GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.226

Relationships and Growth

Shingo Ishiguro

November 2011

GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Relationships and Growth
Shingo Ishiguro ∗Graduate School of Economics
Osaka University
November 17, 2011

Abstract
In this paper we present a dynamic general equilibrium model to investigate how
diﬀerent contracting modes based on formal and relational enforcements endogenously emerge and are dynamically linked with the process of economic development. Formal contracts are enforced by third party institutions (courts), while relational contracts are self-enforcing agreements without any third party involvement.
The novel feature of our model is to demonstrate the co-evolution of these diﬀerent
enforcement modes and market equilibrium conditions, all of which are jointly determined. We then characterize the equilibrium paths of such dynamic processes
and show the time structure of relational contracting (self-enforcing agreement)in the endogenous process of economic development. In particular we show that
relational contracting fosters the emergence of the market-based economy in low
development stages but its role declines as the economy grows and enters high development stages.
Keywords: dynamic general equilibrium, economic development, arm’s length contract, relational contract
JEL Classiﬁcation Numbers: D86, E10, O11∗
Correspondence: Shingo Ishiguro, Graduate School of Economics, Osaka University.
Machikaneyama, Toyonaka, Osaka 560-0043, Japan. e-mail: ishiguro@econ.osaka-u.ac.jp

1

1-7

1

Introduction

Informal contract arrangements, which we call relational contracting in this paper,
are common during the developing stages of economies. These arrangements are notbased on formally written contracts, but rather on long term relationships, implicit
agreements, and a reputation mechanism based on personal ties and connections, as
typically observed in tribal and ancient societies1 as well as in emerging and transition
economies.2
Among other informal contract arrangements, one well-documented example is relationship (insider) lending based on personal relationships between borrowers andlenders, typically banks. Lamoreaux (1994) reported that, during the early 19th century, the New England banks lent a large portion of their funds to the board of directors
and those who had close personal ties with these banks. Lamoreaux then found theevidence that such relationship lending contributed to the economic growth in New
England during that period, in contrast to the view that relationship-based ﬁnancemight be less eﬃcient than market based ﬁnance.3 Similar informal ﬁnancial arrangements were also widely observed during preindustrial stages in other countries such as
China, India, and Islamic Middle Eastern countries.4
Some authors argue that relational contracting, which is based on closed relationships among particular members such as kinship networks, inhibits economic development compared to the market based economy, in which trades are made in an impersonalmanner. For example, one of the reasons that China, India, and the Islamic Middle
East economically lagged behind Europe during the 19th century is thought to be theirstrong reliance on relational contracting, despite that these countries had technological advantages over Europe during the Middle Ages (see, for example, Kumar andMatsusaka (2006)). However, a contrasting view is that relational contracting is not a
substitute but a complement to a market economy in that the former fosters the latter(see the papers contained in Aoki and Hayami (2000)).
The main objective of our paper is to provide a uniﬁed theoretical framework for understanding how and when relational contracting plays positive or negative roles in economic development. Then we characterize the time structures of relational contracting(self-enforcing agreement) in the endogenous process of economic development: whenand in what stages of economic development does relational contracting have tighter
self-enforcing agreement constraint? A novel feature of our model is to embed rela1

See Levi-Strauss (1969), Malinowski (1961), and Mauss (1967) for anthropological studies on reciprocal exchange and gift exchange in tribal societies. See Greif (2006) and Milgrom, North and Weingast(1990) for a discussion on how the merchant trade system functioned as a reputation device in medieval
times.
2
See McMillan and Woodruﬀ (1999) for a discussion of trade credits in Vietnam and Johnson,
McMillan and Woodruﬀ (2002) for a discussion of relational contracting in Russia.
3
As Lamoreaux (1994) emphasized, insider lending was a phenomenon observed not only in New
England but also in other U.S. states during the early 19th century before the U.S. markets expanded.
4
See Kumar and Matsusaka (2009), Greif (2006) and North (1998) for historical evidence on thisissue.

2

tional contracts that are supported by long term relationships into a dynamic generalequilibrium model. Long term relationships have been mostly analyzed in the partial
equilibrium frameworks in the literature of repeated games (see Mailath and Samulaeson (2006)): in a typical repeated game, players play the stage games repeatedly overtime by assuming that the outside markets are exogenously given. However, despitemuch historical evidence that shows the important roles of long-term relationships inthe process of economic development, there are few theoretical studies that consider
the macroeconomic implications of these relationships. These implications are what we
address in this paper.
In our model economy, producers who need to ﬁnance their capital investment havethe incentive to default after they borrow the funds from lenders. We assume that there
are two means by which producers can commit themselves not to default. The ﬁrst
type is the anonymous credit market, in which everyone can borrow and lend at a given
market interest rate in the spot manner. When producers ﬁnance capital investment in
the credit market, they must invest in the veriﬁcation technology in advance, preventing
themselves from defaulting. We call this type of contract an arm’s length (formal)
contract and call producers who engage in an arm’s length contract non-relationshipproducers. The veriﬁcation investment includes activities such as collecting evidence
about accounting data, hiring lawyers and accounting professionals, and establishingthe information disclosure system.
However, there is also a local community in our model economy: each producer
in the community has a personal connection with a particular lender within the samecommunity. The producer and the lender form a long term relationship over successive
generations. We call this type of producer a relationship producer. Each relationshipproducer can engage in relational contracting with a particular lender for ﬁnancing
capital investment without using the outside credit market. This type of contract iswhat we call a relational contract. Because they interact with each other over time,
the relationship producer and the lender can avoid strategic default via a self-enforcing
agreement.
As is well known from the repeated games literature, an implicit agreement is selfenforceable if each party’s deviation from honoring the agreement results in future
losses larger than the one-time gains obtained by the deviation. The novel featureof our model is how it relates the self-enforceability of relational contracting to theendogenous process of economic development in a dynamic general equilibrium framework. The endogenous process of economic development determines the proﬁt of anarm’s length contract, which becomes the deviation payoﬀ for each relationship producer when quitting the current relationship with a lender. This change then aﬀects
the self-enforcing condition of the relational contract. In turn, the change of the selfenforcing condition creates a feedback eﬀect on the equilibrium determination of the
proﬁt of an arm’s length contract through changing the market prices, such as wage
and interest rates. These two-way interactions between the self-enforcing condition ofa relational contract and market equilibrium conditions jointly determine the evolution
processes of the economy.
3The dynamic general equilibrium interactions mentioned above cause non-trivial
eﬀects on the determination of relational contracts over time. When capital accumulation proceeds over time, lowering the interest rates in the anonymous credit market,
the proﬁt of an arm’s length contract becomes more attractive to producers, and theself-enforcing condition of a relational contract becomes tighter. However, a more stringent self-enforcing constraint devalues the relational contract, which then contributes
less to the capital accumulation of the economy as a whole. Thus, in general, it isdiﬃcult to predict whether economic development positively aﬀects and is aﬀected by
the sustainability of relational contracting.
By considering such dynamic general equilibrium eﬀects, we characterize the equilibrium paths of the model economy and show the time structures of relational contracting in the endogenous process of development. In particular we demonstrate thatequilibrium paths involve the structural change from a relationship-based system relying on relational contract to a market-based system relying on an arm’s length (formal)
contract. We show that in any equilibrium path there exists a unique switching period before which the self-enforcing constraint becomes slack but after which it becomesbinding. In periods before the switching period, the economy is in low-developed stages
such that the market interest rates in the credit market are high and the proﬁt of arm’s
length contract is low, and hence only small fraction of non-relationship producers canﬁnance their capital investments. After the economy passes the switching period, selfenforcing condition becomes tighter because interest rates fall, and hence the proﬁt ofan arm’s length contract becomes larger as the anonymous credit market continues toexpand. Thus, a relational contract contributes less to the capital accumulation process
than non-relationship producers who use arm’s length contract do in developed stages.
Our results are consistent with empirical and historical evidence, as described below. First, we show that the co-evolution processes of contract enforcement modes andmarket equilibrium conditions endogenously trigger the dynamic transformation fromthe relationship-based system, in which relational contracting in a local community
contributes more to capital accumulation than does the anonymous credit market, tothe market-based system, in which a relational contract exhibits a smaller contribution to capital accumulation than does an arm’s length contract over time. This result
captures an important argument made by Polanyi (1947): the Western societies experienced the ”great transformation” from nonmarket systems to market-based systems
when diﬀerent goods and services were priced subject to the law of market mechanisms
in the 19th century. Moreover, our result shows that the relationship-based system
plays a positive role in fostering economic growth during low-development stages because it becomes less constrained by self-enforcing conditions during these stages, thus
fostering the emergence of a market-based economy during subsequent stages of development. These results are consistent with the aforementioned historical evidence
that relational contracting complements the rise of a market-based economy (Aoki and
Hayami (2000) and Lamoreaux (1994)).
Second, we also show that relational contracting becomes further constrained by the
self-enforcing condition as the economy grows and enters matured stages of develop4

ment. As the economy becomes richer, it becomes cheaper to invest in the veriﬁcation
technology, thus preventing strategic default and easing the enforcement of formal contracts, while the role of the relational contract declines relative to that of the arm’s
length contract over time. This observation is consistent with the historical fact that the
New England banks that lent to closely related persons (for example, directors of thesebanks) in the early 19th century eventually had begun to lend to “outside” borrowers,
whom they did not personally know well, as the economy changed from capital poor
stages to capital rich stages, thus expanding the anonymous credit market in the late
19th century (Lamoreaux (1994)). A related argument is that the relationship-based
system was dominant in Asian countries such as Korea and Japan after World War II
but it has been changing to the market-based system as capital markets became more
integrated and open to the world (see Rajan and Zingales (2000)). Demirgüç-Kuntand Levine (2004) reported the related evidence that the ratio of bank ﬁnance relative
to equity ﬁnance is negatively associated with per capita GDP levels across countries,
suggesting that bank ﬁnance, which is often characterized as a long-term lending relationship between a particular bank and a ﬁrm, becomes less important and is replacedby market-based ﬁnance in developed countries.
Related literature Although several papers address relational contracting in partial equilibrium frameworks, 5 few studies have attempted to examine its macroeconomic implications via dynamic general equilibrium models. Some papers attempt tocompare informal contracting enforcement, such as reputation, with formal and legal
enforcement in random matching environments. Kranton (1996) focuses on the marketbased monetary exchange and relational (self-enforcing) contracts that emerge in the
Kiyotaki-Wright type of monetary search model. Dhilon and Rigolini (2006) makethe comparison between reputation and legal enforcement by endogenizing the qualityof enforcement institutions. Francois (2011) investigates the evolution of endogenous
institutions, but his analysis focuses on the roles of social norms formed through thechange in endogenous preferences. Francois and Roberts (2003) examine how relational contracting aﬀects long-run economic growth in an R & D-based endogenous
growth model. However, they focus on relational (self-enforcing) contracts but noton the choice between arm’s length and relational contracts. They also address thesteady state analysis of the long-run growth rate and the macroeconomic eﬀects ofproductivity shocks on relational contracting between ﬁrms and workers. Fafchamps(2003) also addresses the dynamic issue of how markets spontaneously emerge in therepeated game setting. Our work here diﬀers from all the research cited above because
our main concern is the dynamic change in the contracting modes and its relation to
the process of economic development. Our new insight is that both the sustainabilityof relational contracting and the evolution of economic development are endogenous
and jointly determined through dynamic general equilibrium eﬀects. Speciﬁcally, we
5See, for example, Baker, Gibbons and Murphy (2002), Itoh and Morita (2007), Levin (2003),
MacLeod and Malcomson (1998), and Ramey and Watson (2003).

5

address the hitherto unaddressed issues of how and when relationship-based economic
systems change to market-based systems during the endogenous process of economic
development.
The remaining sections are organized as follows. In Section 2, we describe an overlapping generations model with a choice between arm’s length and relational contracts.
In Section 3, we derive a self-enforceable incentive compatibility condition for relational
contracts. In Section 4, we deﬁne an equilibrium of the model economy. In Section
5, we characterize the set of equilibrium paths and show that relational contracting
contributes more to economic growth in early stages of development but that its role
becomes more limited as the economy enters mature stages of development. In turn weshow that there exists an equilibrium path with such a feature. All proofs are relegatedto the Appendix.

2

Model

2.1

Economic Environment

We consider an overlapping generations economy. Time is discrete and extends overinﬁnity t = 0, 1, 2, ... In the economy, there is a single ﬁnal good, taken as a numèraire,
which is used for both consumption and investment. The ﬁnal good is produced bylabor and a continuum of intermediate goods with one unit measure (see below for more
speciﬁcations). Every period, a continuum of one unit mass of individuals is newly bornand each individual lives for two periods: young and old. In each generation, one young
individual is born from each old individual, and we use the notation i to denote bothindividual i and the dynasty to which individual i belongs. For simplicity, we assume
that each individual is concerned with only his or her consumption when old.
The newly born individuals consist of workers and intermediate goods producers.
We use the masculine pronoun for intermediate goods producers (or borrowers) andthe feminine pronoun for workers (or lenders). Each young worker is endowed with one
unit of labor and inelastically supplies it to the labor market to earn the market wage
wt . Because every young worker is concerned with her consumption level when old, shewill lend all the wage income to borrowers. Thus, all young workers become lenders.
No old workers are endowed with labor inputs, and hence they simply consume all the
income saved when they are young.
On the other hand, intermediate good i ∈ [0, 1] is produced by producer i, whopossesses the speciﬁc knowledge to produce that intermediate good. Each intermediate
good is produced by investing in capital one period in advance. Speciﬁcally, an intermediate good producer can produce one unit of his intermediate good when old if heinvests in one unit of capital when young.
We assume that capital fully depreciates after one period and that young intermediate good producers are not endowed with labor so that they need to ﬁnance their6

investments when young. Thus, young intermediate goods producers become borrowers.

2.2

Preference

We assume that individual i (worker or producer) has an altruistic preference overthe consumption level of his or her child. More speciﬁcally, consider an individual indynasty i who was born in period t − 1 and whose consumption of the ﬁnal good whenold (in period t) is denoted by Ctt−1 (i) ≥ 0. Then, we assume that the utility U t−1 (i)of an individual in dynasty i, born in period t − 1, depends not only on his/her own
consumption level when old (in period t), Ctt−1 (i), but also on the consumption levelt (i), as follows:of his/her child in period t + 1, Ct+1
t(i),
U t−1 (i) ≡ Ctt−1 (i) + δCt+1(1)

where δ ∈ (0, 1) represents the parameter value measuring the degree to which eachindividual is altruistic about his/her child.
We assume that there is no transfer technology of bequest across generations inthe same dynasty. Then, each individual consumes all of his/her old income for himself/herself such that Ctt−1 (i) is equal to the lifetime income level of individual i bornin period t − 1. We will discuss an extension of the basic model to allow bequests inAppendix B.

