Discussion Papers In Economics
And Business

Holdup, Search and Inefficiency
Shingo Ishiguro
Discussion Paper 07-13

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANHoldup, Search and Inefficiency
Shingo Ishiguro
Discussion Paper 07-13

May 2007

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

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

Abstract
This paper investigates the holdup problem in the dynamic searchmarket where buyers and sellers search for their trading partners andspeciﬁc investments are made after match but before trade. We showthat frictionless (competitive) market imposes severe limitations on
attainable eﬃciencies: Markets with small friction make the holdup
problem more serious than those with large friction because in any
equilibrium, whether stationary or non–stationary, investment must
be dropped down to the minimum level and trade must be delayedwith positive probability.
Keywords: Delay of Trade, Holdup Problem, Search
JEL Classiﬁcation Numbers: C72, C78∗

I am very grateful to Mark Armstrong, V Bhaskar, Leonardo Felli, Philippe Jehiel,
In–Uck Park and Francesco Squintani for useful comments and discussions. Of course allremaining errors are my own responsibility. This research was started during my visit to
University College London. I would like to thank for its great hospitality and support for
my research.
†
Correspondence: Shingo Ishiguro, Graduate School of Economics, Osaka University,
1-7 Machikaneyama, Toyonaka, Osaka 560–0043, Japan. Phone: +81-6-6850-5220, Fax:
+81-6-6850-5256, E-mail: ishiguro@econ.osaka-u.ac.jp1

1

Introduction

In this paper we investigate the holdup problem in the dynamic search market where buyers and sellers search for their trading partners and speciﬁc
investments are made after matching but before trade decisions. In particular we address the issues of whether or not frictionless (competitive) market
helps mitigate the holdup problem. Our answer to this question is negative:
In fact we show that markets with small friction make the holdup problem
more serious and result in signiﬁcant ineﬃciency, which never happens in
the markets with large friction.
What is the holdup problem? Suppose that a seller delivers one indivisible good to a buyer and makes “speciﬁc” investment which has value only
for the trade relationship with the buyer in question. For example the seller
may need to make customized investment which meets the speciﬁc requirements of the buyer. Then, they are locked in “bilateral monopoly” situation
and bargain over trade decision ex post after seller’s speciﬁc investment was
sunk. In the absence of complete contingent contracts the seller expects
that some fraction of the return from his investment will be captured by thebuyer through ex post negotiation, which in turn undermines the seller’s exante investment incentives. This is called the “holdup problem.”
The holdup problem has played the central role to understanding organizational design and boundaries of ﬁrms (See for example Willimason (1975,1985), Klein, Crawford and Alchian (1978), Grossman and Hart (1986), Hart
and Moore (1990) and Hart (1995)). These studies have discussed the solutions to the holdup problem through allocation of ownership rights (suchas vertical integration). Other authors also proposed the mechanism design
approach to resolve the holdup problem (See Aghion, Dewatripont and Rey(1994), Chung (1991) and Rogerson (1992)). However most of the existingstudies have focused on bilateral trade games of particular parties by taking
outside markets as exogenously given. Thus these studies have ignored theinteractions between the holdup problem in bilateral trade and the market environments which govern the processes of trading opportunities suchas how trading parties ﬁnd alternative partners when current negotiation
fails. In other words the reservation values of players, which determine thepayoﬀs of players when they fail to trade with current partners, have been
exogenously ﬁxed in the models.
In this paper we will consider the holdup problem in the market environment with search friction which endogenously determines the players’
reservation values. We will investigate the dynamic search market in which
there are many buyers and sellers and each buyer (seller) searches a seller(buyer) for trading one indivisible good. Each buyer consumes at mostone good and each seller can deliver at most one good. They are randomly
matched with each other with search friction: Some of buyers (sellers) fail tomeet sellers (buyers). After matching a buyer, seller makes “match speciﬁc”
2investment, which has the value only for the current match, before tradedecision of the good. Since they can trade at most one good, all matched
parties leave the market after they trade the goods. If matched parties do
not agree on trading the good, they stay in the market and start searchingfor the next match again.
Alternatively our model can be interpreted as the “marriage market”
with investment choices: 1 Male and female search for their partners andthey make “match speciﬁc investment” for building trust relationship withmatched partner, knowing each other well, cooperating in housework productions and so forth after matching. These investments have no values forall other matches than the current one. After agreeing on marriage, theparties leave the market.
In this dynamic search model the market friction is measured by the
inverse of the discount factor of players and the probability of failing tomeet a partner. Thus the market becomes frictionless as the discount factor
and matching probability go to one. One might think that, since small friction makes the market more competitive, resulting market outcomes become
more eﬃcient as search friction tends to be smaller.
However, in contrast to this intuition, we show that frictionless markets
become highly less eﬃcient than those with large friction. This is because
markets with small search friction make the holdup problem more serious
than those with large friction. In fact we show that in any equilibrium,
whether stationary or non–stationary, sellers’ investments must be dropped
down to the minimum level, zero, and trades must be signiﬁcantly delayedwith positive probability for inﬁnitely many times, although these features
never happen in the markets with large search friction.
This ineﬃciency result follows from the dynamic interactions between
current and future sellers’ investment incentives: Suppose that everyone
expects the future sellers will make high investment. Then the “reservation
values” for the current players become high because they expect high tradevalues will be realized in the future matches. This is more likely to be
the case when search friction becomes small. However, if the reservation
values of the current players are high, ex post surplus in the current match,which is deﬁned as the current trade value minus the sum of the reservation
values of buyer and seller for a given investment, becomes small. Each
matched seller obtains his bargaining payoﬀ from the negotiation over such
ex post surplus after his investment was sunk. When the ex post surplus
is expected to be small, the currently matched sellers will ﬁnd it costly tomake high investment. This will then undermine the sellers’ investments.
On the other hand, suppose that everyone expects the future sellers will
make low investment. Then, the reservation values of the currently matched1

