Discussion Papers In Economics
And Business

Relationships and Growth
Shingo Ishiguro
Discussion Paper 11-31-Rev.

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRelationships and Growth
Shingo Ishiguro
Discussion Paper 11-31-Rev.

May 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRelationships and Growth∗
Shingo Ishiguro†
Graduate School of Economics
Osaka University
May 1, 2013

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 that it demonstrates the co-evolution ofthese 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 in the endogenous
process of economic development. In particular, we show that relational contracting fosters the emergence of a 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∗

I am very grateful to the editor, Philipp Kircher, and three anonymous referees of the journal for
their useful comments and suggestions, which signiﬁcantly improved my paper. Of course, all remaining
errors are my own.
†
Correspondence: Shingo Ishiguro, Graduate School of Economics, Osaka University.
1-7
Machikaneyama, Toyonaka, Osaka 560-0043, Japan. E-mail: ishiguro@econ.osaka-u.ac.jp1

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 via 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 foundevidence that such relationship lending contributed to the economic growth in New
England during that period.3 A related historical fact is that major German banks
such as Commerzbank, Dresdner, and Deutsche grew rapidly in the 19th century by
developing long-term relationships with industry enterprises by oﬀering low interest
rates to them and being represented as board directors on these ﬁrms. Such close andlasting relationships between large banks and ﬁrms contributed to the rapid expansion of the German economy between the late 19th century and the First World War(Allen (2001)). Maurer and Haber (2007) also provide related empirical evidence about
Mexican banks during similar periods (1880-1913): these bankers were engaged in relationship lending by responding to information asymmetry and a large enforcement
cost but they did not loot their own banks.
These facts pose a positive view that relational contracting is not a substitute but
a complement to the market economy in that the former fosters the latter.4 On theother hand, there is also a negative view on relational contracting that it plays a lessimportant role as the economy grows and reaches more developed stages. For example,
the New England banks that lent to closely-related persons (for example, directorsof these banks) 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 to capital-rich, thus expanding the anonymous credit market in the late1

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 Lamoreaux (1994), Allen (2001), and Maurer and Haber (2007) discussed above. For the
positive role of other informal institutions, see the papers contained in Aoki and Hayami (2000). For
example, Greif (2000) discusses the roles of the community trading system in pre-modern Europe andFafchamps (2000) provides evidence about business networks in Sub-Saharan Africa.

2

19th century (Lamoreaux (1994)). A related argument is that the relationship-based
ﬁnancial system was dominant in Asian countries such as Korea and Japan after WorldWar II but it has been recently changing to the market-based ﬁnancial system (seeRajan and Zingales (2000)). Demirgüç-Kunt and Levine (2004) provided the related
evidence that the ratio of bank ﬁnance relative to equity ﬁnance is negatively associatedwith 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 replaced by market-based ﬁnance in developed countries.
The main objective of our paper is to provide a theoretical framework for understanding how and when relational contracting contributes to the process of economic
development. In particular, we characterize the time structures of relational contracting(self-enforcing agreements) that change in the diﬀerent phases of economic development.
A novel feature of our model is to embed relational contracts that are supported bylong-term relationships into a dynamic general equilibrium model. Long-term relationships have been mostly analysed in the partial equilibrium framework in the literatureof repeated games (see Mailath and Samulaeson (2006)): in a typical repeated game,players play the stage games repeatedly over time by assuming that the outside markets
are exogenously given. However, despite much historical evidence that shows the important roles of long-term relationships in the process of economic development, there
are few theoretical studies that consider the macroeconomic implications of long-term
relationships. These are what we address in this paper. The important departure of
our paper from the literature of repeated games is that we investigate how long-term
relationships aﬀect and are aﬀected by the growth path of a macro-economy which isendogenous as well.
In our model economy, producers who ﬁnance their capital investment have the
incentive to default after they borrow the funds from lenders. We consider the twomeans which prevent producers from defaulting. The ﬁrst type is the competitive credit
market, in which everyone can borrow and lend at a given market interest rate in the
spot manner. However, when lenders lend their funds to the credit market, they must
incur some enforcement cost in order to write a formally enforceable contract which
prevents borrowers from defaulting. We call this type of contract an arm’s length orformal contract and call producers (respectively, lenders) who engage in an arm’s length
contract A-producers (respectively, A-lenders). The enforcement cost incurred by Alenders includes costly activities such as collecting evidence about accounting data,
hiring lawyers and accounting professionals, and using the public enforcement agency(courts).
The other way to protect lenders from default is to use implicit and informal contracting arrangements. Speciﬁcally, we suppose that there is a local community in ourmodel economy where each producer in the community has a personal connection witha particular lender within the same community. The producer and the lender forma long-term relationship over successive generations. We call this type of producer
and lender a R-producer and a R-lender, respectively. Each R-producer can engage in
relational contracting with a R-lender for ﬁnancing capital investment without using3the outside credit market. This type of contract is what we call a relational contract.
Because they interact with each other over time, each relationship pair of a R-producer
and a R-lender can avoid the strategic default problem via a self-enforcing agreement.
The advantage of relational contracts over arm’s length contracts is the saving in
the enforcement cost which is inevitable under the latter while its disadvantage is that
relational contracts must be self-enforceable, that is, R-producers must be given the
incentive not to renege on the agreed upon repayment schedules.
As is well known from the repeated games literature, an implicit agreement is selfenforceable if each party’s deviation from honouring the agreement results in future
losses larger than the one-time gains obtained by the deviation. The novelty of ourmodel is how it relates the self-enforceability of relational contracting to the endogenous process of economic development in a dynamic general equilibrium framework.
The proﬁt of an arm’s length contract, which becomes the deviation payoﬀ for eachR-producer when quitting the current relationship, is endogenously determined andaﬀects the self-enforcing condition of relational contract. In turn, the change in the
self-enforcing condition creates a feedback eﬀect on the equilibrium determination of
the proﬁt of arm’s length contract through changes in the market prices, such as wage
and interest rates. These two-way interactions between the self-enforcing condition ofrelational contracts and the market equilibrium conditions jointly lead to the macroeconomic dynamics of the development process.
We then characterize the equilibrium paths of the model economy which involve thedynamic transformation from a relationship-based system relying on relational contracts to a market-based system relying on arm’s length (formal) contracts. In any
equilibrium path, there exists a unique switching period before which the self-enforcing
constraint becomes less stringent so that each R-producer invests and produces morethan each A-producer, but after which it becomes more stringent so that each Aproducer invests and produces more than each R-producer. In particular, we show thatthe output level of a R-producer relative to that of an A-producer, which we call the
relative output of the R-producer, declines over time from the initial period until ithits some critical value. Thus, in the early stages of development, relational contracting contributes more to the expansion of the economy than the arm’s length contract,while the former contributes less than the latter in the matured stages of development.
We also show that the economies which start with less reliance on relational contracting
switch more early from the relationship-based system to the market-based system thanthose which start with more reliance on relational contracting.
The key behind these results is the dynamic general equilibrium interaction between
the self-enforcing condition of relational contracting and the determination of market
prices such as the interest and wage rates. Since relationship pairs of R-producers andR-lenders do not incur the enforcement cost associated with capital investment, they
face a lower opportunity cost to raise capital for production than A-producers do. Thus,
R-producers can invest more in capital than A-producers do when the self-enforcing
condition becomes less stringent. In the early stages of development, the income level isso low that the small saving limits the credit supply, and hence, the market interest rate4

becomes high, resulting in the low capital investment of A-producer. Then, the proﬁtof an arm’s length contract, which becomes the outside option for each R-producer whoquits a relationship, becomes lower so that the self-enforcing condition of a relational
contract becomes more likely to be sustained. This leads to a larger capital investmentof R-producers than A-producers. Since R-producers have lower opportunity cost toraise capital than A-producers, the output expansion of the former relative to thelatter contributes to higher development of the economy as a whole. However, as themarket interest rates fall during the course of development, the self-enforcing condition
becomes more stringent so that R-producers eventually switch to invest less in capitalthan A-producers do.
In such way we show that the economy exhibits the dynamic switching patternsuch that it relies more on relational contracting in the early stages of development
but it switches to rely more on arm’s length contracts in the competitive market in thelater stages of development. This result is related to an argument made by Polanyi(1947) that Western societies experienced the ‘great transformation’ from nonmarket
systems to market-based systems when their economies expanded rapidly in the 19th
century. Moreover, R-producers contribute to economic development by producing
more in the early development stages in which the market economy is so immature that
A-producers produce less. Thus relational contracting plays a positive role in fostering
economic growth during low-development stages and in promoting the emergence of amarket-based economy. These results are consistent with the aforementioned historicalevidence that relational contracting complements the rise of a market-based economy(Aoki and Hayami (2000) and Lamoreaux (1994)). In addition, our result also conﬁrms
the historical fact that the relationship-based system declines as the economy grows andexpands more (see Lamoreaux (1994), Rajan and Zingales (2000) and Demirgüç-Kuntand Levine (2004) as we have already mentioned).
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 compare informal contracting enforcement, such as reputation, with formal and legal enforcementin random matching environments. Kranton (1996) focuses on market-based monetary
exchange and relational (self-enforcing) contracts that emerge in a Kiyotaki-Wrighttype monetary search model. Dhilon and Rigolini (2006) compare reputation and legal
enforcement by endogenizing the quality of enforcement institutions. Francois (2011)
investigates the evolution of endogenous institutions, but his analysis focuses on theroles of social norms formed through the change 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 do not focus on the
choice between arm’s length and relational contracts. They also conduct a steady-state
5See, for example, Baker, Gibbons and Murphy (2002), Itoh and Morita (2007), Levin (2003),
MacLeod and Malcomson (1998), and Ramey and Watson (2003).

5

analysis of the long-run growth rate and study the macroeconomic eﬀects of productivity shocks on relational contracting between ﬁrms and workers. Fafchamps (2003)
addresses the dynamic issue of how markets spontaneously emerge in a repeated gamesetting.
Our paper is also related to the models of competitive economies with endogenous debt constraints that prevent borrowers from defaulting (see, for example, Jeske(2006), Kehoe and Levine (1993), and Hellwig and Lorenzoni (2009) for the models ofinternational borrowing and lending with default risk).
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 which have been not addressed in the existing studies. Our
new insight is that both the sustainability of relational contracting and the evolution of
economic development are endogenous and jointly determined through dynamic generalequilibrium eﬀects. Speciﬁcally, we address the hitherto unaddressed issues of how andwhen relationship-based economic systems change to market-based systems during theendogenous process of economic development.
The remaining sections are organized as follows. In Section 2, we provide a partial
equilibrium model with a choice between arm’s length and relational contracts by taking all the market prices as exogenously given. In Section 3, we turn to the full modelof a dynamic general equilibrium and deﬁne an equilibrium of the model economy. In
Section 4, we characterize the set of equilibrium paths and show that relational contracting contributes more to economic growth in the early stages of development butthat its role becomes more limited as the economy enters the mature stages of development. In Section 5, we discuss an extension of the model which allows endogenous
formation of relationship pairs. All proofs are relegated to the Appendix (with somegiven in the Online Appendix).

2

Model: Partial Equilibrium

Before describing the full model of a dynamic general equilibrium, we will begin witha partial equilibrium model by taking all the market prices as exogenously given. This
will be helpful for understanding the basic structure of relational contracting, which
will be combined in a general equilibrium model later.

2.1

Economic Environment

We consider an overlapping generations (OLG) economy with discrete time t = 0, 1, 2, ....
Every period, a continuum of one unit mass of individuals is newly born; each individual lives for two periods: young and old. In each generation, one young individualis born from each old individual, and we use notation i to denote both individual iand the dynasty to which individual i belongs. For simplicity, we assume that eachindividual is concerned with only his or her consumption when old.
6

The newly born individuals consist of lenders and producers (who become borrowers). We use the masculine pronoun for producers and the feminine pronoun for
lenders. Each young lender born in period t is endowed with her income wt in terms ofthe numèraire good (which corresponds to the ﬁnal good in the full model in Section
3). Because every young lender is concerned with her consumption level when old, shewill lend all the income wt to borrowers and consume all the saved income. In the fullmodel of a general equilibrium in Section 3, we will determine the income of young
lender wt in the labour market equilibrium.
In contrast, producer i can produce a speciﬁc good i (which corresponds to intermediate good i in the full model below) by investing in capital one period in advance.
Speciﬁcally, a producer can produce one unit of his speciﬁc good when old if he invests
in one unit of capital when young. We assume that capital fully depreciates after one
period and that young producers are not endowed with any income so that they needto ﬁnance their investments when young.

2.2

Preference

We assume that individual i (lender or producer) has an altruistic preference over the
consumption level of his or her child. 6 The reason why we introduce the altruistic
preference is to ensure that each individual has a reputational concern as we will see
in more detail below. If an individual reneges on an implicitly agreed upon contract,
his or her child may be punished by not having a better trading opportunity andhence may consume less in the future period, which becomes a utility loss for thedeviating individual in the current period. Without an altruistic concern about thechild’s consumption, any individual never honours implicit promises as will be apparent
in the following analysis.
More speciﬁcally, we consider an individual in dynasty i who was born in periodt − 1 and whose consumption when old (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), butt (i), as follows:
also on the consumption level of his/her child in period t + 1, Ct+1
t
U t−1 (i) ≡ Ctt−1 (i) + δCt+1
(i),

(1)

where δ ∈ (0, 1) represents the parameter value measuring the degree to which eachindividual is altruistic about his/her child. We will often call δ the discount factor when6

One might think why we use the framework of two-period-lived OLG model with altruistic preference instead of using an inﬁnitely-lived agent model. The main reason for this is to simplify the saving
decision of each individual which makes the credit market equilibrium condition easier to be handled.
In an inﬁnitely-lived agent model, we need to keep track of the Euler equation of saving decision as
an additional dynamic equation which would make the dynamic analysis of self-enforcing agreement
more complicated. Our OLG model of a two-period-lived agent with altruistic preference can avoid
such complication while allowing us to incorporate a self-enforcing agreement into a dynamic generalequilibrium in a simpler way.

7

no confusion arises because it plays a similar role to the discount factor in repeatedgames.
To keep the model simple, we assume that there is no bequest transfer across generations in each dynasty. Then, each individual consumes all of his/her old income forhimself/herself such that Ctt−1 (i) is equal to the lifetime income level of individual iborn in period t − 1. We will discuss an extension of the basic model to allow bequests
in the Conclusion (see the Online Appendix).

2.3

Strategic Default and Contracting Modes

Each producer wants to ﬁnance his capital investment for future production fromlenders but there exists a strategic default problem such that any producer can run
away and deny any repayment after he borrows from a lender. Anticipating such a
strategic default, the lenders never lend to the producers.
In this paper, we consider two alternative means to alleviate this problem.
Arm’s Length Contract. In the economy, there is a competitive credit market
where borrowing and lending are made at a given interest rate. The credit market iscompetitive in the sense that everyone takes the market interest rate rt ≥ 0 as given
in any period t. 7 However, the credit market is not perfectly competitive in the sensethat it involves the strategic default problem we have assumed above. This can be
avoided in the credit market as follows.
In the credit market, each young lender must spend 1−λ units of the numèraire good(0 < λ < 1) for lending one unit to a borrower and preventing him from committing
strategic default. Here, 1−λ is called the enforcement cost per unit lending in the credit
market. For example, 1 − λ 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. 8 Then,
when a young lender lends wt to the credit market in period t, she needs to spend(1 − λ)wt for contract enforcement and will thus earn the interest income rt+1 λwt in
the next period t + 1 by lending the remaining amount of λwt .
As long as lenders incur the enforcement cost, the credit market works as thestandard competitive market where lenders and borrowers trade at a given market
interest rate rt in the anonymous and spot fashion. We call such a transaction madein the anonymous credit market an arm’s length contract. We also call a producer
7Here, rt 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 rt 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.
8
We can also allow borrowers to incur some costs preventing themselves from committing strategic
default. For example, each borrower makes some investment in the enforcement technology such that hecan commit himself not to default. Since such a possibility cannot alter the main results substantially,
we will not pursue such a case in what follows.

8

(borrower) and a lender who are engaged in the arm’s length contract in the credit
market, an A-producer and an A-lender, respectively.
We let xt ≥ 0 denote the capital investment (equivalently, the production level) ofan A-producer. We also let pt ≥ 0 denote the price for a speciﬁc good an A-producer
produces. Then, the proﬁt of an old A-producer in period t is given by πt ≡ pt xt − rt xt .
In this section, we take the capital investment choice of A-producers and their proﬁt
πt as exogenously ﬁxed; these will be endogenously determined in the next section.
Relational Contract. Relational contracting becomes an alternative to avoid thestrategic default problem as follows. In the economy, there is a local community wherel (l < 1) producers and l lenders reside. Thus, 1 − l producers and 1 − l lenders areoutside the local community and engage in arm’s length contracts in the credit market.
The producers and lenders in the community are matched with each other in the initialperiod t = 0 in a one-to-one manner. Then, each of the matched l pairs forms apersonal connection and relationship. We assume that such relationships formed in theinitial period can be inherited over successive generations. We call a producer in thecommunity a R-producer and a lender in the community a R-lender respectively.
In the basic model, we will focus on the case that the number of relationship pairs in
the local community does not increase from its initial value l ∈ (0, 1) by assuming that
nobody can enter the local community in order to seek a relationship partner. Further,
as we will see below, all R-producers and R-lenders never quit their relationships inequilibrium, ensuring that the number of relationship pairs is never decreased from its
initial value l ∈ (0, 1) as well. Then, the number of relationship pairs becomes constantat l ∈ (0, 1) over time. The fact that the membership of those who engage in relational
contracting is constant is observed in several places where relational contracting is basedon closed community memberships. 9 In Section 5, we will drop this assumption andturn to the case where the number of relationship pairs evolves over time because theproducers and lenders freely enter the matching market to seek relationship partners.
Any R-producer and R-lender in each pair can always exercise the option to quit
their relationship (called the quitting option) at any time by leaving the community.
We will then assume that if a R-producer (respectively, a R-lender) quits the relationship, his (respectively, her) child cannot also form a relationship with the child of his(respectively, her) partner in the next period. In such a case, not only the current
R-producer and R-lender but also their children have no choice but to engage in anarm’s length contract in the anonymous credit market. 10
In what follows, we let zt ≥ 0 denote the capital investment (equivalently, theproduction level) of a R-producer. Each young R-producer, born in period t − 1, can
9For example, see Fafchamps (2000) for the case of the business community in Zimbabwe.
However, we do not assume permanent dissolution here; that is, we do not assume that once a Rproducer and an R-lender dissolve their relationship, all their descendants cannot re-form the previous
relationships forever. We only assume that it takes at least one period for a dissolved relationship to be
re-formed. We later show that the assumption of one-period dissolution of a relationship is suﬃcientfor each relationship pair to honour the agreed upon relational contracts over time.
10

9

directly ﬁnance his investment zt from his partner, a young R-lender, in exchange formaking repayment Rt in period t. Since the repayment Rt is not secured due to thestrategic default of the R-producer, such an agreement, {zt , Rt }, must be implicit andself-enforceable. We call this type of contract a relational contract.

2.4

Timing within Each Period

Events in each period proceed in the local community as follows (see Figure 1 for thetime line). First, at the beginning of each period, the old R-producer and R-lenderof each relationship pair decide whether to exercise the quitting option. When they
exercise the quitting option, their relationship is dissolved and their children will haveno choice but to engage in an arm’s length contract in the next period. When an old Rproducer and an old R-lender in a relationship do not exercise the quitting option, in thesame period, their children (the young R-producer and R-lender) simultaneously decide
whether to exercise the quitting option. When they do not exercise the quitting option,
they agree on a relational contract, {zt , Rt }, which speciﬁes the capital investment levelzt and the repayment Rt to the R-lender. Because the producer cannot commit himself
to repay Rt , such a relational contract must be self-enforceable.
By exercising the quitting option, the young R-producer and R-lender obtain theiroutside payoﬀs: any young R-producer who quits a relationship in period t − 1 obtains
the outside proﬁt πt while any young R-lender who quits a relationship in period t − 1
lends her endowment wt−1 to the outside credit market and earns the interest income
λrt wt−1 . For the time being, we will treat the outside proﬁt πt of a producer and themarket interest rate rt to be exogenous, although we will endogenize these variables in
the next section.

2.5

Initial Period (t = 0)

In the initial period (t = 0), each old producer (irrespective of an A-producer or a
R-producer) owns an initial capital stock, x0 , which is assumed to be historically given.
11 Because the old producers in the initial period do not need to raise funds for capital
investment, they can produce the goods of x0 without any production costs. There isalso one unit mass of old lenders in the initial period.

2.6

Relational Contracts: Self-Enforcing Conditions

Consider any dynasty of relationships that consist of R-producers and R-lenders in the
successive generations who implicitly agree to enforce a sequence of relational contracts{zt , Rt }∞t=0 from the initial period t = 0 onward. Here, a contract {zt+1 , Rt+1 } isdesigned for t-th generation of the dynasty born in period t. Then we will consider the11

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 do not substantially change even when weallow z0 ̸= x0 .