2.3

Final Good Market

A single ﬁnal good Yt is produced by a continuum of intermediate goods, each of whichis indexed by i ∈ [0, 1] and labor Lt in the following manner:∫ 1
1−α
Yt = ALt
yt (i)α di,(2)0where α ∈ (0, 1), A > 0 and yt (i) denotes the input demand for (equivalent to its
output) intermediate good i in period t.
We assume that there is perfect competition in the ﬁnal goods market. Then,the ﬁnal good ﬁrm chooses the demand for labor Lt and intermediate inputs yt (i) tomaximize its proﬁt:
∫ 1
∫ 1
1−α
α
ALt
yt (i) di − wt Lt −pt (i)yt (i)di,
(3)
0

0

where wage rate wt and price of intermediate good i, pt (i), are taken as given. The
corresponding ﬁrst-order conditions are as follows:
AαL1−αyt (i)α−1 = pt (i)tandA(1 − α)L−α
t

(4)

∫ 1
y(i)α di = wt .
0

7

(5)

2.4

Credit Market with Commitment Problem

Each young intermediate good producer needs to ﬁnance capital investment for production to be performed when old. However, we suppose that any borrowed amountis not veriﬁable. Then, each old intermediate good producer will always default a borrowed amount and refuse to repay lenders. Thus, there must be some commitment
device impelling producers not to default because otherwise lenders would never lend
to producers.
In the economy there are two places where intermediate goods producers ﬁnancetheir capital investments without default.
Credit Market for Arm’s Length Contracts
One is the market place, which we call the credit market, where borrowing and lending
are made at a given interest rate. The credit market is perfectly competitive in the sensethat everyone takes the market interest rate ρt ≥ 0 as given in any period t.6 However,the credit market is not perfectly competitive in the sense that it involves a commitment
problem regarding default, as we have already mentioned: it is not veriﬁable how mucha producer (borrower) actually borrowed from lenders.7 We assume that each borrowercan, however, commit himself not to default his borrowed amount when he invests inthe veriﬁcation technology one period in advance.
Investment in the veriﬁcation technology requires I > 0 units of the ﬁnal good one
period in advance. Here, I includes the costs of making the information disclosure
credible, collecting hard evidence, hiring professionals such as lawyers and accountants
who help write formal contracts, and using outside institutions such as courts. When
he invests in the veriﬁcation technology in advance, a producer can commit himself
not to default when old.8 In what follows, we call an intermediate good producer who
ﬁnances capital investment from the anonymous credit market a non-relationship producer. We let xt ≥ 0 denote the capital investment (equivalently, the production levelof an intermediate good) of a non-relationship producer. Then, if a non-relationshipproducer invests in capital xt and in veriﬁcation technology I in period t − 1, he needs
6Here, ρt denotes the gross interest rate for a spot transaction in the credit market, which speciﬁes
that one unit of borrowing in period t − 1 must result in the repayment of ρt units in the next periodt. Thus, by the nature of spot transactions in the credit market, no borrower can roll a debt obligation
forward in period t − 1 to his child who would repay it in period t + 1.
7
Even when how much an intermediate good producer produced can be observed, it is not veriﬁable
how much he actually borrowed. He might simply insist “I have prepared an amount xt for capital
investment by myself (e.g., from my own pocket or received freely from someone) but I have notborrowed it.”8
We can also allow lenders to incur some veriﬁcation cost when they lend to the credit market.
For example, they monitor whether borrowers in the credit market renege on repayments to them.
Speciﬁcally, we assume that γ ∈ (0, 1) is the cost incurred by each lender to monitor one unit oflending. Thus, each lender obtains an interest return of (1 − γ)ρt from lending one unit to the credit
market. Even when we introduce such veriﬁcation (monitoring) costs on the side of lenders in the credit
market, our main results are not substantially diﬀerent. Thus, we assume away such costs from themodel and simply set γ = 0 in what follows.

8to borrow xt + I units of the ﬁnal good in period t − 1 and repay ρt (xt + I) in period t.
We call such spot contract made in the anonymous credit market an arm’s length
contract. This contract is formally written and enforced by the court.
Local Community
On the other hand, in the economy there is a local community where l (l < 1) intermediate producers and l workers (lenders) reside (thus the remaining 1 − l intermediate
producers and workers are outside the community.) The producers and lenders in thecommunity are matched with each other in the initial period t = 0 in a one-to-one
manner. Then, each of the matched l pairs forms a personal connection and relationship. We assume that such relationships formed in the initial period can be inheritedover successive generations. We call an intermediate good producer in the community
a relationship producer and a lender in the community a relationship lender. They can
avoid the default problem by using long term relationships, as is shown below.
Any (young or old) relationship producer and lender in each pair can always exercise
option to quit their relationship (quitting option) at any time by leaving the community.
We will then assume that if a relationship producer (lender) quits the relationship, his(her) child cannot also form a relationship with the child of his (her) partner in thenext period. In such a case, not only the current relationship producer and lender
but also their children have no choice but to engage in an arm’s length contract in
the anonymous credit market. However, we do not assume the permanent dissolution;that is, we do not assume that once a relationship producer and a lender dissolve theirrelationship, all their descendants cannot form relationships forever. We only assume
that it takes at least one period for a dissolved relationship to be re-formed. We latershow that the assumption of one-period dissolution of a relationship is suﬃcient foreach relationship pair to honor the agreed upon relational contracts over time.
In what follows we let zt ≥ 0 denote the capital investment (equivalently, theproduction level of an intermediate good) of a relationship producer. Each young
relationship producer, born in period t − 1, can directly ﬁnance his investment zt fromhis partner, a relationship lender, in exchange for making repayment Rt in period t.
They can then save the veriﬁcation cost I > 0 by such implicit agreement. However,
because repayment Rt is not secured unless the investment in the veriﬁcation technology
I is made, such an agreement {zt , Rt } must be implicit and self-enforceable. We callthis type of contract a relational contract.

2.5

Timing Within Each Period

Events in each period proceed in the local community as follows. First, at the beginningof each period, old relationship producer and lender of each relationship pair decide
whether to exercise the quitting option. When they exercise the quitting option, theirrelationship is dissolved and their children will have no choice but to engage in anarm’s length contract in the next period. When an old producer and an old lenderin a relationship do not exercise the quitting option, in the same period their children
9

(young relationship producer and lender) decide whether to exercise the quitting option.
By exercising the quitting option, the young relationship producers and lenders canensure at least the payoﬀs obtained by an arm’s length contract in the credit market.
Furthermore, their children born in the next period must engage in an arm’s length
contract because of our assumption that any dissolved relationship pair of producer andlender needs to wait at least one period for their descendants to re-forma relationshippair. When they do not exercise the quitting option, they agree on a relational contract,{zt , Rt }, which speciﬁes the capital investment level zt and the repayment Rt to therelationship lender.9 Because the producer cannot commit himself not to repay Rt ,such a relational contract must be self-enforceable.

2.6

Labor Market

We assume that the labor market is perfectly competitive and that no commitment
problems arise. We let wt ≥ 0 denote the competitive market wage in period t in thelabor market.

2.7

Initial Period (t = 0)

In the initial period (t = 0), each old intermediate good producer (irrespective of nonrelationship or relationship producer) owns an initial capital stock, z0 = x0 , which isassumed to be historically given.10 Because the old producers in this period do notneed to raise funds for capital investment, they can supply intermediate goods z0 = x0
without any production and veriﬁcation costs. There is also one unit mass of old lenders
in the initial period.

3

Arm’s Length and Relational Contracts

3.1

Arm’s Length Contract Producers

Because every non-relationship producer faces the same demand function, (4), for his
intermediate good, we will omit notation i hereafter.
Let πt be the proﬁt of an intermediate good producer who ﬁnances his investment
xt + I via an arm’s length contract in period t deﬁned as follows:πt ≡ pt xt − (xt + I)ρt
= ALt1−α αxαt − (xt + I)ρt ,9(6)Note here that Rt = 0 and Rt+1 > 0 may be possible. Each old relationship producer (borrower)
makes no repayment Rt in the current period, t; however, his child makes the repayment in the next
period, t + 1, implying that each relationship producer (borrower) can roll his debt over to his child,an option unavailable in the case of borrowing by the arm’s length contract.
10
We here assume that all old producers in the initial period own the same amount of initial capitalstock, z0 = x0 , for simplifying the analysis. Our results are not substantially changed even when weallow z0 ̸= x0 .

10where the price of an intermediate good, pt , is given by (4). Here, note that every nonrelationship producer has no other option but to borrow capital investment xt using
an arm’s length contract in the credit market. As such, the non-relationship producer
incurs borrowing cost (xt +I)ρt that includes investment I in the veriﬁcation technology
in addition to capital investment xt for production. Because Lt = 1 holds in the labormarket equilibrium, we set Lt = 1 in what follows.
Each young non-relationship producer born in period t−1 chooses capital investment
level xt to maximize his payoﬀ πt + δπt+1 . Here, note that his consumption level Ctt−1
is equal to the proﬁt πt he earned in period t, whereas the consumption level of hist= πt+1 . Thus, the payoﬀ of a non-relationship producer is given bychild is Ct+1
t−1
t
Ct + δCt+1 = πt + δπt+1 .
We deﬁne by ht a history of all the events observed by a non-relationship producer ina dynasty up to period t. Such a history includes the past investments {x0 , x1 , ..., xt−1 }t−1made in the same dynasty and the past market prices {ws , ρs , ps }s=1
. Let Ht be a setof all such histories up to period t. Then, a strategy for each non-relationship producer
in each dynasty is deﬁned as a mapping from the set of histories to capital investment
levels: σt : Ht → [0, ∞).

3.2

Relationship Producers

Next, we consider a pair consisting of a relationship producer and a relationship lender
who engage in relational contracting. Again, we omit notation i to index individuals
because every relationship pair can be treated symmetrically.
Let Jt be the joint proﬁt of a relationship pair, which is split between the relationshipproducer and lender after an intermediate good is produced. Because each young worker(who becomes a lender) earns market wage wt−1 in period t − 1, the young relationshipproducer who wants to invest zt units of capital in period t − 1 can borrow this amountzt directly from the young relationship lender matching him. He can then save on theveriﬁcation cost ρt I, which would otherwise arise in an arm’s length contract in thecredit market. For the time being, assume that wt−1 > zt . (Below, we will show that
this is actually the case in equilibrium). Then, a relationship producer can borrow
capital investment zt via insider lending from the relationship lender instead of using
an arm’s length contract. The relationship lender then supplies the remaining amount,wt−1 − zt , to the credit market to earn interest income ρt (wt−1 − zt ) in the next period,t. Hence, the joint proﬁt of a relationship pair in period t is given by the sum of therevenue from capital investment pt zt and the interest income ρt (wt−1 − zt ):Jt = pt zt + ρt (wt−1 − zt )= αAztα + ρt (wt−1 − zt ),where the price of intermediate good, pt , is given by (4). The repayment Rt is used to
split this joint proﬁt between the relationship producer and lender.
In the following section we consider three constraints that limit the relational contract: The ﬁrst constraint is the incentive compatibility (IC) condition, according to11which the old relationship producer has no incentives to renege on the agreed upon
repayment Rt . The second constraint is the individual rationality condition for relationship producer (IRP), according to which the young relationship producer is weakly
better oﬀ by agreeing to a relational contract instead of exercising the quitting option. The third constraint is the individual rationality condition for relationship lender(IRL), according to which the young relationship lender is weakly better oﬀ agreeing
to a relational contract instead of exercising the quitting option.

3.3

Incentive Compatibility

Consider any pair of a young relationship producer and a young relationship lender bornin period t − 1. Then, assume that they agree to a relational contract {zt , Rt }: theyoung relationship producer promises to repay Rt to the relationship lender matching
him when old (in period t) in exchange for borrowing zt directly from her. Assume alsothat their relationship in period t is inherited by the next generation in period t+1 who
will agree to a relational contract {Rt+1 , zt+1 }. Assume then that the old relationshipproducer in period t + 1 (who is the child of the old relationship producer in period t)
does not exercise the quitting option but again honors the contracted agreement Rt+1 .
Anticipating the outcome in period t + 1 described above, the old relationshipproducer in period t makes the repayment Rt to the old relationship lender and does
not exercise the quitting option only if the following incentive compatibility constraint,(ICt ), is satisﬁed:
pt zt − Rt + δ{pt+1 zt+1 − Rt+1 } ≥ pt zt + δ max{πt+1 , 0}(ICt ),

where πt+1 denotes the proﬁt an intermediate producer could obtain if he exercised
the quitting option and ﬁnanced his investment via an arm’s length contract in periodt + 1.
We now explain (ICt ) in detail. The right-hand side of (ICt ) denotes the payoﬀ theold relationship producer could obtain if he reneged on repayment Rt and exercised
the quitting option. By doing so, he can save on repayment Rt and capture the whole
revenue pt zt from capital investment zt , but in the next period, (t + 1), his child faces
the dissolution of the relationship with the child of the current lender. In such a case,the child of such a deviating producer would obtain at least proﬁt max{πt+1 , 0} via
an arm’s length contract in the credit market when old (in period t + 1). Here, we
use the operator max{πt+1 , 0} to denote whether proﬁt πt+1 should be non-negative.
This child’s future payoﬀ is evaluated using the altruistic parameter δ > 0 from theviewpoint of the current producer. Thus, the sum of these payoﬀs can be guaranteedby the old relationship producer when he exercises the quitting option. On the other
hand, the left-hand side of (ICt ) denotes the payoﬀ of the old relationship producer
when he makes contracted repayment Rt to the relationship lender matching him inperiod t expecting that the relationship is inherited by the next period generation, in
which case his child also makes repayment Rt+1 in period t + 1. This future payoﬀ
is also evaluated using δ from the viewpoint of the current old producer in period t.
12

Thus, (ICt ) is necessary for the old relationship producer not to renege on repayment
Rt in period t.
The following individual rationality constraint, (IRLt ), of the relationship lendermust also be satisﬁed:
Rt + ρt (wt−1 − zt ) + δ{Rt+1 + ρt+1 (wt − zt+1 )} ≥ ρt wt−1 + δρt+1 wt(IRLt ).

Otherwise, it is a strictly dominant strategy for the young relationship lender to exercise
the quitting option in period t − 1; by exercising the quitting option, the lender and herchild lend their entire wage income to the credit market so as to earn the interest income
corresponding to the payoﬀs on the right-hand side of (IRLt ). Here, the interest incomeof her child, ρt+1 wt , is evaluated by the altruistic parameter δ from the viewpoint of thecurrent old lender. However, if the old relationship lender does not exercise the quitting
option, she would obtain contracted repayment Rt in period t, which appears as theﬁrst term on the left-hand side of (IRLt ), in addition to interest income ρt (wt−1 − zt )from the savings on the remaining income wt−1 − zt after lending zt to the relationshipproducer (note here that we are assuming that wt−1 > zt ). The child of the lenderis paid the contracted amount Rt+1 in addition to the interest income ρt+1 (wt − zt+1 )when old in period t + 1. Here, the payoﬀ of the lender’s child is evaluated by thealtruistic parameter δ > 0 again from the viewpoint of the current old lender.
Combining (ICt ) with (IRLt ), we can derive the following modiﬁed incentive compatibility condition, denoted by (IC∗t ):δ{pt+1 zt+1 − ρt+1 zt+1 − max{πt+1 , 0}} ≥ ρt zt(IC∗t ),

This condition is necessary for relationship pairs in period t to be sustained so that therelational contract is self-enforceable. Using (4), (IC∗t ) can also be expressed asαδ{αAzt+1− ρt+1 zt+1 − max{πt+1 , 0}} ≥ ρt zt(IC∗t ).