See for example Peters and Siow (2002) for the recent model of the marriage market
with investment choices.

3

players become small because they expect low trade values will be realized
in the future matches. However, then the ex post trade surplus in thecurrent match becomes high and hence the current period sellers have more
incentives to make high investment.
Thus the investment incentives of current sellers move in the opposite
direction with those of the future sellers: If future sellers will have more
incentives to make high investment, current sellers have less incentives to
do so, and vice versa. This argument leads to the following results: First, if
there exists a symmetric stationary equilibrium in which all matched sellers
choose the same investment strategy in all periods, it must involve mixed
strategy to randomize between high and low investments. Second, if equilibrium is not stationary, it must exhibit some cyclical feature that equilibrium
investments ﬂuctuate between high and low levels over time. In either case,equilibrium investment must be low with some positive probability. Also,
whenever equilibrium investment goes down to zero, every match leads tono–trade and hence trade must be delayed. This dynamic mechanism makes
the holdup problem more serious as search friction becomes smaller andhence imposes severe limitations on attainable eﬃciencies.
We will also investigate several extensions of the model: First, we introduce heterogeneity of buyers and sellers into the model and then showthat our ineﬃciency result still holds even in such case. Second, we allow
matched parties to write ex ante contracts contingent on trade outcome before investment choice of matched seller. Then we show that there exists
some equilibrium when ex ante contracts are possible, which is less eﬃcientthan the equilibrium when ex ante contracts cannot be written at all, if
search friction becomes small enough. Thus enlarging the set of possible
contracts may make the market eﬃciency worse than the case of no ex ante
contracts. This might lead to the implication that it may not be optimal
to make markets competitive in developing countries with limited contract
enforcement. Third, we examine the two–sided investment case that bothsides of matched parties (buyer and seller) make speciﬁc investments before
trade. Then we ﬁnd that in the symmetric example in which both buyer
and seller have identical preferences the two–sided investment helps recovereﬃciency even when it is lost in the one–sided investment case. However,ineﬃciency arises again when buyer and seller are heterogeneous.
Related Literature
Several recent papers have paid attentions to the holdup problem in market environments and its implications about the eﬀects of competition (See
Acemoglu and Shimer (1999), Cole, Mailath and Postlewaite (2001), Felliand Roberts (2002) and MacLeod and Malcomson (1993)). Some among
these studies show that market competition may help recover the ﬁrst best
eﬃciency. In contrast to this, our result highlights on the negative eﬀects of“competition” on the holdup problem where in our model the competitive4

environment can be interpreted as the market with small search friction.2
Our ineﬃciency result is also in contrast to the recent paper by Che andSákovics (2004) who investigate the holdup problem in a dynamic setting
and show that there exists a Markov perfect equilibrium to attain the efﬁcient outcome when discount factor goes to one.3 De Mezza and Lockwood (2004) also reported that ineﬃciency may arise when search friction
becomes small, although the ineﬃciency in their model comes from the overinvestment (not underinvestment) problem due to coordination failure in thefrictional matching market.
Some papers have also focused on the issues of holdup problem in randomly matching markets (Acemoglu (1996, 2001) and Acemoglu and Shimer(1999)) where investing parties make ex ante investments before matching incontrast to our approach that investments are made after matching. Whichapproach of ex ante or ex post investment is more plausible depends on different economic circumstances we have in mind. Ex ante investments may
ﬁt to the cases of general human capital investment (education) by workers
and physical capital installment by ﬁrms before employment. On the other
hand, ex post investments may ﬁt to the cases of the parties making customized investments (as is often observed in the productions of automobile
parts) and building speciﬁc relationships with matched partners (as in themarriage market).
When we interpret our model as the marriage market as we have mentioned before, our result that competitive markets exaggerate the holdup
ineﬃciency is also in contrast to the papers showing the eﬃciency property
of competitive marriage markets (See for example Peters and Siow (2002)).
We can also relate our result to the literature about delay in bargaining
games. Several papers have shown the phenomena of delay in bargaining
games with complete information (See for example Busch and Wen (1995),
Cai (2000), Jehiel and Moldovanu (1995a, 1995b) and Sákovics (1993)). 4
However, most of the existing studies have assumed that the trade value,
which is a pie to be split between negotiating parties, is exogenously given.
On the other hand, in our model the trade value is endogenously determinedby players’ strategies in equilibrium and this leads to signiﬁcant delay of
trade.5 Frankel (1998) examines the bargaining game in which players exert
2

