 0. This sequence eventually becomes lower
than λγ + ψ. Then the equilibrium path {Qt }∞t=1 must leave the implicit contract regime.
This is a contradiction.
Second, suppose that Qt > Qi . Since Φ is decreasing in Q when λγ + ψ < Q < γ + ψ,
Qt > Qi and Qt+1 ∈ (λγ + ψ, γ + ψ) imply that Qt+1 < Qi . Then we can use the same
argument as in the case of Qt < Qi by replacing Qt by Qt+1 and starting the same steps
from Qt+1 < Qi .
Thus we have established the result that, if Qt ̸= Qi at some t, the equilibrium pathmust enter the mixed or explicit contract regime in some period T . Then, Proposition 1
implies that any equilibrium path must stay in the mixed contract or the explicit contract
regime from period T onward and that it never returns back to the implicit contract
regime. Q.E.D.
Proof of Lemma 6. By the deﬁnition of L̂ (ψ = Φ(ψ)), we have Qe = ψ at L = L̂. Also,
it can be veriﬁed that Qe is decreasing in L under Assumption 3 because the left handside of (24) in the main text decreases with L. Q.E.D.
Proof of Proposition 3. Suppose that Assumption 1-3 hold and that L̂ > L′ . Thensuppose that L > L̂.
Recall the deﬁnition of Q̂ such that Q̂ = Φ(ψ) and then note that Q̂ > ψ whereQ̂ ≡ L(1 − γ)and L > L̂ ≡ ψ/(1 − γ). When Qt+1 < ψ, we have the explicit contract regime in which
γQt= 1 − γ − (1 + (q0 /∆q))ψ + (1 + (q0 /∆q))Qt+1 .
L − Qt48

Then we can verify that dQt+1 /dQt > 0 and d2 Qt+1 /dQ2t > 0.
Also in the mixed contract regime we haveγQt
1 − γ + (−(1/λ))ψ + (1/λ)Qt+1=
L − Qt1 − 1−λλγ (Qt+1 − ψ)from which we obtain
Qt+1 =(1 + (1 − λ)ψ/λγ)γZt − (1 − γ + (−(1/λ))ψ)((1 − λ)/λ)Zt + (1/λ)

where Zt ≡ Qt /(L − Qt ). Then in the mixed contract regime we can verify that()21
dQt+1 /dQt ∝>0
L − λQt
and d2 Qt+1 /dQ2t > 0. Furthermore, in the mixed contract regime Φ(Q) > Q and Φ′ > 0
hold due to Q′ > λγ + ψ (by L > L′ ) and Q̂ > ψ (by L > L̂).
In the explicit contract regime we can also show that Qt+1 > 0 holds at Qt = 0 (i.e.,−1Φ (0) > 0) under Assumption 3. Recall that Q̂ = Φ(ψ). Then, since Q̂ > ψ (i.e.,Φ−1 (ψ) < ψ under L > L̂), there exists some Q ∈ (0, ψ), denoted by Qe , such that
Qe = Φ(Qe ) in the explicit contract regime:
[
(
) ]
q0
Qe
= (1/γ) 1 − γ − (1 + q0 /∆q)ψ + 1 +Qe .
L − Qe∆qSuch Qe is unique because dQt+1 /dQt > 0 and d2 Qt+1 /dQ2t > 0 together with Φ−1 (0) > 0
and Φ(ψ) > ψ.
Since Φ is increasing over [0, λγ + ψ] and Qe is the unique value of Q satisfying Q =Φ(Q) over [0, λγ + ψ], we can see that there exist equilibrium paths {Qt }∞t=1 (i) startingwith Q1 ∈ (λγ + ψ, γ + ψ) in the implicit contract regime in which DE becomes bindingin period t = 0 and (ii) converging to Qe as t → ∞ (Figure 3). Along such process, theequilibrium path {Qt }∞t=1 moves from the implicit contract regime to the mixed contract
regime and then to the explicit contract regime over time. Since Q1 is a jump variable,
there exist multiple paths with diﬀerent initial values of Q1 ∈ (λγ + ψ, γ + ψ).
In the initial period t = 0 the ﬁrm obtains the equilibrium proﬁt as π0 = k0γ [1 −ψ − max{(q0 /∆q)(ψ − Q1 ), 0}] when implementing high eﬀort a = 1 and π0 = sk0γ when
implementing low eﬀort a = 0 respectively. The ﬁrm attains the larger one of these proﬁts.
Here the initial capital k0 is exogenously given. Thus the initial period proﬁt is determined
only by the initial ﬁrm value Q1 .
Finally, we can verify that the ﬂow proﬁt of the ﬁrm becomes positive in any period
in the equilibrium path constructed above.
First, in the implicit contract regime when DE is not binding, the ﬁrm’s ﬂow proﬁt is
given byπt = ktγ − λrt kt − p1 ψt= ktγ (1 − γ − ψ)
> 049