10conditions under which such sequence of relational contracts is self-enforceable by all
the generations of the dynasty.
There are three constraints to be satisﬁed for a relational contract {zt , Rt } to be selfenforceable. The ﬁrst constraint is the incentive compatibility (IC) condition, according
to which the old R-producer has no incentive to renege on the agreed upon repayment
Rt . The second constraint is the individual rationality condition for R-producer (IRP),
according to which the young R-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 R-lender (IRL), according to which the young R-lender isweakly better oﬀ agreeing to a relational contract instead of exercising the quitting
option.
Incentive Compatibility (IC). Under a relational contract {zt , Rt }, the young Rproducer promises to repay Rt to the R-lender matching him when old (in period t)
in exchange for borrowing zt directly from her. For the time being, we suppose thatwt−1 ≥ zt ; this will be shown to be true in any equilibrium (see Lemma 2 below). Afterthe generation in period t − 1 follows the relational contract, the next generation bornin period t will also agree to a relational contract {Rt+1 , zt+1 } and enforce it where the
R-producer (who is the child of the old R-producer in period t) does not exercise thequitting option but honours the contracted agreement Rt+1 when old in period t + 1.
Anticipating the outcome in period t + 1 described above, the old R-producer inperiod t makes the repayment Rt to the old R-lender and does not exercise the quitting
option only if the following IC constraint, (ICt ), is satisﬁed:
pt zt − Rt + δ{pt+1 zt+1 − Rt+1 } ≥ pt zt + δπt+1 ,(ICt )

where pt+1 > 0 denotes the price of a speciﬁc good an old R-producer produces inperiod t + 1.
We now explain ICt in detail. The right-hand side of ICt denotes the payoﬀ the
old R-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 dissolutionof the relationship with the child of the current lender. In such a case, the child of sucha deviating producer would obtain his outside proﬁt πt+1 . This child’s future payoﬀ
is evaluated using the altruistic parameter δ > 0 from the viewpoint of the current
producer. Thus, the sum of these payoﬀs can be guaranteed by the old R-producer
when he reneges on the repayment Rt . In contrast, the left-hand side of ICt denotes
the payoﬀ of the old R-producer when he makes contracted repayment Rt to the Rlender matching him in period t expecting that the relationship is inherited by the next
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. Thus, ICt is necessary for the old R-producer not to renege on repayment
Rt in period t.

11

Individual Rationality (IR). In addition to the above, the following individual rationality constraint, IRLt , of each young R-lender must be satisﬁed: 12
Rt + λrt (wt−1 − zt ) + δ{Rt+1 + λrt+1 (wt − zt+1 )} ≥ λrt wt−1 + δλrt+1 wt .

(IRLt )

Otherwise, the young R-lender would exercise the quitting option in period t − 1; byexercising the quitting option, the lender and her child lend their entire wage income tothe credit market so as to earn the interest income corresponding to the payoﬀs on theright-hand side of IRLt . Here, (1−λ)wt must be spent for the enforcement cost of arm’s
length contract. Further, the interest income of her child, λrt+1 wt , is evaluated by thealtruistic parameter δ from the viewpoint of the current old lender. However, if the
R-lender does not exercise the quitting option, she would obtain contracted repayment
Rt in period t, which appears as the ﬁrst term on the left-hand side of IRLt , in addition
to interest income λrt (wt−1 − zt ) from the saving on the remaining income wt−1 − zt
after lending zt to the R-producer. (Note here that we are assuming that wt−1 ≥ zt .)Here, (1 − λ)(wt−1 − zt ) must be spent for the enforcement cost of arm’s length contract
in the credit market. The child of the lender is then paid the contracted amount Rt+1
in addition to the interest income λrt+1 (wt − zt+1 ) when old in period t + 1. Here, thepayoﬀ of the lender’s child is evaluated by the altruistic parameter δ > 0 again fromthe viewpoint of the current old lender.
Combining ICt with IRLt , we derive the following modiﬁed IC condition, denoted
by IC∗t :δ{pt+1 zt+1 − λrt+1 zt+1 − πt+1 } ≥ λrt zt .
(IC∗t )
This condition is necessary for a relational contract to be self-enforceable.
Next, we consider the individual rationality constraint of each young R-producer(IRPt ):
pt zt − Rt + δ{pt+1 zt+1 − Rt+1 } ≥ πt + δπt+1 .
(IRPt )
The condition IRPt ensures that each young R-producer prefers continuing the relationship to dissolving it. Suppose that some young R-producer exercises the quitting
option when young. Then, he earns the outside proﬁt πt and his child earns the outside
proﬁt πt+1 . The latter is evaluated by δ from the viewpoint of the current producer.
The sum of these payoﬀs corresponds to the payoﬀ each A-producer obtains via an
arm’s length contract, which thus becomes his outside option. However, the left-handside of IRPt denotes the payoﬀ of the young R-producer who does not exercise thequitting option when young; by doing so, he earns pt zt − Rt when old (period t) bymaking repayment Rt to the relationship lender. Following this, his child also continues
the relationship and earns proﬁt pt+1 zt+1 − Rt+1 by making the contracted repayment
12

As long as IRLt is satisﬁed, any old R-lender has no incentive to renege on the repayment Rt as
well when Rt < 0. This follows from the following fact: since the right-hand side of IRLt is larger thanλrt (wt−1 − zt ) + δλrt+1 wt , IRLt implies that Rt + δ{R+1 + λrt+1 (wt − zt+1 )} ≥ δλrt+1 wt , which meansthat every old R-lender wants to make the repayment Rt < 0 and continue a relationship to the next
generation rather than quitting the relationship, provided she has already lent zt to the R-producer
from her income wt−1 when young.

12

Rt+1 to the R-lender when old (period t + 1). Thus, the young R-producer obtains thesum of these payoﬀs by continuing the relationship.
By subtracting the right-hand sides of IRPt and IRLt from their left-hand sides,
the net total surplus of a relationship pair of a young R-producer and a young R-lenderborn in period t − 1 is deﬁned as
T St ≡ pt zt − λrt zt − πt + δ{pt+1 zt+1 − λrt+1 zt+1 − πt+1 }for t ≥ 1.
Here, we assume that the inverse demand function for a speciﬁc good is given by
pt = αAηtα−1 for ηt = zt or η = xt , where A > 0 and α ∈ (0, 1) are the parameters,
which we will derive by using monopolistic competition in the general equilibrium modelin Section 3. Then, we can deﬁne the net gain from relational contracting in period taspt zt − λrt zt − πt = αA(ztα − xαt ) + rt (xt − λzt ).
This net gain increases when the interest rate rt goes up, provided the R-producer’s
investment, zt , is less than (1/λ)xt , which will actually be the case in equilibrium (seeLemma 3 below). This is because a rise in the interest rate reduces the proﬁt of anarm’s length contract πt more than the increase in the opportunity cost of relational
contracting λrt zt when the R-producer’s investment is less than (1/λ)xt . We will see
later that the increase in the interest rate makes relational contract easier to be satisﬁed
so that it becomes more sustainable in the low developed stages in which the interest
rates are high. We will then endogenize the interest rate and show how the endogenouschange in the interest rate plays an important role for determining the self-enforcing
condition of relational contracting, and as a result, the dynamic process of the economy.
In the initial period t = 0, each initial old R-producer earns the proﬁt π0 ≡ αAxα0
because he owns the initial capital x0 while each initial old R-lender earns nothing.
Thus, their joint proﬁt is π0 . Since each initial old R-producer also has the outside
option to earn π0 , the net total surplus T S0 in the initial period t = 0 becomes T S0 ≡
δ{αAz1α − λr1 z1 − π1 }. 13
Then, for a relationship pair of a young R-producer and a young R-lender born inperiod t − 1 to agree on a relational contract {zt , Rt }, it must satisfy both IRPt andIRLt ; that is, the total net surplus of each relationship pair must be non-negative:
T St ≥ 0,

for any t ≥ 0(TSt ).

We can then readily show that there exists a sequence of repayments {Rt }∞t=1 that∗satisfy ICt , IRLt−1 , and IRPt−1 for all t ≥ 1 as long as ICt and TSt−1 are satisﬁed for
all t ≥ 1.14
Finally, we denote byJt ≡ pt zt + λrt (wt−1 − zt )Here, we have IRP0 : π0 − R0 + δ{p1 z1 − R1 } ≥ π0 + δπ1 , and IRL0 : R0 + δ{R1 + λr1 (w0 − z1 )} ≥
δλr1 w0 .
14
It is suﬃcient to set Rt = λrt zt for each t ≥ 1 and R0 such that π0 ≥ R0 ≥ 0.
13

13

and
Vt ≡ Jt + δJt+1
the joint proﬁt and the joint payoﬀ of a relationship pair born in period t − 1, respectively.
Although there are many possible equilibria sustained by diﬀerent relational contracts as known in the Folk Theorem of repeated game theory, we will focus on theequilibrium called the best relational contracting equilibrium (BRCE), in which the initial generation of relationship pairs in each dynasty chooses a sequence of all future
maximize the weighted sum of joint payoﬀs
relational contracts {zt , Rt }∞t=1 so as to∑tof all generations in the same dynasty ∞t=0 β Vt for some weight β ∈ (0, 1), subject∗∞
to the above constraints {(ICt ), (TSt−1 )}t=1 , given the future paths of all the market
15 16 17 Thus, the initial generation in each dynasty puts theprices {rt , wt }∞t=0 .
t
welfare weight β on the t-th generation
same dynasty when
the opti∑∞of the∑∞ solvingtt−1
mal relational contracts. Then, since t=0 β Vt = J0 + (δ + β) t=1 β Jt and sinceJ0 = π0∑is exogenously given, the above optimization problem is equivalent to maxit−1 J subject to IC∗ and TS
mizing ∞tt−1 for all t ≥ 1. The results we establish int
t=1 β
what follows do not depend on the particular values of the welfare weight β ∈ (0, 1).18
Then, we can show the following result on the optimal relational contracts.
Lemma 1. (i) Each relationship pair born in period t−1 agrees to a relational contract{zt , Rt } and follows it if and only if IC∗s and TSs are satisﬁed for all s ≥ t. (ii) In
the optimal relational contract, zt ≤ λ̂xt holds in any period t where λ̂ ≡ λ1/(α−1) > 1,and λ̂xt maximizes the joint proﬁt Jt . (iii) The optimal relational contract involves adownward distortion of capital investment zt < λ̂xt as compared to the investment level
maximizing the joint proﬁt Jt only if IC∗t is binding.
Lemma 1 (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 relationship pairs ineach dynasty chooses the relational contracts that maximize the weighted sum of the15

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.
16
In addition, we need to impose the condition pt zt ≥ Rt ≥ −λrt (wt−1 − zt ) for all t ≥ 1 andπ0 ≥ R0 ≥ 0. This ensures that the consumption levels of each old R-producer and each old R-lender
are non-negative in each period; this is called condition NNCt . However, we can solve the optimal
relational contracts without NNCt , and then, can check that the optimal relational contract satisﬁes
NNCt later by adjusting the repayment Rt appropriately and using the other conditions such as IC∗tand TSt . Also, IC∗t and T S0 imply that pt zt + λrt (wt−1 − zt ) ≥ λrt wt−1 + πt for all t which thenensures that every relationship cannot gain from engaging in arm’s length contract in the credit market
in each period t on the equilibrium path in which ICt and TS0 are satisﬁed for all t ≥ 1.
17
This formulation is equivalent to maximizing the sumPof R-producers’ payoﬀs in each dynastytsubject to ICt , IRPt−1 , and IRLt−1 for all t ≥ 1: max{Rt ,zt } ∞t=0 β {pt zt − Rt } subject to ICt , IRPt−1
and IRLt−1 for t = 1, 2, ....
18
If one might think that it is reasonable to suppose that the initial generation has the same welfare
weight as his or her altruistic preference parameter δ, we can set β = δ.

14∑
t−1 J . Lemmas 1 (ii) andjoint proﬁts of all generations in the same dynasty ∞tt=1 β(iii) then show that the optimal relational contract involves a lower capital investment
than the one maximizing the joint proﬁt Jt without IC∗t ; that is, zt < λ̂xt . This occurs
only if IC∗t is binding.
One of the key variables which aﬀect the optimal relational contract is the interest
rate rt in the anonymous credit market. As we have already mentioned, high interest
rates make relational contract easier to be self-enforceable so that R-producers invest
more than A-producers do when zt ≤ (1/λ)xt holds, which will be shown to be true inany equilibrium below. This implies that relational contract becomes less constrainedin the early stages of development in which the anonymous credit market is immature sothat the interest rates are high but it becomes more constrained in the well-developed
stages with low interest rates. In the next section, we will address this issue by turning
to the full model of a dynamic general equilibrium in which all the market prices
such as {wt , rt }∞t=0 are endogenously determined together with the optimal relational
contract we have shown above. The novel part of our model is that we investigate howthese market prices aﬀect and are aﬀected by the self-enforcing conditions of relational
contracts, resulting in the dynamic process of the whole economy.

3

Model: General Equilibrium

In this section we provide a dynamic general equilibrium model which is kept as simple
as possible in order to incorporate the market equilibrium conditions into the partial
equilibrium model presented in the previous section. To this end, we need at least
three things: (i) the income level each lender earns wt is endogenously determined,(ii) the interest rate rt is endogenously determined, and (iii) the price of each speciﬁc
good pt is endogenous and some positive proﬁt for the production of that good can be
guaranteed because otherwise the total surplus of relational contract cannot be positiveand hence relational contract will never become sustainable. The ﬁrst and second parts
are addressed by introducing the credit and labour market clearing conditions. The lastpart is addressed by considering a monopolistic competition for speciﬁc goods (modelled
as intermediate goods below) which ensures a positive proﬁt for each producer who
obtains monopolistic power over a diﬀerentiated good he produces.

3.1

Markets

In the economy, there is a single ﬁnal good, taken as a numèraire, which is used forboth consumption and investment. The ﬁnal good Yt is produced by a continuum ofintermediate goods, each of which is indexed by i ∈ [0, 1], and labour Lt in the following
manner:∫1

Yt = AL1−αtηt (i)α di,(2)0where α ∈ (0, 1), A > 0, and ηt (i) denotes the input demand for intermediate good iin period t (which is equivalent to its output).
15

Each intermediate good i is produced by producer i, either an A-producer or a
R-producer, who possesses the speciﬁc knowledge to produce that intermediate good.
The speciﬁc good we have considered in the partial equilibrium model in Section 2 isinterpreted as an intermediate good in the general equilibrium model in this section.
Here, as we have assumed in the partial equilibrium model, producer i (an A-producer
or a R-producer) can produce one unit of intermediate good i when old by investing
one unit of capital when young.
We also assume that each young lender, born in each period t, is endowed with
one unit of labour and inelastically supplies it to the labour market in order to earn
the wage income wt . The labour supplied by these young lenders (workers) is used for
producing the ﬁnal good of the economy.
We assume that there is perfect competition in the ﬁnal good market. Then, theﬁnal good ﬁrm chooses the demand for labour Lt and intermediate inputs ηt (i) tomaximize its proﬁt:
∫ 1
∫ 1
1−α
α
pt (i)ηt (i)di,(3)
ηt (i) di − wt Lt −
ALt
0

0

where the wage rate wt and price of intermediate good i, pt (i), are taken as given. The
corresponding ﬁrst-order conditions are as follows:
AαLt1−α ηt (i)α−1 = pt (i)andA(1 − α)L−α
t

(4)

∫ 1
ηt (i)α di = wt .

(5)0Equation (4) corresponds to the inverse demand function for a speciﬁc good we haveused in the partial equilibrium model in the previous section.
We also assume that the labour market is perfectly competitive and we let wt ≥ 0denote the competitive market wage in period t in the labour market.
Finally, as we have explained in the previous section, there exists the credit market
where a producer and a lender can enter into a formal credit contract, which we callthe arm’s length contract, in the spot manner. However, here, the lender must incur
the enforcement cost to prevent the borrower from defaulting: each lender must spend
1 − λ ∈ (0, 1) unit of the ﬁnal good as the enforcement cost for lending one unit to thecredit market.

3.2

Arm’s Length Contract Producers

Because every A-producer faces the same demand function, (4), for his intermediate
good, we will omit notation i hereafter.
We then obtain πt , the proﬁt of an intermediate good producer (an A-producer)
who ﬁnances his investment xt via an arm’s length contract in period t, as follows:πt ≡ pt xt − rt xt= ALt1−α αxαt − rt xt ,16

(6)

where the price of an intermediate good, pt , is given by (4) and ηt (i) = xt for eachA-producer i. Because Lt = 1 holds in the labour market equilibrium, we set Lt = 1
in what follows.
Each young A-producer born in period t − 1 chooses the capital investment levelxt to maximize his payoﬀ πt + δπt+1 . Here, note that his consumption level Ctt−1 isequal to the proﬁt πt he earned in period t, whereas the consumption level of his childtt= πt+1 . Thus, the payoﬀ of a young A-producer is given by Ctt−1 + δCt+1=
is Ct+1
πt +δπt+1 . Since each A-producer takes the market prices such as wt and rt as given andhis capital investment xt aﬀects only his own proﬁt πt , we will focus on the equilibrium
in which a young A-producer chooses his capital investment xt so as to maximize only
his own proﬁt πt .19
We then deﬁne such optimal investment by xt = x(rt ) as a function of the interest
rate rt , which satisﬁes the following ﬁrst-order condition:
Aα2 xα−1= rt .
t(7)

max πt = Aα(1 − α)xαt(8)Thus, we havext

where xt satisﬁes (7). In what follows, we will often omit the argument rt from x(rt )and simply write xt for xt = x(rt ) when no confusion arises.

3.3

Deﬁnition of Equilibrium Paths

Now, we provide a formal deﬁnition of an equilibrium path in this model economy.
Deﬁnition. A sequence {xt , zt , wt , rt }∞t=0 is said to be an equilibrium path of the economy if the following conditions are satisﬁed.
(i) Each young A-producer born in period t − 1 chooses his capital investment xt so
as to maximize his own proﬁt πt , which satisﬁes the optimality condition (7).
(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∑ thet weighted
sum of the joint payoﬀs of all generations in the same dynasty ∞t=0 β Vt subject∗∞
to {(ICt ), (TSt−1 )}t=1 .
(iii) The labour market equilibrium (LMEt ): Setting Lt = 1 in (5) determines themarket wage wt as
wt = A(1 − α)(lztα + (1 − l)xαt ).
19However, in general, there may exist strategic interactions between successive generations in thesame dynasty of A-producers in that they may make the current capital investment choices contingent
on the observed history about the capital investments chosen in the past periods in the same dynasty.
However, we can show that no generality is lost by conﬁning our attention to the equilibrium in which
each A-producer chooses xt in order to maximize only his own proﬁt πt in any period t no matter what
histories are observed up to period t (see the Online Appendix).

17

(iv) The credit market equilibrium (CMEt−1 ):λwt−1 = λlzt + (1 − l)xt .
Here, the initial capital z0 = x0 is given.
Condition (i) yields the optimal capital choice of each young A-producer taking
market prices wt and rt as given. Condition (ii) is the optimal choice with regard to
relational contract {zt , Rt } for every relationship pair, as we have explained in Section
2. Condition (iii) gives LMEt which is the labour market clearing condition: market
wage wt is determined by clearing the labour market (Lt = 1) using the labour demand
function (5) with the labour supply equal to one unit mass of young workers in every
period. Condition (iv) gives CMEt−1 which is the credit market clearing conditionin period t − 1. Each young R-producer lends λ(wt−1 − zt ) to the anonymous credit
market in period t−1 after she makes a relationship lending zt to the young R-producer
matching her. Here, note that we are supposing wt−1 ≥ zt and that each R-lendermust spend (1 − λ)(wt−1 − zt ) for the enforcement cost of the arm’s length contract.
In addition to this supply of credit, 1 − l young A-lenders outside the community havenothing but to lend λwt−1 each to the credit market. Thus, the total credit supply inthe anonymous credit market is given by lλ(wt−1 − zt ) + (1 − l)λwt−1 . Conversely, 1 − lyoung A-producers ﬁnance their capital investment xt each from the credit market.
Thus, the total demand in the credit market is given by (1 − l)xt . Then, the credit
market of period t − 1 clears if CMEt−1 holds.
In this model economy, the initial condition is given by initial capital stock z0 = x0
owned by each initial old intermediate good producer irrespective of a R-producer oran A-producer. Thus, every initial old intermediate good producer produces z0 = x0
without any production costs. Then, the market wage in the initial period (t = 0) is
determined by the labour market equilibrium LME0 : w0 = A(1 − α)xα0 .

4Characterization of Equilibrium Dynamics: From Relationships to Markets

In this section, we show that relational contracting contributes to economic growth
in low-development stages while its value declines as the economy grows and enters
high-development stages.