Also, most of the papers cited above have focused on the classical holdup problem
which assumes no contracts are possible before investment choices. In our paper wewill examine the extension of the basic model to allow matched parties to write ex ante
contracts before investments are made. Acemoglu and Shimer (1999) also consider arelated issue but they assume wage posting by ﬁrms before ex ante investment and searchdecisions.
3
In their model a buyer and a seller repeatedly interact with each other and there areno elements of switching partners in markets.
4
See also Jehiel and Moldovanu (2004) for the phenomena of gradualism in contribution
and bargaining games.
5
Merlo and Wilson (1995) consider the situation in which a pie to be split between bar-5

creative eﬀorts to expand the set of proposals, although the main focus wason premature agreement which means that players agree on trade before
total pie is larger. Also, most of the above papers have focused on repeatednegotiations between particular players while we will consider the dynamic
matching market in which a buyer and a seller interact each other only once
after they match together and they will not meet again in the future.6
The remaining sections of the paper are organized as follows: In Section
2 we will set up the basic model of dynamic search market with the holdup
problem. In Section 3 we will characterize the search market equilibria
and in Section 4 we will show that small search friction makes the holdup
problem worse and causes serious ineﬃciency. In Section 5 we will extend
the basic model in several directions.

2

Model

2.1

Search Market

We consider the dynamic search market in which buyers and sellers are randomly matched with each other. Time is discrete and extends over inﬁnity,t = 0, 1, 2... There is a single consumption good. At initial period (t = 0)
there are a unit mass of buyers and a unit mass of sellers in the market.
Seller is indexed by i ∈ [0, 1] and buyer by j ∈ [0, 1]. Each buyer canconsume at most one good and each seller can produce at most one good.
Each buyer (seller) matches a seller (buyer) with probability α ∈ (0, 1). If
matched parties agree on trading the good, they leave the market because
they can trade at most one good. There is no new entry into the market atall. Thus in every period the measure of buyers is same as that of sellers.7
The underlying assumption is here that the matching technology exhibits
the constant returns to scale.

gainers is changed over time, although such change follows exogenous stochastic process.
6
Samuelson (1992) also considers the bargaining in a dynamic matching market andshows that delay of trade is observed as an equilibrium outcome. However, in Samuelson(1992) the asymmetric information between matched parties causes delay of trade whilein our model there are no uncertainty but delay is observed.
7
We can also consider alternative scenarios about matching process: First, we may
assume that there are exogenous new inﬂows of players into the market at rate α ∈ (0, 1)
in each period. Then the population sizes of buyers and sellers can be kept at constantequal to a unit measure over time in any stationary equilibrium. Second, we may assume
that all the players who left the market are replaced by the same measure of new players.
In that case the population sizes of buyers and sellers become constant over time even in
non–stationary equilibrium. In any modeling choice the important point is that all players
who traded leave the market. This makes the “outside option” relevant for trade decision.
Many pagers in the literature about bargaining and contract choices in the search markets
follow this approach. See for example Inderst (2001), Osborne and Rubinstein (1990) andRubinstein and Wolinsky (1990) for the related models sharing the same approach.