due to Assumption 2. When DE is binding, it becomes
πt = ktγ − λrt kt − ψt
= ktγ − Vt+1
= ktγ − Qt+1 )> ktγ (γ + ψ − Qt+1 )≥ 0because of Assumption 2 and γ + ψ ≥ Qt+1 when DE is binding.
In the mixed contract regime the ﬂow proﬁt of the ﬁrm is given by
πt = ktγ − rt kt + (1 − λ)rt xt − ψt= (1 − γ)ktγ − ψt + (1 − λ)rt xt= ktγ (1 − γ − ψ) + (1 − λ)rt xt> 0
due to Assumption 2.
In the explicit contract regime the ﬁrm’s expected ﬂow proﬁt is given by
πt = ktγ − rt ktγ − (1 + q0 /∆q)ψt + (q0 /∆q)Vt+1= ktγ {q1 [1 − γ − (Qt+1 + (1/∆q)(ψ − Qt+1 ))] + (1 − q1 ) [1 − γ − Qt+1 ]}where γktγ−1 = rt . Note here that the ﬁrm must pay bt + vt = Vt+1 + (1/∆q)(ψt − Vt+1 )when the old worker’s veriﬁable signal is realized as σt = σh while it must pay bt = Vt+1
when σt = σl respectively. In either case of σt = σh or σt = σl the ﬁrm’s realized proﬁt
must be non-negative. Since ψ > Qt+1 , it then suﬃces to show that
1 − γ − (Qt+1 + (1/∆q)(ψ − Qt+1 )) ≥ 0
which is rewritten by
Qt+1 ≥ Q ≡ψ − ∆q(1 − γ)
> 0.
1 − ∆qHere Q > 0 holds due to Assumption 3. Also Q < ψ is satisﬁed due to Assumption 1.
Since the equilibrium sequence Qt is decreasing over time (Qt+1 ≥ Qt ) and converges to
Qe , it is enough to obtain Qe ≥ Q. By Lemma 6, Qe is decreasing in L and Qe = ψ at
L = L̂. Also Qe → Q∗ when L → ∞ where Q∗ is deﬁned as the ﬁrm value making the
expected ﬂow proﬁt q1 [1−γ −(Q+(1/∆q)(ψ −Q))]+(1−q1 )[1−γ −Q] zero. Then Q > Q∗
holds. Thus, we can ﬁnd some L > L̂ such that for all L ∈ (L̂, L) we have ψ > Qe > Q
because Q∗ < Q < ψ. Q.E.D.
Proof of Proposition 4. Suppose that Assumption 1-4 and L > max{L̂, L′ } are satisﬁed.
By Proposition 2, we know that any equilibrium path must be either Qt = Qi for allt ≥ 1 or Qt ̸= Qi for some period t.
50

(i) In the former case (Qt = Qi for all t ≥ 1) the equilibrium path must stay in theimplicit contract regime forever from the initial period t = 0. Then it follows from CME
that (L − Qi )ktγ = λkt+1 for t = 0, 1, 2, ... and hence kt → k i as t → ∞. If the initial
capital k0 satisﬁes (L−Qi )k0γ > λk0 , we have kt ≤ k i for all t ≥ 0 because (L−Qi )k γ > λkγ
> λkt−1 so that kt > kt−1 and hence gt > 0 forfor all k < k i . Then, λkt = (L − Qi )kt−1
all t ≥ 1.
(ii) Second, when Qt ̸= Qi holds for some t in an equilibrium path, such path must enter
the mixed or explicit contract regime from some period T onward and never returns backto the implicit contract regime anymore. Since L > max{L̂, L′ }, Φ(Q) > Q holds for all
Q ∈ [ψ, λγ + ψ]. Thus, if the equilibrium path lies in the mixed contract regime in someperiod, the ﬁrm value Qt declines from that period over time and eventually enters the
explicit contract regime in which Qt < ψ. Then, Qt → Qe holds as t → ∞, which also
implies that kt → k e in such limit.
Furthermore, when the economy enters the mixed contract regime, Qt = Φ(Qt+1 )satisﬁes)(
γQtψ
1 + 1−λλγ
L−Qt − η
Qt+1 =
1−λ Qt1
λ L−Qt + λwhere we deﬁne η ≡ 1 − γ + (1 − p1 − (1/λ)). Then we have1−

1−λ(Qt+1 − ψ) =λγ=

L − Qt[η + (1/λ)Qt+1 ]
γQt
(
)


γQt
1−λ
1
+
ψ
λγ
L−Qt − η
L − Qt η+γQt(1 − λ) Qt + 1
L−Qt

where the last bracket term can be re-written by
(
)

 [
]
γQt
1 + 1−λψη(1 − λ) + γ + 1−λψλγ
L−Qt − η
λ
η +
=
Qt
Qt
L − λQt(1 − λ) L−Q+1t

Thus CME in the mixed contract regime implies that
[
]
1
1−λ
γ
kt−1 =
1−(Qt+1 − ψ) kt(L − Qt )λγ[]η(1 − λ) + γ + 1−λψλ=kt
L − λQt
so that[kt]
L − λQt
γ
= γ
kt−1
1−λ
η(1 − λ) + γ + λ ψγ≡ G(Qt )kt−1

51

The right hand side of the above expression is decreasing in Qt . Since Qt decreases overtime in the mixed contract regime, we can show that kt+1 /ktγ = G(Qt+1 ) > G(Qt ) =γ
kt /kt−1, which implies that (gt+1 + 1) > (gt + 1)γ for gt ≡ kt /kt−1 − 1. Then gt increases
whenever gt < 0 and gt+1 > 0 holds as long as gt > 0. This shows the desired result.
In the explicit contract regime CME implies thatγkt = (L − Qt )kt−1.