4.1

Preliminary Results

We begin by showing the preliminary results that will be useful for characterizing theequilibrium paths.
First, we have thus far assumed that each R-producer does not invest in capital
more than the funds available to the R-lender matching him, i.e., wt−1 ≥ zt , in any
equilibrium. Now we show that this is actually the case in any equilibrium. If this is18

not the case in some period t, we have zt > wt−1 and hence each R-producer needs toﬁnance the remaining amount zt − wt−1 from the credit market after borrowing wt−1
directly from the R-lender matching him. Then, the joint proﬁt of the relationshippair is changed to Jt = pt zt − rt (zt − wt−1 ) which is decreasing in the interest rate rt .
If each R-producer demands high capital investment zt > wt−1 in the credit market,
the equilibrium interest rate rt must go up in order to clear the credit market. This
however reduces the joint proﬁt Jt and lowers the capital investment of the R-producer
zt until it restores the condition wt−1 ≥ zt in the credit market equilibrium. We thus
obtain the following.
Lemma 2. In any equilibrium path zt ≤ wt−1 must hold in any period t.
Second, we show that IC∗t is always binding in any equilibrium path.
Lemma 3. In any equilibrium, zt ≤ (1/λ)xt holds in any period t. Then, the IC
condition IC∗t becomes binding in any period t in any equilibrium.
Recall that the net gain from relational contracting in period t is deﬁned as αA(ztα −αxt )+rt (xt −λzt ) which is decreasing in rt when zt > (1/λ)xt . If each R-producer invests
more than (1/λ)xt (i.e., zt > (1/λ)xt ) in some period t, the interest rate must go up
in order to meet such high capital investment demand in the credit market equilibrium(CMEt−1 ). This in turn reduces the net gain from a relational contract, lowering thecapital investment of the R-producer so that as a consequence zt ≤ (1/λ)xt is restored. When zt ≤ (1/λ)xt holds, we have that zt < λ̂xt so that (IC∗t ) must be binding
in equilibrium due to Lemma 1.

4.2

Rise and Fall of Relational Contracts

Next, we characterize equilibrium paths. In particular, we investigate how relationshiplending contributes to the process of economic development relative to market lending
based on arm’s length contracts. Then, we show that each R-producer who enters into
relational contract invests and hence produces more than each A-producer who enters
into arm’s length contract in the early stages of development, but the former invests
less than the latter in later development stages.
Thanks to Lemma 2 and 3, any equilibrium path of the economy can be described∗
as a sequence {zt , xt , rt , wt }∞t=0 which satisﬁes binding ICt and CMEt :(IC∗t )αδ{αAzt+1− λrt+1 zt+1 − πt+1 } = λrt zt , t = 1, 2, ...,

λwt = λlzt+1 + (1 − l)xt+1 , t = 1, 2, ...,

(CMEt )

where rt = α2 Axtα−1 (see (7)), wt = A(1 − α)(lztα + (1 − l)xαt ) (see LMEt ), and
λw0 = λlz1 + (1 − l)x1
19

(CME0 )

for a given initial market wage w0 = (1 − α)Axα0 .
By substituting rt = α2 Axtα−1 into IC∗t , using πt = Aα(1−α)xαt (see (8)), and deﬁning a new variable yt ≡ zt /xt for the relative output of a R-producer which measures
the ratio between capital investments (output) of relational and arm’s length contracts,
we can re-write the above equilibrium conditions as follows:)(
xt+1 ααδ{yt+1− λαyt+1 − (1 − α)} = λαyt , t = 1, 2, ...,(9)xt
λ(1 − α)Axαt (lytα + (1 − l)) = xt+1 (λlyt+1 + (1 − l)), t = 1, 2, ...,

(10)

λw0 = x1 (λly1 + (1 − l)).

(11)

and
Note that yt ≤ λ−1 holds in any period t due to Lemma 3.
Since w0 is exogenously given, the above equations (9) and (10) fully determine anequilibrium path {xt , yt }∞t=1 once we ﬁx the value of (x1 , y1 ) in the ﬁrst period t = 1.
The ﬁrst period values of (x1 , y1 ) can be freely chosen as long as they satisfy the initialcondition (11). Thus, there are multiple equilibrium paths even though the initialcondition x0 and hence w0 are exogenously ﬁxed.
We ﬁrst show that the relative output of R-producer yt declines over time until
it hits some critical value and that there exists a switching period T before which
yt > 1 holds but after which yt < 1 holds. Such a switching period T is unique for agiven y1 . Thus, the economy relies more on relational contracts in the early stages ofdevelopment (for t ≤ T ) but more on arm’s length contracts in the subsequent stagesof development (for t > T ) in the sense that each young R-producer invests more thaneach young A-producer in the former stages but not in the latter stages.
To derive this result, we introduce some notations and make conditions on theprimitives of the model.
We ﬁrst deﬁne the following output of A-producers:[λ(1 − α)A(λ−α l + (1 − l))
x≡
1−l

] 1

1−α

(12)

and we then show that this becomes the upper bound for capital investment (productionlevel) which A-producers can attain in any equilibrium path in which they do notstagnate below the initial investment level x0 . To see why, note that using CMEt fort ≥ 1, 1/λ ≥ yt and zt+1 ≥ 0, we have thatλ(1 − α)Axαt (λ−α l + (1 − l)) ≥ (1 − l)xt+1(13)

in any period t ≥ 1. Further, CME0 and λ < 1 imply that λ(1−α)Axα0 (λ−α l+(1−l)) ≥λw0 = x1 (ly1 + (1 − l)) ≥ (1 − l)x1 . Thus, the above inequality (13) must be satisﬁed
for all periods t ≥ 0. When we consider the development process of the economy inwhich each A-producer invests more in capital than the initial level x0 , it is necessary
20to have x ≥ x0 because otherwise we can show that xt ≤ x0 holds for all t ≥ 1. 20
This implies that the market economy never emerges in the sense that A-producers
always stagnate below the production level in the initial period, which is contrary to
our interest for considering the process of economic development. Thus, we need thefollowing condition for investigating the process such that the economy takes oﬀ fromthe initial development state.
Assumption 1. x ≥ x0 .
Given Assumption 1, we can readily see from (13) that xt ≤ x is satisﬁed in any
period t ≥ 0, that is, x becomes an upper bound for capital investment which eachA-producer chooses in any period.21
Next, we deﬁne the following value of the altruistic preference parameter (the discount factor), which plays a critical role for determining equilibrium paths.
() ( −α)αλlλ + (1 − l))
δ̃ ≡
.
(14)
λ1−α − λ(1 − l)2
Then, we make the following assumption. 22
Assumption 2. δ > δ̃.
Assumption 2 can ensure a large enough gain from relational contracting (the lefthand side of IC∗t ) which makes it possible for a relationship pair to invest more in capitalthan an A-producer when the interest rate is high. Since the interest rates are high inthe early stages of development in which wage income and hence saving is low, eachyoung R-producer has more incentive to choose a higher investment than each young
A-producer in the early stages but not in the well-developed stages under Assumption
2.
More formally, we show the following main result of this paper.
Proposition 1. Suppose that Assumptions 1 and 2 hold. Then, any equilibrium pathhas the following features: if y1 > 1, then
• the relative output of R-producer yt decreases over time until it hits some criticalvalue less than one, denoted by ŷ < 1, and thus, y1 > y2 > · · · > ŷ;Suppose that x0 > x. Then, by the deﬁnition of x, we have λ(1 − α)A(λ−α l + (1 − l))xα0 ≤ (1 − l)x0 .
≥
(1
−
l)x
,
we
have
x
≥
x
. Next, using
Then, since CME0 implies that λ(1 − α)A(λ−α l + (1 − l))xα
1
0
1
0
−α
l + (1 − l))xα
CMEt , we have (1 − l)x0 ≥ λ(1 − α)A(λ−α l + (1 − l))xα1 ≥ (1 − l)x2 ,0 ≥ λ(1 − α)A(λimplying that x0 ≥ x2 . Repeating this, we obtain xt ≤ x0 for all t ≥ 1.
21
−1
Note that x = x1−α xα ≥ x1−α xαl + (1 − l))/(1 − l)]xα0 ≥ x1 due to CME0 .
0 = [λ(1 − α)A(λRepeating this, we get x ≥ xt for all t ≥ 0.
22Here, for 1 > δ to be satisﬁed, Assumption 2 requires that 1 > δ̃, which is more likely to be satisﬁed
when the number of relationship pairs l is small and the enforcement cost 1 − λ is large. More precisely,
if we let l → 0 in δ̃, then we can show that 1 > δ̃ holds when λ < (1/(1 + α))1/α .
20

21• there exists a switching period T such that yt > 1 holds for all t ≤ T but yt < 1holds for all t > T ;• the switching period T is unique for a given y1 .
Proposition 1 states that relational contracting contributes to the process of economic
development relative to arm’s length contracts more in the early stages than the laterstages of development in that the relative output of R-producer yt declines over time
until it hits some critical value ŷ < 1. In particular, there exists a unique switching
period T before which relationship lending yields more capital investment and output
than the market lending based on arm’s length contracts, but after which the former
yields smaller capital investment and output than the latter (see Figure 2). Proposition
1 thus conﬁrms the historical argument that the Western society drastically changed
from a non-market system, which relies more on personal connections, kinship networks,
and community, to a market system in the 19th century as the Western economies grewfaster (Polanyi (1947)).
Proposition 1 supports the positive view of relational contracting that the nonmarket system (the relationship-based system) is not impediment to economic growthin the early periods but has a positive eﬀect of promoting the emergence of a market
system in subsequent stages. Speciﬁcally, the expansion of the relative output of Rproducer yt can save the enforcement cost associated with arm’s length contracts inthe whole society, contributing to economic growth. Lamoreaux (1994) reports supportive evidence that the relationship-based lending became eﬀective in ﬁnancing andpromoting economic growth in New England in the early 19 century where the ﬁnancial
market was so immature that market-based ﬁnancing was diﬃcult. Another related evidence is the positive role of large German banks such as Commerzbank, Dresdner, andDeutsche in having ﬁnanced the rapid expansion of German industries between the late
19th century and the early 20th century. These large banks formed close and lasting
relationships with industry enterprises by oﬀering low interest rates and being represented as board directors on these ﬁrms (Allen (2011)). Maurer and Haber (2007) also
empirically support the positive view of relationship lending in that Mexican bankers
during 1888–1913 were largely involved in relationship lending because they optimally
responded to a large enforcement cost but they did not loot their own banks at theexpense of outside shareholders. 23
Proposition 1 also shows the other side of relational contracting: the relationshipbased system eventually declines and has serious limitations as the economy becomesricher. Speciﬁcally, our result shows that each R-producer who engages in relationshiplending eventually switches to invest less than each A-producer who engages in market
lending in the later stages of development. Lamoreaux (1994) found evidence that
insider lending practices, which had served as an important ﬁnancing device in New
23
Fafchamps (2000) also discusses that business networks based on personal relationships perform avariety of valuable functions in Sub-Saharan Africa.

22

England in the early 19th century, became less important and were eventually replacedby market-based ﬁnance as the United States transitioned from a capital-poor economy
to a capital-rich economy in the late 19th century. Demirgüç-Kunt and Levine (2004)
also reported a related fact that bank ﬁnance, which is sometimes characterized asa long-term relationship between particular banks and ﬁrms, becomes less popular
relative to market-based ﬁnance, such as equity, in a more-developed country (seeRajan and Zingales (2000) for a related argument).
The intuition behind Proposition 1 is understood as follows. The income levelsin the early stages of development are so low that total saving is limited and hence
the market interest rates are high. Since each relationship pair extracts the net gainfrom relational contracting deﬁned as αA(ztα − xαt ) + rt (xt − λzt ) in period t and its
investment zt is less than (1/λ)xt (Lemma 3), a rise in the interest rate leads to alarge net gain from relational contracting, which makes the IC condition (IC∗t ) more
likely to be satisﬁed. This leads to higher capital investment by a R-producer over an
A-producer in the low-development stages in which the interest rates are expected tobe high in the credit market. However, as the economy develops well in the course
during which relationship pairs save on enforcement cost, the anonymous credit market
expands and the interest rates fall over time. Thus, each R-producer who expects low
interest rates and tight IC∗t invests less in capital and produces less than an A-producer
in the subsequent stages in which lower interest rates make arm’s length contracts more
proﬁtable.
As we have already mentioned, equilibrium paths are not unique because there
are multiple candidates for the equilibrium values of y1 and x1 in the ﬁrst period aslong as they satisfy (11): λw0 = x1 (λly1 + (1 − l)). Thus, the diﬀerences in theequilibrium output in the ﬁrst period (x1 , y1 ) cause the diﬀerent evolution patterns of
the whole economy with diﬀerent switching periods. Which economies with larger orsmaller relative output of R-producer y1 in the ﬁrst period switch more early from therelationship-based system to the market-based system? We now address this issue, andgive the following result.
Proposition 2. Suppose that Assumptions 1 and 2 hold. Take any two equilibrium
′
′′
′′ ∞
paths {yt′ }∞t=1 and {yt }t=1 with the corresponding switching periods T and T where
′
′′
′
′′
′′
′
y1 ≥ y1 . Then, yt ≥ yt holds in any period t ≥ 1 so that T ≥ T .
Proposition 2 shows that the economy which starts with less reliance on relational
contracts (smaller y1 ) in the ﬁrst period continues to rely less on relational contracts inevery future period and switches more early from the relationship-based phase to themarket-based phase than the economy which starts with more reliance on relational
contracts in the ﬁrst period (see Figure 3). This result has the implication that thedegree to which the market-based system is currently dominant in a country depends
on the extent to which that country has relied on it in the early development phases.
We can also investigate how the switching period T changes with exogenous variables of the model economy such as the discount factor of individuals δ and the en23

forcement cost of an arm’s length contract 1 − λ.
Suppose for example that δ increases, that is, individuals become more altruistic
about their children. This change raises the gain from relational contracting so thatIC∗t becomes more likely to be satisﬁed. However, since IC∗t always binds in any periodt (Lemma 3), such an increase in the gain from relational contracting must be oﬀset
by the reduction of the relative output yt+1 of a R-producer in a future period in orderto keep IC∗t binding in equilibrium. Thus, as long as the relative output y1 in theﬁrst period does not rise by an increase in the discount factor δ, the relative output inthe second period y2 must decrease in order to keep IC∗1 binding. Repeating this, therelative outputs of R-producers yt in all the future periods must decrease, resulting ina decrease in the switching period T . We can also see a similar eﬀect of the rise in theenforcement cost 1 − λ. Suppose that λ decreases. Then, the relationship pairs facelower opportunity costs to invest in capital such that IC∗t becomes more easily satisﬁed.
Again, such an increase in the gain from relational contracting must be oﬀset by the
reduction in the relative output yt+1 in the future period in order to keep IC∗t binding
in equilibrium.
More formally, by letting ζ ≡ (δ, 1 − λ), we use {yt (ζ)}∞
t=1 to denote an equilibrium
path of the relative output of R-producer by making its dependence of δ and λ explicit.
Let T (ζ) be the corresponding switching period. Then, we show that the economy
more early switches from a relationship-based system to a market-based system when
the discount factor and the enforcement cost increase as long as the ﬁrst period relative
output of R-producers y1 does not increase by such a change. 24
Proposition 3. Suppose that Assumptions 1 and 2 hold. Let ζ ′ > ζ ′′ . Then, T (ζ ′ ) ≤T (ζ ′′ ) if y1 (ζ ′ ) ≤ y1 (ζ ′′ ).

4.3

Long-Run Behaviour

We have shown that the economy switches from the development stages which rely
more on the relationship-based system to the development stages which rely moreon the market-based system. We have also shown the time structure of relational
contracting such that its value relative to an arm’s length contract declines over time
until the relative output of the R-producer yt hits some critical value ŷ < 1. However,
Proposition 1 does not say anything about how the relative output yt behaves over timeafter it goes below the critical value ŷ.
To investigate the long-run behaviour of equilibrium dynamics, we deﬁne the steadystate of the economy, denoted by (ỹ, x̃), as yt+1 = yt = ỹ and xt+1 = xt = x̃ in (9) and24

Recall that there is a freedom to choose the relative output y1 in the ﬁrst period t = 1 fordetermining an equilibrium path. By such indeterminacy, we cannot know how y1 changes with ζ.
If y1 increases with ζ, it may be possible that we have T (ζ ′ ) > T (ζ ′′ ) even when ζ ′ > ζ ′′ .

24

(10) which satisfyδ{ỹ α − λαỹ − (1 − α)} = λαỹ,(15)

λ(1 − α)Ax̃ (lỹ + (1 − l)) = x̃(λlỹ + (1 − l)).

(16)

α

α

Recall that our model economy is characterized by a two-dimensional system of nonlinear diﬀerence equations (9) and (10). Then, we show that the economy eventuallyreaches the steady state where each A-producer invests in more capital and hence
produces more than each R-producer does.
Proposition 4. Suppose that Assumptions 1 and 2 hold. Then, (i) the steady sate of
the economy (x̃, ỹ) exists and is unique. (ii) The economy eventually converges to thesteady state (ỹ, x̃) in the long run where ỹ < 1.
Proposition 4 states that the relationship-based system declines in the long run as
the economy develops well, although it contributes to economic development more thanthe market-based system does in the early development stages.
We can also conduct a comparative statics exercise about the steady state of theeconomy (ỹ, x̃).
Corollary. Suppose that Assumption 2 holds. Then, the relative output of R-producerỹ in the steady state is decreasing in the discount factor δ and the enforcement cost
1 − λ. Suppose also that λ is so small that λ ≤ α(1 − l)/(1 − α). Then, the output of
A-producer x̃ is also decreasing in δ and 1 − λ.
When individuals become more altruistic about their children (δ increases), the net
gain of relational contracting (the left-hand side of (15)) increases as well. However,since IC∗t always binds in the steady state, the relative output of a R-producer ỹ mustthen go down in order to make IC∗t binding. Furthermore, the decrease in the relative
output of a R-producer reduces not only the credit demand (the right-hand side of (16))
but also saving (the left-hand side of (16)) because the wage income becomes lower.
When the enforcement cost 1 − λ is large (λ is small), the former eﬀect is dominatedby the latter eﬀect so that the net credit supply decreases and thus the output of an
A-producer x̃ becomes smaller as well. The comparative statics result about the change
in the enforcement cost λ is similarly explained as well.

5

Formation of New Relationships

5.1

Matching and Separation

We have thus far focused on the case where the measure of those who engage in relational contract, denoted by l ∈ (0, 1), is exogenously ﬁxed and does not change overtime. By restricting to such a case, we have investigated how the investment (output)
25

levels of A-producers and R-producers change over time. Put diﬀerently, our analysis
has been limited to the changes in the ‘extensive’ margin of relational and arm’s length
contracts by assuming that its ‘intensive’ margin, which is the relative measures of
those who engage in these contracting modes, is exogenous.
In this section, we will extend our basic model to allow producers and lenders tofreely ﬁnd their partners and form relationship pairs. Then, we will investigate thechanges in both the intensive and extensive margins of the two contracting modes
together.
For this purpose, we introduce two twists into the basic model as follows. First, weabandon the assumption that there is a local community where particular producers
and lenders have personal relationships. Instead, we allow every individual, irrespectiveof a producer or a lender, to enter the matching market for seeking a partner andforming a relationship pair. The producers and lenders who enter the matching market
are randomly matched with each other (we will specify the matching function below).
Second, each relationship pair is resolved with exogenous probability, in which case theseparated producer and lender enter the matching market for meeting another partners.
More speciﬁcally, we denote by lts the measure of relationship pairs of young Rproducers and young R-lenders whose predecessors have started the relationships fromperiod s and are not still dissolved in the beginning of period t ≥ s. Each of those lts
relationship pairs will continue to be matched and carried over the next generation inperiod t + 1 with probability γt ∈ (0, 1), while the relationship will be dissolved withprobability 1 − γt . We also introduce an uncertainty into the exogenous separation
probability 1 − γt ∈ (0, 1) such that the set of possible separation shocks∑
is given by1niiΓ ≡ {γ , ..., γ } and γt = γ is realized with probability q ∈ (0, 1) where i=n q i = 1
and γ 1 < · · · < γ n ≡ 1 (here we set γ n = 1 without loss of generality). Such
uncertainty allows us to generate an endogenous process that relationship pairs can beheterogeneous with respect to the exogenous separation shock γt ∈ Γ they face when
making relational contracts. Then, we can address the dynamic issue of how eachrelationship switches from engaging in relational contract to arm’s length contract insome periods although it relies on the former in other periods.25
We assume that a young R-producer and a young R-lender of relationship pair, bornin period t − 1, know how much likely is their relationship to continue to their children
in the next period t, that is, they know the realization of γt before making a capital
investment decision. Thus a relational contract in period t can be contingent on therealization of the exogenous separation shock γt . However, we assume that they do notknow how likely is it that the relationship in the next generation will be resolved, thatis, they do not know the exogenous separation shock γt+1 such that the relationship of
their children will be resolved.
The events in a period proceeds as follows (see Figure 4 for the time line). When a25

Without any such uncertainty about γt , every formed relationship pair engages in either relational
contract or arm’s length contract from the outset when the relationship starts. Then we cannot address
the dynamic issue of how the relationships engaging in relational contracts endogenously switch to arm’s
length contracts over time.