6

After matching a buyer, matched seller makes “match speciﬁc” investment a ∈ A ⊂ ℜ+ which aﬀects both the buyer’s beneﬁt from consuming thegood b(a) (cooperative investment) and the production cost of seller (selﬁshinvestment) d(a). We will call v(a) ≡ b(a)−d(a) the trade value. The seller’s
investment is “match speciﬁc” in the sense that each matched seller’s investment has the value only for the trade with the currently matched buyer.
Thus it has no values in all other matches. For example the seller needs
to make customized investment which meets some speciﬁc requirements ofthe buyer. In fact customized inputs are often required for productionsof automobiles. Also in the marriage market matched partners need tomake “match speciﬁc” investments for building trust relationships with thematched partners, knowing each other well, cooperating in housework productions and so forth. These investments are however worthless for all othermatches. 8
Note here that we make no restrictions on the seller’s investment a ∈ A.
a is allowed to be both discrete and continuous.
The seller personally incurs the cost of investment c(a). We assume thatboth v and c are increasing and that v(0) > 0 and mina∈A c(a) = c(0) = 0.
As in the traditional approach to the holdup problem, we assume that seller’s
investment a ∈ A is observable to the buyer who matched the seller inquestion but not veriﬁable. We also assume that no contracts are possible
before seller’s investment choice a ∈ A. We will discuss how ex ante contracts
aﬀect the results in Section 5. Note also here that we make no restrictions
on v and c except that they are increasing.
After seller’s investment choice, the matched parties decide whether or
not to trade the good. We here assume the simple bargaining process: With
probability β ∈ (0, 1) the seller makes a take–it–or–leave–it oﬀer to thebuyer about trade decision and payment while with probability 1 − β thebuyer makes a take–it–or–leave–it oﬀer to the seller about trade decisionand payment. This is equivalent to assuming the Nash bargaining solution
which has been commonly used in the literature.
All players are risk neutral and discount their future payoﬀs at the common discount factor δ ∈ (0, 1).
The outside payoﬀ of each player is normalized to zero. Thus each player
obtains the payoﬀ of zero if he or she does not trade the good in the current
period.
8
We can explicitly introduce some “speciﬁcity” aspects of trade into the model. In
each match some random state η is independently drawn from [0, 1]. Then the buyer who
has drawn η needs the good with speciﬁcation η. Then trade value is realized as v(a)when the matched seller delivers the good with speciﬁcation η and chooses the investment
a to enhance such value. On the other hand, when the seller does not deliver the goodmeeting the buyer’s required speciﬁcation η the trade value is given by v ≡ v(0) > 0
irrespective of his investment. Here we can interpret the minimum investment level a = 0
as the “general” investment which attains the minimum trade value v whatever buyer’s
required speciﬁcation.

7

Each period is divided into three dates. The timing of events in a period
is as follows:
date 0 Buyers and sellers randomly match. The matched parties make alump–sum transfer, which is used to distribute the ex ante total surplus
between them as we will explain below. A matched seller makes a take–it–or–leave–it oﬀer of lump–sum transfer with probability γ ∈ (0, 1)
while a matched buyer does so with probability 1 − γ.
date 1 The matched seller makes match speciﬁc investment a ∈ A.
date 2 The matched parties bargain to decide whether or not they trade thegood and how much they make payments. If they agree on trading
the good, the seller produces and delivers the good to the buyer who
consumes it. Then they leave the market. Otherwise they stay in themarket and search for the next match.
The friction in the search market described above is inversely related
to the discount factor δ and the matching probability α ∈ (0, 1): If thesevalues are large, the market is said to have small friction because the tradesof the goods are quickly made and many matches can form. Then our mainconcern is to investigate how equilibrium outcomes in the search market
change when the market friction goes to zero.
As we have said, our model can be also interpreted as the marriage market where male and female search for each other and make “match speciﬁc”
investment after matching. Then they leave the market after they agree on
marriage.

2.2

The Static Benchmark

First we consider the static benchmark which corresponds to the case thatthe market friction is extremely high (δ = 0) in the above dynamic searchmodel.
In this static case all players obtain their outside payoﬀ, zero, when they
do not trade the goods. Thus ex post surplus, which is deﬁned as the tradevalue v(a) minus the reservation values the parties will obtain under no trade
agreement, is simply given by the trade value v(a) itself.
Since seller can extract β fraction of ex post surplus v(a), he obtains thefollowing payoﬀ:
βv(a) − c(a).
(1)
The seller chooses the speciﬁc investment a ∈ A to maximize this payoﬀ.
Let as ∈ A be the static equilibrium investment:as ∈ arg max βv(a) − c(a).
a∈A

8

(2)

We deﬁne the ex ante surplus asS(a) ≡ v(a) − c(a),(3)

and make the following assumption:
Assumption 1. as is unique and as ̸= 0.
Assumption 1 ensures that βv(as ) − c(as ) > βv(0).

3Search Market Equilibria

Next we turn to the dynamic search market model. Since matching is random and the set of players is continuum, it is the measure zero event thateach player can meet the same partner again as he or she has previously
matched. Also, the market is anonymous in the sense that any player cannot observe the past history about what decisions the matched partner has
made. Thus each matched seller simply chooses his current period investment in order to maximize his payoﬀ by taking his own and the buyer’s
reservation values as given. These assumptions are all standard in the literature of dynamic matching markets.
The main diﬀerence of this dynamic model from the static game is thatthe reservation values of players are endogenous. Here the reservation valueof a player means the discounted present value of his or her future payoﬀs to
be obtained when he or she does not trade the good in the current period.
Consider a matched pair of a buyer and a seller in period t. Then let
Utb and Uts denote the reservation values of the buyer and the seller, which
they will obtain when they do not trade the good in the current period t.
Then the ex post surplus from this match is deﬁned as the trade value
v(a) minus these reservation values:v(a) − Utb − Uts .