Since Qt decreases over time when Qt < ψ, we have (gt+1 + 1) > (gt + 1)γ again, yielding
the desired result as well. Q.E.D.
Proof of Proposition 5. When λ → 0, we have L̂ > L′ . Then, under Assumption 1-3
and L > L̂, there are the steady states only in the implicit contract regime (Qi , k i ) andthe explicit contract regime (Qe , k e ). Then SW i > SW e holds as λ → 0 because Qi and
Qe are independent of λ.
Next we take λ close to 1. When L > L′ still holds, the long run steady states are
only the implicit contract steady state and the explicit contract steady states again. In
this case the long run welfare in the implicit contract regime becomesγ

lim SW i = (L − Qi ) 1−γ (1 + Qi )λ→1When L′ > L, the implicit contract steady state no longer exists but the mixed contract
steady state (Qm , k m ) exists in addition to the explicit contract steady state. The steadystate welfare in the mixed contract regime SW m becomes(lim SW m = lim

λ→1

λ→1

L − Qmm
1 − 1−λλγ (Q − ψ)

) γ

1−γ

(1 + Qm ) = (L − Qm )γ/(1−γ) (1 + Qm )where Qm → Q∗ as λ → 1 and recall that Q∗ satisﬁesγQ∗= 1 − γ − ψ + Q∗ .
L − Q∗On the other hand, the steady state welfare in the explicit contract regime still exists andis given by SW e which is independent of λ. Also Qe < Qm holds.
We now compute from SW (Q) ≡ (L − Q)γ/(1−γ) (1 + Q) thatγγγ−1(L − Q) 1−γ (1 + Q) + (L − Q) 1−γ
1−γ
{
}
γ
γ
−1
1−γ
= (L − Q)−(1 + Q) + L − Q
1−γSW ′ (Q) = −The bracket term in the above expression of SW ′ (Q) is decreasing in Q. Thus
SW ′′ (Q) < 0 at any Q satisfying SW ′ (Q) = 0 (note that L > Q must hold in any
steady state. Thus L ̸= Q.) This shows that SW is a quasi-concave function. Also, we52have SW ′ (Qe ) ≤ 0 if and only if (1 − γ)L − γ ≤ Qe . Since the realized proﬁt of the ﬁrm
must be non-negative in the steady state in the explicit contract regime, it must be thatQe ≥ Q for the cut oﬀ value Q deﬁned in the proof of Proposition 3. This implies that
γQ
L−Q

≥

[

]
1 − γ − (1 + q0 /∆q)ψ + (q0 /∆q)Q + Q≥ Q
because the expected ﬂow proﬁt, expressed by the above bracket term, becomes nonnegative at Q. Thus, Q ≥ L−γ which shows that Qe ≥ Q ≥ L−γ. Then Qe ≥ (1−γ)L−γ
is satisﬁed. This implies that SW ′ (Qe ) ≤ 0. Since Qe ≤ Qm → Q∗ and Qe ≤ Qi , weobtain that SW (Qe ) = SW e > SW i = SW (Qi ) and SW e > SW (Qm ) = SW m . Q.E.D.

8

Appendix B: Extensions

In this appendix we investigate several extensions of the model.

8.1

Separable Wage

A wage of an old worker can be written by W (s, σ) in more general form, which depends
on the perfect but unveriﬁable signal s = a ∈ S ≡ {0, 1} and the imperfect but veriﬁable
signal σ ∈ Σ ≡ {σh , σl } in the interdependent way. We however show that we can restrict
our attention to the additive separable form W (s, σ) = b(s)+v(σ) without loss of generality
as we have considered in the main text.
To show this, suppose that W (s, σ) satisﬁes DE, WIC and LL. Then deﬁne a new wage
scheme as follows:vt (σ) ≡ min Wt (s, σ)s∈Sfor an explicit wage scheme and∑bt (s) ≡q(σ|s)[Wt (s, σ) − min Wt (s, σ)]
s∈S

σ∈Σ

for an implicit wage scheme respectively. Here, q(σ|s) ∈ (0, 1) denotes the probability ofthe veriﬁable signal σ being realized conditional on an unveriﬁable signal (eﬀort) s = a
respectively. By construction, vt (σ) ≥ 0 and bt (s) ≥ 0 hold for all s ∈ S and σ ∈ Σ.
Then we have∑∑q(σ|s)[bt (s) + vt (σ)] =q(σ|s)Wt (s, σ)
σ∈Σ

σ∈Σ

for any perfect unveriﬁable signal (eﬀort) s = a so that the new scheme still implements
the same eﬀort as the original one does. Also, the expected wage of the new wage scheme
is same as the original one as well.