Next, we introduce the individual rationality constraint of the relationship producer(IRPt ):
pt zt − Rt + δ{pt+1 zt+1 − Rt+1 } ≥ max{πt , 0} + δ max{πt+1 , 0}

(IRPt ).

The condition (IRPt ) ensures that the relationship producer prefers continuing therelationship to dissolving it. Suppose that the producer exercises the quitting option
when young. Then, he earns proﬁt max{πt , 0} via an arm’s length contract when old(period t). Furthermore, his child earns proﬁt max{πt+1 , 0} via an arm’s length contract
when old in period t + 1, which is evaluated by δ from the viewpoint of the current
producer. The sum of these payoﬀs corresponds to the payoﬀ that each non-relationshipproducer obtains via an arm’s length contract, which thus becomes his outside option.
However, the left-hand side of (IRPt ) denotes the payoﬀ of the relationship producer
who does not exercise the quitting option when young; by doing so, he earns pt zt − Rt
when old (period t) by making repayment Rt to the relationship lender. His child also
13

continues the relationship and earns proﬁt pt+1 zt+1 − Rt+1 by making the contracted
repayment Rt+1 to the relationship lender when old (period t + 1). Thus, the young
relationship producer obtains the sum of these payoﬀs by continuing the relationship.
Then, by subtracting the right-hand sides of (IRPt ) and (IRLt ) from their left-handsides and using (4), the net total surplus of a pair of a relationship producer and a
relationship lender born in period t − 1 is deﬁned as
T St−1 ≡ pt zt − ρt zt − max{πt , 0}
+ δ{pt+1 zt+1 − ρt+1 zt+1 − max{πt+1 , 0}}= αAztα − ρt zt − max{πt , 0}α
+ δ{αAzt+1− ρt+1 zt+1 − max{πt+1 , 0}}.

(7)

We impose the following condition, which ensures that each relationship pair born inperiod t − 1 ﬁnds it optimal to sustain the relationship rather than dissolve it:
T St−1 ≥ 0(TSt−1 ).

We can then readily show that there exists a sequence of repayments {Rt }∞t=1 that∗satisfy (ICt ), (IRLt ) and (IRPt ) for all t ≥ 1 as long as (ICt ) and (TSt−1 ) are satisﬁed
for all t ≥ 1.11
Although there are many possible equilibria sustained by diﬀerent relational contracts, as in the Folk Theorem of repeated game theory, we will focus on the equilibrium,
called the Best Relational Contracting Equilibrium (BRCE), in which the initial generation of relationship pairs in each dynasty chooses a sequence of all future relational∞
generations in thecontracts {zt , R∑t }∞t=1 tso as to maximize the sum of joint∗payoﬀs of allsame dynasty t=0 δ Jt subject to the constraints {(ICt ), (TSt−1 )}∞t=1 , given the future12∞∞
paths of all the market prices {ρt }t=1 and {wt }t=0 . Along such an equilibrium paththe relationship pair born in any subsequent period, say T ≥ 1, has the incentive tofollow the relational contract designed for period T , {zT , RT }, provided that all future
generations in the same dynasty will also do so.13

It is suﬃcient to set Rt = ρt zt for each t ≥ 1.
See Acemoglu, Golosov and Tsyvinski (2008) for a related treatment of self-enforcing agreements
in a dynamic general equilibrium model, although they consider a diﬀerent model from ours.
13∞This can be readily seen byPthe following fact: let {zt , Rt }t=1 be the optimal relational contracts
that are the solutions to max t δ t Jt subject to {(IC∗t ), (TSt−1 )}∞t=1 . If a relationship pair in some
generation born in period T − 1 would be better oﬀ by agreeing to a relational contract diﬀerent fromthe original one, {zT′ , RT′ } ≠ {zT , RT }, it must satisfy (IC∗T ) and (TST −1 ) and have larger joint payoﬀ′′
JT −1 + δJT than the joint payoﬀ obtained under the original relational contract JT −1 + δJT . Then,′ ∞′if the relational contracts {zt , Rt }∞t }t=1 , wheret=1 designed in the initial period are replaced by {zt , RP
t
′
′
′
′
(zT , RT ) ̸= (zT , RT ) but (zt , Rt ) = (zt , Rt ) for any t ̸= T , the sum of the joint payoﬀs ∞t=0 δ Jt is∞
improved. However, this result contradicts the fact that the relational contract {zt , Rt }t=1 maximizesP∞t∗
t=0 δ Jt subject to (ICt ) and (TSt−1 ) for all t ≥ 1.
11

12

14

4

Equilibrium Paths of the Economy

We let θt ∈ [0, 1] denote the fraction of non-relationship producers who can produce
their intermediate goods in period t. Because there is the veriﬁcation cost I > 0 foreach non-relationship producer to borrow in the credit market, some such producers
may decide not to ﬁnance capital investment and hence decide to shut down production.
Now we provide a formal deﬁnition of an equilibrium path in this model economy:Deﬁnition. A sequence {xt , zt , wt , ρt , θt }∞t=0 is said to be an equilibrium of the economy
if the following conditions are satisﬁed:(i) Each young non-relationship producer chooses a strategy σt : Ht → [0, ∞) thatmaps from the set of observed histories Ht to a capital investment level xt ≥ 0 so
as to maximize his payoﬀ πt + δπt+1 ,(ii) The initial generation of the relationship pairs in each dynasty chooses a sequence of future relational contracts {zt , Rt }∞t=1 so as to maximize∑∞ t the sum of
the joint payoﬀs of all generations in the same dynasty
t=0 δ Jt subject to,{(IC∗t ), (TSt−1 )}∞t=1(iii) The labor market equilibrium (LME): Lt = 1, and the market wage wt is determined by:
wt = A(1 − α)[lztα + θt (1 − l)xαt ],(iv) The credit market equilibrium (CME) is
wt−1 = lzt + θt (1 − l)(xt + I),
where the initial capital z0 = x0 is given.
Here, condition (i) yields the optimal capital choice of each young non-relationshipproducer taking market prices wt and ρt as given. Condition (ii) is the optimal choicewith regard to relational contract {zt , Rt } for every relationship pair, as explained
above. The third condition (LME) is the labor market clearing condition: market wage
wt is determined by clearing the labor market (Lt = 1) using the labor demand function(5) and that the labor supply is given by one unit mass of young workers in every period.
The fourth condition (CME) is the credit market clearing condition. Each relationshipproducer lends wt−1 − zt to the anonymous credit market in period t − 1 after she
makes a relationship lending zt to the relationship producer. In addition to this supplyof credit, 1−l non-relationship lenders outside the community have nothing but to lend
wt−1 to the credit market. Thus, the total supply in the anonymous credit market isgiven by l(wt−1 − zt ) + (1 − l)wt−1 . Conversely, 1 − l non-relationship producers ﬁnancetheir capital and veriﬁcation investments xt + I from the credit market. However,
only θt fraction of them can borrow the fund from the credit market. Thus, the total
15

demand in the credit market is given by θt (1 − l)(xt + I). Then, the credit market
clears if (CME) holds.
In this model economy, the initial condition is given by initial capital stock z0 = x0
owned by each initial old intermediate good producer. Thus, every initial old intermediate goods producer produces x0 without any production and veriﬁcation costs. Then,θ0 = 1 so that the market wage in the initial period (t = 0) is determined by LME:
w0 = A(1 − α)xα0 .

5From Relationships to Markets

5.1

Characterization of Equilibrium Paths

In this section we show the characterization result that relational contracting contributes to economic growth in low development stages while its value declines as the
economy grows and enters high development stages.
We begin by showing the three preliminary results that will be useful for characterizing equilibrium paths below.
First, Lemma 1 describes the optimal behavior of non-relationship producers. Because the utility function of each non-relationship producer depends on his child’s
consumption level, there may be intergenerational strategic interactions between a nonrelationship producer’s choice of capital investment xt and his child’s capital investment
choice xt+1 . However, the following lemma shows that it is suﬃcient to focus only on
the equilibrium in which each non-relationship producer acts to maximize only his own
proﬁt πt no matter the history.
Lemma 1. In any equilibrium, every non-relationship producer born in period t − 1chooses capital investment level xt to maximize only his own proﬁt πt no matter thehistory observed up to period t.
Thanks to Lemma 1, equilibrium investment of each non-relationship producer isdetermined by maximizing πt . We then deﬁne such optimal investment as x(ρt ), which
satisﬁes the following ﬁrst-order condition:
Aα2 xα−1= ρt .
t(8)π(ρ) ≡ max π = Aα(1 − α)x(ρ)α − ρI.

(9)Thus, we havex

Because maxx π is decreasing in ρ, we can ﬁnd a unique d > 0 such that π(d) = 0.
Then, the equilibrium interest rate in the credit market must be bounded above by
d. Let x ≡ x(d). Because π(d) = Aα(1 − α)xα − dI = 0 and α2 Axα−1 = d, we havex = αI/(1 − α). Then, ρt ≤ d is equivalent to xt ≥ x.
By (CME), ρt = d becomes the equilibrium interest rate when wt−1 − lzt < (1 −l)(x + I) (see Figure 1). In this case, θt fraction of non-relationship producers decide16

to ﬁnance the investment x + I, while the remaining fraction decide to shut down. In
either case, each non-relationship producer obtains zero proﬁt because π(d) = 0.
Second, Lemma 2 describes the optimal behavior of relationship pairs.
Lemma 2. (i) Relational contract {zt , Rt } in period t is sustained (self-enforceable) if
and only if (IC∗s ) and (TSs ) are satisﬁed for all s ≥ t. (ii) zt ≤ x(ρt ) in any period t.
(iii) zt < x(ρt ) only if (IC∗t ) is binding.
Lemma 2 (i) implies that a sequence of relational contracts {zt , Rt }∞t=1 is selfenforceable if and only if (IC∗t ) and (TSt−1 ) are satisﬁed for all t ≥ 1. Among all
relational contracts that satisfy this requirement, the initial generation of relationshippairs in each dynasty chooses the relational contracts∑∞thatt maximize the sum of thejoint payoﬀs of all generations in the same dynasty t=0 δ Jt . Lemma 2 (ii) and (iii)then show that the optimal relational contracts may involve a downward distortion ofcapital investment zt < x(ρt ) relative to the one maximizing the joint payoﬀ Jt without(IC∗t ). Such downward distortion occurs only if (IC∗t ) is binding.
Third, we show that in any equilibrium path each relationship producer never invests
in capital more than the funds available to his matching lender wt−1 in any period t.
Otherwise, zt > wt−1 holds in some period t, and hence each relationship producer needs
to ﬁnance the remaining amount zt − wt−1 from the credit market after borrowing wt−1
directly from the relationship lender matching him. Then, the relationship producer
must incur the veriﬁcation cost ρt I, but this cannot be optimal relational contract
because lowering capital investment from zt to wt−1 can avoid the veriﬁcation cost andimprove the joint payoﬀs of some generations in the dynasty of relationship pairs.
Lemma 3. In any equilibrium path zt ≤ wt−1 must hold in any period t.
Next, we turn to characterize equilibrium paths. The features of equilibrium paths
depend on the altruistic parameter δ ∈ (0, 1), which represents how much each individual cares about his or her child. Here, δ plays a role similar to that of the discountfactor in the repeated games. Thus, (IC∗t ) becomes less stringent as δ becomes larger,
in which case the choice of capital investment by relationship pair zt is never constrained and hence is given as the same level chosen by the non-relationship producer(zt = xt = x(ρt )).
As a benchmark, we ﬁrst consider the case that (IC∗t ) is never binding in any
period. Thus, the capital investment choice of any relationship pair is not constrainedby (IC∗t ), and hence zt = xt , as shown in Lemma 2. Then, by using wt = (1 − α)[lAztα +(1 − l)θt Axαt ] together with (CME), we can readily verify that a sequence of capital
investments and fractions of active non-relationship producers {xt , θt }∞t=0 obeys thefollowing dynamic equation:(l + (1 − l)θt )(1 − α)Axαt = lxt+1 + (1 − l)(θt+1 xt+1 + I)
where θt < 1 occurs only when xt = x = x(d).
17

(10)

We consider the following assumptions.
Assumption 1. (1 − α)Axα > x + I.
Assumption 2. x∗ > x where x∗ is deﬁned as a unique x satisfying (1 − α)Axα =x + (1 − l)I.
Assumptions 1 and 2 state that the productivity of the economy A is so large thatthe total investment x+I can be covered and non-relationship producers can eventuallyescape from the lowest production level x as the economy reaches the steady state x∗
without (IC∗t ). Without these requirements, the economy cannot take oﬀ from a low
production level x for each non-relationship producer, even when we ignore (IC∗t ).
Now we can show the following result.
Proposition 1. Suppose that Assumptions 1 and 2 hold. Then, if (IC∗t ) is neverbinding in any period t in an equilibrium path, the economy converges to a uniquesteady state x∗ in that equilibrium.
According to Proposition 1, if the altruistic parameter δ is so large that (IC∗t ) is
not binding at all, the economy eventually converges to a unique steady state x∗ in thelong run. However, this scenario is never the case when δ is not large, as we will show
in the following lemma.
Lemma 4. Suppose that Assumptions 1 and 2 hold. Suppose also that δ < x∗ /I. Then,zt < xt must hold so that (IC∗t ) must be binding in some period t in any equilibrium
path.
Lemma 4 is intuitive because if zt = xt holds in any period t, the equilibrium pathof the economy is unique and converges to the steady state x∗ because of Proposition
1. However, then (IC∗t ) becomes δρt+1 I ≥ ρt xt in a neighborhood of the steady statex∗ because zt = xt for all t and π(ρt+1 ) > 0 for all xt close to x∗ . Thus, (IC∗t ) can be
α−1expressed as δxt+1
I ≅ δ(x∗ )α−1 I ≥ xαt ≅ (x∗ )α for xt+1 ≅ xt ≅ x∗ , implying δI ≥ x∗ .
This expression violates δ < x∗ /I, and hence (IC∗t ) must be binding in some period t
in any equilibrium.
In what follows we rule out the trivial case that (IC∗t ) never binds in any period t.
We thus assume that δ < x∗ /I, as Lemma 4 shows. We also suppose that δ is not sosmall that (IC∗t ) becomes slack when non-relationship producers operate at the lowest
production level x. Otherwise, we may reach the other polar case that (IC∗t ) is always
binding at the lowest production level zt = x in any period. What we will address inthe following is to determine when (IC∗t ) changes from non-binding case to binding caseor vice versa and how such timing interacts with the process of economic development.
Thus, we will rule out the case that (IC∗t ) is always binding at the lowest production

18

level x. Let xt+1 = xt = x. Then, we can see that (IC∗t ) becomes slack at zt+1 = zt = x:
δ{αAxα − dx} > dx

(11)where the proﬁt of non-relationship producer becomes π(d) = 0 at xt+1 = x. The above
inequality is equivalent to δ(1 − α)Axα > α2 Axα , that is, δ > α/(1 − α). When thiscondition is satisﬁed, (IC∗t ) will be not binding if non-relationship producers operate at
the lowest production level x.
Thus we assume the following:
Assumption 3. α/(1 − α) < δ < x∗ /I.
We will also slightly strengthen Assumptions 1 and 2 as follows.
Assumption 4. (1 − l)(1 − α)Axα > x + I.
Assumption 5. x > x where x is the value of x that is a unique solution to1(δI/xα ) 1−α = (1 − α)Axα − (1 − l)I.