(4)

After the investment has been sunk, as long as this surplus is non–negative,
the parties have the incentive to trade the good in the current period t. Then,with probability β the seller makes a take–it–or–leave–it oﬀer of payment
and obtains the payoﬀ v(a) − Utb . On the other hand, with probability 1 − β
the buyer makes a take–it–or–leave–it oﬀer of payment and hence the seller
obtains the payoﬀ Uts . If the ex post surplus (4) is negative, they will not
trade the good and will stay in the market for searching the next matches,in which case they simply obtain their reservation values.
Thus the seller’s payoﬀ from ex post negotiation is given by β max{v(a)−Utb − Uts , 0} + Uts . Then the seller’s net payoﬀ subtracting his investment
cost c(a) can be written by
β max{v(a) − Utb − Uts , 0} + Uts − c(a).
9(5)

The seller will choose investment a ∈ A to maximize this payoﬀ. Morespeciﬁcally, the mixed strategy of matched seller i in period t is represented
by a probability measure fti ∈ ∆(A) where ∆(A) is the set of all probability
measures on A. Let also Efti [ ·] denote the expectation operator according
to fti .
We give a deﬁnition of equilibrium in the search market as follows:
Deﬁnition of Equilibrium. A sequence {ft , Uts , Utb }∞t=0 is said to be an
equilibrium in the search market if the following conditions are satisﬁed:(i) Symmetry and Optimality of Seller’s Investment: In period t eachmatched seller follows the symmetric strategy ft to maximize his payoﬀ:∫{β max{v(a) − Uts − Utb , 0} + Uts − c(a)}dftA∫≥{β max{v(a) − Uts − Utb , 0} + Uts − c(a)}df ′ ∀f ′ ∈ ∆(A).
A(ii) The Reservation Values: The buyer’s and seller’s reservation values
Utb and Uts are determined as follows:b
Utb = δ{αubt+1 + (1 − α)Ut+1
},

and
s
Uts = δ{αust+1 + (1 − α)Ut+1}

where ubt+1 and ust+1 are the payoﬀs of a matched buyer and a matchedseller respectively:bss
ust+1 ≡ γ max{Eft [S(a)] − Ut+1
− Ut+1, 0} + Ut+1
,

and
b
s
b
ubt+1 ≡ (1 − γ) max{Eft [S(a)] − Ut+1
− Ut+1, 0} + Ut+1.

Condition (i) says that in period t each matched seller optimally chooses
symmetric strategy ft to maximize his payoﬀ, given the reservation values
Utb and Uts . Since all sellers are identical, it is natural to focus on thesymmetric equilibrium in which all matched sellers follow the same strategy
in each period. However, note that this does not necessarily mean that thesellers’ equilibrium strategy is stationary because it may depend on timeindex t.
Condition (ii) says that the reservation value of a buyer (resp. seller) Utb(resp. Uts ) in period t is the discounted present value of his or her expected
future payoﬀs which come from the payoﬀ when matching ubt+1 (resp. ust+1 )10bs ) when not matching respectively.
and the reservation value Ut+1
(resp. Ut+1
The former payoﬀ is realized with probability α ∈ (0, 1), while the latter
payoﬀ is realized with probability 1 − α. The buyer’s (resp. seller’s) payoﬀ
ubt+1 (ust+1 ) when matching is equal to γ fraction of ex ante total surplusbsmax{Eft+1 [S(a)] − Ut+1
− Ut+1, 0}s ). This is because theb(resp. Ut+1
plus her (his) reservation value Ut+1
seller has the chance of making a lump–sum transfer oﬀer to the buyer withprobability γ at date 0 in each period and hence he can extract γ fraction of
the ex ante surplus. Note that we are taking max{·, ·} operator to express
these payoﬀs because the ex ante surplus may be negative, in which caseboth parties simply obtain their reservation values Utb and Uts .
The reservation values of players Utb and Uts depend on what investments and trade decisions the future matched players will do. Then, since
the investment choices by the currently matched sellers are aﬀected by thereservation values, the current sellers’ investments are indirectly linked withthose of the future sellers. This intertemporal linkage between the current
and future sellers’ investments choices plays the central role in the following
analysis.
Note also that the sum of the reservation values of a buyer and a sellerb
Ut +Uts matters for determining the equilibrium investment of matched sellers (how to distribute the ex ante surplus does not aﬀect seller’s investment
incentives). Once the total value Utb + Uts is determined, we can decompose
it into the separate values Utb and Uts by splitting it appropriately according
to ex ante bargaining power γ. Thus in what follows we will mostly focus
on the determination of the total value Utb + Uts but not each of it.

We ﬁrst show the following two lemmas both of which make the characterization of equilibria much simpler.
Lemma 1. In any equilibrium any matched seller never chooses a ∈/ {0, as }in any period t, i.e., ft (A \ {0, as }) = 0 ∀ t.
Proof. See Appendix.
Lemma 1 follows from Assumption 1 and the fact that the payoﬀ functionof matched seller (5) attains its maximum at a = 0 or a = as (or both). In
particular we can readily show that each seller optimally chooses the static
investment as if and only ifv(as ) − c(as )/β ≥ Utb + Uts .