53

Finally, we check that the new wage scheme satisﬁes DE. Since the original wage
scheme W satisﬁes DE, we have
Vt+1 ≥ Rt + Wt (s, σ) − min Wt (s, σ)s∈Sfor each σ ∈ Σ. This states that the ﬁrm has no incentive to renege on the worker’s wage
Wt (s, σ) and the repayment Rt to the ﬁnancier by claiming that the worker’s unveriﬁable
signal s ∈ S was the one such that the wage is the lowest, i.e., mins Wt (s, σ), rather thanpaying the agreed upon one Wt (s, σ), for each realization of the two signals s ∈ S andσ ∈ Σ.
Then, by using the deﬁnition of bt and taking the expectation about the above inequality DE over σ, we obtain∑
Vt+1 ≥ Rt +q(σ|s)[Wt (s, σ) − min Wt (s, σ)]
s∈S

σ∈Σ

= Rt + bt (s)≥ Rt + bt (s) − min bt (s).
s∈Swhich implies that the new wage scheme bt satisﬁes DE: under the bonus scheme bt , theﬁrm can renege on bt and Rt by claiming that the worker’s signal s was the one such thatthe bonus is the lowest, i.e., mins bt (s), which is however less proﬁtable than paying theagreed upon implicit wage bt (s) for each realization of s ∈ S.

8.2

Equity Financing

We have assumed in the main text that the ﬁrms borrow the remaining capital investment
kt − xt from the credit market which is not covered by insider lending. Alternatively, theﬁrms can issue new equities to ﬁnance some part of kt −xt . However, such equity ﬁnancing
is irrelevant as long as the no-arbitrage condition holds.
We denote by nt the number of equities issued by a ﬁrm (owned by young entrepreneurs)in the end of period t − 1 for ﬁnancing some kt′′ of kt − xt which will be used for productionin period t while the remaining kt′ ≡ kt − xt − kt′′ is borrowed from the credit market. n0
denotes the number of existing equities held by initial old individuals in period 0, whichis exogenously given. An equity contract issued in period s speciﬁes a stream of dividends{dst }∞
has the ﬁnancial claim to receive dst in period
t=s over time such that each shareholder∑tt ≥ s from the ﬁrm. Note that Πt = s=0 ns dst holds where Πt = ktγ − rt kt′ − Rt − E[b∑ t + vt ]denotes the proﬁt of a ﬁrm which should be shared between shareholders owning ts=0 ns
total equities of the ﬁrm in period t. Here E[bt + vt ] denotes the expected wage paid toan employed old worker.
The period t value of an equity issued in period s ≤ t, denoted by Vts , satisﬁes the
no-arbitrage condition:srt Vts = dst + Vt+1.
(B1)
Consider the total value of equities of a ﬁrm owned by the existing (young) individuals
in period t − 1 before they issue new equities for ﬁnancing a capital investment kt′′ for the54

productionin period t. We deﬁne such total equity value of the existing shareholders as∑sṼt ≡ t−1
s=0 ns Vt . The young shareholders who own the ﬁrm in period t − 1 behave tomaximize their total payoﬀs represented by the total value Ṽt .
By multiplying (B1) by nst and summing over s from s = 0 to s = t, we obtainrtt∑ns Vts =s=0t∑

ns dst +

s=0

t
∑

s
ns Vt+1.

s=0which can be written byrt (Ṽt + nt Vtt )= Πt +t∑sns Vt+1

s=0= Πt + Ṽt+1
Since newly issued equities are used to ﬁnance kt′′ , we obtain nt Vtt = kt′′ . By using this,the above expression can be further written by
rt Ṽt = Πt − rt kt′′ + Ṽt+1= ktγ − rt (kt′′ + kt′ ) − Rt − E[bt + vt ] + Ṽt+1= ktγ − rt (kt − xt ) − Rt − E[bt + vt ] + Ṽt+1
which is exactly same as the value Vt deﬁned in the main text. The young shareholders
owing the ﬁrm in period t − 1 thus has the same objective Vt to maximize as we haveconsidered in the main text.
Finally, we check that the credit market equilibrium (CME) is not essentially changed
from the one in the main text. Suppose that each of 1 − Nf young entrepreneurs born inperiod t − 1 purchases lts units of equities issued in period s. Note that (1 − Nf )lts = N ns
must hold because the total demand in period t − 1 for the equity issued in period s shouldbe equal to its total supply. Since the period t − 1-price∑of an equity issued in period s isgiven by Vts , each of those young entrepreneurs spends ts=0 lts Vts for purchasing the total
equities in period t − 1. Also the young entrepreneurs born in period t − 1 must spend′′kt′ ≡ kt −∑xt t − kt new capital investment by market lending per ﬁrm. Thus, since there aretotally s=0 ns units of equities per ﬁrm which∑t have been issued up to period t − 1, each′of those young entrepreneurs must spend kt / s=0 ns per equity he or she owns. Since heor she owns lts units for period t-equity issued in period s, he or she must spendt∑(kt − xt − kt′′ )lts ∑ts=0 nss=0

for ﬁnancing new capital investment by market lending. Thus, the young entrepreneurs
born in period t − 1 have the excess credit demand as follows
{
}
t
t
′′ )
∑
∑
(k
−
x
−
k
t
t
t
(1 − Nf ) wt−1 −lts Vts −lts ∑t
n
s=0 s
s=0
s=0
55

which is equivalent tott∑∑(kt − xt − kt′′ )s s(1 − Nf )wt−1 −(1 − Nf )lt Vt −(1 − Nf )lts ∑t
s=0 nss=0
s=0= (1 − Nf )wt−1 − Nt∑ns Vts − N (kt − xt − kt′′ )s=0= (1 − Nf )wt−1 − N