(12)

Assumption 4 means that the productivity of the economy measured by the parameter value A is so large that the economy can cover the minimum investment costx + I even when only non-relationship producers operate a the lowest scale x. We willshow later that this assumption can ensure that the economy eventually takes oﬀ fromthe lowest production scale x as it develops over time. Assumption 5 is also satisﬁed
when the productivity parameter A of the ﬁnal good production is large relative to theveriﬁcation cost I (note that x = αI/(1 − α) and thus it is independent of A.)
It can be readily seen that x∗ > x holds because δ < x∗ /I under Assumption 3.
Thus, Assumption 5 is stronger than Assumption 2 when Assumption 3 holds. Because
Assumptions 4 and 5 imply Assumptions 1 and 2 as long as we keep Assumption 3,
Proposition 1 and Lemma 4 still hold.
Now we proceed to show the general characterization result of equilibrium pathsof the economy. To this end, we combine (LME) with (CME) to obtain the following
dynamic equation:(1 − α)A[lztα + (1 − l)θt xαt ] = lzt+1 + (1 − l)θt+1 (xt+1 + I)

(CMEt )

where θ < 1 occurs only when xt = x, and xt > x implies that θt = 1. Further, notethat xt = x(ρt ) holds by (8).
Additionally, the choice of capital investment zt by a relationship pair must satisfy
the following optimality condition in Lemma 2: zt = xt holds if (IC∗t )αδ{αAzt+1− ρt+1 zt+1 − π(ρt+1 )} ≥ ρt zt

19(IC∗t )

is not binding, and zt ≤ xt otherwise.
These two dynamic conditions govern the evolution of {zt , xt , ρt , θt }∞t=0 given theinitial condition z0 = x0 . Before showing a formal result, it would be helpful to describe
these equilibrium conditions diagrammatically. In Figure 2 (a), we depict possible combinations of xt+1 and zt+1 that satisfy both (CMEt ) and (IC∗t ). To simplify argument,
here we assume that θt = θt+1 = 1 such that xt > x and xt+1 > x. The downward sloping straight line corresponds to (CMEt ): l(1 − α)Aztα + (1 − l)(1 − α)Axαt =lzt+1 + (1 − l)(xt+1 + I) given (xt , zt ). The upward sloping curve corresponds to (IC∗t )α −ρ
with equality given (zt , xt ): δ{αAzt+1
t+1 zt+1 − π(ρt+1 )} = ρt zt (see more details
about this result in Lemma A1 in Appendix). The shaded area in Figure 2 (a) is theregion for possible (zt+1 , xt+1 ) for which (IC∗t ) is satisﬁed. Then, a straight line BD
corresponds to possible equilibrium values of zt+1 and xt+1 , given (zt , xt ). If (IC∗t ) is notbinding, the equilibrium must be the intersection between BD line and 45 degree line,
a point B. If (IC∗t ) is binding, the equilibrium must be a point D. Once (zt+1 , xt+1 ) is
determined as shown above, (zt+2 , xt+2 ) in the next period can be found in the similarway.
It seems that it is diﬃcult to characterize equilibrium paths deﬁned above. In
general, we cannot rule out the case that there exist multiple paths of capital accumulation and that they are not monotonic over time. Due to such complication ofequilibrium paths, it is not easy to see the time structures of relational contracting inequilibrium paths: in what stages of an equilibrium path do the incentive compatibility(self-enforcing agreement) constraints become binding or slack? Even so, we can show
the following characterization result about the time structure of relational contracting.
Proposition 2. Suppose that Assumptions 3-5 are satisﬁed. Then, in any equilibrium
path there exists a unique switching period T ≥ 0 such that(i) zt = xt holds for all t ≤ T and, in particular, (IC∗t ) is not binding in any periodt ≤ T − 1 (if T ≥ 2), and(ii) (IC∗t ) is binding so that zt < xt for all t ≥ T + 1
where T satisﬁes xT ≤ x for x deﬁned in Assumption 5.
Proposition 2 states that in any equilibrium path the capital investment chosen
by relationship pairs must be downward distorted zt < xt in subsequent stages ofthe equilibrium path (t ≥ T + 1), but has no distortions zt = xt in early stages(t ≤ T ). In other words, in any equilibrium path the time structure of relational
contracting must possess a unique switching period that triggers the change from therelationship-based system, in which (IC∗t ) does not constrain capital accumulation, tothe market-based system, in which (IC∗t ) becomes tighter, and hence the value of a
relational contract becomes lower relative to arm’s length contract (zt < xt ). The
switching period T must occur before the economy hits a unique critical production20

level x, that is, xT ≤ x. Above such critical production level, the equilibrium pathmust have downward distortion of the relational contract.
Although the formal proof of Proposition 2 is relegated to the Appendix, we explain
the main logic of this result by using diagrams. The key to the result is the dynamic
interactions between (CME) and (IC∗t ) over time.
First, we show that if downward distortion does not occur in some period t, zt = xt ,then this is the case in any period before t, i.e., zs = xs for all s ≤ t. To see this result,simplify the argument by assuming that θt = θt+1 = 1 (see the proof in Appendix for
more details). Assume that zt = xt holds in some period t. Then, (CME) in period t
becomes(1 − α)Axαt = lzt+1 + (1 − l)(xt+1 + I)
(CMEt )
When we set zt = xt in Figure 2(a), (zt+1 , xt+1 ) must lie on the segment of (CMEt ),denoted by BD in Figure 2(a). Then, from Figure 2(a), we have (δIα2 A/ρt xt )1/(1−α) ≥(1 − α)Axαt − (1 − l)I, which is equivalent to xt ≤ x. Next, we turn to the previous
period t − 1. Because x∗ > x and x ≥ xt , we can ﬁnd a unique x̃t−1 such that
(1 − α)Ax̃αt−1 = xt + (1 − l)I (see Figure 3). Here, x̃t−1 < x. Thus, xt = (1 − α)Ax̃αt−1 −(1 − l)I < (δIα2 A/ρ(x̃t−1 )x̃t−1 )1/(1−α) . Also, because (CME) in period t − 1 becomesα(1 − α)A(lzt−1+ (1 − l)xαt−1 ) = xt + (1 − l)I,

(CMEt−1 )as well as zt−1 ≤ xt−1 , we have xt−1 ≥ x̃t−1 ≥ zt−1 . Thus, ρt−1 ≡ ρ(xt−1 ) ≤ ρ(x̃t−1 ).
Then, because ρt−1 zt−1 ≤ ρ(x̃t−1 )x̃t−1 , (IC∗t−1 ) must be slack when zt = xt (a point E
in Figure 2(b)):δ{αAxαt − ρt xt − π(ρt )} > ρt−1 zt−1 .
Repeating this process backward, we can show that (IC∗s ) becomes slack in any periods ≤ t − 1 if capital investment of relationship pairs have no distortions (zt = xt ) insome period t.
Second, by Lemma 3 we already know that any equilibrium path must possess theproperty that (IC∗t ) is binding in some period, for example T + 1. By combining thiswith the above result, we can show that (IC∗t ) must be binding for all t ≥ T + 1, but
is slack for all t ≤ T . To see this last result, assume that (IC∗m ) is binding again after(IC∗T ) is binding but (IC∗t ) is slack for all t such that T < t < m. However, if (IC∗t ) isslack at t where m < t, downward distortion never occurs in period t (zt = xt ), andthen (IC∗m ) must be slack as well, resulting in a contradiction. Thus, there must exist
a unique switching period T such that (IC∗t ) becomes slack in any period t ≤ T but
must be binding in any period t ≥ T + 1.
Although Proposition 2 states that relational contracting becomes less constrainedin the early stages of an equilibrium path than in subsequent stages, it does not address
how the value of relational contracting is related to the development levels of theeconomy. To understand this implication, we make the following additional condition:
Assumption 6. w0 < (1 − l)(x + I).

21

Assumption 6 says that the initial wage w0 is not so large enough that all nonrelationship producers can ﬁnance their capital and veriﬁcation investment x1 + I inperiod 1. Thus, only some fraction of these producers can ﬁnance the investment for
future production, and the market interest rate is given by its upper bound ρ1 = d inperiod 1.
Then we can show the following result.
Proposition 3. Suppose that Assumptions 3-6 hold. Then, in any equilibrium paththere must exist some periods T ∗ ≥ 0 and T ∗∗ ≥ 0 where T ∗ ≤ T ∗∗ such that• zt = xt = x and θt < 1 are satisﬁed in any period t ≤ T ∗ , and• (IC∗t ) is binding in any period t ≥ T ∗∗ in which zt < xt , xt > x and θt = 1 hold.
Proposition 3 shows that relational contract becomes less constrained in low development stages of the economy than in developed stages. Proposition 3 thus captures thefeature discussed by several historians (see for example Lamoreaux (1994)) that relational contracting contributes more to economic growth in low development phases, inwhich the anonymous market does not work well for all non-relationship producers tooperate (θt < 1 and xt = x), than in high development stages, in which the market wellfunctions (θt = 1 and xt > x). Also, this result is consistent with the argument that
relational contracting is not a substitute for the market based economy but plays arole to complement it and foster economic development (see Aoki and Hayami (2000)):
relational contracting has no downward distortion in low development stages, whereas
all non-relationship producers cannot ﬁnd a way to ﬁnance their capital investments.
Thus, in these stages, a relational contract rather than an arm’s length contract contributes to capital accumulation in the economy, which in turn lowers the interest rates
over time, resulting in the increase of the proﬁt of an arm’s length contract. Thus,
after the economy passes a critical development point, all non-relationship producers
can start ﬁnancing their capital investments.
The intuition behind Proposition 3 is as follows: Assumption 6 ensures that θt < 1holds for all small t, and hence non-relationship producers operate at the lowest level
x in early stages of equilibrium path. In these stages, (IC∗t ) becomes more likely tobe slack so that relational contract has no downward distortion (Proposition 2). Conversely, because the productivity of the economy A is large enough that the minimum
scale of capital investment and the veriﬁcation cost x + I can be covered under Assumption 4, we can also show that, as time goes by, the economy eventually takes oﬀ
from the low production level x. This result is due to the fraction of non-relationshipproducers who can ﬁnance capital investment tends to increase over time. Then, theincrease in the proﬁt of arm’s length contract makes (IC∗t ) harder to satisfy so that
relational contract eventually exhibits a downward distortion as the economy grows.
However, it is not obvious how much relational contracting becomes diﬃcult sustained as the economy enters well-developed stages because the general equilibrium
22

eﬀect manifests: when (IC∗t ) becomes tighter, capital investment of relationship producers zt declines, negatively aﬀecting capital accumulation of the economy as a whole.
However, this eﬀect makes the interest rates larger, by which the proﬁt of an arm’s
length contract becomes lower, and hence the self-enforcing condition of relational contract becomes more likely to be satisﬁed, contrary to the ﬁrst eﬀect that (IC∗t ) becomestighter.
Even though this general equilibrium eﬀect makes the long run features of equilibrium paths diﬃcult to predict, in the next proposition we show that the downward
distortion of capital investment choice by relationship producers persists and has apositive lower bound in the long run.
To see this eﬀect, we deﬁne x̂ as the largest value of x that satisﬁes(1 − l)(1 − α)Ax̂α = x̂ + (1 − l)I.

(13)

By Assumption 4, we can verify that such x̂ exists and x̂ > x.
Then we can show the following result.
Proposition 4. Suppose that Assumptions 3, 4 and δ < x̂/I hold. Then in any
equilibrium path, the downward distortion of capital investment by relationship pairs
xt − zt has the positive lower bound in the long run, that is,
xt − zt ≥ x̂ − δI > 0
for all large t.
Proposition 4 shows that the downward distortion of capital investment chosen
by relationship producers never disappears and that it still persists in the long run.
The degree to which relational contracting distorts capital investment depends on thealtruistic parameter δ and the veriﬁcation cost I. When δ (or/and I) is lower and thus
the gain of supporting relational contract becomes smaller, the downward distortion
becomes larger.
Proposition 4 conﬁrms the empirical and historical fact that the relationship-based
system, which relies on the use of personal ties, kinships, and informal connections,
eventually declines and has serious limitations as the economy becomes richer: Lamoreaux (1994) found the evidence that insider lending practices, which had served an
important ﬁnancing device in New England in early 19th century, became less important and were eventually replaced by the market-based ﬁnance as the United Statestransitioned from a capital poor economy to a capital rich economy in late 19th century.
Demirgüç-Kunt and Levine (2004) also reported the related fact that the bank ﬁnance,
which is sometimes characterized as long term relationship between particular banks
and ﬁrms, becomes less popular relative to the market-based ﬁnance, such as equity, in
more developed country (see Rajan and Zingales (2000) for a related argument).
Although our characterization results above highlight the role of relational contracting in diﬀerent phases of economic development, these results do not necessarily rule23

out the case that (IC∗t ) is binding in any period t ≥ 1 (thus T = 0 holds in Proposition
2). To address this issue, we show that there exists an equilibrium path that has thefollowing feature: there exists some T ≥ 1 such that in early stages of developmentt ≤ T (IC∗t ) is not binding, and hence no downward distortion occurs (zt = xt ), whilein subsequent stages, t ≥ T + 1 (IC∗t ) becomes binding, and hence the economy involves downward distortion zt < xt . We call this type of equilibrium path a switchingequilibrium path.

5.2

Existence of Switching Equilibrium Path

To show the existence of the switching equilibrium path deﬁned above, we make thefollowing additional assumptions:
Assumption 7. min{l, (1 − l)}(1 − α)Axα > x + (1 − l)I.
}{
Assumption 8. x∗ > max x, (δI/xα )1/(1−α) where x∗ is deﬁned as x satisfying(1 − α)Axα − (1 − l)I = (1 − l) (δI/xα )1/(1−α)

Assumption 7 is essentially same as Assumption 4: the economy can eventually takeoﬀ from the lowest production level x for each intermediate good of non-relationshipproducer. Assumption 8 seems complicated, but we can observe that this condition issatisﬁed when the veriﬁcation cost I is not large (see Appendix for more details).
Then, we can show the following existence result.
Proposition 5. Suppose that Assumptions 3 and 6-8 hold. Then, there exists theswitching equilibrium path with T ≥ 1 such that
• (IC∗t ) is not binding and zt = xt = x with θt < 1 in any period t ≤ T ,• (IC∗t ) is binding such that zt < xt and xt > x (θt = 1) in any period t ≥ T + 1.
The proof of Proposition 5 is constructive: First we set xt = zt = x for all t ≤ T
for some switching period T ≥ 1. Thus, no downward distortion of relational contract
occurs and the production level of non-relationship producers is given at the lowest scale
x. Put diﬀerently, the interest rate in the credit market hits its upper bound ρt = dsuch that only some fraction of non-relationship producers can ﬁnance their capital
investment in any period t ≤ T . We can then see that (IC∗t ) becomes δ{αAxα −dx} ≥ dx
in any period t ≤ T , which is satisﬁed by Assumption 3. After the switching period(t ≥ T + 1), we sequentially construct a path {zt , xt }∞t=T +1 as follows: (i) xt > x. (ii)(IC∗t ) holds as equality. (iii) (CME) holds with θt = θt+1 = 1. Assumption 7 ensuresthat the economy can take oﬀ from the low production scale: xt > x holds for all24

t ≥ T + 1 when we take T to be large. By using Assumption 8, we can then ﬁnd a pair(zt+1 , xt+1 ) such that both (IC∗t ) as equality and (CME) in period t are satisﬁed, given(zt , xt ).