(IC)

Otherwise, seller chooses the minimum investment a = 0. The reason forthis is as follows: Suppose that each matched seller chooses as . Then, since11any seller can guarantee at least Uts by choosing a = 0, we must have
β max{v(as ) − Utb − Uts , 0} ≥ c(as ) which then implies (IC). On the other
hand, each matched seller prefers a = as to a = 0 if β max{v(as ) − Utb −Uts , 0} − c(as ) ≥ β max{v(0) − Utb − Uts , 0}. This inequality holds by (IC)and deﬁnition of as .
By Lemma 1 without loss of generality we can write the equilibrium
strategy of each matched seller by a probability xt ∈ [0, 1] to choose a = asand the remaining probability 1 − xt to choose a = 0 in period t. Thusin what follows we will refer to xt for denoting the equilibrium strategy ofmatched sellers instead of using ft .
Lemma 2 In any equilibrium trade is (resp. not) realized in any period inwhich a = as (resp. a = 0) is chosen.
Proof. See Appendix.
Lemma 2 follows from Lemma 1 and the fact that trade occurs if andonly if the ex post surplus (4) is non–negative.
The equilibrium features crucially depend on the degree of search friction
which is represented by the discount factor δ ∈ (0, 1). We deﬁne the cutoﬀ
value of the discount factor, denoted δ ∗ ∈ (0, 1), to satisfy the following
equation:{}δ ∗ αS(as )sβ v(a ) −= c(as ).
(6)
1 − δ∗αBy Assumption 1, c(as ) > 0 and β < 1, such δ ∗ ∈ (0, 1) uniquely exists.
Then we ﬁrst show the following result:
Proposition 1. Suppose that δ ∈ (0, δ ∗ ). Then equilibrium is unique andhas stationary property such that every matched seller chooses the static
equilibrium investment as with probability one in every period.
Proof. See Appendix.
Proposition 1 states that when search friction is large the equilibriumoutcome coincides with the static one.

4

Small Search Friction and Ineﬃciency

4.1

Low Investment and Delay of Trade

Next we will turn to the case of small search friction (large discount factor
δ). For any equilibrium proﬁle e ≡ {xt , Utb , Uts }∞t=0 , we deﬁne the following

12

set:
L(e) ≡ {t ∈ {0, 1, 2...} | xt < 1}.

(7)

L(e) is the set of the periods in which matched sellers choose the minimum
investment, zero, with some positive probability.
Then we show the following result which has the diﬀerent feature fromthe case of large search friction.
Proposition 2. Suppose that δ ∈ (δ ∗ , 1). Then in any equilibrium e theminimum investment level, zero, must be chosen with positive probability forinﬁnitely many times, i.e.,supL(e) = +∞.
Proof. See Appendix.
Given an equilibrium e, we also deﬁne the following set:D(e) ≡ {t ∈ {0, 1, 2, ...} | trade does not occur with positive probability in period t.}.
Then the direct corollary from Proposition 2 is the following:
Corollary. Suppose that δ ∈ (δ ∗ , 1). Then in any equilibrium e trade must
be delayed with positive probability for inﬁnitely many times, i.e.,supD(e) = +∞.
Proof. By Proposition 2 we know that xt < 1 must happen for inﬁnitely
many times when δ ∈ (δ ∗ , 1). Then, by Lemma 2, since trade does notoccur when a = 0 was realized, no–trade outcome must happen with positive
probability for inﬁnitely many times as well. Q.E.D.
The intuition behind these results can be explained as follows: If there
exists some equilibrium in which the minimum investment a = 0 is chosenwith positive probability only for ﬁnite times, then by Lemma 1 the equilibrium investment must coincide with the static equilibrium one as withprobability one from some period T onward. However, if at = as for allt ≥ T , the sum of the reservation values of a buyer and a seller at period T
must be stationary and equal to
UTb + UTs =δαS(as ).
1 − δ(1 − α)This is because the total surplus S(as ) will be realized in every match fromperiod T onward. Since each matched seller can obtain at least UTs bychoosing a = 0, the following inequality must be satisﬁed for each seller tochoose as :β{v(as ) − UTb − UTs } + UTs − c(as ) ≥ UTs ,13which is reduced to{β v(as ) −δαS(as )1 − δ(1 − α)}− c(as ) ≥ 0.