26

relationship was not dissolved in the previous period and the separation shock of γt wasobserved to them, a young R-producer and a young R-lender of the relationship, bornin period t − 1, have the following options: one is to continue the relationship by implementing the agreed upon relational contract and the other is to go to the anonymous
credit market for earning the proﬁts of arm’s length contract without dissolving therelationship.26 When a R-producer and a R-lender become old, they have the following
options: one is to go to the matching market for seeking another partners for theirchildren (the details of the matching market will be given below) and the other is tokeep the relationship carried over the next generation. All the relationship pairs of old
R-producers and old R-lenders who were fully separated have nothing but to go to thematching market as well as all old A-producers and old A-lenders do so. Then, if amatch between an old producer and an old lender in the matching market is successful,their children will start new relationships. Otherwise, their children must engage inarm’s length contract in the anonymous credit market.
The matching market is modelled as follows: when there are lp,t old producers andll,t old lenders in the matching market for seeking relationship partners, we assume
that they are randomly matched with each other according to the matching function
M (lp,t , ll,t ) where M exhibits the constant returns to scale. Since every existingre∑lationship is dissolved exogenously with the same average probability µ ≡ i q i γ i ,27
lp,t = ll,t always holds in any period t in any equilibrium. Thus, each old producer(respectively, lender) is matched with an old lender (respectively, producer) in thematching market with probability M (lp,t , ll,t )/li,t = m ≡ M (1, 1) for i = p, l.
We denote by {zts (γt ), Rts (γt ), dst (γt )}t≥s a sequence of relational contracts fromperiod t onward, which is agreed by a relationship pair matched in period s (see Figure
5 for the time line). Here, capital investment zts (γt ) and repayment Rts (γt ) depend on therealization of the separation shock γt ∈ Γ because, by the above assumption, the young
R-producer and the young R-lender, born in period t − 1, know the separation shock γtin period t before they make a capital investment decision in period t−1. dst (γt ) ∈ {0, 1}
is the indicator function which takes one only when the young relationship pair decides
to engage in a relational contract instead of going to the outside credit market after
having observed the separation shock γt .

5.2

Equilibrium with Formation of New Relationships

Given the above modiﬁcation of the basic model, we show that relational contracting
contributes to economic development in the early stages but it declines in the laterstages in terms of not only its extensive margin but also its intensive margin: each
26

There is also the third option that the young R-producer and young R-lender fully dissolve therelationship and engage in arm’s length contract in the anonymous credit market. However, we canshow that this case never happens on equilibrium path because they ﬁnd it optimal to continue therelationship rather than voluntarily dissolve it.
27
Also all relationships are never voluntarily dissolved in any equilibrium as we have noted in footnote
26.

27

R-producer invests less than each A-producer does over time as well as the measure of
the relationships who engage in relational contracts becomes smaller over time.
In this subsection we will give an informal and intuitive argument for this result(see Appendix for more detailed analysis).
∑∑
The fraction of the relationship pairs ts=0 ni=1 q i dst (γ i )lts that engage in relational
contract rather than go to the outside credit market changes over time because thedecision to implement a relational contract dst (γ) ∈ {0, 1} is an endogenous object.
Then we will see how such implementation decisions {dst (γ)}t≥s evolve over time bymodifying the incentive compatibility condition as follows: let Jts denote the sum of
expected proﬁts of an old R-producer and an old R-lender in period t whose predecessors
have started the relationship from period s. If their parents decided to go to thematching market instead of continuing the relationship in period t − 1, they would
obtain the sum of the expected proﬁts in period t as mJtt +(1−m)(λrt wt−1 +πt ) because
in the matching market each of their parents ﬁnds a diﬀerent partner with matchingprobability m in which case they would obtain the sum of expected proﬁts Jtt by startinga new relationship formed in period t while each of their parents fails to match a newpartner with probability 1 − m in which case they would obtain the sum of proﬁts
λrt wt−1 + πt by engaging in arm’s length contract in the credit market. Thus, the net
gain from relational contracting in period t ≥ s becomes Jts −mJtt −(1−m)(λrt wt−1 +πt )for the relationship that has lasted from period s.
Then, we can show that the previous IC constraint IC∗t is replaced by the new one,called A-ICst (γ), for a relationship that has lasted from period s and faces the exogenous
separation shock γ ∈ Γ in period t ≥ s as follows:t+1sγδ{Jt+1− mJt+1− (1 − m)(πt+1 + λrt+1 wt )} ≥ λrt zts (γ).

(A-ICst (γ))

As in the previous IC, the gain from relational contracting, captured by the left handside of the above inequality, must be at least as large as the opportunity cost of investingin capital λrt zts (γ) under relational contract, captured by its right hand side. The
main diﬀerence from the original IC∗t is here that the net gain from honouring therelational contract (the left-hand side of A-ICst (γ)) is multiplied by the probability of
the relationship being continued γ ∈ Γ and the probability of meeting a new partner inthe matching market m. Since the relationship is resolved with exogenous probability
1 − γ, the net gain from relational contracting is realized with probability γ. Further,t+1
since the R-producer and R-lender could obtain the sum of expected proﬁts mJt+1+(1−m)(πt+1 +λrt+1 wt ) even if they separated from each other, such outside gain should
be subtracted from the net surplus the relationship pair can extract from the current
match.
t+1s− mJt+1
Here, the net gain from relational contracting Jt+1
− (1 − m)(λrt+1 wt +πt+1 ) depends on how many separation shocks γt+1 ∈ Γ will result in the decision to
implement the agreed upon relational contract in period t + 1. When many separation
shocks cause the future decisions to go to the outside credit market, i.e., dst+1 (γ) =0 for many γ ∈ Γ, such gain from relational contracting becomes so small that it
becomes diﬃcult to enforce the agreed upon relational contract in the current period.
28

Also, as we have seen in the basic model, since the interest rates fall over time as theoutside credit market expands, the gain from relational contracting tends to be small
during the development process. By combining these eﬀects, A-ICst (γ) becomes more
stringent for lower γ as the economy proceeds to develop well over time. Eventually, therelationships facing more severe separation shocks (lower γ) decide not to implement
relational contract but to switch to arm’s length contract in the credit market in thelater stages of development.
More formally, we can show the following result.
Proposition 5. Suppose that q n > m. Then, there exist some ŷ ∈ (0, 1/λ) andλ ∈ (0, 1) such that, for all λ ∈ (0, λ), in any equilibrium path the relational contract
agreed by each relationship pair formed in period s ≥ 0 has the following dynamic
pattern:(i) Intensive Margin: there exists a sequence of the cut-oﬀ periods {T s (γ i )}ki=1 forsome k (1 ≤ k ≤ n) such that dst (γ i ) = 0 for all t ≥ T s (γ i ) where T s (γ 1 ) ≤ · · · ≤T s (γ k ).
(ii) Extensive Margin: the relative output of R-producer, deﬁned by yts (γ i ) ≡ zts (γ i )/xt
for i ≤ k, decreases over time as long as the relationship chooses a higher relative
output yts (γ i ) than the cut-oﬀ value ŷ, i.e., as long as yts (γ i ) > ŷ.
Proposition 5 shows that relational contracting contributes to economic development in the early stages but its role declines over time in both intensive and extensive
margins: the relationships that face a separation shock γ i switch to engage in arm’s
length contract after the cut-oﬀ period T s (γ i ) and such switching becomes more early
for the relationships that face more severe separation shocks (i.e., the cut-oﬀ period
T s (γ) decreases with γ). Also, in the development stages in which each R-producer
invests more than an A-producer, its relative output declines over time as shown in
Proposition 1.

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 does not function well to support capital
investment. In subsequent periods, producers ﬁnd it proﬁtable to use arm’s length contracts because the economy is so well-developed that the market size is large and theinterest rate falls. Thus, 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 valuable29

for relating our theoretical results to the historical evidence on relationship-based ﬁnancing. This is one of the modelling 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 as ﬁrm-worker relationships, inter-ﬁrm relationships, and government-public relationships.
We conclude the paper by brieﬂy discussing the role of bequest transfers between
successive generations. In the main text, we have assumed that each old individual has
no technology to give his or her child bequest. 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 toﬁnance a part of capital investment from the bequest he has received from his parent.
Thus, A-producers can reduce the enforcement cost and relationship pairs are more
likely to satisfy IC∗t by using bequest to ﬁnance capital investment. However, we canshow that no old individuals bequeath to their children at all, as in our basic setting,as long as the altruistic parameter value δ belongs to a certain small range (a moredetailed analysis is relegated to the Online Appendix).

7

Appendix: Proofs for Lemmas and Propositions

In this section we will give the formal proofs for the lemmas and propositions presented
in the main text.
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 must be satisﬁed for all s ≥ t. Otherwise, IC∗t̃ or TSt̃ is violated in someperiod t̃ ≥ t, which implies that at least one of ICt̃ , IRLt̃ and IRPt̃ is violated so thatthe relationship pair born in period t̃−1 is dissolved. By anticipating this and using the
backward induction argument, every relationship pair born in any period before t̃ − 2would not have the incentive to maintain the relationship. On the other hand, if bothIC∗s and TSs are satisﬁed for all s ≥ t, then we can show that there exists a subgame
perfect equilibrium in which every relationship pair born in period s ≥ t − 1 honours
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 hold for all s ≥ t. Then, we can ﬁnd a sequence
28
of relational contracts {zs , Rs }∞s=t such that all ICs , IRPs , and IRLs are satisﬁed.
∞
Take a sequence of relational contracts {zs , Rs }s=t which satisﬁes ICs , IRPs andIRLs for all s ≥ t. Then, we can show that the following strategies played by Rproducers and R-lenders can constitute a subgame perfect equilibrium in a continuation
For example, we can set Rs = λrs zs in every period s ≥ t ≥ 1 and R0 ∈ (0, π0 ), which satisfy
all ICs , IRPs , and IRLs with some zs in every period s ≥ t ≥ 0. Also, note that IC∗t implies thatpt zt > λrt zt for t ≥ 2 and TS0 implies that p1 z1 > λr1 z1 . Thus we can ﬁnd Rt to ensure non-negative
consumption NNCt that pt zt ≥ Rt ≥ −λrt (wt−1 − zt ) for all t ≥ 1 as well as π0 ≥ R0 ≥ 0.
2830

equilibrium from period t onward:• Period s ≥ t:
– The young R-producer and young R-lender of each relationship agree tothe relational contract {zs+1 , Rs+1 }29 if their parents agreed and honoured
the relational contract {zs , Rs } in the previous period. Otherwise, they
simultaneously exercise the quitting option.
• Period s + 1:
– The old R-producer honours to make the repayment Rs+1 and does not
exercise the quitting option, provided his and the R-lender’s parents did not
exercise the quitting option in the previous period.
– The old R-lender does not exercise the quitting option, provided her andthe R-producer’s parents did not exercise the quitting option in the previous
period.
The above strategies specify the trigger strategy-like feature as follows: if a young
R-producer and a young R-lender agree to the relational contract {zs+1 , Rs+1 } in periods and they actually behave according to that contract in period s + 1, their children
will decide to continue the relationship in period s + 1 and follow the agreement of
the relational contract {zs+2 , Rs+2 } in the next period s + 2. Suppose that this is not′′} ≠ {zs+1 , Rs+1 } or
, Rs+1the case. That is, they agree to a diﬀerent contract {zs+1they agree to the speciﬁed contract but some of them does not implement it (e.g., theR-producer reneges on the repayment Rs+1 ) in period s + 1. If this were the case, theirchildren would play a continuation equilibrium in which they simultaneously decide
to exercise the quitting option in period s + 1. Here, note that we always have acontinuation equilibrium in which any R-producer and any R-lender of any relationshippair simultaneously exercise the quitting option because the relationship pair is resolved
as long as at least one partner decides to exercise the quitting option.
Then, because ICs , IRLs , and IRPs are satisﬁed in any period s ≥ t, the R-producer
and R-lender never make any proﬁtable deviation from agreeing the speciﬁed relational
contract {zs+1 , Rs+1 } and implementing it, ensuring that the relationship is inheritedby the next generation.
(ii) First, the optimal relational contract should solve the following problem:max∞{zt }t=1∞∑

β t−1 Jt

t=1

subject to IC∗t and TSt−1 , t = 1, 2, .... Here, IC∗t , t ≥ 1, implies that TSt holds for allt ≥ 2. Also, TS1 is satisﬁed when TS0 and IC∗1 hold. Thus only the relevant TSt is TS0 .
29
For example, we can suppose that a young R-producer oﬀers {zs+1 , Rt+1 } to the young R-lendermatching him.

31

∑
t
This shows that the optimal relational contract should maximize ∞t=0 β Jt subject to∗
ICt for t ≥ 1 and TS0 .
Second, note that λ̂x(rt ) maximizes αAztα − λrt zt over zt ≥ 0. Suppose now thatzt > λ̂x(rt ) in some period t. Then, if we can slightly decrease zt , we can still keepIC∗t , IC∗t−1 and TS0 . Note that zt aﬀects only IC∗t , IC∗t−1 and possibly TS0 . The slightdecrease of zt makes IC∗t easier to be satisﬁed while it increases the left hand side ofIC∗t−1 and TS0 (when considering the choice of z1 ) because λ̂x(rt ) maximizes the left
hand side of IC∗t−1 and TS0 (when considering z1 ). But then such change increases thejoint proﬁt Jt = αAzt −λrt zt +λrt wt−1 due to the deﬁnition of λ̂x(rt ). Thus zt ≤ λ̂x(rt )holds.
(iii) Suppose that zt < λ̂x(rt ) but IC∗t is not binding in some period t. Then, if we
can slightly increase zt toward λ̂x(rt ), 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 becauseof zt < λ̂x(rt ). This does not violate TS0 as well. But then such change increases thejoint proﬁt Jt due to the deﬁnition of λ̂x(rt ). Q.E.D.
Proof of Lemma 2.
Suppose contrary to the claim that zs > ws−1 holds in some period s in some equilibrium path. In such equilibrium each R-producer must ﬁnance the remaining capital
investment zs − ws−1 > 0 from the credit market after he borrows ws−1 from theR-lender matching him.
There are two cases: (i) zs−1 ≤ ws−2 and (ii) zs−1 > ws−2 .
Case (i): zs−1 ≤ ws−2 . In case (i), the R-producer in period s − 2 does not ﬁnance fromthe credit market. Thus, in case (i), ICs−1 for R-producer and IRLs−1 for R-lendershould be modiﬁed tops−1 zs−1 − Rs−1 + δ{ps zs − Rs − rs (zs − ws−1 )} ≥ ps−1 zs−1 + δπs(IC′s−1 )Rs−1 + λrs−1 (ws−2 − zs−1 ) + δRs ≥ λrs−1 ws−2 + δλrs ws−1

(IRL′s−1 )andCombining these conditions, IC∗s−1 must be changed toδ{αAzsα − rs zs − πs + (1 − λ)rs ws−1 } ≥ λrs−1 zs−1 .

(IC∗s−1 -(i))

Also, in case (i) the joint proﬁt in period s is given by Js = αAzsα − rs zs + rs ws−1
because capital investment zs − ws−1 > 0 must be ﬁnanced from the credit market.
Case (ii): zs−1 > ws−2 . In case (ii), the R-producer in period s − 2 ﬁnances theremaining amount zs−1 − ws−2 from the credit market after borrowing ws−2 directlyfrom the R-lender. Thus, in case (ii), ICs−1 for R-producer and IRLs−1 for R-lendershould be modiﬁed tops−1 zs−1 − Rs−1 − rs−1 (zs−1 − ws−2 ) + δ{ps zs − Rs − rs (zs − ws−1 )}≥ ps−1 zs−1 − rs−1 (zs−1 − ws−2 ) + δπs32

andRs−1 + δRs ≥ λrs−1 ws−2 + δλrs ws−1

(IRL′s−1 )Combining these conditions, IC∗s−1 must be changed toδ{αAzsα − rs zs − πs + (1 − λ)rs ws−1 } ≥ λrs−1 ws−2 .

(IC∗s−1 -(ii))

In case (ii) the joint proﬁt in period s is given by Js = αAzsα − rs zs + rs ws−1 again.
In both cases (i) and (ii) the credit market equilibrium in period s − 1 is modiﬁedas(1 − l)λws−1 = l(zs − ws−1 ) + (1 − l)xs

(CME′ )

because l R-producers require the capital demand zs − ws−1 > 0 each in the credit
market in addition to the credit demand xs of A-producers each while (1 − l) A-lenders
supply the capital ws−1 each. Then CME′ is written by (l+(1−l)λ)ws−1 = lzs +(1−l)xs .
Since zs > ws−1 by our supposition, this yields (l + (1 − l)λ)zs > (l + (1 − l)λ)ws−1 =lzs + (1 − l)xs , which in turn implies that zs > λzs > xs .
However, the joint proﬁt of the relationship pair in period s is given by Js =αAzsα − rs zs + rs ws−1 which is maximized at xs = x(rs ). Then Js can increase by
reducing zs slightly because zs > xs = x(rs ). Since IC∗s−1 −(i), IC∗s−1 −(ii) and TS0
are more likely to be satisﬁed when zs decreases toward x(rs ), the original relational
contract cannot be optimal as long as the incentive compatibility constraint in period
s is not violated when zs is slightly decreased.
We verify this last point.
Case (1): zs+1 ≤ ws . In this case, by using a similar argument and noting thatzs > ws−1 by our supposition, we can show that the incentive compatibility constraint
in period s is modiﬁed asαδ{αAzs+1− λrs+1 zs+1 − πs+1 } ≥ λrs ws−1

Case (2): zs+1 > ws . In this case we modify the incentive compatibility constraint inperiod s + 1 asαδ{αAzs+1− rs+1 zs+1 − πs+1 + (1 − λ)rs+1 ws } ≥ λrs ws−1

Either case (1) or (2) above is not aﬀected by zs . Thus a slight decrease in zs does not
violate these constraints. Thus we prove that zs > ws−1 never happens in any period
in any equilibrium. Q.E.D.
Proof of Lemma 3.
By CMEt we haveλl(wt − zt+1 ) + λ(1 − l)wt = (1 − l)xt+1
where wt ≥ zt+1 due to Lemma 2. Thus, the above equality implies that λ(1 − l)wt ≤(1 − l)xt+1 and hence λwt ≤ xt+1 . Since zt+1 ≤ wt holds by Lemma 2 again, we haveλzt+1 ≤ λwt ≤ xt+1 which shows that xt+1 ≥ λzt+1 .
33

Since λ̂ > λ−1 , we then obtain λ̂xt+1 > λ−1 xt+1 ≥ zt+1 which implies that λ̂xt+1 >zt+1 . Then (IC∗t+1 ) is binding due to Lemma 1. Q.E.D.
Proof of Proposition 1.
Suppose that Assumption 1 and 2 are satisﬁed. By Assumption 1, we know that xt ≤ xholds in any period t in any equilibrium path.
By using (10) (i.e., CMEt ) in the main text, we obtain()α
xt+1 αα−1 λ(1 − α)A(lyt + (1 − l))= xt+1xt
λlyt+1 + (1 − l)λ(1−α)A(lytα + (1 − l))≥ xα−1λlyt+1 + (1 − l)1−llytα + (1 − l)=lλ−α + (1 − l) λlyt+1 + (1 − l)(1 − l)2≥lλ−α + (1 − l)where the ﬁrst inequality follows from the fact that xt ≤ x for all t and the second
inequality from 0 ≤ yt ≤ λ−1 (Lemma 2) respectively.
Then, by using the above fact, we can write (9) (i.e., IC∗t ) in the main text as)(
xt+1 ααδ{yt+1− λαyt+1 − (1 − α)}λαyt =
xt(1 − l)2αδ{yt+1− λαyt+1 − (1 − α)}.
≥lλ−α + (1 − l)Now, we will show that the right hand side of this inequality, denoted by F (yt+1 ), is
greater than its left hand side λαyt when yt+1 = yt = 1. To see this, deﬁne
f (y) ≡y α − λαy − (1 − α)yover [y, 1/λ] where y > 0 is deﬁned as y which satisfying y α − λαy − (1 − α) = 0.
In any equilibrium yt ≥ y must be satisﬁed. Then we can show that f (y) = 0 and
f (λ−1 ) = λ1−α − λ > 0. Also we can verify that f ′ (y) = (1 − α)y −2 (1 − y α ) so thatf ′ = 0 at y = 1 and f ′ > (<)0 for y < (>)1. Then we can show that f (y) > f (λ−1 ) for
all y ∈ [1, 1/λ).
()
2Since Assumption 2 (δ ≥ δ̃) implies that δ (λ−α(1−l)
f (λ−1 ) > λα, we havel+(1−l))
(
)
(
)
2
2
δ (λ−α(1−l)
f (y) > δ (λ−α(1−l)
f (λ−1 ) > λα for all y ∈ [1, 1/λ) so that F (y) >l+(1−l))l+(1−l))λαy for all y ∈ [1, 1/λ). Then, if the economy starts with y1 > 1, any equilibrium path{yt }∞
t=1 which must satisfy F (yt+1 ) ≤ λαyt decreases over time until it hits ŷ < 1 suchthat F (ŷ) = λαŷ. Here, such ŷ is unique and ŷ < 1 holds due to δ ≥ δ̃. Also, since yt
monotonically decreases over time until it hits ŷ < 1, there exists a switching period T
34such that yt > 1 for all t < T while yt < 1 for all t > T .
Uniqueness: ﬁnally, we show that the switching period T deﬁned above is uniqueonce we ﬁx the relative output of R-producer y1 in the ﬁrst period (t = 1). To see this,we show that equilibrium path {yt }∞t=1 of the relative output is uniquely determined
for a given y1 . Then, since any equilibrium path {yt }∞t=1 is decreasing over time whenyt < ŷ, the switching period T must be unique for a given y1 .
Fix a (x1 , y1 ) which satisﬁes CME0 : λw0 = x1 (λly1 + (1 − l)). By using (9), we
obtain
(
)
xt+1
1/α
α
δ 1/α [yt+1− λαyt+1 − (1 − α)]1/α = (λα)1/α yt .
(A1)
xt
Also we can re-write (10) as
xt+1
lytα + (1 − l) α−1= λ(1 − α)Axxt
λlyt+1 + (1 − l) t
which we substitute into (xt+1 /xt ) of (A1) in order to obtain
λ1−1/α (1 − α)Aδ 1/α1/α
α − λαy[yt+1α1/α xt1−α
t+1 − (1 − α)]= α−1/α.
−1/αλlyt+1 + (1 − l)ly
+ (1 − l)y
t