{ t−1
∑

}
ns Vts + nt Vtt + (kt − xt − kt′′ )s=0= (1 − Nf )wt−1 − N {Ṽt + kt′′ + (kt − xt − kt′′ )}= (1 − Nf )wt−1 − N (Ṽt + (kt − xt )).
This is essentially same as the excess credit demand of the young entrepreneurs which wehave obtained in the main text.
Since the young ﬁnanciers born in period t − 1 have the excess credit demand asNf (wt−1 − mλxt ) where m stands for the number of the ﬁrms each ﬁnancier lends to, it is
not changed from the one given in the main text. Since Nf m = N holds, the total excess
credit demand in the economy in period t − 1 is given by
wt−1 − N {kt − (1 − λ)xt + Ṽt }which must be zero in CME. This condition is same as the CME given in the main text.
Hence, equity ﬁnancing does not aﬀect the results obtained in the main text.

8.3

Endogenous Endowment

For simplicity, in the main text we have assumed that young capitalists and old workers
can access to the private technology which produces the outputs by themselves: young
capitalists can produce wt = LAt units of good which becomes their endowment where
At = N yt denotes the social knowledge embodied in the aggregate outputs yt . Old workers
can secretly access to the private technology which produces ψAt units of good by spending
endowed labor skill.
In this section we will provide a microeconomic foundation for this setting.
In stead of assuming the private technology, we introduce a perfectively competitive
labor market in which young capitalists and old workers supply their endowed labor units
to earn the competitive market wage, which becomes the sources for the credit supply ofthe economy. In addition to the prefect competitive labor market there is another labormarket in which old workers are hired by the ﬁrms as considered in the basic model. They
arrive at each ﬁrm every period and one worker among those who visited a ﬁrm is hiredby the ﬁrm. The employed old workers work by choosing eﬀort a ∈ {0, 1} as in the mainmodel.
We assume that each young capitalist (entrepreneur or ﬁnancier) is endowed with one
unit of labor while each old worker is endowed with B > 0 units of labor. They can56inelastically supply their endowed labor units to the perfectly competitive labor market
to earn the market wage, denoted by wt . The old workers who are employed by the ﬁrms
can decide whether to choose high eﬀort (at = 1) by spending B units of labor for the
production of the ﬁrm or to choose low eﬀort (at = 0) by secretly spending the endowed
labor to the perfectly competitive labor market to earn Bwt outside incomes.
We then slightly change the basic model as follows: There is the ﬁnal good sector inwhich many ﬁnal good producers produce the ﬁnal good, of which price is normalized tounity. The ﬁrms modeled in the main text are reinterpreted as the ones producing theideas to be used for the production of the ﬁnal good. In what follows, when we refer to“ﬁrm”, it means the ﬁrm producing the ideas while, when we refer to “producer”, it means
the producer producing the ﬁnal good respectively. We suppose that the ﬁnal good sector
is the decentralized matching market in which the ﬁnal good producers and the ﬁrms
producing the ideas are randomly matched with each other. When they are matched in
the end of period t − 1, the ﬁrm invests in capital kt to produce the ideas yt = h(at )ktγ
in period t. In the beginning of period t, the ﬁnal good producer who is matched with aﬁrm expects the ideas developed by the matched ﬁrm, denoted by ŷt , and then employs
the competitive labor Lt from the perfectly competitive labor market. By combining theideas yt with the employed labor Lt , a matched pair of a ﬁnal good producer and a ﬁrm
can produce the ﬁnal good outputs Yt = yt Lt . At the production stage, the ﬁnal good
producer is assumed to observe the ﬁrm’s ideas yt and then enter the bargaining stage inwhich they split the ﬁnal good outputs Yt according to a constant share; the ﬁrm obtainsβ ∈ (0, 1) share of Yt while the ﬁnal good producer obtains the remaining share 1 − βrespectively.
The ﬁnal good producer matched with a ﬁrm thus obtains the proﬁt (1−β)Yt −wt Lt =(1 − β)ŷt Lt − wt Lt . Then, the proﬁt maximization yields (1 − β)ŷt = wt . In the rational
expectation equilibrium we have yt = ŷt so thatwt = (1 − β)yt = (1 − β)ktγ
provided at = 1 as considered in the main text.
The ﬁrm matched with a ﬁnal good producer obtains the share of the ﬁnal good outputs
as βYt = βktγ because Lt = 1 + (1 − N )B = 1 holds in the competitive labor market. Notethat all young capitalists supply their endowed labor to the competitive labor market but
1 − N old workers who are not matched with the ﬁrms work in the competitive labormarket. However, since we assume N = 1, Lt = 1 holds. Then, the ﬁrm matched with aﬁnal good producer earns the proﬁt as
πt = βktγ − rt (kt − xt ) − λrt xt − (bt + q0 vt )which is essentially same as the one derived in the main text.
Each old worker who is matched with a ﬁrm earns the expected wage as
bt + q0 vt = ψt + (q0 /∆q) max{ψt − Vt+1 , 0}
where ψt ≡ Bwt = B(1 − β)ktγ .
57