However this inequality is not satisﬁed when δ > δ ∗ . Thus each matchedseller deviates and chooses a = 0 with certainty, which gives him the higher
payoﬀ UTs than the equilibrium payoﬀ obtained by choosing as . Hence theminimum investment a = 0 must be chosen with positive probability forinﬁnitely many times in any equilibrium. Then by Lemma 2 trade must be
delayed with positive probability for inﬁnitely many times as well.
We can also relate this result to the literature about delay in bargaining
games. Our above result sheds a new light on equilibrium delay in theview point of the holdup problem and search friction. The key factor tocause delay in our model is that the trade value v(a) is endogenous and it isaﬀected by matched seller’s investment a ∈ A. This is the main diﬀerence
from the standard bargaining games which have mostly assumed that trade
surplus is exogenous.
When trade value is endogenous, players’ expectation about the future
investment decisions plays the important role: If they expect the future
sellers will invest much and hence the future trade values will be high, thereservation values of buyer and seller Utb and Uts become high as well. Thenthe current negotiation surplus v(a)−Utb −Uts becomes small so that current
sellers invest little, which then leads to no–trade outcome in the current
match. On the other hand, if players expect the future sellers will invest
little, then the current net negotiation surplus becomes high and hence thecurrent sellers invest much, which leads to trade. This argument shows thatin any equilibrium there must exist both the event in which investment is
low and trade is not realized and the event in which investment is high andtrade occurs with non–trivial probabilities. By this reason, delay of tradeoccurs with some positive probability in our model.
Next we will focus on some particular equilibrium which possesses thefeature of low investment and delayed trade as identiﬁed in Proposition 2.
We can show that there are many equilibria which exhibit this feature forsome large discount factors.
Proposition 3. Suppose that δ ∈ (δ ∗ , 1). Then there exists a unique stationary equilibrium in which every matched seller chooses the static investment
as and the minimum one, zero, with probabilities x∗ ∈ (0, 1) and 1 − x∗respectively in every period where x∗ satisﬁesv(as ) − c(as )/β =14αδx∗ S(as ).
1 − δ(1 − αx∗ )Proof. Suppose that the sum of the reservation values U b and U s is givenbyαδx∗ S(as )Ub + Us =.
1 − δ(1 − αx∗ )Note here that v(as ) > U b + U s > v(0) by Assumption 1. Also, suchx∗ ∈ (0, 1) is uniquely determined.
Then every matched seller is indiﬀerent for choosing as and zero because
β{v(as ) − U b − U s } + U s − c(as ) = U s .
Thus we can suppose that each matched seller randomizes his investment
choice by putting the probability x∗ ∈ (0, 1) on a = as and the probability
1 − x∗ on a = 0 respectively. Also, such sellers’s behavior can be consistent
with the sum of the reservation values U b and U s we have deﬁned above.
Q.E.D.
When equilibrium is stationary in that every matched seller follows thesame investment strategy in every period, it must be unique and involve themixed strategy of seller’s investment choice randomizing between as andzero.
We can also show that there exist non–stationary equilibria as well.
Proposition 4. For any positive integer K ≥ 1, there exist some δ K andδ K such that for all δ ∈ (δ K , δ K ) a non–stationary equilibrium arises where
each matched seller chooses equilibrium investment at in period t as follows: s a for t = τ K + τ − 1 (τ = 1, 2, ...),at =
0 otherwise
Proof. See Appendix.
The equilibrium shown in Proposition 4 is non–stationary in the sensethat equilibrium investments of matched sellers cyclically change over time.
Following the matched sellers’ investment as in some period, equilibrium
investments become zero and hence trades are not realized in all subsequent
K periods. In the next period sellers’ investment will be returned back tothe static equilibrium one as again, which leads to trade agreements for all
matches in that period. Thus equilibrium investment and trade have cyclicalpatterns with period K.
Also, if there exists a non–stationary equilibrium with period K cycle forsome range of discount factors, then there also exist non–stationary equilibria with cycle k (k ≤ K − 1) for the same range of the parameters ofthe model as well. This is because we have (δ K , δ K ) ⊂ (δ K−1 , δ K−1 ): If
we have δ ∈ (δ K , δ K ) such that a period K cycle equilibrium exists, then15δ ∈ (δ K−1 , δ K−1 ) and hence period K − 1 cycle equilibrium also exists as
well.
Proposition 3 and 4 show that there exist multiple equilibria even forthe same levels of market friction (δ ∈ (0, 1)). The next natural question iswhat equilibrium can attain the highest eﬃciency in the market among all
possible equilibria. We will now explore this issue.