(A2)

t

We denote by g(yt+1 ; δ) the left hand side of (A2) and by ψ(yt , xt ) the right hand sideof (A2) respectively. Then we can show that
∂g
∂yt+1

∝ (1 − l)(ytα−1 − λ) + λ(1 − α)l(1 − λyt )> 0

due to yt ≤ λ̂ ≡ λ1/(α−1) .
By using CMEt−1 , i.e., λwt−1 = xt (λlyt +(1−l)), and substituting xt = λwt−1 /(λlyt +(1 − l)) into ψ(yt , xt ), we obtainψ̃(yt , wt−1 ) ≡α1/α (λwt−1 )1−α
α−1/α

(lyt

−1/α

+ (1 − l)yt)(λlyt + (1 − l))1−α

where we can verify that ψ̃ is increasing in yt and wt−1 .
Also, by using LMEt and CMEt , we also obtainwt = λ(1 − α)Axαt (lytα + (1 − l))
)α
(
λwt−1
(lytα + (1 − l))= λ(1 − α)Aλlyt + (1 − l)≡ w̃(yt , wt−1 )Here we can veriy that w̃ is increasing in yt :∂ w̃∝ (1 − l)lα(ytα−1 − λ) > 0∂yt35

due to yt < λ̂. w̃ is also increasing in wt−1 as well. From this, we can show the following:deﬁne w1 = ŵ(y1 ) ≡ w̃(y1 , w0 ) which is uniquely determined once we ﬁx y1 because w0
is exogenously given. Next we deﬁne w2 = ŵ(y2 , y1 ) ≡ w̃(y2 , w1 ) = w(y2 , ŵ(y1 )) whereŵ(y2 , y1 ) is increasing in both y1 and y2 because w̃(y2 , w1 ) is increasing in y2 and w1
as well as w1 = ŵ(y1 ) is increasing in y1 . Thus ŵ(y2 , y1 ) is uniquely determined once
we ﬁx y2 and y1 . Then we can deﬁne wt = ŵ(yt , yt−1 , ..., y1 ) in the similar way whereŵ is increasing in each argument.
By substituting wt−1 = ŵ(yt−1 , yt−2 , ..., y1 ) into ψ̃(yt , wt−1 ), we haveφ(yt , yt−1 , ..., y1 ) ≡ ψ̃(yt , ŵ(yt−1 , ..., y1 ))which is increasing in each argument because ψ̃ is increasing. Then the equilibrium
path {yt }∞t=1 recursively satisﬁes g(yt+1 ; δ) = φ(yt , ..., y1 ) for a given y1 . Since g(y2 ; δ)is increasing with y2 and g(y; δ) = 0, the equilibrium value of y2 which satisﬁesg(y2 ; δ) = φ(y1 ) must be unique for a given y1 . Also, the equilibrium value of y3
which satisﬁes g(y3 ; δ) = φ(y2 , y1 ) must be also unique. Repeating this, it must be thatyt+1 which satisﬁes g(yt+1 ; δ) = φ(yt , ..., y1 ) must be unique for a given y1 . Q.E.D.
Proof of Proposition 2.
In the proof of Proposition 1 we have shown that an equilibrium path {yt }∞t=1 is determined by solving g(yt+1 ; δ) = φ(yt , yt−1 , ..., y1 ) recursively. Here recall that φ isincreasing in each argument and g is increasing in yt+1 . Take any two equilibrium
′′ ∞
′
′′
paths {yt′ }∞t=1 and {yt }t=1 with y1 ≥ y1 . Then, since g(y2 ; δ) = φ(y1 ) holds, we havey2′ ≥ y2′′ when y1′ ≥ y1′′ . Also, since g(y3 ; δ) = φ(y2 , y1 ) and yt′ ≥ yt′′ for t = 1, 2, we have′′′y3′ ≥ y3′′ . Repeating this, we can see from g(yt+1 ; δ) = φ(yt , yt−1 , ..., y1 ) that yt+1
≥ yt+1
because ys′ ≥ ys′′ for all s ≤ t − 1.
Since yt′ ≥ yt′′ holds in any period t and these equilibrium paths decrease over time
until they hit the same critical value ŷ < 1, the corresponding switching periods T ′ andT ′′ must have the property that T ′ ≥ T ′′ . Q.E.D.
Proof of Proposition 3.
First, suppose that δ ′ > δ ′′ for a given λ. Then, let ζ ′ ≡ (δ ′ , 1 − λ) > ζ ′′ ≡ (δ ′′ , 1 − λ).
Note that g(y; δ) which was deﬁned in the proof of Proposition 1 is increasing in bothy and δ. Thus, since g(y2 (ζ); δ) = φ(y1 (ζ)) and φ is increasing, we have y2 (ζ ′ ) ≤ y2 (ζ ′′ )if y1 (ζ ′ ) ≤ y1 (ζ ′′ ). In the recursive way we can show that yt+1 (ζ ′ ) ≤ yt+1 (ζ ′′ ) holds
for any t ≥ 1 because g(yt+1 (ζ); δ) = φ(yt (ζ), ..., y1 (ζ)) holds, φ is increasing in eachargument and ys (ζ ′ ) ≤ ys (ζ ′′ ) for all s ≤ t.
Next, suppose that λ′ < λ′′ for a given δ. Then, let ζ ′ ≡ (δ, 1−λ′ ) > ζ ′′ ≡ (δ, 1−λ′′ ).
We verify that ψ̃ is increasing in λ while g is decreasing in λ. We can also show thatŵ(yt , ..., y1 ) is increasing in λ for all t ≥ 1. Thus φ is increasing in λ as well. Since
g is decreasing and φ is increasing in λ, yt (ζ ′ ) ≤ yt (ζ ′′ ) holds for all t ≥ 1 wheny1 (ζ ′ ) ≤ y1 (ζ ′′ ). Q.E.D.
Proof of Proposition 4.
36

We ﬁrst show that there exists a unique steady state. For this purpose, it suﬃces toshow that there exists a unique ỹ which satisﬁes (15) because then there exists a uniquex̃ satisfying (16). Let D(y) ≡ δ{y α − λαy − (1 − α)} − λαy over the domain y ∈ [y, 1/λ].
Then we can show that D(y) = −λαy < 0 and that D(1) = α(δ(1 − λ) − λ) > 0
due to Assumption 2. Also D(λ−1 ) = δ(λ−1 − 1) − α > 0 due to Assumption 2 andD′′ = α(α − 1)δy α−2 < 0. Thus D is strictly concave with D(y) < 0, D(1) > 0 andD(λ−1 ) > 0. Then there exists a unique ỹ ∈ (y, 1/λ) such that D(ỹ) = 0 where ỹ < 1holds.
Next, we show that the economy eventually converges to a unique steady state (ỹ, x̃)deﬁned in (15) and (16) in the main text under Assumption 1 and 2.
We already know from Proposition 1 that yt < 1 holds for all large t. Thus in what
follows we suppose that yt < 1 without loss of generality.
By substituting the following CMEtλ(1 − α)Axαt (lytα + (1 − l)) = xt+1 (λyt+1 + (1 − l))into the following IC∗t
(

xt+1
xt

)α
α
δ{yt+1− λαyt+1 − (1 − α)} = λαyt ,(CMEt )(IC∗t )

we obtainλ(1 − α)Axα−1t+1

α − λαyδ{yt+1λαyt
t+1 − (1 − α)}= α.
λlyt+1 + (1 − l)lyt + (1 − l)We deﬁne[λ(1 − α)A(ly α + (1 − l))h(y, y ) ≡λly ′′ + (1 − l)
′′

(A3)

] 1

1−α

.

From CMEt together with y < yt < 1, we have λ(1 − α)Axαt (ly α + (1 − l)) ≤xt+1 (λl + (1 − l)) which implies that xt is eventually larger than h(y, 1):[λ(1 − α)A(ly α + (1 − l))xt ≥ h(y, 1) ≡λl + (1 − l)

1
] 1−α

(A4)

for all large t. Then (A3) implies that for all large t:λ(1 − α)Ah(y, 1)α−1

α − λαyδ{yt+1λαyt
t+1 − (1 − α)}≥ αλlyt+1 + (1 − l)lyt + (1 − l)which is equivalent to()α − λαyλl + (1 − l) δ{yt+1λαyt
t+1 − (1 − α)}≥ α.
ly α + (1 − l)λlyt+1 + (1 − l)lyt + (1 − l)(A5)for all large t. Letting H(yt+1 ) denote the left hand side of (A5), we can verify that
H(1) > λα that is equivalent to δ > (ly α + (1 − l))λ/(1 − λ) implied by δ > δ̃. Also,37

H(y) = 0 < λαy. Thus there exists some y, denoted by y ∗ (1), such that y ∗ (1) ∈ (y, 1)
and H(y ∗ (1)) = λαy ∗ (1) (if such y ∗ (1) are multiple, we take the smallest one). Then
(A5) shows that yt ≥ y ∗ (1) eventually holds for all large enough t.
Next, by using CMEt and y < yt < 1, we haveλ(1 − α)Axαt ≥ xt+1 (λly + (1 − l))which implies that[λ(1 − α)A
xt+1 ≤ h(1, y) =λly + (1 − l)

] 1

1−α

(A6)

for all large t. By combining this with (A3), we obtainλ(1 − α)Ah(1, y)α−1

α − λαyδ{yt+1λαyt
t+1 − (1 − α)}≤ α.
λlyt+1 + (1 − l)lyt + (1 − l)which is equivalent to(λly + (1 − l))α − λαyδ{yt+1λαyt
t+1 − (1 − α)}≤ αλlyt+1 + (1 − l)lyt + (1 − l)

(A7)

for
all large t. When
( α(1−λ)(λly+(1−l))) we set yt+1 = yt = 1, the left hand side of (A7) is equal toδ
which is larger than λα due to δ > δ̃. Thus there exists some y,λl+(1−l)denoted by y ∗∗ (1), such that y ∗∗ (1) ∈ (y, 1) and the both sides of (A7) are equal at
yt+1 = yt = y ∗∗ (1) (if such y ∗∗ (1) are multiple, we take the largest one). Then we canverify from (A7) that yt ≤ y ∗∗ (1) holds for large enough t.
We have thus established that y ∗ (1) ≤ yt ≤ y ∗∗ (1) holds for all large enough t.
Next, since y ∗ (1) ≤ yt ≤ y ∗∗ (1) for all large t, we can verify that xt ≥ h(y ∗ (1), y ∗∗ (1))
and xt ≤ h(y ∗∗ (1), y ∗ (1)) for all large t. Then, by using the similar step to (A5), we
can show thatα − λαyλly ∗∗ (1) + (1 − l) δ{yt+1λαyt
t+1 − (1 − α)}≥ α∗αl(y (1)) + (1 − l)λlyt+1 + (1 − l)lyt + (1 − l)(A8)for all large t. Here the left hand side of (A8) is less than its right hand side at
yt+1 = yt = y ∗ (1) if and only if
λly ∗∗ (1) + (1 − l)λl + (1 − l)< αl(y ∗ (1))α + (1 − l)ly + (1 − l)which holds because of y ∗∗ (1) < 1 and y ∗ (1) > y. Also the left hand side of (A8) islarger than its right hand side at yt+1 = yt = 1 if and only if()()
λly ∗∗ (1) + (1 − l)δα(1 − λ)> λα(A9)l(y ∗ (1))α + (1 − l)λl + (1 − l)

38

)
)(
( −α
+(1−l)
λ∗∗∗−1
which holds due to δ > δ̃ ≥ lλ (1−l)
21−λ , y (1) > 0 and y (1) < 1 < λ . Thus
there exists some y ∗ (2) ∈ (y ∗ (1), 1) such that the left hand side of (A8) is equal to its
right hand side at yt+1 = yt = y ∗ (2) (if such y ∗ (2) are multiple, we take the smallest
one). Then we obtain from (A8) the result that yt ≥ y ∗ (2) for all large enough t.
Since xt ≤ h(y ∗∗ (1), y ∗ (1)) for all large t, by a similar argument to (A7), we canalso show thatα − λαyλly ∗ (1) + (1 − l) δ{yt+1λαyt
t+1 − (1 − α)}≤ αl(y ∗∗ (1))α + (1 − l)λlyt+1 + (1 − l)lyt + (1 − l)(A10)

for all large t. The left hand side of (A10) is larger than its right hand side at yt+1 =yt = y ∗∗ (1) if and only if
λly + (1 − l)λly ∗ (1) + (1 − l)>∗∗αl(y (1)) + (1 − l)l + (1 − l)which is satisﬁed because of y ∗ (1) > y and 1 > y ∗∗ (1). Thus there exists some y ∗∗ (2) ∈(y, y ∗∗ (1)) such that the left hand side of (A10) is equal to its right hand side at
yt+1 = yt = y ∗∗ (2). Thus (A10) implies that yt ≤ y ∗∗ (2) holds for all large t.
We have thus shown that y ∗ (2) ≤ yt ≤ y ∗∗ (2) holds for all large enough t.
Repeating this process, we can ﬁnd a sequence {y ∗ (m), y ∗∗ (m)}∞m=0 such that∗y (m) < y ∗∗ (m), where y ∗ (0) = y and y ∗∗ (0) = 1, andλly ∗∗ (m − 1) + (1 − l) δ{(y ∗ (m))α − λαy ∗ (m) − (1 − α)}λαy ∗ (m)=l(y ∗ (m − 1))α + (1 − l)λly ∗ (m) + (1 − l)l(y ∗ (m))α + (1 − l)
(A11)
and
λly ∗ (m − 1) + (1 − l) δ{(y ∗∗ (m))α − λαy ∗∗ (m) − (1 − α)}λαy ∗∗ (m)=l(y ∗∗ (m − 1))α + (1 − l)λly ∗∗ (m) + (1 − l)l(y ∗∗ (m))α + (1 − l)(A12)for each m = 1, 2, ... Here, y ∗ (m) ≤ y ∗ (m + 1) while y ∗∗ (m + 1) ≤ y ∗∗ (m) for eachm ≥ 1. Then, for each m ≥ 1, there exists some T (m) such that y ∗ (m) ≤ yt ≤ y ∗∗ (m)holds for all t ≥ T (m).
We now deﬁne the limit point of (y ∗ (m), y ∗∗ (m)), denoted by (y ∗ , y ∗∗ ), satisfying(A11) and (A12) at y ∗ (m) = y ∗ (m − 1) = y ∗ and y ∗∗ (m − 1) = y ∗∗ (m) = y ∗∗ :andλly ∗∗ + (1 − l)
δ{(y ∗ )α − λαy ∗ − (1 − α)} = λαy ∗λly ∗ + (1 − l)(A13)

λly ∗ + (1 − l)
δ{(y ∗∗ )α − λαy ∗∗ − (1 − α)} = λαy ∗∗ .
λly ∗∗ + (1 − l)(A14)Here, note that, since y ∗ (m) is increasing and bounded above, the sequence {y ∗ (m)}converges to y ∗ . Also, note that, since y ∗∗ (m) is decreasing and bounded below, thesequence {y ∗∗ (m)} converges to y ∗∗ .
39

Let S ≡ {(y ∗ , y ∗∗ ) ∈ [y, 1/λ]2 | (A13) and (A14) are satisﬁed.}. Then S ̸= ∅ because the steady state of the economy (ỹ, ỹ) satisﬁes (A13) and (A14) by its deﬁnition.
Now we show that limt→∞ yt ∈ [y ∗ , y ∗∗ ] for any (y ∗ , y ∗∗ ) ∈ S. Take any (y ∗ , y ∗∗ ) ∈ S.
Take also any ε > 0. Then, since (y ∗ (m), y ∗∗ (m)) is convergent, there exists some large
m′′ such that y ∗ − y ∗ (m′′ ) ≤ ε and y ∗∗ (m′′ ) − y ∗∗ ≤ ε. For such m′′ , we can ﬁnd alarge T (m′′ ) such that for all t ≥ T (m′′ ), y ∗ (m′′ ) ≤ yt ≤ y ∗∗ (m′′ ), which implies thaty ∗ − ε ≤ yt ≤ y ∗∗ + ε. Since ε > 0 is arbitrary, we have limt→∞ yt ∈ [y ∗ , y ∗∗ ]. Since(y ∗ , y ∗∗ ) ∈ S is arbitrary, by taking (ỹ, ỹ) ∈ S we establish that limt→∞ yt = ỹ. Q.E.D.
Proof of Corollary.
First, we consider the eﬀect of δ on ỹ and x̃.
We can see from (15) that∂ ỹỹ α − λαỹ − (1 − α)=−<0∂δδα(ỹ α−1 − λ) − λαbecause ỹ is the smallest root to (15) and thus the denominator is positive.
Next we deﬁnely α + (1 − l)ρ(y) ≡
λly + (1 − l)where we verify that ρ′ has the same sign as λly α (α − 1) + (1 − l)(αy α−1 − λ) and thatρ′′ < 0 at any y satisfying ρ′ (y) = 0. Thus ρ is strictly quasi-concave function. Alsoρ′ (0) = +∞ and ρ′ (1) ≥ 0 if and only if λ ≤ α(1 − l)/(l(1 − α) + (1 − l)) as we haveassumed. Thus ρ′ > 0 for all y ∈ [0, 1] under our assumption.
Then, we can show from (16) that ∂ x̃/∂δ < 0 because the increase in δ reduces ỹ.
Second, we consider the eﬀect of λ. We can readily see from (15) that∂ ỹαỹ + δαỹ
=
>0
α−1
∂λ
δα(ỹ
− λ) − λαAlso, from (16) we have x̃ = [λ(1 − α)Aρ(ỹ)]1/(1−α) . Then we can compute thatdx̃∂ x̃ ∂ x̃ ∂ ỹ
=
+
>0
dλ
∂λ ∂ ỹ ∂λwhere ∂ x̃/∂λ has the same sign as ∂/∂λ(λ/(λlỹ + (1 − l))) > 0, ∂ x̃/∂y > 0 if λ ≤α(1 − l)/(l(1 − α) + (1 − l)), and ∂ ỹ/∂λ > 0. Q.E.D.
Proof of Proposition 5.
Fix any equilibrium path. In the equilibrium each relationship pair matched in periods agrees to a sequence of relational contracts {zts (γ), Rts (γ), dst (γ)}t≥s from period s onward and its t-th generation implements the corresponding agreement {zts (γ), Rts (γ), dst (γ)}
for the realization of exogenous
separation shock γ ∈ Γ in period t.
∑ts the total measure of the relationship pairs of young
We deﬁne by lt ≡l
s=0 t
R-producers and young R-lenders in period t whose parents were not separated in the40

previous period. Thus there are 1 − lt remaining young R-producers (or young Rlenders) who engage in arm’s length contracts in the credit market in period t. Then
we can derive the law of motion governing the process of lt . Since lt is the sum of
all the relationships with∑ (0 ≤s s ≤ t) which are not dissolved in∑ diﬀerent birth dates
for t ≥ s + 1 is the measure ofperiod t, we have lt = ts=0 lts . Here lts = i q i γ i lt−1
the relationships which have been carried over from period s and ltt is the measure of
newly matched relationships in period t. In the beginning of period t there are 1 − lt−1
old A-producers and 1 −∑lt−1 old A-lenders who engaged in arm’s length contracts inthe credit market and i q i (1 − γ i )lt−1 relationships are exogenously separated. All
these old producers and lenders enter the matching market in period t for ﬁnding
the new partners for their children. Since each old producer (respectively, lender) ismatched with an old lender (respectively,
with probability m ∈ (0, 1) in the∑ i producer)timatching market, lt = m(1 − lt−1 + i q (1 − γ )lt−1 ) relationships of old R-producers
and old R-lenders are newly matched in period t. Their children
will start the new∑t−1
s , we obtain
relationships from period t. Then, by using the fact that lt−1 = s=0
lt−1∑t−1 s∑t−1 ss . Thus, we can deriveµlt−1 = s=0 µlt−1 = s=0 lt because lts = µlt−1
lt =

t−1
∑

lts + ltt

s=0

= µlt−1 + ltt= µlt−1 + m(1 − µlt−1 )= m + (1 − m)µlt−1mwhich converges to a unique steady state l∗ ≡ 1−(1−m)µ. Then we deﬁne l ≡ max{l0 , l∗ },which implies that lt ≤ l for all t.
Next, we derive the necessary conditions for equilibrium relational contract to besustainable. Take a relationship pair of a young R-producer and a young R-lendermatched in period s who agree on a sequence of relational contracts {zts (γ), Rts (γ), dst (γ)}s≥t
from period s onward. Here, recall that dst (γ) ∈ {0, 1} and that dst (γ) = 1 holds when
they implement the period t-relational contract (zts (γ), Rts (γ)) and dst (γ) = 0 holds
when they decide to go to the outside credit market respectively.30 Let Γst ≡ {γ ∈
Γ | dst (γ) = 1} be the set of the exogenous separation shocks which induce the implementation of the relational contract in period t instead of going to the outside credit
market.
We deﬁne by
usp,t (γ) ≡ pt zts (γ) − Rts (γ) − rt max[zts (γ) − wt−1 , 0],
the proﬁt of an old R-producer in period t when dst (γ) = 1 where the term rt max[zts (γ)−
wt−1 , 0] reﬂects the case that the R-producer invests zts (γ) more than wt−1 owned by30
In any equilibrium any R-producer and R-lender never voluntarily dissolve their relationship as we
will see below.