6

Conclusion

In this paper, we have investigated a dynamic general equilibrium model that takes into
account the dynamic change in the contract enforcement modes from relational contracts to arm’s length contracts over time. We have shown that relational contracting
plays an important role in sustaining production in the early stages of the development
process in which arm’s length contracting is not widely used. In subsequent periods,
non-relationship producers ﬁnd it proﬁtable to use arm’s length contracts because theeconomy is so well developed that the market size is large and the interest rate falls.
However, as the economy enters its mature stages, relationship-based systems decline
and may be partially replaced by market-based systems. We have focused on relational
contracting between borrowers and lenders, which becomes valuable for relating ourtheoretical results to the historical evidence on relationship-based ﬁnancing. Of course,
this model is one of the modeling choices that capture relational contracting in dynamic
general equilibrium frameworks. It is important to investigate how the development
process is dynamically linked with long-term relationships in diﬀerent contexts such asﬁrm-worker relationships, inter-ﬁrm relationships, and government-public relationships.
We conclude the paper by discussing the role of bequest transfers between successive generations. In the main text we have assumed that each old individual has notechnologies to give their children bequest at all. One might think that old individuals will bequeath their children when bequest is available because they care about theconsumption levels of their children. The possibility of bequest allows each producer
to ﬁnance a part of capital investment from the bequest he has received from his parent. Thus non-relationship producers can reduce the veriﬁcation cost and relationshippairs can make (IC∗t ) more easily satisﬁed because the opportunity cost to use the arm’s
length contract becomes lower. However, we can show that no old individuals bequeath
their children at all, as in our basic setting, as long as the altruistic parameter value δ
belongs to a certain small range (a more detailed analysis is relegated to Appendix).

7

Appendix A: Proofs for Lemmas and Propositions

Proof of Lemma 1.
(i) For a relationship pair born in period t − 1 to be inherited by the next generation,(IC∗s ) and (TSs−1 ) must be satisﬁed for all s ≥ t. Otherwise, (IC∗t̃ ) or (TSt̃−1 ) isviolated in some period t̃ ≥ t, which implies that the relationship pair born in periodt̃ − 1 is dissolved. By anticipating this and using the backward induction argument,
every relationship pair born in any period before t̃ − 2 would not have the incentive tomaintain the relationship. On the other hand, if both (IC∗s ) and (TSs−1 ) are satisﬁed
25

for all s ≥ t, then we can show that there exists a subgame perfect equilibrium inwhich every relationship pair born in period s ≥ t − 1 honors the relational contract{zs , Rs }, which satisﬁes all (ICs ), (IRPs ), and (IRLs ) for all s ≥ t. To see this, assume
that (IC∗s ) and (TSs−1 ) hold for all s ≥ t. Then, we can ﬁnd a sequence of relational
14 Given
contracts {zs , Rs }∞s=t such that all (ICs ), (IRPs ), and (IRLs ) are satisﬁed.
such a sequence of relational contracts {zs , Rs }∞s=t , we can show that the following
strategies played by relationship producers and lenders can constitute a subgame perfectequilibrium in a continuation equilibrium from period t onward:• Period s ≥ t:
– The young relationship producer and young relationship lender agree ona relational contract {zs+1 , Rs+1 }. At this stage, the young relationshiplender lends zs to the relationship producer if their parents honored relational
contract {zs , Rs } speciﬁed in the previous period. Otherwise, they exercise
the quitting option.
• Period s + 1:
– The old relationship producer honors repayment Rs+1 and does not exercise
the quitting option, provided his and the relationship lender’s parents did
not exercise the quitting option in the previous period.
– The old relationship lender does not exercise the quitting option, provided
her and the relationship producer’s parents did not exercise the quitting
option in the previous period.
The above strategies specify the trigger-like feature in which the young relationshiplender agrees on continuing the relationship provided the producer in the previous
generation made the contractual repayment, Rs ; if this was not the case, she would
revert to the punishment strategy by exercising the quitting option. Here, note that
we always have a continuation equilibrium in which any relationship producer and any
relationship lender of any pair simultaneously exercise the quitting option. Then, we
can readily see that, because (ICs ), (IRLs ), and (IRPs ) are satisﬁed in any period s ≥ t,
all relationship producers and lenders optimally follow these strategies from period t
onward, ensuring that every relationship is inherited by the next generation.
(ii) Note ﬁrst that x(ρt ) maximizes αAztα −ρt zt over zt ≥ 0. Suppose now that zt > x(ρt )in some period t. Then, if we can slightly decrease zt , we can still keep (IC∗t ) and(IC∗t−1 ). Note that zt aﬀects only (IC∗t ) and (IC∗t−1 ). The slight decrease of zt makes(IC∗t ) easier to be satisﬁed while it increases the left hand side of (IC∗t−1 ) because x(ρt )maximizes the left hand side of (IC∗t−1 ). But then such change increases the joint payoﬀ
Jt = αAzt − ρt zt + ρt wt−1 due to the deﬁnition of x(ρt ). Thus zt ≤ x(ρt ) holds.
14
For example, we can set Rs = ρs zs in every period s ≥ t, which satisﬁes all (ICs ), (IRPs ), and(IRLs ) with some zs in every period s ≥ t.

26

(iii) Suppose that zt < x(ρt ) but (IC∗t ) is not binding in some period t. Then, if we
can slightly increase zt toward x(ρt ), we can still keep (IC∗t ) and (IC∗t−1 ). The slightincrease of zt does not violate (IC∗t ) while it increases the left hand side of (IC∗t−1 )because of zt < x(ρt ). But then such change increases the joint payoﬀ Jt due to thedeﬁnition of x(ρt ). Q.E.D.
Proof of Lemma 2.
Take an equilibrium path in which each non-relationship producer uses a strategy σt :Ht → [0, ∞) which maps from, Ht , the set of all previous observed histories up toperiod t to the current capital investment level xt ≥ 0.
Then we will show that xt = x(ρt ) holds so that every non-relationship producer
acts to maximize his own proﬁt πt in every period irrespective of observed histories
ht ∈ Ht . We denote by σ t such strategy deﬁned as σ(ht ) = x(ρt ) for all ht ∈ Ht .
To show this, suppose that there exists an equilibrium with σt ̸= σ t for some non–relationship producer in some period t. Thus σ(ht ) ̸= x(ρt ) for some ht ∈ Ht . We thendenote by {xs }∞s=t the equilibrium sequence of capital investments of non-relationshipproducers from period t onward according to the equilibrium strategies {σs }∞s=t .
In the proof of Lemma 2, with a slight abuse of notation we will denote by π(xt )the proﬁt of a non-relationship producer who chooses xt in period t.
Since that non-relationship producer could always choose x(ρt ) in period t, it must
be the case thatπ(xt ) + δπ(xt+1 ) ≥ π(x(ρt )) + δπ(x′t+1 )where his child will choose x′t+1 in period t + 1 following the choice of his parent x(ρt )according to his equilibrium strategy σt+1 . The above inequality yields
δ{π(xt+1 ) − π(x′t+1 )} ≥ π(x(ρt )) − π(xt ).

(A1)

Also, for x′t+1 to be the optimal choice by the young non–relationship producer inperiod t + 1 following x(ρt ) chosen by his parent, we must haveπ(x′t+1 ) + δπ(x′′t+2 ) ≥ π(x(ρt+1 )) + δπ(x′t+2 )′′because he could always choose x(ρt+1 ) following his parent choice x(ρt ) where σt+2′′
denotes the choice of his child in period t + 2 following xt+1 . Here xt+2 is the choice inperiod t + 2 following x(ρt+1 ). This then yields

δ{π(x′′t+2 ) − π(x′t+2 )} ≥ π(x(ρt+1 )) − π(x′t+1 ).
Since π(x(ρt+1 )) ≥ π(xt+1 ), we have
δ{π(x′′t+2 ) − π(x′t+2 )} ≥ π(xt+1 ) − π(x′t+1 ).

(A2)

Combining inequalities (A1) with (A2), we have
δ 2 {π(x′′t+2 ) − π(x′t+2 )} ≥ π(x(ρt )) − π(xt ).
27

(A3)

In period t + 2 the young non–relationship producer in the dynasty in question mustchoose x′t+2 following the choice x(ρt+1 ) by his parent in period t + 1. Thus, since hecould always choose x′′t+2 instead of x′t+2 in period t + 2, we must haveπ(x′t+2 ) + δπ(x′′t+3 ) ≥ π(x′′t+2 ) + δπ(x′t+3 )where x′′t+3 and x′t+3 denote the choices by the non–relationship producer in period t+3
following x′t+2 and x′′t+2 respectively. Thus
δ{π(x′′t+3 ) − π(x′t+3 )} ≥ π(x′′t+2 ) − π(x′t+2 )≥ π(x′′t+2 ) − π(x′t+2 )Combining this with (A3), we have
δ 3 {π(x′′t+3 ) − π(x′t+3 )} ≥ π(x(ρt )) − π(xt )Repeating this argument for all periods s + t ≥ t, we obtainδ s {π(x′′t+s ) − π(x′t+s )} ≥ π(x(ρt )) − π(xt ), ∀ s ≥ 0.
Since π(x(ρt )) ̸= π(xt ) by our supposition (σ(ht ) = xt ̸= x(ρt )), there exists some
ε > 0 such that π(x(ρt )) − π(xt ) ≥ ε. The left hand side of the above inequality isbounded above by δ s π(ρt+s ) because π(xt+s ) ≥ 0 and π(x(ρt+s )) = maxx π(x) given
ρt+s . Note here that π(xt+s ) ≥ 0 because, by the spot transaction nature of arm’s
length contract in the anonymous credit market, each non-relationship producer must
make the repayment ρt+s (xt+s + I) from what he earns pt+s xt+s when he is old.
Then, if π(x(ρt+s )) is bounded above, the left hand side of the above inequality
goes to zero by letting s → ∞ and noting δ < 1, which is a contradiction. Thus it
suﬃces to show that π(x(ρt )) < +∞ for any equilibrium interest rate ρt ≥ 0. Thisis equivalent to the condition that ρt ≥ ρ for all t for some ρ > 0 because thenπ(x(ρ)) ≤ maxx α(Ax)α − ρx < +∞. Suppose that ρt = 0 for some period t or
ρt → 0. In either case ρt < d which implies θt = 1. Also x(ρt ) must go to inﬁnity.
Since π(xt ) + δπ(xt+1 ) ≥ π(x(ρt )) + δπ(x′t+1 ) must hold in period t (see (A1)), either
π(xt ) → ∞ or π(xt+1 ) → ∞ or both must hold due to π(x(ρt )) → ∞ and π(x′t+1 ) ≥ 0,
which implies xt → ∞ or xt+1 → ∞ with ρt+1 = 0. In the former case (xt → ∞)(CME) must imply wt−1 = lzt + (1 − l)(xt + I) → ∞, which holds only when wt−1 → ∞so that xt−1 → ∞ or zt−1 → ∞. We can deal with the latter case xt+1 → ∞ in a similar
way. However, then wt−2 → ∞ by (CME) in period t − 1. Repeating this, w0 → ∞which is however impossible. Thus ρt ≥ ρ must hold in any period t for some ρ > 0.
This then establishes the lemma. Q.E.D.
Proof of Lemma 3.
Suppose contrary to the claim that zs > ws−1 in some period s in some equilibrium
path. In such equilibrium each relationship producer must incur the veriﬁcation costρs I because all his capital investment zs cannot be ﬁnanced by the relationship lender28

who owns only ws−1 and hence the remaining amount zs −ws−1 must be borrowed fromthe credit market.
There are two cases: (i) zs−1 ≤ ws−2 and (ii) zs−1 > ws−2 . In case (i), the relationship producer in period s − 2 does not ﬁnance from the credit market. Thus, incase (i), (ICs ) for relationship producer and (IRLs ) for relationship lender should be
modiﬁed tops−1 zs−1 − Rs−1 + δ{ps zs − Rs − ρs (zs − ws−1 ) − ρs I} ≥ ps−1 zs−1 + δπs
andRs−1 + δρs−1 (ws−2 − zs−1 ) + δRs ≥ ρs−1 ws−2 + δρs ws−1

(IC′s )
(IRL′s )Combining these conditions, (IC∗s ) must be changed toδ{αAzsα − ρs (zs + I) − π(ρs )} ≥ ρs−1 zs−1 .

(IC∗s -(i))

In case (ii), the relationship producer in period s − 2 ﬁnances the remaining amountzs−1 − ws−2 from the credit market after borrowing ws−2 directly from the relationshiplender. Thus, in case (ii), (ICs ) for relationship producer and (IRLs ) for relationshiplender should be modiﬁed tops−1 zs−1 − Rs−1 − ρs−1 (zs−1 − ws−2 − I) + δ{ps zs − Rs − ρs (zs − ws−1 ) − ρs I}≥ ps−1 zs−1 − ρs−1 (zs−1 − ws−2 − I) + δπs
andRs−1 + δRs ≥ ρs−1 ws−2 + δρs ws−1

(IRL′s )Combining these conditions, (IC∗s ) must be changed toδ{αAzsα − ρs (zs + I) − π(ρs )} ≥ ρs−1 ws−1 .