The credit market equilibrium is modiﬁed as follows: one unit mass of young capitalists
earn the competitive market wage wt−1 in period t − 1 which become the savings of theeconomy. Thus, the credit market clears whenwt−1 = kt − (1 − λ)xt + Vt
which is same as in the main text. Here, wt−1 = (1 − β)ktγ .
By deﬁning the ﬁrm value as Vt /βktγ , we can obtain the equilibrium equations governing
the ﬁrm values {Qt }∞t=1 in the diﬀerent regimes which are similar to what we have derived
in the main text, by replacing L by (1 − β)/β and deﬁning ψ ≡ B(1 − β)/β.
We can then ensure Assumption 1 made in the main text by taking β ∈ (0, 1) close
to 1 and/or small B so that 1 − γ > ψ ≡ (1 − β)B/β. Also, Assumption 2 and 3
are all satisﬁed for small ∆q, given ψ. We have also made the conditions L̂ > L′ in
Proposition 3. This is modiﬁed as ψ/(1 − γ) > (1 + λγ)(λγ + ψ) which is satisﬁed aswell for small B (and/or β close to 1) and small λ ∈ (0, 1). The condition L > L̂
made in Proposition 3 then holds when 1 − γ > B, by replacing L by (1 − β)/β because(1 − β)/β > L̂ ≡ ψ/(1 − γ) = B(1 − β)/β(1 − γ) which is rewritten by 1 − γ > B.

8.4

Contingent Repayment

In the main text we have assumed that the repayment to a ﬁnancier is not contingent
on the worker’s performance signals. This is a realistic assumption because it is rarely
observed in the real world that the repayment to a bank varies with the task performancesof workers who are employed by the client ﬁrm borrowing from the bank. It is hence
reasonable to assume that ﬁnancier’s repayment is independent of the worker’s signals.
However, in a theoretical interest it is possible to allow the ﬁnancier’s repayment Rt to becontingent on worker’s signals s = a ∈ {0, 1} and σ ∈ {σl , σh }. In this subsection we willconsider such extension.
Let denote by Rt (s, σ, x′′t ) the repayment to a ﬁnancier in period t, which is contingent
on the worker’s signals s = a and σ, when the ﬁnancier provides a relation-speciﬁc capital
x′′t . Suppose that the ﬁnancial contract speciﬁes the repayment Rt (s, σ, x′′t ) ≥ 0 to theﬁnancier and the relation-speciﬁc capital xt to be provided by the ﬁnancier. 28
Deﬁne
R̃t (s, σ) ≡ Rt (s, σ, xt ) − min Rt (s, σ, z) ≥ 0
σ,zand set a new repayment scheme:
Rt′′ (a) ≡{Eσ [R̃t (a, σ)|a] if z = xt0otherwise

The repayment Rt (s, σ, x′′t ) must be non-negative. Otherwise, for some s, σ and xt , the ﬁnancier isrequired to make a positive payment to the ﬁrm. However, then the ﬁnancier reneges on that payment
because, even in doing so, there are no ways to punish the ﬁnancier: the ﬁnancier can always refuse
to pay and secure at least the payoﬀ rt (wt−1 − λxt ) while, if she makes a payment Rt < 0, she earns
Rt + rt (wt−1 − λxt ) which is less than the former payoﬀ.
28

58

Here Eσ [·|a] denotes the expectation over σ conditional on the perfect but unveriﬁable
signal s = a.
Then, since the original scheme Rt (·, ·, xt ) implements the provision of relation-speciﬁc
capital xt , the new repayment scheme does so:λrt (xt − z) ≤ Eσ [Rt (a, σ, xt ) − Rt (a, σ, z)|a]≤ Eσ [Rt (a, σ, xt ) − min Rt (a, σ, z)|a]σ,z= Eσ [R̃t (a, σ)|a]for all z ̸= xt . Thus, λrt (xt − z) ≤ Eσ [R̃t (a, σ)|at ] = Rt′′ (at ) for all z ̸= xt . This is thedesired result. In particular, we have Rt′′ (at ) ≥ λrt xt by setting z = 0, which implies
that the new repayment scheme satisﬁes IR of the ﬁnancier. Also, the new repayment
scheme achieves a lower expected repayment than the original one because Rt′′ (at ) =Eσ [R̃t (at , σ)|at ] ≤ Eσ [Rt (at , σ, xt )|at ] (note that minσ,z Rt (s, σ, z) ≥ 0.) Then, generality
is not lost by focusing on the repayment scheme Rt (a) which is contingent only on theperfect unveriﬁable signal s = a ∈ {0, 1} of the worker but not on the veriﬁable signal
σ ∈ Σ.
Then, when the old worker chooses ât ∈ {0, 1} in equilibrium, the ﬁnancier can fully
anticipate that she will surely face the repayment Rt (ât ) on the equilibrium path. Thus,
the IR of the ﬁnancier is given by Rt (ât ) ≥ λrt xt which is same as one in the main text.
Also, DE constraint is not changed because Vt+1 ≥ Rt (ât ) + bt must hold following theworker’s eﬀort choice ât on the equilibrium path. Thus, the equilibrium contract is notchanged at all even when we allow the repayment of ﬁnancier Rt to be contingent on thesignals of the employed old worker.

8.5

Dropping Assumption 1

When Assumption 1 is dropped, the ﬁrms always use insider lending but never resort tomarket lending at all. Thus Assumption 1 is necessary for market lending to emerge in an
equilibrium path. This is shown as follows: the ﬂow proﬁt deﬁned in the proof of Lemma
1, Π(kt , xt ), is always increasing in xt , given kt , when Assumption 1 is not satisﬁed. Thus
the ﬁrm always chooses xt = kt so that it never uses market lending at all.