41the R-lender, in which case he must borrow the extra amount zts (γ) − wt−1 from theoutside credit market. Also, we deﬁne byusl,t (γ) ≡ Rts (γ) + λrt max[wt−1 − zts (γ), 0]
the proﬁt of an old R-lender in period t when dst (γ) = 1 where the term λrt max[wt−1 −zts (γ), 0] reﬂects the case that the young R-lender lends the remaining amount wt−1 −zts (γ) to the credit market after making the relationship lending zts (γ) to the young
R-producer. Then we also deﬁne by
s
Vp,t
≡

n
∑

q i {dst (γ i )usp,t (γ i ) + (1 − dst (γ i ))πt }

i=1

and
s
Vl,t
≡

n
∑

q i {dst (γ i )usl,t (γ i ) + (1 − dst (γ i ))λrt wt−1 }i=1

the expected proﬁts of an old R-producer and an old R-lender respectively.
Suppose that γ ∈ Γst is realized where dst (γ) = 1. Then, by following the agreed
upon relational contract, the old R-producer obtains the following payoﬀ:
t+1
s
s
Up,t
(γ) ≡ usp,t (γ) + γδVp,t+1+ (1 − γ)δ{mVp,t+1+ (1 − m)πt+1 }when facing the separation shock γ ∈ Γ in period t. With probability γ, the relationshipscontinues to the next period in which case his child will obtain the expected proﬁt Vp,t+1.
With probability 1 − γ, the relationship is exogenously separated in which case the old
R-producer go to the matching market for seeking a diﬀerent old lender. Then, withprobability m, match is successful so that his child can start a new relationship andt+1
obtain the expected proﬁt Vp,t+1
while with probability 1 − m match is not successful
so that his child has nothing but to go to the outside credit market and earn the arm’s
length contract proﬁt πt+1 .
On the other hand, if the old R-producer reneged on the repayment Rts (γ) andquitted the relationship (and then went to the matching market) in period t, he would
obtain the following deviation payoﬀ:
t+1
t
Ũp,t
(γ) ≡ pt zts (γ) − rt max[zts (γ) − wt−1 , 0] + δ{mVp,t+1+ (1 − m)πt+1 }.

Thus, the following incentive compatibility constraint must be satisﬁed:
ssUp,t(γ) ≥ Ũp,t(γ)(ICst (γ))

Second, for any γ ∈ Γ, the young R-producer must be not better oﬀ by quitting therelationship, which yields the proﬁt πt , and going to the matching market for seeking
t+1
a new lender when old, which yields the expected proﬁt mVp,t+1+ (1 − m)πt+1 to hischild. This individual rationality constraint is given by
t+1
dst (γ)usp,t (γ) + (1 − dst (γ))πt ≥ πt + δ{mVp,t+1+ (1 − m)πt+1 }42(IRPst (γ))

where the left hand side denotes the proﬁt of the young R-producer who obtains usp,t (γ)
for γ ∈ Γst but πt for γ ∈/ Γst .
Third, a similar individual rationality constraint for the young R-lender must be
satisﬁed: for all γ ∈ Γ,
t+1
dst (γ)usl,t (γ) + (1 − dst (γ))λrt wt−1 ≥ λrt wt−1 + δ{mVl,t+1+ (1 − m)λrt+1 wt } (IRLst (γ))

By combining ICst (γ) with IRLst (γ) for γ ∈ Γst (thus dst (γ) = 1), we obtain themodiﬁed incentive compatibility condition:t+1sγδ{Jt+1− mJt+1− (1 − m)(λrt+1 wt + πt+1 )} ≥ λrt min[zts (γ) − wt−1 , 0] + λrt wt−1
(A-ICst (γ))
sss+ Vl,t+1.
≡ Vp,t+1
where Jt+1
Also, by combining IRLst (γ) with IRPst (γ) for any γ ∈ Γ (thus any dst (γ) ∈ {0, 1}),the total net surplus of relationship must be non-negative:

T Sts (γ) ≡ dst (γ)(usp,t (γ) + usl,t (γ)) + (1 − dst (γ))(λrt wt−1 + πt ) − (λrt wt−1 + πt )t+1s
+ γδ{Jt+1− mJt+1− (1 − m)(λrt+1 wt + πt+1 )}≥0(TSst (γ))

Note that the young R-producer (respectively, the young R-lender) obtains his(her) expected payoﬀ of period t, which takes into account his (her) concern about
t+1
s + µδV s
the child’s consumption in period t + 1, as Vp,tp,t+1 + (1 − µ){mVp,t+1 + (1 −t+1s + µδV ssm)πt+1 } (respectively, Vl,tl,t+1 + (1 − µ){mVl,t+1 + (1 − m)λrt+1 wt }). Let Ṽtbe the sum of these expected payoﬀs. Then, as in the basic model, we suppose thateach relationship pair formed in period s chooses a sequence of relational contracts{zts (γ), Rts (γ), dst (γ)}t≥s from period s onward in order to maximize the weighted sumof the expected payoﬀs in the future generations of the same dynasty, Ws = Ṽss +µβWs+1 + (1 − µ)β × Constant where the last term Constant denotes the future payoﬀ
attained when the relationship is separated, which is not aﬀected by the choice ofrelational contracts {zts (γ), Rts (γ), dst (γ)}t≥s by the relationship formed in period s andhence we can ignore it. Here, some remarks on the set of constraints are in order. First,
as we have noted, any young R-producer and any young R-lender of any relationshipnever fully dissolve their relationship in which case they must engage in arm’s length
contract and go to the matching market for seeking another partners when old. Thisis because they can always attain the same payoﬀs under full dissolution by using
a relational contract without dissolution.31 Second, TSst (γ) can be always satisﬁed
31
Suppose that they dissolve relationship for some γ ∈ Γ in period t (thus going to the credit market
when young in period t and going to the matching market when old in period t + 1). Then, thet+1
R-producer (respectively, R-lender) obtains πt + δ{mVp,t+1+ (1 − m)πt+1 } (respectively, λrt wt−1 +t+1
δ{mVl,t+1 +(1−m)λrt+1 wt }). However, they can always attain at least these payoﬀs without dissolving
the relationship by using a relational contract which speciﬁes dst (γ) = 0 and mimics the equilibriumt+1(γ)}τ ≥t+1 if Jt+1≥
contract used by the relationship matched in period t + 1, {zτt+1 (γ), Rτt+1 (γ), dt+1
τ
λrt+1 wt + πt+1 and dst+1 (γ) = 0 (i.e., going to the outside credit market) otherwise.

43

because dst (γ) = 1 should be chosen only when usp,t (γ) + usl,t (γ) ≥ λrt wt−1 + πt . Thus
we consider only A-ICst (γ) for γ ∈ Γst as the relevant constraint.
Finally, the credit market and labour market equilibrium conditions are modiﬁed
as follows:
(
)
t
∑
∑
∑
i s i sis i
ssi
q dt (γ )lt max[wt − zt+1 (γ ), 0] +λ(1 − lt )wt + λq (1 − dt (γ ))lt wt
s=0= (1 − lt )xt+1 +

i

(
t
∑
∑
s=0

i
s
q i dst (γ i )lts max[zt+1(γ i ) − wt , 0] +∑i)q i (1 − dst (γ i ))lts xt+1i

(A-CMEt )and(

wt = (1 − α)A (1 − lt−1 )xαt +

(
t−1 ∑
∑
s=0

s
q i dst−1 (γ i )lt−1
zts (γ i )α +

i

∑

))
s
q i (1 − dst−1 (γ i ))lt−1xαti(A-LMEt )Here the left hand side of A-CMEt is the total credit supply while its right hand sideis the total credit demand in the credit market. There are (1 − lt ) young A-lenders(respectively, A-producers) who supply wt each (respectively, demand xt+1 each) while
R-lenders (respectively, R-producers) who implement the relational contracts supplys (γ), 0] each (respectively, demand max[z s (γ) − w , 0] each).
the credit of max[wt − zt+1t
t+1
All other R-lenders (respectively, R-producers) behave as A-lenders (respectively, Aproducers) do so, i.e., R-lenders (respectively, R-producers) lend (respectively, borrow)
all their wage income wt (respectively, the investment demand xt+1 ).
Then, by using A-ICst (γ), A-CMEt and A-LMEt together, we can show the following
claims (the formal proofs are given in Online Appendix).
s (γ) for all s, all t ≥ s and all
Lemma A1. Suppose that λ ≤ 1 − l. Then wt ≥ zt+1sγ ∈ Γt .

Lemma A2. Suppose that λ ≤ 1 − l. Then, zts (γ) ≤ (1/λ)xt for all γ ∈ Γst so that
A-ICst (γ) always binds for γ ∈ Γst .
Lemma A3. (i) If dst (γ) = 1, then dst (γ ′ ) = 1 for all γ ′ > γ. (ii) Suppose thatλαδ > (1−α)(1−m). Then, dst (γ n ) = 1 must be satisﬁed for all t ≥ T for some period T ≥ s
Lemma A4. In any equilibrium path()
xt+1
≥ λ(1 − α)A(1 − l)xtα−1
xt
and xt ≤ max{x0 , X} for all t ≥ 0 where() 1(1 − α)A 1−α
X≡λ
.
1−l
44

Lemma A1 shows that any R-producer never invests more than the fund owned by
the R-lender matching him as we have seen in Lemma 2. Thus A-ICst (γ) is simpliﬁed
to the condition derived in the main text:
t+1sγδ{Jt+1− mJt+1− (1 − m)(λrt+1 wt + πt+1 )} ≥ λrt zts (γ)(A-ICst (γ))
∑
i
s
s
s
s (γ i )−λr
= i q i max[pt+1 zt+1
where Jt+1
t+1 zt+1 (γ ), πt+1 ] by optimal choice of dt+1 (γ) ∈{0, 1}. Lemma A2 states that such A-ICst (γ) becomes always binding for γ ∈ Γst which
implies that yts (γ i )/γ i = yts (γ j )/γ j for all γ i , γ j ∈ Γst . Then, it follows from Lemma
A2 that zts (γ)/xt ≤ 1/λ for all γ ∈ Γst . In addition, by using rt = α2 Axtα−1 , we have
pt zts (γ) − λrt zts (γ) ≤ αAxαt {λ−α − α}. This shows that Jtt ≤ αAxt (λ−α − α) due to
λ−α − α > 1 − α. Lemma A3 (i) says that the relationships facing lower separation
probability (higher γ) implement the relational contracts more likely than those facing
higher separation probability. Lemma A3 (ii) also shows that every relationship eventually implements the relational contract when facing the least severe separation shock
γ n = 1. Finally, Lemma A4 states that the output growth rate of A-producers xt+1 /xt
is bounded below and that the output level of each A-producer xt is bounded above.
Now we assume that λ ≤ λ′′ where δ = λ′′ α/(1 − m)(1 − α).
By Lemma A3, dst (γ n ) = 1 for all t ≥ for some period T ≥ s. This implies that
A-ICst (γ n ) is satisﬁed for all t ≥ T and that Γst ̸= ∅ for all t ≥ T . Then, due to Lemma
A2, A-ICst (γ) must hold as equality for all γ ∈ Γst for all t ≥ T .
t+1
Recall that Jt+1
≤ αAxαt+1 (λ−α − α), πt = αA(1 − α)xαt and rt = α2 Axα−1. Then,tssby using the fact that A-ICt (γ) binds for all γ ∈ Γt and all t ≥ T for some T ≥ s, we
obtain()
xt+1 α sδ{Jt+1 − m(λ−α − α) − (1 − m)(1 − α)} ≤ λαzts (γ)/γ
xt∑iss (γ i )α −λrs= i q i max[αAzt+1
for all γ ∈ Γst and all t ≥ T . Note that Jt+1
t+1 zt+1 (γ ), πt+1 ]
j
j
s
s
i
i
s
s
and zt+1 (γ )/γ = zt+1 (γ )/γ for i ̸= j. Deﬁne ỹt ≡ zt (γ)/γ for all γ ∈ Γt whereỹt ≤ 1/λ by Lemma A2. Here we omit superscript s from ỹt to denote the birth date ofa relationship becasue our result can be applied to any relationship which starts fromany period.
Then, the above inequality can be written by

λαỹt ≥ δ[λ(1 − α)A(1 − l)(max{x0 , X})α−1 ]α}{∑αq i max[(γ i )α ỹt+1− λαỹt+1 , 1 − α] − m(λ−α − α) − (1 − m)(1 − α) .
×
i

(A15)
Let K(ỹt+1 ) be the right hand side of A15 which is a function of ỹt+1 ∈ [0, 1/λ]. Then we
can readily verify that K(1/λ) ≥ δ[λ(1−α)A(max{x0 , X})α−1 ]α {(q n −m)(λ−α −1)} > 0
due to q n > m and hence that there exists some λ′ ∈ (0, 1) such that K(1/λ) > αλ45holds for all λ ∈ (0, λ′ ). Then deﬁne λ ≡ min{λ′′ , λ′ , 1 − l} and let λ ∈ (0, λ). Thus,
Lemma A1-A3 can be applied for all λ ∈ (0, λ) as well as K(1/λ) > αλ. Then, since
K(0) < 0 and K(1/λ) > αλ for λ ∈ (0, λ), there exists some ŷ ∈ (0, 1/λ) such thatK(ŷ) = λαŷ. If ỹt > ŷ, ỹt decreases over time according to A15 and eventually ỹt ≤ ŷ
holds.
Also, we can deﬁne y i ∈ [0, 1/λ], i = 1, 2, ..., n, such that fi (y) ≡ (γ i )α y α − γ i λαy =1 − α as long as fi (1/λ) > 1 − α. Since fi (y) > fj (y) for all i > j, y i < y j for alli > j. Let I ≡ {i = 1, 2, ..., n | fi (1/λ) ≥ 1 − α}. Then we can verify that for all i ∈ I,
fi (y) ≥ 1 − α if and only if y ∈ [y i , 1/λ]. On the other hand, fi (y) < 1 − α for ally ∈ [0, 1/λ] for all i ∈/ I. Then deﬁne k such that y k−1 ≥ ŷ ≥ y k (deﬁne k = 1 if ŷ < y n
and k = n if ŷ > y 1 ).
Then, we can ﬁnd some period T s (γ i ) ≥ T for i ≥ k − 1 such that ỹt decreases overtime when ỹt > ŷ according to the dynamic constraint A15 and eventually ỹt ≤ y i for
all t ≥ T s (γ i ) so that dst (γ i ) = 0 for all t ≥ T s (γ i ). Since y i > y j for all i < j, we haveT s (γ j ) ≤ T s (γ j ) for i > j where k − 1 ≥ i > j. If ỹt < ŷ for all t ≥ s, then deﬁne
T s (γ i ) = s for all i ≤ k − 1.
Finally, we show that yt (γ i ) decreases over time whenever yt (γ i ) > ŷ. By the above
result, we know that K(ỹt ) ≤ λαỹt for all ỹt > ŷ which implies that ỹt decreases overtime as long as ỹt > ŷ. This gives the desired result. Q.E.D.

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48

YoungOld

Relational Contract{zt , Rt }Relational Contract{zt+1 , Rt+1 }Period t − 1

Period t

Quit or Not

invest zt

Quit or Not

Repay
Rt

Repay
Rt−1
or Quitor QuitFigure 1: Time Line49

invest zt+1
Repay
Rt+1
or Quit

yt

y1

1

ŷ

t

T

1

Figure 2: Equilibrium path of the relative output of R-producer yt
yt

y1′
y1′′
1

ŷ

1

T ′′

t

T′

Figure 3: Diﬀerent equilibrium paths with diﬀerent ﬁrst period values y150

Young

Implement relational contractor
Go to the credit market

Old

Continue relationshipγt
1 − γtSeparationand go to the matching market
New relationships formed

Figure 4: Matching and Separation
Period t − 1

1 − γtγtγtis realized
and observed
Agree to{zt (γt ), Rt (γt )}Period t

Separation
Continue

γt+1
is realized
and observed Agree to{zt+1 (γt+1 ), Rt+1 (γt+1 )}or go to the credit market

or go to the credit market

Figure 5: Time Line for Relational Contract51

Online Appendix (Not for Publication)

Appendix B: Equilibrium capital investment by A-producers
In the main text we have so far assumed that each young A-producer born in periodt − 1 chooses a capital investment xt so as to maximize only his own proﬁt πt . Because
the utility function of each A-producer depends on his child’s consumption level, one
might think that there may be intergenerational strategic interactions between an Aproducer’s choice of capital investment xt and his child’s capital investment choice xt+1 .
However, the following claim shows that it is suﬃcient to focus only on the equilibrium
in which each A-producer acts to maximize only his own proﬁt πt no matter what
histories are observed.
Suppose that each young A-producer in each dynasty, born in period t, can observe
all the capital investments {x1 , x2 , ..., xt−1 } chosen in the same dynasty and all themarket prices {w0 , ..., wt−1 , r1 , ..., rt−1 , p0 , ..., pt−1 } in the past periods. Then we let ht
be such a history observed up to period t. Let Ht be also the set of all these histories
observed to a young A-producer up to period t.
Proposition B1. In any equilibrium, every A-producer born in period t − 1 chooses
capital investment level xt to maximize only his own proﬁt πt no matter the history
observed up to period t.
Proof. Take an equilibrium path in which each A-producer uses a strategy σt : Ht →[0, ∞) which maps from, Ht , the set of all previous observed histories up to period t to
the current capital investment level xt ≥ 0.
Then, by recalling that x(rt ) maximizes the per period proﬁt πt of an A-producer,
we will show that every A-producer acts to maximize his own proﬁt πt and hence chooses
xt = x(rt ) in every period irrespective of observed histories ht ∈ Ht . We denote by σ tsuch strategy deﬁned as σ(ht ) = x(rt ) for all ht ∈ Ht .
To show this, suppose that there exists an equilibrium with σt ̸= σ t for some Aproducer in some period t. Thus σ(ht ) ̸= x(rt ) for some ht ∈ Ht . We then denote by{xs }∞
s=t the equilibrium sequence of capital investments of A-producers from period t
onward according to the equilibrium strategies {σs }∞s=t .
In the proof of the claim, we will denote by π(xt ) the proﬁt of an A-producer whochooses xt in period t.
Since the A-producer could always choose x(rt ) in period t, it must be the case thatπ(xt ) + δπ(xt+1 ) ≥ π(x(rt )) + δπ(x′t+1 )where his child will choose x′t+1 in period t + 1 following the choice of his parent x(rt )according to his equilibrium strategy σt+1 . The above inequality yields
δ{π(xt+1 ) − π(x′t+1 )} ≥ π(x(rt )) − π(xt ).
1(B1)

Also, for x′t+1 to be the optimal choice by the young A-producer in period t+1 following
x(rt ) chosen by his parent, we must haveπ(x′t+1 ) + δπ(x′′t+2 ) ≥ π(x(rt+1 )) + δπ(x′t+2 )because he could always choose x(rt+1 ) following his parent choice x(rt ) where x′′t+2
denotes the choice of his child in period t + 2 following x′t+1 . Here x′t+2 is the choice inperiod t + 2 following x(rt+1 ). This then yields
δ{π(x′′t+2 ) − π(x′t+2 )} ≥ π(x(rt+1 )) − π(x′t+1 ).
Since π(x(rt+1 )) ≥ π(xt+1 ) for any xt+1 ̸= x(rt+1 ), we have
δ{π(x′′t+2 ) − π(x′t+2 )} ≥ π(xt+1 ) − π(x′t+1 ).

(B2)

Combining inequalities (B1) with (B2), we have
δ 2 {π(x′′t+2 ) − π(x′t+2 )} ≥ π(x(rt )) − π(xt ).