(IC∗s -(ii))

However, we show that the second case (ii) never happens in equilibrium. This isbecause the left hand side of (IC∗s -(ii)) is bounded above by
δ{αAxαs − ρ(xs + I) − π(ρs )} = 0
and hence from (IC∗s -(ii) )we must have ρs−1 ws−1 = 0, which is however impossible.
Thus we consider only the ﬁrst case (i). Then, since the left hand side of (IC∗s -(i)) isbounded above by zero, we must have zs−1 = 0 (due to ρs−1 > 0) which happens only
when zs = xs . Also, by (CME) in period s − 1, we have ws−1 = lzs + (1 − l)θs (xs + I) =lxs + (1 − l)θs (xs + I). Since ws−1 < zs and xs = zs , we must have θs < 1 so thatxs = x and ρs = d. Thus xs = zs = x and the joint payoﬀ of relationship pair in periods becomes Js = αAxα − d(x + I) = π(d) = 0. Next, since zs−1 = 0, from (IC∗s−1 (i)) we have −δπ(ρs−1 ) ≥ ρs−2 zs−2 which implies that ρs−1 = d (hence π(ρs−1 ) = 0)
and zs−2 = 0. Thus the joint payoﬀ of relationship pair in period s − 1 is given by
Js−1 = ps−1 zs−1 − dzs−1 = 0. Repeating this process, we have zt = 0 and Jt = 0 for allt = 1, 2, ..., s.
29

We then consider the deviation of relational contracts as follows:
Case A: s ≥ 2. Then set zs′ = ws−1 > 0 and zt′ = ε > 0 for any t where 2 ≤ t ≤ s whilezt′ = zt for all other t. We can then show that this deviation contract satisﬁes (IC∗t )and (TSt−1 ) for any period t. Since ρt = d for all t = 2, 3, ..., s, (IC∗s ) holds when ε > 0
is small becauseαδ{αAws−1− dws−1 − π(d)} ≥ dzs−1 = dε
where π(d) = 0 and ws−1 < zs = xs = x. Also, in any period t where 1 ≤ t ≤ s − 1,(IC∗t ) holds as well when ε > 0 is small because
δ{αAεα − dε − π(d)} ≥ dε
where π(d) = 0. Clearly, the joint payoﬀs of relationship pairs in period t (2 ≤ t ≤ s)
are increased and hence (TSt−1 ) are satisﬁed in these periods as well. Such deviation
improves the payoﬀs of the generations from t = 2 to t = s in a dynasty of relationshippairs. Thus the original relational contract cannot be optimal.
Case B: s = 1. Thus w0 < z1 . If θ1 = 1, then we have from (CME) in period 1 thatw0 = lz1 + (1 − l)(x1 + I) < z1 and hence z1 > x1 + I which is a contradiction. Thus
θ1 < 1 so that ρ1 = d. Then J1 = αAz1 − d(z1 + I) which is not larger than π(d) = 0.
Thus J1 ≤ 0. However, if we consider the deviation of relational contract as z1′ = w0 ,(IC∗1 ) still holds because δ{αAz2α − ρ2 z2 − π(ρ2 )} ≥ ρ1 z1 > ρ1 w0 while the joint payoﬀ
is increased to J1′ = αAw0α − dw0 > 0 because w0 < z1 ≤ x1 = x. Since (TS0 ) isclearly satisﬁed, such deviation makes the initial and second generations in a dynasty
of relationship pairs better oﬀ. Q.E.D.
Proof of Proposition 1.
Suppose that Assumption 1 and 2 are satisﬁed. Suppose also that (IC∗t ) is never bindingin any period t. Then, zt = xt holds in any period t. Thus the equilibrium path ofcapital accumulation zt = xt follows(l + (1 − l)θt )(1 − α)Axαt = lxt+1 + (1 − l)θt+1 (xt+1 + I), t = 0, 1, 2...

(A4)

where θt < 1 holds only when xt = x.
First we show that θs = 1 for some period s. If this is not the case, θt < 1 for all t.
Then (A4) becomes(l + (1 − l)θt )(1 − α)Axα = lx + (1 − l)θt+1 (x + I), t = 0, 1, 2...

(A5)

Then, by Assumption 1 we can show that the sequence {θt }∞t=1 which satisﬁes (A5)must have θs > 1 for some period s. This is a contradiction.
Second, we show that θs = 1 implies θs+1 = 1. Suppose contrary to the claim thatθs = 1 but θs+1 < 1. Then we have from (A4) and xs ≥ x that
(1 − α)Axα ≤ lx + (1 − l)θs+1 (x + I)
< lx + (1 − l)(x + I)
= x + (1 − l)I
< x+I
30

which contradicts to Assumption 1.
By these results, we know that θt = 1 for all t ≥ s for some period s. Thus, forlarge t, (A4) can be written by(1 − α)Axαt = lxt+1 + (1 − l)(xt+1 + I)
from which xt converges to a unique steady state x∗ . Q.E.D.
Proof of Lemma 4.
By Proposition 1, xt converges to x∗ when zt = xt holds in any period t.
However, when t is so large that xt ≅ x∗ , (IC∗t ) can be written by
δ{αA(x∗ )α − ρ∗ (x∗ ) − π(ρ∗ ) ≥ ρ∗ x∗
where ρ∗ satisﬁes x∗ = x(ρ∗ ). However, such (IC∗t ) is modiﬁed as δI ≥ x∗ which does
not meet Assumption 3. Thus zt < xt must hold in some period t. Q.E.D.
Proof of Proposition 2
We prove Proposition 2 by the following series of claims.
Lemma A1. Suppose that xT +1 > x and xT > x in some period T . Then, if zT = xT ,we must have xT ≤ x where x is deﬁned in Assumption 5.
Proof. Let xT +1 > x and xT > x in some period T . Suppose also that zT = xT . Then,since π(ρT +1 ) > π(d) = 0, (IC∗T ) can be written by
δ{αAzTα+1 − ρT +1 zT +1 − π(ρT +1 )} ≥ ρT xT(IC∗T )

From this we can derive the relationship between zT +1 and xT +1 as follows:
dzT +1dρT +1 /dxT +1 (xT +1 − zT +1 + I)
>0=−dxT +1α2 AzTα−1+1 − ρT +1
for zT +1 < xT +1 because dρT +1 /dxT +1 = Aα2 (α − 1)xα−2
T +1 < 0 and the denominator ispositive for zT +1 < xT +1 = x(ρT +1 ).
We also have (CME) in period T :(1 − α)AxαT = lzT +1 + (1 − l)(xT +1 + I)
because θT = θT +1 = 1 and zT = xT .
Thus, combining (IC∗T ) with (CME) in period T , we can verify that zT +1 and xT +1
which satisfy both these conditions exist only if((1 − α)AxαT − (1 − l)I ≤31

δα2 AI
ρT xT)1/(1−α)(A6)where the right hand side can be written by (δI/xT α )1/(1−α) (see Figure 2 (a)). In
other words inequality (A6) implies that xT ≤ x by deﬁnition of x. Q.E.D.
Lemma A2. Suppose that zT = xT in some period T . Then (IC∗t ) is not binding sothat zt = xt holds in any period t ≤ T − 1.
Proof. Let zT = xT . We obtain (IC∗T ) as follows:
δ{αAzTα+1 − ρT +1 zT +1 − π(ρT +1 )} ≥ ρT xT
Then we show that (IC∗t ) is not binding in any period t ≤ T − 1 in what follows.
CASE 1: xT = x. Then ρT = d. Now we show that (IC∗T −1 ) is not binding, i.e.,
δ{αAxα − dx} ≥ ρT −1 zT −1 where note that zT = xT = x.
We also have (CME) in period T − 1:l(1 − α)AzTα−1 + (1 − l)θT −1 AxαT −1 = lx + (1 − l)θT (x + I).
From this we can show that θT −1 < 1. To see this, since zT −1 ≥ 0 and xT −1 ≥ x, wex+(1−l)I
have (1 − l)θT −1 Axα ≤ x + (1 − l)I which implies that θT −1 ≤ (1−l)(1−α)Ax
α < 1 dueto Assumption 4. Thus θT −1 < 1 which implies that xT −1 = x and hence ρT −1 = d.
However, then by using Assumption 3 (δ > α/(1 − α)), we obtain
δ{αAxα − dx} > dx
≥ dzT −1
= ρT −1 zT −1
because x = xT −1 ≥ zT −1 . This shows that (IC∗T −1 ) must be slack.
CASE 2: xT > x. Thus ρT < d. Now we will show that (IC∗T −1 ) is not binding in what
follows.
CASE 2-1: xT +1 > x (thus ρT +1 < d). In this sub-case we have xT +1 > x and xT > x(note that we are in CASE 2). Thus we can apply Lemma A1 so that xT ≤ x.
We also have (CME) in period T − 1 as l(1 − α)AzTα−1 + (1 − l)θT −1 (1 − α)AxαT −1 =xT + (1 − l)I because zT = xT > x by our supposition in Lemma A2 so that θT = 1.
We will consider further two sub-cases:
Case (A): θT −1 = 1. We then deﬁne x̃T −1 as(1 − α)Ax̃αT −1 = xT + (1 − l)I(A7)Since zT −1 ≤ xT −1 , by using (CME) in period T −1 above, we can see that xT −1 ≥ x̃T −1
and zT −1 ≤ x̃T −1 . Then, by using xT ≤ x < x∗ , we can show that x̃T −1 < xT andhence x̃T −1 < x. Also we have ρT −1 = ρ(xT −1 ) ≤ ρ(x̃T −1 ).

32

Since x̃T −1 < x, the following inequality is satisﬁed:(δI)1/(1−α)> (1 − α)Ax̃αT −1 − (1 − l)Ix̃αT −1

which is equivalent to the condition that zT = xT satisﬁes
δ{αAxαT − ρT xT − π(ρT )} > ρ(x̃T −1 )x̃T −1 .
Then we can show that
δ{αAzTα − ρT zT − π(ρT )} = δ{αAxαT − ρT xT − π(ρT )}> ρ(x̃T −1 )x̃T −1≥ ρT −1 zT −1
because ρ(x̃T −1 ) ≥ ρ(xT −1 ) = ρT −1 and zT −1 ≤ x̃T −1 . Thus (IC∗T −1 ) is not binding.
CASE (B): θT −1 < 1. In this case xT −1 = x. Thus (CME) in period T − 1 becomesl(1 − α)AzTα−1 + (1 − l)θT −1 Axα = xT + (1 − l)I which impliesxT= l(1 − α)AzTα−1 + (1 − l)θT −1 (1 − α)Axα − (1 − l)I≤ (1 − α)Axα − (1 − l)I

because zT −1 ≤ xT −1 = x. ThusxT ≤ (1 − α)Axα − (1 − l)I.

(A8)

On the other hand, by noting that zT = xT by our supposition, (IC∗T −1 ) can be also
written byδρT I = δ{αAxαT − ρT xT − π(ρT )}≥ dzT −1
because ρT −1 = d. If this is binding, we must have δρT I = dzT −1 ≤ dxT −1 = dx. ThusxT ≥ (δI/xα )1/(1−α) is satisﬁed due to ρT = α2 AxTα−1 .
By combining this with (A8), we obtain(δI/xα )1/(1−α) ≤ xT ≤ (1 − α)Axα − (1 − l)I
so that (δI/xα )1/(1−α) ≤ (1−α)Axα −(1−l)I which implies that x ≥ x. This contradicts
to Assumption 5. Thus (IC∗T −1 ) is not binding.
CASE 2-2: xT +1 = x. In this case ρT +1 = d.
(CME) in period T becomes (1 − α)AxαT = lzT +1 + (1 − l)θT +1 (x + I) because ofzT = xT > x by our supposition and hence θT = 1. Then, since zT +1 ≤ xT +1 = x
and xT > x, this can be written by (1 − α)Axα < x + (1 − l)I which contradicts to33

Assumption 4. Q.E.D.
We have thus established that zt = xt holds in any period t ≤ T if zT = xT . In
particular we have shown that (IC∗t ) is not binding in any period t ≤ T − 1. ((IC∗T )may bind even when zT = xT .) Also, by Lemma 4 we know that zt < xt must hold insome period t. Then we can show that, if zT +1 < xT +1 holds so that (IC∗t ) is binding insome period T + 1, zt < xt holds in any period t ≥ T + 1. If this is not the case, there
exists some period m such that T + 1 < m and zm = xm . However, by Lemma A2, if
zm = xm , then zt = xt holds as well for all t < m. Thus zT +1 = xT +1 , a contradiction.
Thus we have shown that there exists a unique T such that zt = xt holds in any
period t ≥ T whereas zt < xt holds in any period t ≥ T + 1. Q.E.D.
Proof of Proposition 3.
We will ﬁrst show the following lemma.
Lemma A4. Suppose that Assumption 4 holds. Then in any equilibrium path there
exists some period τ ≥ 1 such that xt > x and hence θt = 1 hold for all t ≥ τ .
Proof.
(CME) in period t:l(1 − α)Aztα + (1 − l)θt (1 − α)Axαt = lzt+1 + (1 − l)θt+1 (xt+1 + I)

(CME)

where θt < 1 only when xt = x.
First we show that θs = 1 in some period s. This is a similar result to Proposition
1 but we need some modiﬁcation because zt ̸= xt may happen in this case.
Suppose contrary to the claim that θt < 1 for all t. Then xt = x (and hence ρt = d)for all t. Since by Proposition 2 we know that (IC∗t ) becomes binding for all t ≥ T forsome T , we then haveαδ{αAzt+1− dzt+1 } = dzt
for all t ≥ T . From this, zt monotonically decreases over time.
We deﬁne a unique z such that (1 − α)Az α = z. Then, by using (CME) above, wehaveθt (1 − α)Axα ≤ θt+1 (x + I).
(A10)
for all large t because, if t is large, then zt+1 < zt < z and hence (1 − α)Aztα − zt+1 ≥(1 − α)Az α − z = 0 for all large t. From (A10) and Assumption 4, we can verify that θt
unboundedly increases over time and hence θt > 1 for some period t, a contradiction.
Thus θs = 1 holds in some period s.

34

Second, we show that θs = 1 implies θs+1 = 1. Suppose that θs = 1 but θs+1 < 1.
Then, by substituting xs+1 = x ≥ zs+1 into (CME) in period s, we havel(1 − α)Azsα + (1 − l)(1 − α)Axαs = lzs+1 + (1 − l)θs+1 (x + I)
≤ lx + (1 − l)θs+1 (x + I)
≤ lx + (1 − l)(x + I)
≤ x + (1 − l)I
Then, since zs ≥ 0 and xs ≥ x, we obtain(1 − l)(1 − α)Axα ≤ x + (1 − l)I
which contradicts to Assumption 4. Thus θs = 1 implies θs+1 = 1.
Then we have established the result that there exists some period s ≥ 1 such thatθt = 1 for all t ≥ s (hence xt > x for all t ≥ s). Q.E.D.
Now we move back to the proof of Proposition 3.
Suppose that Assumption 6 holds. Then, if θ1 = 1, we have w0 = lz1 + (1 − l)(x1 +I) ≥ (1 − l)(x + I) which contradicts to Assumption 6. Thus θ1 < 1 must hold. By
combining this with Proposition 2, we can ﬁnd some T ∗ ≥ 0 such that zt = xt andθt < 1 hold in any period t ≤ T ∗ . On the other hand, Lemma A4 shows that θt = 1
for all large t. Then, by Proposition 2 we can ﬁnd some T ∗∗ ≥ 0 such that T ∗∗ ≥ T ∗ ,(IC∗t ) is binding and θt = 1 in any period t ≥ T ∗∗ + 1. Q.E.D.
Proof of Proposition 4
By Lemma A4, we know that θt = 1 when t is large. Thus π(ρt ) > 0 for large t. In
what follows we take t to be large so that θt = 1. When t is so large enough that πt > 0,α −ρ(IC∗t ) implies δ{π(ρt+1 ) + ρt+1 I − π(ρt+1 )} ≥ δ{αAzt+1
t+1 zt+1 − π(ρt+1 )} ≥ ρt zt
and hence δρt+1 I ≥ ρt zt . This can be further re-written by δ (xt /xt+1 )1−α I ≥ zt due
to ρt = α2 Axtα−1 .
On the other hand, due to (CME) in period t, we havel(1 − α)Aztα + (1 − l)(1 − α)Axαt = lzt+1 + (1 − l)(xt+1 + I)
which implies that xt+1 ≥ (1 − l){(1 − α)Axαt − I} because of zt ≥ 0 and xt+1 ≥ zt+1 .
By combining this with δ (xt /xt+1 )1−α ≥ zt , we can show that(
ψ(xt ) ≡ δ

xt(1 − l)[(1 − α)Axαt − I]