8.6

Sub-Optimality of Implementing Low Eﬀort at = 0

In the main text we have assumed that the ﬁrms always implement high eﬀort at = 1 from
employed old workers. In this subsection we will show that it becomes never optimal forthe ﬁrm to induce old workers to choose low eﬀort when the worker’s productivity in thatcase s ≡ h(0) is suﬃciently small.
Speciﬁcally we make the following assumption:
Assumption B1. (1 − γ)(s/λ)γ/(γ−1) > (1 + q0 /∆q)ψ + s.

59

Assumption B1 is satisﬁed when s is small enough. Then we can show the following.
Proposition B1. Suppose that Assumption B1 holds. Then, in any equilibrium path
at = 1 must hold for all t ≥ 1. 29
Proof. Suppose that there exists an equilibrium path in which the ﬁrms implement low
eﬀort at = 0 from matched old workers in some period t. Then, the ﬁrm’s ﬂow proﬁt inperiod t becomesΠ(kt , xt ) ≡ sktγ − rt kt + (1 − λ)rt xt
in the equilibrium in which the ﬁrms implement low eﬀort at = 0 because the ﬁrms pay
zero wage for implementing at = 0. Here Vt+1 ≥ λrt xt so that xt ≤ x̂t ≡ Vt+1 /λrt . SinceΠ is increasing in xt , we have xt = kt when kt ≤ x̂t and xt = x̂t when kt > x̂t respectively.
Thus the ﬁrm’s ﬂow proﬁt is written by{ γ
skt − λrt kt
if kt ≤ x̂t
π(kt ) ≡γ
skt − rt kt + (1 − λ)(Vt+1 /λ) if kt > x̂t
The ﬁrms choose kt to maximize π(kt ) in the equilibrium. There are three cases whichmight happen in the equilibrium.
Case 1: The ﬁrms choose the equilibrium capital kt = x̂t . That is, rt > γsx̂tγ−1 > λrt .
Then the equilibrium proﬁt becomes π t = sx̂γt − Vt+1 .
Suppose that some ﬁrm deviates from the equilibrium by implementing high eﬀort
at = 1 from matched old workers and borrowing all capital zt from market lending. Suchdeviation yields at least the ﬂow proﬁt as follows:
πt′′ ≡ max ztγ − rt zt − ψt − max{(q0 /∆q)(ψt − Vt+1 ), 0}zt= (1 − γ)ztγ − ψt − max{(q0 /∆q)(ψt − Vt+1 ), 0}
where ψt = ψyt = ψx̂γt in the supposed equilibrium and zt satisﬁes γztγ−1 = rt . Sincesγ x̂γ−1> λrt = λγztγ−1 , we obtain zt > (s/λ)1/(γ−1) x̂t . Thus the deviation proﬁt becomest
πt′′ ≥ (1 − γ)(s/λ)γ/(γ−1) x̂γt − ψt − max{(q0 /∆q)(ψt − Vt+1 ), 0}≥ (1 − γ)(s/λ)γ/(γ−1) x̂γt − (1 + (q0 /∆q))ψt= (1 − γ)(s/λ)γ/(γ−1) x̂γt − (1 + (q0 /∆q))ψx̂γtwhich can be larger than π t under Assumption B1.
29
In the initial period t = 0 the ﬁrms may implement low eﬀort from employed old workers (a0 =0). However, this does not change the equilibrium path of the ﬁrms values {Qt }∞t=1 : the credit market
equilibrium in t = 0 is written by Ly0 = k1 − (1 − λ)x1 + V1 so that L = (k1 − (1 − λ)x1 )/y0 + Q1 . Also
the initial ﬁrm value is given by Q1 = (y1 /r1 y0 )(π1 /y1 + Q2 ) where the term (y1 /r1 y0 ) depends only on
Q1 and ψ1 by using the above credit market equilibrium. Thus the initial output y0 and hence initial old
workers’ eﬀort a0 = 0 do not aﬀect the equilibrium paths of the ﬁrm values.