(B3)

In period t + 2 the young A-producer in the dynasty in question must choose x′t+2
following the choice x(rt+1 ) by his parent in period t + 1. Thus, since he could 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 A-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 )Combining this with (B3), we have
δ 3 {π(x′′t+3 ) − π(x′t+3 )} ≥ π(x(rt )) − π(xt )Repeating this argument for all periods s + t ≥ t, we obtainδ s {π(x′′t+s ) − π(x′t+s )} ≥ π(x(rt )) − π(xt ), ∀ s ≥ 0.
Since π(x(rt )) ̸= π(xt ) by our supposition (σ(ht ) = xt ̸= x(rt )), there exists some
ε > 0 such that π(x(rt )) − π(xt ) ≥ ε. The left hand side of the above inequality
is bounded above by δ s π(x(rt+s )) because π(xt+s ) ≥ 0 and π(x(rt+s )) = maxx π(x)given rt+s . Note here that π(xt+s ) ≥ 0 because, by the spot transaction nature ofarm’s length contract in the credit market, each A-producer must make the repayment
rt+s xt+s from what he earns pt+s xt+s when he is old.
Then, if π(x(rt+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(rt )) < +∞ for any equilibrium interest rate rt ≥ 0. This is2equivalent to the condition that rt ≥ r for all t for some r > 0 because then π(x(r)) ≤maxx α(Ax)α − rx < +∞. Suppose that rt = 0 for some period t or rt → 0. In eithercase x(rt ) must go to inﬁnity. Since π(xt ) + δπ(xt+1 ) ≥ π(x(rt )) + δπ(x′t+1 ) must holdin period t (see (B1) above), either π(xt ) → ∞ or π(xt+1 ) → ∞ or both must hold dueto π(x(rt )) → ∞ and π(x′t+1 ) ≥ 0, which implies xt → ∞ or xt+1 → ∞ with rt+1 = 0.
In the former case (xt → ∞) CMEt−1 must imply λwt−1 = λlzt + (1 − l)xt → ∞, whichholds only when wt−1 → ∞ so that xt−1 → ∞ or zt−1 → ∞. We can deal with thelatter case xt+1 → ∞ in a similar way. However, then wt−2 → ∞ by CMEt−2 in periodt − 1. Repeating this, w0 → ∞ which is however impossible. Thus rt ≥ r must hold inany period t for some r > 0. This then establishes the claim. Q.E.D.

Appendix C: Proofs for Lemma A1-A4
In this subsection we will give the proofs for Lemma A1-A4 which are used to prove
Proposition 5 in the main text.
s (γ) for all s, all t ≥ s and all
Lemma A1. Suppose that λ ≤ 1 − l. Then wt ≥ zt+1γ ∈ Γst .

Proof. Suppose contrary to the claim that zts (γ) > wt−1 for some period t and some γ ∈Γst . Then, the relationship’s joint proﬁt of period t becomes pt zts (γ) − rt (zts (γ) − wt−1 ).
Also the right hand side of A-ICst (γ) must be λrt wt−1 which is independent of zts (γ).
Thus the choice of zts (γ) does not aﬀect A-ICst (γ) but changing zts (γ) slightly toward xt
can increase the joint proﬁt of period t, pt zts (γ)−rt (zts (γ)−wt−1 ), as long as zts (γ) ̸= xt .
Thus zts (γ) = xt must hold. On the other hand, A-CMEt−1 can be written byR
λ(1 − lt−1 )wt−1 + St−1
= (1 − lt−1 )xt + DtR

where
StR ≡ λ

(
t−1 ∑
∑
s=0

and
DtR ≡

}
s
q i lt−1[dst−1 (γ i ) max[wt−1 − zts (γ i ), 0] + (1 − dst−1 (γ i ))wt−1 ]

i

{
t−1 ∑
∑
s=0

}
q i [dst−1 (γ i ) max[zts (γ i ) − wt−1 , 0] + (1 − dst−1 (γ i ))xt ] .

iRR ≤ λlSince St−1
t−1 wt−1 and Dt ≥ 0, A-CMEt−1 implies that λwt−1 ≥ (1 − lt−1 )xt .
Since lt ≤ l for all t, λwt−1 ≥ (1 − l)xt . But, by assumption that λ ≤ 1 − l, we haveλwt−1 ≥ (1 − l)xt ≥ λxt which shows that wt−1 ≥ xt , contradicting to zts (γ) = xt >wt−1 . Q.E.D.
Lemma A2. Suppose that λ ≤ 1 − l. Then, zts (γ) ≤ (1/λ)xt for all γ ∈ Γst so that
A-ICst (γ) always binds for all γ ∈ Γst .

3

Proof. Suppose that A-ICst (γ) is not binding for some period t and some γ ∈ Γst .
Lemma A1 shows that wt−1 ≥ zts (γ) holds for all γ ∈ Γst and for all t ≥ s. Then therelationship’s joint proﬁt of period t becomes pt zts (γ) − λrt (zt (γ) − wt−1 ) for such γ.
This proﬁt is maximized at zts (γ) = λ̂xt . Thus, as long as A-ICst (γ) is slack for someγ at t, zts (γ) = λ̂xt must hold by the optimality of relational contract: otherwise, aslight change of zts (γ) toward λ̂xt can increase the joint proﬁt without violating all theconstraints.
Due to Lemma A1, A-CMEt−1 can be written byλ(1 − lt−1 )wt−1 + λ

t−1 ∑
∑
s=0

+

t−1 ∑
∑
s=0

s
q i dst−1 (γ i )lt−1(wt−1 − zts (γ i ))isq i (1 − dst−1 (γ i ))lt−1(λwt−1 − xt )i

= (1 − lt−1 )xt
Now we show that xt ≥ λwt−1 . Suppose not. Then, the left hand side of the above
A-CMEt−1 cannot be less than λ(1 − lt−1 )wt−1 so that λ(1 − lt−1 )wt−1 ≤ (1 − lt−1 )xt
must hold, implying that xt ≥ λwt−1 , a contradiction. Thus xt ≥ λwt−1 must hold.
But then (1/λ)xt ≥ wt−1 ≥ zts (γ) which shows that λ̂xt ≥ (1/λ)xt > zts (γ) = λ̂xt , acontraction. Q.E.D.
Lemma A3. (i) If dst (γ) = 1, then dst (γ ′ ) = 1 for all γ ′ > γ. (ii) Suppose thatλαδ > (1−α)(1−m). Then, dst (γ n ) = 1 must be satisﬁed for all t ≥ T for some period T ≥ s
Proof. (i) Suppose that dst (γ ′ ) = 1 but dst (γ ′′ ) = 0 for some γ ′′ > γ ′ in some periodt. Then, modify the original equilibrium contract only for such period t and γ ′′ asd˜st (γ ′′ ) = 1 and z̃ts (γ ′′ ) = zts (γ ′ ) + ε for ε ≥ 0. Note that dst (γ ′ ) = 1 implies thatpt zts (γ ′ ) − λrt zts (γ ′ ) ≥ πt > 0. Thus zts (γ ′ ) > 0 and hence A-ICst (γ ′ ) implies thats − mJ t+1 − (1 − m)(λr
s ′
′′
′
Jt+1
t+1 wt + πt+1 ) > 0. Then, since A-ICt (γ ) holds and γ > γ ,
t+1
t+1
t+1
s − mJ′s
we can see that γ ′′ δ{Jt+1
t+1 − (1 − m)(λrt+1 wt + πt+1 )} > γ δ{Jt+1 − mJt+1 −s′s′′(1 − m)(λrt+1 wt + πt+1 )} ≥ λrt zt (γ ). Thus A-ICt (γ ) can be slack at z̃ts (γ ′′ ) for asmall enough ε ≥ 0.
Case 1: wt−1 > zts (γ ′ ). Then, take ε > 0 that satisﬁes z̃ts (γ ′′ ) ≤ wt−1 . Thus z̃ts (γ ′′ ) >zts (γ ′ ). But, then we obtainpt z̃ts (γ ′′ ) + λrt (wt−1 − z̃ts (γ ′′ )) > pt zts (γ ′ ) + λrt (wt−1 − zts (γ ′ ))≥ λrt wt−1 + πt
where the ﬁrst inequality follows from that zts (γ ′ ) < λ̂xt and pt zt − λrt zt is increasing
in zt ≤ λ̂xt , and the second inequality from dst (γ ′ ) = 1. This shows that the modiﬁed
4contract can improve the joint surplus of period t, a contradiction to dst (γ ′′ ) = 0.
Case 2: wt−1 = zts (γ ′ ). Then, z̃ts (γ ′′ ) = wt−1 + ε ≥ wt−1 for ε ≥ 0 in which case thejoint proﬁt of period t becomesΦt (ε) ≡ pt z̃ts (γ ′′ ) + rt (wt−1 − z̃ts (γ ′′ ))= pt (wt−1 + ε) − rt ε= αA(wt−1 + ε)α − rt εwhere Φt (0) = pt wt−1 ≥ λrt wt−1 + πt due to dst (γ ′ ) = 1. Thus, if pt wt−1 > λrt wt−1 + πt ,then we set ε = 0 which shows that the joint surplus of period t, Φt (0), can be positive.
Next, suppose that pt wt−1 = λrt wt−1 + πt which is re-written by pt wt−1 − λrt wt−1 = πtαα − λr w
so that αAwt−1
t t−1 = πt ≡ maxz αAz − rt z. This implies that wt−1 < xtα−1αbecause xt maximizes αAx − rt x over x ≥ 0. However, then Φ′t (0) = α2 Awt−1− rt =α−1
α−1
2
α A[wt−1 − xt ] > 0. We can thus set a small enough ε > 0 yielding a posituve joint
surplus of period t, Φt (0) > 0, contradicting to dst (γ ′′ ) = 0 again.
λα(ii) Suppose that δ > (1−m)(1−α).
Suppose contrary to the claim that dst (γ n ) = 0 for all t ≥ s for some period s. Then,by above (i), dst (γ) = 0 for all γ ∈ Γ for all t ≥ s. Then, we will show that this cannot
be an equilibrium becasue there are the deviation contracts which are proﬁtable forsome relationships.

Case 1: pt ztτ (γ) − λrt ztτ (γ) ≤ πt for all τ > s, for all t ≥ τ and for all γ ∈ Γ. Then,
Jtt = λrt wt−1 + πt for all t ≥ s, which implies that there exist no relationships engaging
in relational contracts at all from period s onward in all the future periods. Thus,
A-CMEt implies that λwt = xt+1 for all t ≥ s, which shows that xt converges tox∗ ≡ [λ(1 − α)A]1/(1−α) as t → ∞. We will however show that Case 1 never happens.
If not, set a new contract oﬀered at period s as follows: z̃ts (γ) ≡ xt + ε (ε > 0) andλαd˜st (γ) = 1 for all t ≥ s and for all γ satisfying δ ≥ γ(1−m)(1−α)(such γ exists by}{λα. Then,assumption). Set d˜st (γ) = 0 for all other γ. Let Γ̃ ≡ γ ∈ Γ | δ ≥ γ(1−m)(1−α)
ssince λwt−1 = xt for all t > s, we have wt−1 > xt for all t > s. Thus z̃t (γ) ≤ wt−1
holds for all γ ∈ Γ̃ if we take small enough ε > 0. However, then we obtainpt z̃ts (γ) − λrt z̃ts (γ) > pt xt − λrt xt≥ pt xt − rt xt= πt
for all γ ∈ Γ̃ where the ﬁrst inequality follows from pt z − λrt z is increasing in z < λ̂xt
expected joint proﬁt of period t ≥ s thatand z̃ts (γ) = xt +ε. Thus the∑∑ is attained by the
i
i
s
s
s
˜
new contract becomes Jt ≡ γ i ∈Γ̃ [pt z̃t (γ ) + λrt (wt−1 − z̃t (γ ))] + γ i ∈/ Γ̃ [λrt wt−1 + πt ]which can be larger than the equilibrium proﬁt λrt wt−1 + πt for all t ≥ s. This can be5then a proﬁtable deviation if the new contract is feasible. The feasibility will be shown
for all large t ≥ s, i.e., A-ICst (γ) for γ ∈ Γ̃ is satisﬁed for all large enough t ≥ s. To see
λαthis, note that xt → x∗ as t → ∞ and that δ ≥ γ(1−m)(1−α)
implies that, when t → ∞,
t+1
s
˜
− mJ
we have γδ{J− (1 − m)(λrt+1 wt + πt+1 )} > γδ(1 − m)πt ≥ λrt (xt + ε) for
t+1

t+1

t+1
a small enough ε where note that Jt+1
= λrt+1 wt + πt+1 . Thus A-ICst (γ) holds for allsγ ∈ Γt for all large enough t ≥ s. Thus the new contract becomes a proﬁtable deviation
for all large t ≥ s.

Case 2: πt ztτ (γ ′′ ) − λrt ztτ (γ ′′ ) > πt for some τ ≥ s, some t ≥ τ and some γ ′′ ∈ Γ. Then,consider the new contract in such period s by mimicking the equilibrium contract usedby the relationship pair formed in such future period τ > s as follows: i) dst (γ) = 0
for all t where s ≤ t ≤ τ − 1, ii) {ztτ (γ), Rtτ (γ), dτt (γ)}t≥τ where set dst (γ) = 1 if
pt ztτ (γ) − λrt ztτ (γ) > πt and set dst (γ) = 0 if pt ztτ (γ) − λrt ztτ (γ) ≤ πt . Note that there
exist some γ ∈ Γ and some period t for which pt ztτ (γ) − λrt ztτ (γ) > πt by assumption.
However, then the above modiﬁed contract is feasible and strictly improves the joint
proﬁt in any period t whenever pt ztτ (γ)−λrt ztτ (γ) > πt . This is a contradiction. Q.E.D.
Lemma A4. In any equilibrium path)(xt+1
≥ λ(1 − α)A(1 − l)xtα−1
xt
and xt ≤ max{x0 , X} for all t where(X≡λ(1 − α)A
1−l

) 1

1−α

.

s (γ) for all γ ∈ Γs ), A-CME implies that
Proof. By using Lemma A1 (wt ≥ zt+1t
t+1
λwt ≥ (1 − lt )xt+1 . Then, by using A-LMEt and Lemma A2, we obtain
(
(
))
t−1 ∑
∑
s
λwt = λ(1 − α)A (1 − lt−1 xαt +q i lt−1[dst (γ i )zts (γ i )α + (1 − dst (γ i ))xαt ](≤λ(1 − α)Axαt(1 − lt−1 ) +(≤ λ(1 − α)Axαt

s=0
i
(
t−1
∑ ∑

(1 − lt−1 +))s
q i lt−1[dst (γ i )(1/λ)α + (1 − dst (γ i ))]

s=0
i
t−1
∑∑

)

s
q i lt−1(1/λ)α )s=0i= λ(1 − α)Axαt ((1 − lt−1 ) + (1/λ)α lt−1 )≤ λ(1 − α)A(1/λ)α xαt= λ1−α (1 − α)Axαt
6

which in turn implies that λ1−α (1 − α)Axαt ≥ λwt ≥ (1 − l)xt+1 for all t. Thus,(
xt ≤ X ≡ λ(1 − α)A
1−l

) 1

1−α

for all t when x0 ≤ X. When x0 > X, xt ≤ x0 for all t. In either case we havext ≤ max{x0 , X} for all t.
ss (γ) ≤ (1/λ)x
Second, by using Lemma A1 and Lemma A2 (zt+1
t+1 for all γ ∈ Γt+1 ),A-CMEt shows that
λwt = λ
≤ λ

t ∑
∑
s=0 i
t ∑
∑
s=0

i

s
q i dst (γ i )lts zt+1
(γ i ) + (1 − lt )x+1 +

t ∑
∑
s=0

q i lts (1/λ)xt+1 +

t ∑
∑
s=0

q i (1 − dst (γ i ))lts xt+1i

q i lts xt+1 + (1 − lt )xt+1i= xt+1
which yields xt+1 ≥ λwt . Then, we obtain xt+1 ≥ λwt = λ(1 − α)A(1 − lt−1 )xαt ≥λ(1 − α)A(1 − l)xαt because lt ≤ l for all t. Thus,()
xt+1≥ λ(1 − α)A(1 − l)xα−1.
t
xt
Q.E.D.
Appendix D: Bequest transfer
In the main text we have 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 introduction of bequest may aﬀect the analysis because the bequest can be used to ﬁnancecapital investment. However, in what follows we will show that individuals 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 the conditions under which
the results we have already established in the main text remain true even when we
introduce the bequest technology into the basic model as long as δ is in a certain small
range.
First, consider a young A-producer born in t − 1 who receives bequest bt−1 from his
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 an A-producer born in period t − 1. Given this,the proﬁt of an old A-producer is given as follows:{pt xt + λ(bt−1 − xt )rt − bt if bt−1 ≥ xt
(D1)
π(xt , bt |bt−1 ) =pt xt − (xt − bt−1 )rt − bt
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 λrt (bt−1 − xt ) in the credit market. Note here
7

that lending of one unit in the credit market must be accompanied with the enforcement
cost 1 − λ. When bt−1 < xt , the A-producer needs to use the external ﬁnancing andincur the borrowing cost rt (xt − bt−1 ).
We suppose that each individual observes the past capital investments and bequest
transfers made in the dynasty to which he or she belongs as well as he or she observes
the past market prices. Let Ht be a set of all these histories observed to an A-producer
up to period t. A strategy of a young A-producer is then deﬁned as a mapping fromHt the set of observed histories up to period t to capital investment level xt ≥ 0 andbequest transfer to his child bt ≥ 0. Let σt = (xt , bt ) : Ht → [0, ∞)2 be such strategy.
We will denote by π(σt |bt−1 ) the proﬁt of an A-producer who uses a strategy σt =(xt , bt ) in period t, given the bequest bt−1 received from his parent. Then a young Aproducer 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, given the previous period
choice bt−1 .
In what follows we will impose the upper bound on the market interest rate rt ,which we denote by d > 0. In order to make exposition simple, for the time being,
we will maintain the condition of rt ≤ d in any equilibrium path. We will give more
precise conditions on the primitives of the model which ensure this restriction later.
Then we show the following result:
Proposition D1. Suppose that δ < 1/(1 + d). Then in any equilibrium every Aproducer born in period t − 1 chooses σt = σ t ≡ (xt , 0) which gives his child no bequest bt = 0 and makes capital investment xt to maximize only his own current proﬁtπ(xt , 0|bt−1 ) given any bequest received from his parent bt−1 no matter the history.
Proof. Suppose contrary to the claim that there exists an equilibrium with {σt }∞t=1 inwhich some A-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)

(D2)

′
denotes the strategy chosen by the child of the A-producer who deviates
where σt+1
from his equilibrium play σt to σ t = (xt , 0) where xt maximizes π(xt , 0|bt−1 ).
Let σ̂t ≡ (xt , 0) be the strategy deﬁned by setting bequest transfer equal to zero in′≡ (x′t+1 , 0).
the original strategies σt = (xt , bt ). Let also σ̂t+1
Then, by using (D1), (D2) can be written by′π(σ̂t |bt−1 ) − bt + δ{π(σ̂t+1 |bt ) − bt+1 } ≥ π(σ t |bt−1 ) + δ{π(σ̂t+1|0) − b′t+1 }.

8

Since π(σ̂t+1 |bt ) ≤ π(σ̂t+1 |0) + rt+1 bt due to λ < 1, the above inequality implies that−(1 − δrt+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 |bt−1 ) − π(σ̂t |bt−1 )}′+ δ{π(σ̂t+1|0) − π(σ t+1 |0)} − δb′t+1= ∆t − δ∆t+1 − δb′t+1′ |0) ≥ 0. Since
where ∆t ≡ π(σ t |bt−1 ) − π(σ̂t |bt−1 ) ≥ 0 and ∆t+1 ≡ π(σ t+1 |0) − π(σ̂t+1
σt ̸= σ t , ∆t > 0 or/and bt > 0 must hold.
Since bt+1 ≥ 0, the above inequality implies−(1 − δrt+1 )bt − ∆t ≥ −δ∆t+1 − δb′t+1(D3)′
to be the strategy chosen by the child of the A-producer after the latter
For σt+1
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)

(D4)

′
′′
denote the strategies chosen by the child of the A-producer inand σt+2
where σt+2′
and σ t+1 respectively. Here the left hand side of (D4) isperiod t + 1 following σt+1
bounded above by′π(σ̂t+1|0) − b′t+1 + δ{π(σ t+2 |0) + rt+2 b′t+1 − b′′t+2 }
′′
′
′′
′′
′
′′
′′ |b′
due to π(σt+2
t+1 ) = π(σ̂t+2 |bt+1 ) − bt+2 ≤ π(σ̂t+2 |0) + rt+2 bt+1 − bt+2 ≤ π(σ t+2 |0) +rt+2 b′t+1 − b′′t+2 .
Then, by a similar argument to (D3), we can show that−(1 − δrt+2 )b′t+1 − ∆t+1 ≥ −δ∆t+2 − δb′t+2 .

(D5)′
to be chosen after the deviation σ t+1 , we must have
Similarly, for σt+2−(1 − δrt+3 )b′t+2 − ∆t+2 ≥ −δ∆t+3 − δb′t+3(D6)

1{∆t+2 − δ∆t+3 − δb′t+3 }1 − δrt+3(D7)

which implies that−b′t+2 ≥because of 1 > δd ≥ δrt+3 .
By applying the same condition to period t + 4, we obtain−b′t+3 ≥1{∆t+3 − δ∆t+4 − δb′t+4 }.
1 − δrt+4
9

Then we show that1{∆t+3 − δ∆t+4 − δb′t+4 }1 − δrt+4
δrt+4
δ
≥∆t+3 −(∆t+4 + b′t+4 )1 − δrt+4
1 − δrt+4
δ(∆t+4 + b′t+4 )≥ −1 − δrt+4−∆t+3 − b′t+3 ≥ −∆t+3 +due to 1 < δrt+4 ≤ δd, b′t+4 ≥ 0 and ∆t+i ≥ 0, i = 3, 4. Then (D7) must imply that−b′t+2 ≥1
1 − δrt+3{∆t+2 −}
δ2′(∆t+4 + bt+4 ) .
1 − δrt+4

By substituting this into b′t+2 in (D5), using the similar argument repeatedly and notingthat rt ≤ d for all t, we can show that−(1 − δrt+2 )b′t+1 − ∆t+1 ≥ −δ∆t+2 − δb′t+2

{
}
δ2
δ
′
∆t+2 −(∆t+4 + bt+4 )≥ −δ∆t+2 +1 − δrt+3
1 − δrt+4()
δ2
δ 2 rt+3∆t+2 − δ=(∆t+4 + b′t+4 )1 − δrt+3(1 − δrt+3 )(1 − δrt+4 ))T(δ(∆t+T +1 + b′t+T +1 )≥ −δ1 − δdfor arbitrary integer T .
Since T is arbitrary and ∆s and b′s are bounded above for all s ≥ 0,32 the right handside converges to zero by taking T → ∞, given δ/(1 − δd) < 1 which is equivalent toδ < 1/(1 + d). Thus the right hand side of (D5) is bounded below from zero. However,since the left hand side of (D5) is strictly negative because 1 > δd ≥ δrt+1 and bt > 0
or/and ∆t > 0, this is a contradiction. Q.E.D.
We also show that every A-lender has no incentives to give their children positive
bequest in any equilibrium when δ < 1/(1 + d).
Proposition D3. Suppose that δ < 1/(1 + d). Then A-lenders leave no bequest 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 A-lender in some dynasty gives a positive bequest
32

Note here that b′s ≤ maxxs ≥0 π(xs , 0|0) < +∞ and ∆s ≤ maxxs π(xs , 0|0) < +∞.