)1−αI ≥ ztHere xt − ψ(xt ) is increasing in xt because ψ is decreasing.
Let x̂ be the value of x satisfying (1 − l)(1 − α)Ax̂α = x̂ + (1 − l)I. From (CME)for large t we obtain (1 − l)(1 − α)Axαt ≤ xt+1 + (1 − l)I. Then, since xt > x for large t(Lemma A4) and Assumption 4 holds, we can verify that xt goes beyond x̂ as t tends
35

to be large enough. Thus xt ≥ x̂ for large t. Then, since xt − ψ(xt ) is increasing, we
obtain xt − zt ≥ xt − ψ(xt ) ≥ x̂ − ψ(x̂) = x̂ − δI > 0 for large t. Q.E.D.
Proof of Proposition 5.
We construct the switching equilibrium path described in the proposition as follows.
First, we set z1 = x1 = x and θ1 < 1 such thatw0 = lx + (1 − l)θ1 (x + I)
Such θ1 < 1 exists under Assumption 6. Then, given the initial value θ1 deﬁned in this−1
way, we sequentially deﬁne a sequence of {θt }Tt=1
such that(l + (1 − l)θt−1 )(1 − α)Axα = lx + (1 − l)θt (x + I), t = 2, 3, ..., T − 1
where T is deﬁned such that θT −1 ≤ 1 but θT > 1. Under Assumption 7 the above
T −1sequence {θt }t=1
is increasing over time and θT > 1 for some T . Then we deﬁne an
T −1equilibrium path {zt , xt , θt }t=1
from t = 1 to t = T − 1 such that zt = xt = x andθt < 1 satisfying the above equation. We call the phase with t ≤ T − 1 low development
phase. In the low development phase (IC∗t ) is slack under Assumption 3 (δ > α/(1−α)):δ{αAx − dx} > dx. Also (TSt−1 ) is satisﬁed in any period t ≤ T − 1 because π(d) = 0
and αAxα − dx > 0 for all t ≤ T − 1.
Second, we consider the switching period t = T . Deﬁne θT = 1 and set (zT , xT )satisfying both (IC∗T −1 ) as equality
δ{αAzTα − ρT zT − π(ρT )} = dx

(IC∗T −1 )and (CME) in period T − 1:(l + (1 − l)θT −1 )(1 − α)Axα = lzT + (1 − l)(xT + I)(CMET −1 )To show the existence of (zT , xT ) such that both (CMET −1 ) and (IC∗T ) as equality
hold with x < xT < x∗ and zT < xT , note that the following inequality is satisﬁed
x < (l + (1 − l)θT )(1 − α)Axα − (1 − l)I < (1 − l) (δI/xα )1/(1−α) .

(A11)

To see this, note ﬁrst that under Assumption 7 the ﬁrst inequality of (A11) holds.
Second, due to Assumption 8 x < x∗ holds so that, by deﬁnition of x∗ , we have
(1 − α)Axα − (1 − l)I < (1 − l) (δI/xα )1/(1−α) . Since θT < 1, the second half ofinequality (A11) also holds.
Thus, from (A11) we can see that x < xT < (δI/xα )1/(1−α) (see Figure 4 (a)). By
Assumption 8 we have (δI/xα )1/(1−α) ≤ x∗ . Then we can ﬁnd a unique (zT , xT ) suchthat zT < xT and x < xT < x∗ with (CMET −1 ) and (IC∗T −1 ) as equality are satisﬁed.
Next we consider period T +1. Given (zT , xT ) deﬁned above, we will ﬁnd (zT +1 , xT +1 )which satisﬁes (CME) in period T :l(1 − α)AzTα + (1 − l)(1 − α)AxαT = lzT +1 + (1 − l)(xT +1 + I)
36

(CMET )

and (IC∗T ) with equality:
δ{αAzTα+1 − ρT +1 zT +1 − π(ρT +1 )} = ρT zT .

(IC∗T )

Since xT < x∗ holds by deﬁnition of x∗ , we have (1 − α)AxαT − (1 − l)I < (1 −l) (δI/xαT )1/(1−α) which implies that1{l(1 − α)AzTα + (1 − l)AxαT − (1 − l)I}1−l
1{(1 − α)AxαT − (1 − l)I}<
1−l
< (δI/xαT )1/(1−α)

xT +1 <and hencexT +1 < (δI/xαT )1/(1−α)= (δIα2 A/ρT xT )1/(1−α)< (δIα2 A/ρT zT )1/(1−α)

due to zT < xT . Thus such (zT +1 , xT +1 ) satisfying the above two equations (CMET )and (IC∗T ) exists where zT +1 < xT +1 (see Figure 4 (b)). We can also show that x < xT +1
because Assumption 7 impliesx ≤ (1 − l)(1 − α)Axα − (1 − l)I≤ l(1 − α)AzTα + (1 − l)(1 − α)AxαT − (1 − l)I= lzT +1 + (1 − l)xT +1< xT +1
Thus we can ﬁnd (zT +1 , xT +1 ) which satisﬁes (CMET ) and (IC∗T ) with equality wherezT +1 < xT +1 andx < xT +1 < (δI/xαT )1/(1−α) < (δI/xα )1/(1−α) .
Thus xT +1 < x∗ due to Assumption 8.
Repeating this process for all t ≥ T , we can ﬁnd a sequence of {zt , xt }∞t=T such that∗(CME) in period t and (ICt ) with equality are satisﬁed as well as zt < xt and x < xt
hold. We also verify that these paths satisfy (TSt−1 ) for all t ≥ T as follows: (TST −1 )holds because JT −1 = αAxα − dx > π(d) = 0 and JT = αAzTα − ρT zT > π(ρT ) due to(IC∗T ). Similarly, since Jt > π(ρt ) holds for all t ≥ T + 1, (TSt−1 ) is satisﬁed for allt ≥ T + 1.
∗
We have thus established that a sequence {zt , xt , wt , ρt , θt }∞t=1 exists such that (ICt ),(TSt−1 ) and (CME) are all satisﬁed in any period t. We can then ﬁnd a sequence of
repayments {Rt }∞t=1 which satisﬁes (ICt ), (IRLt ) and (IRPt ) in any period t, given
37the constructed market variables {wt , ρt , θt }∞t=1 . From this, we can readily show thatthe trigger-like strategy of relationship producers and lenders as described in the proof
of Lemma 1 (i) sustains the constructed relational contracts {zt , Rt }∞t=1 as a subgame
perfect equilibrium.
Finally we can show that wt−1 ≥ zt holds in any period t in the equilibrium path
constructed above. In the low development phases (t ≤ T − 1) we have wt−1 = (1 −α)Axα ≥ x = zt due to Assumption 7. On the other hand, in any period t ≥ T we have
wt−1 = lzt + (1 − l)(xt + I) which implies that wt−1 > zt + (1 − l)I > zt . Q.E.D.

8

Parametric Conditions on Assumption 8
1

Let X ≡ (δI/xα ) 1−α . Then Assumption 8 requires (i) x∗ ≥ X and (ii) x∗ > x.
(i) x∗ ≥ X. To see this, note that such inequality is equivalent to (1−α)AX α −(1−I) ≤1
1(1 − l)(δI/X α ) 1−α . By substituting x = (α/(1 − α))I and X = (δI/xα ) 1−α into thisinequality, we obtain(1 − α)Aδ

α
1−α

(

α
1−α

)− α2

1−α

(
I ≤I +δ
α

α
1−α

) α2

1−α

I.

Here the left hand is increasing and concave with respect to I while the right hand side
is linear with I. Thus we can ﬁnd some I > 0 such that the inequality is satisﬁed for
all I < I.
(ii) x∗ > x. By deﬁnition of x∗ , this inequality is equivalent to (1 − α)Axα − (1 − l)I <α(δI/xα ) 1−α . By substituting x = (α/(1 − α))I into this condition, we have(
(1 − α)A

α
1−α

)α
α

I <I +δ

1
1−α

(

α
1−α

)− α

1−α

I.

Here the left hand side is increasing and concave with respect to I while the right handside is linear with I. Thus we can ﬁnd some I˜ > 0 such that the inequality is satisﬁed
˜
for all I < I.
˜
Therefore Assumption 8 is satisﬁed for all I ≤ min{I, I}.

9Appendix B: Bequest transfer

We have so far assumed that any individual cannot give his/her child bequest at all.
We now allow each individual to bequeath his/her child when old. The introductionof bequest may aﬀect the analysis because the bequest can be used to ﬁnance capital
investment. However, in what follows we will show that non–relationship producers
38

have no incentives to bequeath their children in any equilibrium when the altruistic
preference parameter δ > 0 is in some small range. Then we can show that the results
we have already established in the main text remain true even when we introduce thebequest technology into the basic model as long as δ is in a certain small range.
Consider a non–relationship producer born in t − 1 who receives bequest bt−1 fromhis parent and wants to invest xt in capital. Suppose also that he gives his child bequest
bt ≥ 0. Then let (xt , bt ) be a choice of a non–relationship producer born in period t − 1.
Given this, the proﬁt of a non–relationship producer is given as follows:{pt xt + (bt−1 − xt )ρt − bt
if bt−1 ≥ xt
(B1)
π(xt , bt |bt−1 ) =pt xt − (xt − bt−1 )ρt − bt − ρt I if bt−1 < xt
When bt−1 ≥ xt , the bequest bt−1 can be used for ﬁnancing xt and the remaining
amount bt−1 − xt can be saved to earn ρt (bt−1 − xt ). In this case the veriﬁcation costρt I is not necessary. When bt−1 < xt , the non-relationship producer needs to use theexternal ﬁnancing and incur the veriﬁcation cost ρt I. We suppose that each individual
observes the past capital investments and bequest transfers made in the dynasty towhich he or she belongs to as well as he or she observes the past market prices. Let
Ht be a set of all these histories observed to a non-relationship producer up to periodt. A strategy of a non-relationship producer is then deﬁned as a mapping from Ht theset of observed histories up to period t to capital investment level xt ≥ 0 and bequest
transfer to his child bt ∈ [0, bt ] where bt ≡ π(x(ρt ), 0|bt−1 ). Let σt : Ht → [0, bt ] × [0, ∞)be such strategy. Note that σt = (xt , bt ). Here each non-relationship producer cannot
give his child bequest more than the maximum current proﬁt.
With a slight abuse of notation, we will denote by π(σt |bt−1 ) the proﬁt of nonrelationship producer who uses a strategy σt in period t, given the bequest bt−1 received
from his parent. Then a non–relationship producer born in period t − 1 obtains his
payoﬀ as π(σt |bt−1 ) + δπ(σt+1 |bt ) and chooses his strategy σt to maximize this payoﬀ
subject to bt ∈ [0, bt ] (the bequest must be paid from the realized income), given theprevious period choice bt−1 .
Then we show the following result:
Proposition B1. Suppose that δ < 1/(1 + d). Then in any equilibrium every non–relationship producer born in period t − 1 chooses σt = σ t ≡ (x(ρt ), 0) which maximizes
only his own proﬁt π(xt , bt |0) given no bequest of his parent bt−1 = 0 where x(ρt ) ≡arg maxxt π(xt , 0|0) = α(Axt )α − ρt xt no matter the history.
Proof. Suppose contrary to the claim that there exists an equilibrium with {σt }∞t=1 inwhich some non–relationship producer born in some period t − 1 chooses σt ̸= σ t . For
σt to be an equilibrium choice, it must be that′π(σt |bt−1 ) + δπ(σt+1 |bt ) ≥ π(σ t |bt−1 ) + δπ(σt+1
|0)

(B2)

′
denotes the strategy chosen by the child of the non–relationship producer
where σt+1
who deviates from his equilibrium play σt to σ t = (x(ρt ), 0).

39

Let σ̂t ≡ (xt , 0) be the strategy deﬁned by setting bequest transfer equal to zero in′
the original strategies σt = (xt , bt ). Let also σ̂t+1≡ (x′t+1 , 0).
Then, by using (B1), (B2) can be written by−(1 − δρt+1 )bt − δbt+1 ≥ {π(σ t |bt−1 ) − π(σ̂t |bt−1 )}′+ δ{π(σ̂t+1|0) − π(σ̂t+1 |0)} − δb′t+1≥ {π(σ t |0) − π(σ̂t |0)}′+ δ{π(σ̂t+1|0) − π(σ t+1 |0)} − δb′t+1= ∆t − δ∆t+1 − δb′t+1′ |0). Since σ ̸= σ ,where ∆t ≡ π(σ t |0) − π(σ̂t |0) ≥ 0 and ∆t+1 ≡ π(σ t+1 |0) − π(σ̂t+1
tt∆t > 0 or/and bt > 0 must hold.
Since bt+1 ≥ 0, the above inequality implies−(1 − δρt+1 )bt − ∆t ≥ −δ∆t+1 − δb′t+1(B3)′
to be the strategy chosen by the child of the non–relationship producer
For σt+1
after the latter chose σ t , it must be that′′′′π(σt+1|0) + δπ(σt+2|b′t+1 ) ≥ π(σ t+1 |0) + δπ(σt+2
|0)

(B4)

′
′′
denote the strategies chosen by the child of the non–relationshipand σt+2
where σt+2′
producer in period t + 1 following σt+1
and σ t+1 respectively. Then, by a similarargument to (B3), we can show that−(1 − δρt+2 )b′t+1 − ∆t+1 ≥ −δ∆t+2 − δb′t+2 .

(B5)The left hand side of (B5) cannot be larger than −δb′t+1 − ∆t+1 due to δ ≤ 1 − δd ≤1 − δρt+2 . By using this fact and combining (B3) with (B5), we can derive−(1 − δρt+1 )bt − ∆t − (1 − δ)∆t+1 ≥ −δ∆t+2 − δb′t+2 .

(B6)

When we apply (B5) to period t + 2, we derive−b′t+2 ≥1{∆t+2 − δ∆t+3 − δb′t+3 }1 − δρt+3(B7)

where 1 > δd ≥ δρt+3 due to δ < 1/(1 + d) and ρt+3 ≤ d. Then, by using this and(B6), we obtain−(1 − δρt+1 )bt − ∆t − (1 − δ)∆t+1 ≥δ 2 ρt+3
δ2∆t+2 −(∆t+3 + b′t+3 )1 − δρt+3
1 − δρt+3

which implies−(1 − δρt+1 )bt − ∆t − (1 − δ)∆t+1 ≥ −40

δ2(∆t+3 + b′t+3 )1 − δd(B8)

because ∆t+2 ≥ 0 and 1 > δd ≥ ρt+3 . Then, by applying (B7) to any period T ≥ t andusing ∆t+i ≥ 0 for any i = 1, 2, ..., the right hand side of (B8) cannot be smaller than(−δδ1 − δd)T(∆t+T +1 + b′t+T +1 ).