60

Case 2: The ﬁrms choose the equilibrium capital kt < x̂t . Then, sγktγ−1 = λrt . The
equilibrium proﬁt becomes π t = (1 − γ)sktγ . Also ψt = ψyt = ψktγ holds.
Consider the same deviation by some ﬁrm as in Case 1. The deviating ﬁrm chooses
capital zt satisfying γztγ−1 = rt = (s/λ)γktγ−1 . Thus zt = (s/λ)1/(γ−1) kt . Then thedeviation proﬁt is at least
πt′′ ≥ (1 − γ)(s/λ)γ/(γ−1) ktγ − (1 + (q0 /∆q))ψktγ
which is larger than the equilibrium proﬁt π t = (1 − γ)sktγ under Assumption B1.
Case 3: The ﬁrms choose the equilibrium capital kt > x̂t . Then the ﬁrms choose sγktγ−1 =rt and earns the ﬂow proﬁt π t = (1 − γ)sktγ + (1 − λ)(Vt+1 /λ). Since kt > x̂t , we then have
π t = (1 − γ)sktγ + (1 − λ)rt x̂t≤ (1 − γ)sktγ + (1 − λ)rt kt= (1 − γ)sktγ + (1 − λ)sγktγ= (1 − γλ)sktγ
The same deviation as in Case 1 gives the capital choice zt satisfying γztγ−1 = rt = γsktγ−1 .
Thus the deviation yields at least the following ﬂow proﬁtπt′′ ≥ (1 − γ)sγ/(γ−1) ktγ − (1 + (q0 /∆q))ψt
where ψt = ψktγ . Thus the deviation proﬁt can be larger than the equilibrium one if(1 − γ)sγ/(γ−1) − (1 + (q0 /∆q))ψ > (1 − γλ)swhich is satisﬁed under Assumption B1. Q.E.D.
Next we show the following result.
Proposition B2. Suppose that Assumption 1-3 and L̂ > L′ hold. Then, for suﬃciently
small s, there exists some L > L̂ such that for all L ∈ (L̂, L) there exists the equilibrium
path which has the same features as in Proposition 3 and at = 1 for all t ≥ 1.
Proof. Take the equilibrium path considered in Proposition 3. Then we show that theﬁrms have no incentive to implement low eﬀort in any period in such equilibrium path,when s is suﬃciently small. If some ﬁrm deviates to implement low eﬀort at = 0 frommatched old workers and invests in capital kt′′ , it would obtain the proﬁt as π(kt′′ ) deﬁned
in the proof of Proposition B1. Then, we obtain
π(kt′′ ) ≤ F (rt ) ≡ max sk γ − λrt k.
k

Case 1: In the implicit contract regime with non-binding DE the equilibrium ﬁrm’s ﬂow
proﬁt becomes πt = (1 − γ − ψ)ktγ where the equilibrium capital kt satisﬁes γktγ−1 = λrt .
61

Given this equilibrium interest rate, the upper bound for the deviation proﬁt F (rt ) canbe written by
F (rt ) = F̃ (kt ) ≡ (1 − γ)s1/(1−γ) λγ/(γ−1) ktγ
where k attaining F (rt ), denoted by k t , satisﬁes sγk tγ−1 = λrt so that k t = (s/λ)1/(1−γ) kt .
However, then F̃ (kt ) ≤ πt = (1 − γ − ψ)ktγ holds if
1 − γ − ψ ≥ (1 − γ)s1/(1−γ) .

(B6)When DE is binding in the implicit contract regime, kt = x̂t ≡ Vt+1 /λrt where rt >γktγ−1 > λrt is satisﬁed. In this case the capital k t attaining F (rt ) satisﬁes sγk γ−1=t
γ−11/(1−γ)kt . Then we obtain
λrt > λγkt
so that k t < (s/λ)
F (rt ) = F̃ (kt )= s(1 − γ)k γt≤ (1 − γ)s(s/λ)γ/(1−γ) ktγ≤ πt ≡ ktγ − Vt+1
= ktγ (1 − Qt+1 )≤ ktγ (1 − γ − ψ)if
1 − γ − ψ ≥ (1 − γ)λγ/(γ−1) s1/(1−γ) .

(B7)Case 2: In the mixed contract regime the equilibrium ﬂow proﬁt is given by πt ≡ (1 −γ)ktγ − ψt + (1 − λ)Vt+1 /λ where γktγ−1 = rt . The upper bound for the deviation proﬁt
F (rt ) is then written by
F (rt ) = F̃ (kt )≡ s(1 − γ)k γt= (1 − γ)λγ/(γ−1) s1/(1−γ) ktγ
because k t satisﬁes sγk γ−1= λrt = λγktγ−1 . The right hand side of the above expressiont
is smaller than the equilibrium proﬁt πt if (B7) holds.
Case 3: In the explicit contract regime the equilibrium proﬁt is given by
πt = [1 − γ − (1 + (q0 /∆q)))ψ + (q0 /∆q)Qt+1 ]ktγ
and the equilibrium capital (interest rate) satisﬁes γktγ−1 = rt . Then the upper bound forthe deviation proﬁt F̃ (kt ) is lower than the equilibrium proﬁt if
1 − γ − (1 + (q0 /∆q))ψ + (q0 /∆q)Qt+1 ≥ (1 − γ)λγ/γ−1 s1/(1−γ)(B8)

which is rearranged as{}
Qt+1 ≥ Q∗ ≡ (∆q/q0 ) (1 + (q0 /∆q))ψ − (1 − γ) + (1 − γ)λγ/(γ−1) s1/(1−γ)
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(B9)

Since the ﬁrm value Qt decreases over time and converges to the steady state Qe in theequilibrium path considered in Proposition 3, (B9) is satisﬁed if Qe ≥ Q∗ holds. Thiscondition is equivalent toγQ∗> (1 − γ)λγ/(γ−1) s1/(1−γ) + Q∗ .
L − Q∗
When s → 0, Q∗ → L − γ. Thus, it suﬃces to obtain Qe > L − γ which is satisﬁed as
we have seen in the proof of Proposition 3 (recall that this was ensured by L < L so thatQe ≥ Q in the proof of Proposition 3).
Summarizing all the above results, in the equilibrium path derived in Proposition 3
the ﬁrms have no incentive to deviate for implementing low eﬀort at = 0 from matched
old workers in any period. Q.E.D.

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