10bt > 0 in some period t where bt ∈ [0, wt ]. For this to be an equilibrium choice, theequilibrium payoﬀ of that lender in period tλrt (wt−1 + bt−1 ) − bt + δ{λrt+1 (wt + bt ) − bt+1 }must not be less thanλrt (wt−1 + bt−1 ) + δ{λrt+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 − δλrt+1 )bt − δbt+1 ≥ −δb′t+1(D8)

For b′t+1 to be an optimal choice by the child of the deviating lender in period t + 1, wemust have that
λrt+1 wt − b′t+1 + δ{λrt+2 (wt+1 + b′t+1 ) − b′t+2 }≥ λrt+1 wt + δ{λrt+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 − δλrt+2 )b′t+1 ≥ −δb′′t+2(D9)
By combining (D8) with (D9), we obtain−(1 − δλrt+1 )bt ≥ −δb′t+1
≥ −

δ2
b′′
1 − δλrt+2 t+2

where the ﬁrst inequality follows from bt+1 ≥ 0 and the second inequality from (D9)respectively where 1 > δdδ ≥ δλrt+2 due to 1/(1 + d) > δ and 1 > λ. Repeating
the similar argument over period t + 3, the right hand side of the above inequality isbounded below from
δ3
−
b′′
(D10)
(1 − δλrt+2 )(1 − δλrt+3 ) t+3
Repeating this process for any period t + T ≥ t, (D10) is bounded below from−δT
b′′t+T
T −1
Πi=1(1 − δλrt+i )which is further bounded below from(−(1 − δλd)δ1 − δλd11)Tb′′t+Tbecause of d ≥ rt+i , i ≥ 1. Since T can be arbitrary, b′′t+T ≤ wt+T < +∞, andδ < 1 − δλd, this converges to zero by taking T → ∞. Thus the right hand side of (D8)must converge to zero but its left hand side is strictly negative because of bt > 0, which
is a contradiction. Q.E.D.
Next we consider how the opportunity of bequest transfer aﬀects the choice ofthe optimal relational contracts in equilibrium. When a young R-producer receives abequest bpt−1 from his parent in period t − 1, he can use this to invest zt in capital.
When a young R-lender receives a bequest blt−1 from her parent in period t − 1, shewill have the total fund wt−1 + blt−1 which will be invested in capital investment for the
R-producer matching her.
We denote by Vtp and Vtl the proﬁts of a R-producer and a R-lender without repayment Rt and bequest transfer bit , i = p, l, leaving to their children respectively.
There are three possible cases which may occur in equilibrium.
Case (1). bpt−1 ≥ zt . Then a R-producer invests zt in capital by himself and saves
the remaining amount bpt−1 − zt . Then we have Vtp = pt zt + λrt (bpt−1 − zt ) and Vtl =λrt (wt−1 + blt−1 ).
Case (2). zt > bpt−1 . There are two sub-cases:
Case (2-1). wt−1 +bt−1 ≥ zt where bt−1 ≡ bpt−1 +blt−1 . In this case a young R-producer
borrows zt − bpt−1 from the R-lender matching him for capital investment zt . Then we
have Vtp = pt zt and Vtl = λrt (wt−1 + bt−1 − zt ).
Case (2-2). zt > wt−1 + bt−1 . Then the R-producer needs to borrow zt − wt−1 − bt−1
from the credit market after receiving wt−1 + blt−1 from the R-lender and using his own
fund bpt−1 . Thus we have Vtp = pt zt + rt (wt−1 + bt−1 − zt ) and Vtl = 0.
Thus the incentive compatibility condition for an old R-producer becomesp− Rt+1 − bpt+1 } ≥ Vtp − b̂pt + δπ(xt+1 , b̂pt+1 |b̂pt )Vtp − Rt − bpt + δ{Vt+1(BICt )

where b̂pt denotes the bequest transfer of an old R-producer after quitting from therelationship and engaging in arm’s length contracts, and π(xt+1 , b̂pt+1 |b̂pt ) denotes the
proﬁt of an A-producer (deﬁned by (D1)). The individual rationality constraint for ayoung R-lender becomesl
Vtl + Rt − blt + δ{Vt+1+ Rt+1 − blt+1 } ≥ λrt (wt−1 + blt−1 ) − b̂lt + δ{λrt+1 (wt + b̂lt ) − b̂lt+1 }(BIRLt )where b̂lt denotes the bequest transfer of a R-lender after quitting relationship. Thus,by combining these conditions together, we have the modiﬁed incentive compatibility
condition:− bt + Vtl + δ{Jt+1 − bt+1 − π(xt+1 , b̂pt+1 |b̂pt ) − [λrt+1 (wt + b̂pt ) − b̂lt+1 ]}(BIC∗t )≥ λrt (wt−1 + blt−1 ) − b̂tpl .
+ Vt+1
where bt ≡ bpt + blt and b̂t ≡ b̂pt + b̂lt and Jt+1 ≡ Vt+1

12

Also the individual rationality constraint for a young R-producer isp− Rt+1 − bpt+1 } ≥ π(xt , b̂pt |bpt−1 ) + δπ(xt+1 , b̂pt+1 |b̂pt )Vtp − Rt − bpt + δ{Vt+1(BIRPt )By combining BIRLt with BIRPt , we obtain the net total surplus condition:
Jt − bt + δ{Jt+1 − bt+1 } ≥ π(xt , b̂pt |bpt−1 ) + δπ(xt+1 , b̂pt+1 |b̂pt )+ λrt (wt−1 + blt−1 ) − b̂lt + δ{λrt+1 (wt + b̂lt ) − b̂lt+1 } (BTSt )Let Vt ≡ Jt − bt + δ{Jt+1 − bt+1 } denote the joint payoﬀ of a young R-producer
and a young R-lender born in period t − 1. Then, each relationship pair chooses a
to maximize the weighted sum of the joint payoﬀssequence {zt , bpt , blt }∞t=0 in order ∑∑∞∞t V = J − b + (δ + β)
t−1 (J − b ) subject to BIC∗ and BTS
β
t
0
0
t
t
t−1 fort
t=0
t=1 β
t = 1, 2, ...
We ﬁrst show the following lemma.
Lemma D1. In any equilibrium path zt ≤ λ̂xt holds in any period t.
Proof. We consider the equilibrium choice of zt . If above Case (1) is applied tothe equilibrium choice of zt in period t, then zt does not aﬀect BIC∗t . zt aﬀects only
the left hand sides of BIC∗t−1 , BTSt and BTSt−1 through the term Jt . Since Jt =pt zt + λrt (wt−1 + bt−1 − zt ) in Case (1), zt must maximize Jt subject to bpt ≥ zt . Thuszt ≤ λ̂xt must be satisﬁed. If Case (2-1) is applied to zt , then zt aﬀects the left handside of BIC∗t through the term λrt zt as well as the left hand sides of BICt−1 , BTSt andBTSt−1 through the term Jt . Since Jt = pt zt + λrt (wt−1 + bt−1 − zt ) in Case (2-1),
the equilibrium choice of zt must maximize Jt subject to BIC∗t . Thus zt ≤ λ̂xt mustbe satisﬁed as well. Finally, if Case (2-2) is applied to zt , then zt aﬀects only BTSt ,BTSt−1 and BIC∗t−1 through the term Jt . Thus, since Jt = pt zt + rt (wt−1 + bt−1 − zt )in Case (2-2), zt must maximize Jt in equilibrium, resulting in zt = xt < λ̂xt .
Thus we have established that zt ≤ λ̂xt in any period t. Q.E.D.
We then show the following lemma.
Lemma D2. Let x̃ be deﬁned as(x̃ ≡λ(1 − α)A(1 − l)lλ̂ + (1 − l))1/(1−α).

Then, it must be that rt ≤ d ≡ α2 A[min{x̃, x0 }]α−1 in any period t in any equilibrium
path.
Proof. There are three distinct cases of the credit market equilibrium depending onwhether or not each producer invests more than what he owns.
13Case (i): wt−1 + bt−1 ≥ zt and xt > b̂pt−1 . In this case the credit market equilibrium
becomesλl(wt−1 + bt−1 − zt ) + λ(1 − l)(wt−1 + b̂lt−1 ) = (1 − l)(xt − b̂pt−1 )which implies that λwt−1 ≤ λlzt + (1 − l)xt ≤ (λlλ̂ + (1 − l))xt due to Lemma D1(zt ≤ λ̂xt ).
Note that both wt−1 + bt−1 ≥ zt and xt ≤ b̂pt−1 are never compatible with each otherin the credit market equilibrium.33 Thus, the remaining cases are the following:
Case (ii): zt > wt−1 + bt−1 and xt > b̂pt−1 . In this case the credit market equilibrium
becomesλ(1 − l)(wt−1 + b̂pt−1 ) = l(zt − wt−1 − bt−1 ) + (1 − l)(xt − b̂pt−1 )which implies that (λ(1 − l) + (1 − l))wt−1 ≤ (lλ̂ + (1 − l))xt .
Case (iii): zt > wt−1 + bt−1 and xt ≤ b̂pt−1 . In this case the credit market equilibrium
becomesλ(1 − l)(wt−1 + b̂lt−1 ) + λ(1 − l)(b̂pt−1 − xt ) = l(zt − wt−1 − bt−1 )which implies that (λ(1 − l) + l)wt−1 ≤ (lλ̂ + (1 − l)λ)xt .
By Case (i)-(iii) above, it must be that λwt−1 ≤ (lλ̂ + (1 − l))xt which together with
LMEt implies that λ(1 − α)A(1 − l)xαt−1 ≤ (lλ̂ + (1 − l))xt . Thus xt ≥ min{x̃, x0 } mustbe satisﬁed. Since rt = α2 Axtα−1 , we conclude that rt ≤ d in any period t. Q.E.D.
In what follows we ﬁx δ such that δ < 1/(1 + d) for d being deﬁned in Lemma
D2. Then we show that introducing bequest transfer never expands the set of incentive
compatible relational contracts under the following condition.
Assumption D1. min{x0 , x̃} ≥ (1/(1 − α))1/α (δα2 A)1/(1−α) .
In particular, we show that under Assumption D1, the equilibrium relational contract which maximizes the weighted sum of expected payoﬀs
all the generations in∑of∞teach dynasty uses no bequest transfers at all. Recall that∑∞ t=0t β Vt is the weighted
sum
joint payoﬀs of generations in a dynasty where t=0 β Vt = J0 − b0 + (δ +∑of∞β) t=1 β t−1 Jt .
Proposition D4. Suppose that β ≤ δ < min{1/(1 + d), 1/d − β} and that Assumption
D1 holds. Then the optimal relational contract involves no bequest transfers at all evenwhen bequest transfer is allowed.
33

If this is the case, credit demand is zero while credit supply is positive.

14

Proof. Suppose that δ < 1/(1+d). Then Proposition D2 and D3 show that b̂pt = b̂lt = 0
in any period t. Thus, (BIC∗t ) is simpliﬁed to−bt + Vtl + δ{Jt+1 − bt+1 − π(xt+1 , 0|0) − λrt+1 wt } ≥ λrt (wt−1 + blt−1 )(BIC∗t )Here, we have−bt + δ{Jt+1 − bt+1 − π(xt+1 , 0|0) − λrt+1 wt } ≥ 0(BIC∗t in Case (1))−bt + δ{Jt+1 − bt+1 − π(xt+1 , 0|0) − λrt+1 wt } ≥ λrt (zt − bpt−1 ) (BIC∗t in Case (2-1))−bt + δ{Jt+1 − bt+1 − π(xt+1 , 0|0) − λrt+1 wt } ≥ λrt (wt−1 + blt−1 ) (BIC∗t in Case (2-2))
Next we show that BTSt for t ≥ 1 is implied by other constraints in any periodt ≥ 1. First note that BIC∗t−1 implies that Jt − bt ≥ π(xt , 0|0) + λrt wt . Also we canshow the following claim:
Claim 1. The choice of bt ≥ 0 which maximizes −b + δπ(xt , 0|b) becomes zero (b = 0).
Proof. Recall that π(xt , 0|b) = pt xt −rt (xt −b) if xt > b and π(xt , 0|b) = pt xt +λrt (b−xt )if b ≥ xt respectively. Here xt must be chosen in order to maximize π(x, 0|b) over x ≥ 0
in equilibrium due to Proposition D1. Let x(rt ) be the maximizer of pt xt − rt xt andx(λrt ) be the maximizer of pt xt − λrt xt . Then xt ∈ {x(rt ), b, x(λrt )} must be satisﬁed
in any equilibrium. Thus π(xt , 0|b) varies with b only when x(rt ) ≤ b ≤ x(λrt ). Since−1 + δα2 Abα−1 ≤ −1 + δrt ≤ −1 + δd < 0 over b ∈ [x(rt ), x(λrt )]. Thus b = 0 is the
optimal choice to maximize −b + δπ(xt , 0|b). Q.E.D.
By using this claim, we can show that for any b′ ≥ 0 and b′′ ≥ 0, π(xt+1 , 0|0) +λrt+1 wt ≥ π(xt+1 , 0|b′ ) − (1/δ)b′ + λrt+1 (wt + b′′ ) − (1/δ)b′′ holds. Thus, π(xt+1 , 0|0) +λrt+1 wt ≥ π(xt+1 , 0|bpt ) − (1/δ)bpt + λrt+1 (wt + blt ) − (1/δ)blt . Then, BIC∗t implies that−bt +Vtl +δ{Jt+1 −bt+1 −π(xt+1 , 0|bpt )−λrt+1 (wt +blt )} ≥ λrt (wt−1 +blt−1 ) which showsthat Jt+1 − bt+1 ≥ π(xt+1 , 0|bpt ) + λrt+1 (wt + blt ) for all t ≥ 1. Since BIC∗t+1 also implies
that Jt+2 − bt+2 ≥ π(xt+2 , 0|0) + λrt+2 wt+1 , BTSt is satisﬁed in any period t ≥ 2.
Finally we show that BTS1 holds as well. Since J2 − b2 ≥ π(x2 , 0|0) + λr2 w1 ,BTS1 is satisﬁed if J1 − b1 ≥ π(x1 , 0|bp0 ) + λr1 (w0 + bl0 ). To see this, consider BTS0 :J0 − b0 + δ{J1 − b1 } ≥ π0 + δπ(x1 , 0|0) + δr1 w0 . Here any initial old lender has no
incomes at all and any initial old producer earns the proﬁt π0 . Thus J0 = π0 . Since
J0 = π0 ≥ b0 ≥ bp0 and −b + δπ(x1 , 0|b) is maximized at b = 0 again (Claim 1 above),
BTS0 implies that −b0 + δ{J1 − b1 } ≥ −bp0 + δπ(x1 , 0|bp0 ) + δr1 w0 which in turn impliesthatδ{J1 − b1 − π(x1 , 0|bp0 )} ≥ bl0 + δλr1 w0≥ δr1 bl0 + δλr1 w0= δλr1 (w0 + bl0 )15

because 1 > δd ≥ δr1 . Thus J1 − b1 ≥ π(x1 , 0|bp0 ) + λr1 (w0 + bl0 ) and BTSt is satisﬁed
for all t ≥ 1.
Thus we have established that only relevant constraint of BTSt is BTS0 :−b0 + δ{J1 − b1 } ≥ δπ(x1 , 0|0) + δr1 w0(BTS0 )

where b̂p0 = 0 and J0 = π0 .
Recall that Vt ≡ Jt − bt + δ(Jt+1 − bt+1 ) be the joint payoﬀ of a R-producer and a Rblt }∞lender in t-th generation. Then the equilibrium relational contract {zt , bpt ,∑t=0 must∞t
maximize the weighted
sumof
expected
payoﬀsofallfuturegenerations
t=0 β Vt =∑∞ t−1J0 − b0 + (δ + β) t=1 β (Jt − bt ) subject to BIC∗t , t ≥ 1, and BTS0 .
First, observe that bp0 = bl0 = 0 must hold because −b0 + δJ1 is maximized at b0 = 0
due to 1 > δd ≥ δrt and BIC∗1 is relaxed by reduction of bl0 . Since δ < 1/d−β also holds
by assumption, −b0 + (δ + β)J1 is maximized at b0 = 0. Thus b0 = 0 becomes optimal.
Second, blt = 0 must be also satisﬁed for t ≥ 1. To see this, note that reduction of blt
from any positive level makes all the relevant constraints BIC∗t , BIC∗t−1 , BIC∗t+1 and
BTS0 satisﬁed because −blt + δJt+1 is increased due to 1 > δd and rt ≤ d. Since thejoint payoﬀ −blt + βJt+1 is improved by reducing blt > 0 due to 1 ≥ βd and d ≥ rt ,blt = 0 becomes optimal.
Finally, consider the equilibrium choice of bpt for t ≥ 1. We ﬁrst show the following
claim.
Claim 2. Case (2-2) (zt > wt−1 + bt−1 ) never happens in any period t.
Proof. Suppose that Case (2-2) occurs in some period t. In this case the equilibrium
choice of zt must maximize Jt = pt zt + rt (zt − wt−1 − bt−1 ) because this makes all the
relevant constraints relaxed while increasing the joint payoﬀ. Thus zt = xt must be
satisﬁed. However, then CMEt implies that λ(1 − l)wt−1 = l(zt − wt−1 − bt−1 ) + (1 − l)xt
where b̂lt−1 = b̂pt−1 = 0 and hence λwt−1 > xt . Thus we must have wt−1 > xt = zt which
however contradicts to zt > wt−1 + bt−1 . Q.E.D.
Case (2-1) is only the case that bpt > 0 may happen in equilibrium because bpt > 0
relaxes BIC∗t in Case (2-1). Now, we use the change of variable z̃t+1 ≡ zt+1 − bpt .
Then, note that bpt aﬀects only −bpt + δJt+1 in all the relevant constraints throughthe term pt+1 (bpt + z̃t+1 ). If Case (1) or (2-1) is applied to period t + 1, then Jt+1 =pt+1 (bpt + z̃t+1 )+λrt+1 (wt − z̃t+1 ). Since maximizing −bpt +δJt+1 and choosing bpt = 0 can
relax all the relevant constraints while maximizing −bpt + βJt+1 improves the objectivefunction, it suﬃces to show that bpt = 0 maximizes −bpt +δJt+1 and −bpt +βJt+1 . Taking
the ﬁrst order derivative of the former, we obtain −1 + δα2 A(bpt + z̃t+1 )α−1 . If this ispositive for some period t + 1, we have z̃t+1 + bpt < (δα2 A)1/(1−α) . Then we obtainpt+1 (z̃t+1 + bpt ) − πt+1
≤ αA(δα2 A)α/(1−α) − (1 − α)αAxαt+1≤ 016

because xt+1 ≥ (1/(1 − α))1/α (δα2 A)1/(1−α) which holds due to Assumption D1 andthe fact that xt ≥ min{x0 , x̃} for all t ≥ 1. However, then the net total surplus
Jt+1 − bt+1 − πt+1 − λrt+1 wt becomes strictly negative so that BIC∗t never holds, acontradiction. Thus bpt = 0 which maximizes −bpt +δJt+1 must be satisﬁed in any period
t in any equilibrium. Also, since β ≤ δ as we assumed, −bpt + βJt+1 is maximized at
bpt = 0. Thus, the equilibrium relational contract must involve bpt = 0 for all t ≥ 1 as
well. Q.E.D.
Finally we make the remark on the conditions we have so far made above: β ≤δ < 1/(1 + d), δ + β < 1/d and Assumption D1. We show that there is an open set ofthe parameter values for which these conditions hold together with Assumption 1 and2 made in the main text. To see this, assume that x0 is small so that x0 < x̃. Thend = α2 Ax0α−1 . Then Assumption D1 is equivalent to δ ≤ (1 − α)(1−α)/α /d. Since d isindependent of λ and l, Assumption 2 can be consistent with δ < 1/(1+d), δ +β < 1/d,β ≤ δ and Assumption D1 if λ and l are both small. Assumption 1 also holds when x0
is small.

17