Since T is arbitrary and ∆s and bs are bounded above for all s ≥ 0,15 this term converges
to zero by taking T → ∞, given δ/(1 − δd) < 1 which is equivalent to δ < 1/(1 + d).
Thus the right hand side of (B8) is bounded below from zero. However, since the left
hand side of (B8) is strictly negative because 1 > δd ≥ δρt+1 and bt > 0 or/and ∆t > 0,
this is a contradiction. Q.E.D.
We also show that every non-relationship lender has no incentives to give theirchildren positive bequest in any equilibrium when δ < 1/(1 + d).
Proposition B2. Suppose that δ < 1/(1 + d). Then non-relationship lenders leave nobequest to their children at all in any equilibrium.
Proof. Suppose that δ < 1/(1 + d). Suppose also contrary to the claim of that there
exists some equilibrium in which some non-relationship lender in some dynasty gives a
positive bequest bt > 0 in some period t where bt ∈ [0, wt ]. For this to be an equilibrium
choice, the equilibrium payoﬀ of that lender in period tρt (wt−1 + bt−1 ) − bt + δ{ρt+1 (wt + bt ) − bt+1 }must not be less thanρt (wt−1 + bt−1 ) + δ{ρt+1 wt − b′t+1 }which can be obtained by oﬀering no bequest b′t = 0 which is then followed by the
choice of his child b′t+1 in the next period t + 1. This condition thus yields−(1 − δρt+1 )bt − δbt+1 ≥ −δb′t+1(B9)

For b′t+1 to be an optimal choice by the child of the deviating lender in period t + 1, wemust have thatρt+1 wt − b′t+1 + δ{ρt+2 (wt+1 + b′t+1 ) − b′t+2 }≥ ρt+1 wt + δ{ρt+2 wt+1 − b′′t+2 }where b′t+2 is the bequest choice by the lender in t + 2 following his parent’s choice b′t+1
whereas b′′t+2 is the choice by the same lender following no bequest given by his parent.
This is simpliﬁed to−(1 − δρt+2 )b′t+1 ≥ −δb′′t+2
(B10)
15

Note here that bs ≤ π(x(ρs ), 0|0) < +∞ and ∆s ≤ π(x(ρs ), 0|0) < +∞.

41

By combining (B9) with (B10), we obtain−(1 − δρt+1 )bt ≥ −δb′t+1
≥ −

δ2
b′′
1 − δρt+2 t+2

where the ﬁrst inequality follows from bt+1 ≥ 0 and the second inequality from (B10)respectively where 1 > δdδρt+2 due to 1/(1 + d) > δ. Repeating the similar argument
over period t + 3, the right hand side of the above inequality is bounded below from
−

δ3
b′′
(1 − δρt+2 )(1 − δρt+3 ) t+3(B11)

Repeating this process for any period t + T ≥ t, (B11) is bounded below from−δT
b′′t+T−1
ΠTi=1(1 − δρt+i )which is bounded below from(−(1 − δd)

δ1 − δd)Tb′′t+T

because of d ≥ ρt+i . Since T can be arbitrary and b′′t+T ≤ wt+T < +∞, this converges
to zero by taking T → ∞. Thus the right hand side of (B9) must be zero but its left
hand side is strictly negative because of bt > 0, which is a contradiction. Q.E.D.
Next we show that introducing bequest transfer never expands the set of incentive
compatible relational contracts. Then the optimal relational contract which maximizes
the sum of all the generations in each dynasty uses no bequest transfers at all.
Proposition B3. Suppose that δ < 1/(d + 1). Then the optimal relational contract
involves no bequest transfers at all even when bequest transfer is allowed.
Proof. We consider the incentive compatibility condition for an old relationship producer to honor an agreed upon repayment Rt in period t, called (IC-Bt ):pt zt − Rt + ρt bpt−1 − bpt + δ{pt+1 zt+1 − Rt+1 + ρt+1 bpt − bpt+1 }≥ pt zt + ρt bpt−1 + δ max{πt+1 , 0}
where bpt ≥ 0 denotes the bequest transferred from the relationship producer to hischild in period t. The relationship producer is given a bequest bpt−1 from his parent
in period t − 1, lends it to the credit market and receives the corresponding interest
income ρt bpt−1 in period t. By honoring an agreed upon repayment Rt and making abequest bpt to his child in period t, the relationship producer obtains the payoﬀ of the42

left hand side of the above inequality while he can obtain the payoﬀ of its right handside by reneging on the repayment Rt , making no bequest transfer bpt = 0 to his childand exercising the quitting option. In the latter case the relationship will be dissolvedin period t+ in which his child will obtain the payoﬀ corresponding to the arm’s length
contract max{πt+1 , 0}. As we have shown in Proposition B1, the continuation payoﬀ
πt+1 after the deviation is equal to πt+1 = π(xt+1 , 0|0) because bt = 0 is chosen by any
non-relationship producer who engages in arm’s length contract in any period t.
Also, we consider the individual rationality constraint that a young relationshiplender agrees on a relational contract {zt , Rt }, called (IRL-Bt ):Rt + ρt (wt−1 + blt−1 − xt ) − blt + δ{Rt+1 + ρt+1 (wt + blt − zt+1 ) − blt+1 }≥ ρt (wt−1 + blt−1 ) + δρt+1 wt+1
where blt ≥ 0 denotes the bequest transferred from an old relationship lender to her childin period t. The relationship lender is given a bequest blt−1 from her parent in periodt − 1 and thus owns total income wt−1 + blt−1 in period t − 1. That lender then lends xt
to the relationship lender matching her and the remaining amount wt−1 + blt−1 − zt tothe credit market which gives her ρt (wt−1 + blt−1 − zt ). The right hand side of the above
inequality means the payoﬀ of the relationship lender by exercising the quitting option
and making no bequest transfer to her child. As we have shown in Proposition B2,
in any continuation equilibrium after an old lender engages in arm’s length contract,
all her descendants will make no bequest transfers to the future generations. Thus nobequest appears in the last term of the right hand side of the above inequality.
By combining (IC-Bt ) with (IRL-Bt ) and deﬁning total bequest b̃t ≡ bpt + blt , weobtain−b̃t + δ{pt+1 zt+1 − ρt+1 zt+1 − max{πt+1 , 0} + ρt+1 b̃t − b̃t+1 } ≥ ρt zt(IC-B∗ )Here, if 1 > dδ holds, then the left hand side of (IC-B∗ ) is bounded above byδ{pt+1 zt+1 − ρt+1 zt+1 − max{πt+1 , 0}}because −b̃t (1 − δρt+1 ) ≤ −b̃t (1 − δd) ≤ 0 (by ρt+1 ≤ d) and b̃t+1 ≥ 0 are satisﬁed.
Thus (IC-B∗ ) implies the incentive compatibility condition (IC∗t ) given in the main
text without bequest transfer. This shows that the set of relational contracts satisfying
bequest transfers if δ < 1/(1 + d).
(IC∗t ) is not expanded even when we allow ∑t∗
Next we consider the maximization of ∞t=0 δ Jt subject to (ICt ) and (TSt−1 ) forpl
all t ≥ 1 when allowing bequest transfers bt and bt for relationship producer and lender.
Here the joint payoﬀ of a relationship producer and a relationship lender Jt is given by
Jt = pt zt + ρt (wt−1 − zt + b̃t−1 ) − b̃t + δ{pt+1 zt+1 + ρt+1 (wt − zt+1 + b̃t )}where b̃t = blt + bpt denotes the total bequest transfer of the relationship producer andlender. However, if δ < 1/(1 + d), then maximizing Jt with respect to b̃t yields b̃t = 0.
43

Also, we already know that the set of possible relational contracts satisfying (IC∗t ) is
not expanded by allowing bequest transfers as we have shown above. In addition tothis, since Jt does not increase in b̃t , the net total surplus T St−1 does not increase in b̃t
as well. Thus setting b̃t = 0 makes both (IC∗t ) and (TSt−1 ) easier to be satisﬁed. From
this, the optimal relational contract must have blt = bpt = 0 in any period t. Q.E.D.
Proposition (B1)-(B3) show that our characterization results given in the main text
(Proposition 2-4) still remain true even when we introduce bequest transfer into thebasic model if the altruistic preference δ is so small that δ < 1/(1 + d) holds.
Next we will show that there actually exists an equilibrium in which all non–relationship producers and relationship pairs choose zero bequest if δ is small. Thus
Proposition 5 still holds. To show this, we take the equilibrium described in Proposition
4. Then, ﬁrst we consider non–relationship producers and their strategies as follows:
every young non-relationship producer born in period t − 1 always chooses zero bequest
bt = 0 and the optimal investment xt = x(ρt ) deﬁned by α2 Aα xtα−1 = ρt whatever
choices of his or her parent. Then, given such strategy of all future generations, each
non–relationship producer in the current period, say t − 1, will choose bt = 0 and xt tomaximize his own proﬁt π(xt , 0|bt−1 ) whatever choice σt−1 = (bt−1 , xt−1 ) by his parent
in the previous period. To show this, we deﬁne by π̃(bt−1 ) the maximum of π(xt , 0|bt−1 )over xt ≥ 0. We then show that each non–relationship producer chooses the capital investment xt as follows: (i) xt = x(ρt ) where α2 Ax(ρt )α−1 = ρt if bt−1 < b t−1 or if bt−1 >bt−1 . Here bt−1 is deﬁned as bt−1 such that αAbαt−1 = maxxt pt xt − (xt − bt−1 )ρt − ρt I
and bt−1 is deﬁned as bt−1 ≡ x(ρt ). (ii) xt = bt−1 if bt−1 ∈ [bt−1 , bt−1 ]. In summary weobtain Aα(1 − α)x(ρt )α + bt−1 ρt − ρt I if bt−1 < bt−1
if bt−1 ∈ [bt−1 , bt−1 ]Aαbαt−1π̃(bt−1 ) ≡αAα(1 − α)x(ρt ) + bt−1 ρt
if bt−1 > bt−1
Each old non–relationship producer in period t − 1 chooses bt−1 ≥ 0 to maximize his
payoﬀ −bt−1 + δπ̃(bt−1 ). Then, we can show that the optimal bequest level bt−1 which
maximizes −bt−1 +δπ̃(bt−1 ) becomes zero if (i) δρt −1 ≤ 0 for bt−1 ≥ bt−1 or bt−1 ≤ bt−1and (ii) δα2 δ α−1 − 1 ≤ 0 for bt−1 ∈ [bt−1 , bt−1 ]. Since ρt ≤ d, condition (i) is reduced
α−1
to δd ≤ 1. Condition (ii) holds if δα2 bt−1
≤ 1. Here, since bt−1 is given by b satisfying
αAbα = α(1 − α)Ax(ρt−1 )α + ρt−1 b − ρt−1 I(B12)

where the right hand side is equal to maxxt−1 Aαxαt−1 − ρt−1 (xt−1 − b + I). We will
write b(ρt−1 ) for bt−1 . Then we can verify that−(x(ρt−1 ) − bt−1 ) − I
∂bt−1
=
<0
∂ρt−1
α2 Abα−1t−1 − ρt−144

because b is the smallest root of (B9) so that x(ρ) > b and the denominator is positive.
Thus bt−1 = b(ρt−1 ) ≥ b(d) because ρt−1 ≤ d. Then we have
2 α
δα2 Abα−1
t−1 − 1 ≤ δα A b(d) − 1≤ 0
if δ is so small that δα2 Aα b(d)α−1 ≤ 1. To be consistent with Assumption 3, we needto have (B12) Since b(d) satisﬁes b(d)α−1 = d/α(1 − α)A, (B12) holds for some δ ifα/(1 − α) < 1/α2 Ab(d)α−1 = A1−α /d. The latter condition is satisﬁed for small α
or/and large A.
In what follows we thus assume
{ 1−α
}
α
A
1
≤ δ ≤ min
,
(B13)
1−α
d
1+d
Given these parametric conditions, we have established the result that all non–relationshipproducers give no bequest at all. Similarly, we can show that every non–relationshiplender never bequeaths to her child. To see this, note that the payoﬀ of non–relationshiplender who bequeaths bt−1 to her child is given by δρt bt−1 − bt−1 because the child obtains bt−1 to earn the additional interest income ρt bt−1 which is evaluated by δ in theview point of the current period non–relationship lender. Maximizing δρt bt−1 − bt−1
with respect to bt−1 ≥ 0 yields bt−1 = 0 because of δd ≤ 1.
Next we consider any pair of a relationship producer and a relationship lender.
Then suppose that they play the following strategy: they agree on a relational contract{zt , Rt } deﬁned in the main text but leave no bequest to their children at all whatever
bequest level they have received from their parents in the previous period. Then we
can show that they will actually choose zero bequest when δ is small as follows: Letb̃t−1 ≥ 0 be the total bequest given by the relationship producer and lender to theirchildren in period t − 1. Then we have b̃t−1 + wt−1 > xt due to wt−1 > xt and b̃t−1 ≥ 0
so that the total proﬁt of a relationship pair born in period t − 1 is given by
J(zt ; b̃t−1 ) ≡ pt zt + ρt (b̃t−1 + wt−1 − zt )where they do not need to incur the veriﬁcation cost I > 0. Since the young relationshipproducer and lender in the next generation of period t + 1 will choose xt+1 whatever b̃t ,the current old relationship pair chooses the bequest b̃t to maximize their joint payoﬀsδJ(zt+1 ; b̃t ) − b̃t which gives b̃t = 0 again, provided δ is so small that δd ≤ 1. Thus,
the equilibrium path described in Proposition 5 still exists where no bequest transfers
occur as long as (B13) is satisﬁed.

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46

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47

ρt

θt

1 − θtd(1 − l)(xt + I)0

wt−1 − lzt(1 − l)(x + I)

Figure 1: Credit Market Equilibrium

48

zt+1
45o

(IC∗t )BD

0

(δIα2 A/ρt zt )1/(1−α)

xt+1x(CMEt )

(1 − α)A(lztα + (1 − l)xαt ) − (1 − l)I
Figure 2 (a): Equilibrium Determination of (xt+1 , zt+1 )

49

zt
45o

∗ )
(ICt−1

E(δIα2 A/ρ(x̃t−1 )x̃t−1 )1/(1−α)(δIα2 A/ρt−1 zt−1 )1/(1−α)

0

x

xt
(CMEt−1 )α + (1 − l)xα ) − (1 − l)I(1 − α)A(lzt−1
t−1

Figure 2 (b): (IC∗t−1 ) is not binding when zt = xt .

50

xt + (1 − l)I(1 − α)Axαt−1

0

x̃t−1

xt

x

x∗

Figure 3

51

zT
45o

(IC∗T −1 )(δI/xα )1/(1−α)

0

x

xT

x∗

xT

(CMET −1 )1α1−l {(l + (1 − l)θT −1 )(1 − α)Ax } − I

Figure 4 (a): Construction of Switching Equilibrium Path52

zT +1
45o(IC∗T )

(δα2 IA/ρT zT )1/(1−α)

zT +1

xT +1
0

x

xT +1
(CMET )1αα1−l {(1 − α)A(lxT + (1 − l)zT ) − (1 − l)I}

Figure 4 (b): Construction of Switching Equilibrium Path

53