4.2

Welfare Analysis

In this subsection we will conduct the welfare analysis about how the changeof search friction aﬀects the equilibrium outcomes and hence the market
welfare.
We deﬁne the social welfare of the search market as the discounted
present value of the sum of buyers’ and sellers’ payoﬀs evaluated at initial period t = 0. Then, given an equilibrium path e ≡ {ft , Utb , Uts }∞t=0 , wecan derive the social welfare under this equilibrium e for a given δ ∈ [0, 1)
as follows:
W (δ|e) ≡ α max{Ef0 [S(a)], U0b + U0s } + (1 − α)(U0b + U0s )

(8)

where
b
s
b
s
Utb +Uts = δ{α max{Eft+1 [S(a)], Ut+1
+Ut+1}+(1−α)(Ut+1+Ut+1)}, t = 0, 1, 2, ...
(9)Note here that the ex ante surplus in period t is given by max{Eft [S(a)], Utb +Uts } because every matched pair will be dissolved and will obtain their reservation values when the net ex ante surplus Eft [S(a)] − Utb − Uts is negative.
By using Lemma 1 and 2, we know that in any equilibrium each matchedseller chooses as with probability xt ∈ [0, 1] only in which case trade occurs
and the trade value S(as ) is realized. In all other cases trade values are neverrealized. By using this fact, we can rewrite the social welfare as follows:

W (δ|e) = αx0 S(as ) + (1 − αx0 )(U0b + U0s ),

(10)

where
b
s
Utb + Uts = δ{αxt+1 S(as ) + (1 − αxt+1 )(Ut+1+ Ut+1)}, t = 0, 1, 2, ... (11)Then we give the following deﬁnition.
Deﬁnition. An equilibrium e = {xt , Utb , Uts }∞t=0 is said to be constrained
eﬃcient if it attains the highest social welfare among all possible equilibria.
Otherwise, it is said to be ineﬃcient.
We ﬁrst show the following lemma.
Lemma 3. Consider a sequence ê = {x̂t , Ûtb , Ûts }∞t=0 satisfying the following:
16

(i) t = 0: x̂0 = 1.
(ii) ∀ t ≥ 1: x̂t = x∗ ∈ [0, 1] where x∗ = 1 if δ ∈ (0, δ ∗ ) and x∗ ∈ [0, 1]
satisﬁes{}δαx∗ S(as )sβ v(a ) −= c(as )1 − δ(1 − αx∗ )if δ ∈ (δ ∗ , 1).
(iii) The reservation values:
Ûtb + Ûts =αx∗ S(as ),1 − δ(1 − αx∗ )t = 0, 1, 2, ...

Then ê can be an equilibrium.
Proof. Since x∗ = 1 for all δ ∈ (0, δ ∗ ), ê coincides with the unique static
equilibrium shown in Proposition 1 for any δ ∈ (0, δ ∗ ).
Next suppose that δ ∈ (δ ∗ , 1). Then, given Ûtb and Ûts above, eachmatched seller faces the following payoﬀ function in period t:{}δαx∗ S(as )uS (a) = β max v(a) −,0+ Ûts − c(a).
1 − δ(1 − αx∗ )Then, by deﬁnition of x∗ , maxa∈A uS (a) = uS (0) = uS (as ). Thus eachmatched seller is willing to randomize between a = as and a = 0 withprobabilities x∗ and 1 − x∗ respectively in any period t ≥ 1. Also it is
optimal for each matched seller to choose as with certainty (x̂0 = 1) ininitial period t = 0.
Note that when as was realized trade occurs because v(as ) > Ûtb + Ûts
for any t while when a = 0 was realized trade never happens because v(0) <v(as ) − c(as )/β = Ûtb + Ûts for any t. Thus trade occurs with probabilityx∗ per match and hence the surplus S(as ) is realized only with probability
αx∗ . Thus the above seller’s strategy leads to the reservation values Ûtb + Ûts
deﬁned above. Q.E.D.
Then we show the following result.
Proposition 5. The equilibrium ê shown in Lemma 3 is constrained eﬃcient.
Proof. See Appendix.
The intuition behind Proposition 5 can be explained as follows: First,
as we have shown, every matched seller chooses either the static equilibrium
investment as or the minimum one, zero, in any equilibrium. In particular
17

each matched seller chooses as in period t if and only if (IC) is satisﬁed as wehave noted. Then (IC) shows that the discounted present value of the total
future payoﬀs of a buyer and a seller in the market from period t + 1 onward(which is evaluated at period t), i.e., Utb + Uts , is bounded from above byv(as ) − c(as )/β. The equilibrium identiﬁed in Lemma 3, ê, can attain thisupper bound and implement the highest investment incentive of matchedsellers while keeping (IC)s hold as equalities in any period t.
From Proposition 5 we can derive the interesting implication about theeﬃciency property of the search market. Proposition 5 shows that the social
welfare in the constrained eﬃcient equilibrium ê is given by
W (δ|ê) ≡ αS(as ) + (1 − α)δαx∗ S(as ),1 − δ(1 − αx∗ )

(12)
∗

s

δαx S(a )where x∗ = 1 for all δ ∈ (0, δ ∗ ) and x∗ satisﬁes v(as ) − c(as )/β = 1−δ(1−αx∗)
for δ ∈ [δ ∗ , 1) respectively. Note here that W (δ|ê) is increasing in δ ∈ (0, δ ∗ )but constant at W (δ|ê) = αS(as )+(1−α)[v(as )−c(as )/β] over all δ ∈ [δ ∗ , 1).
We take any ineﬃcient equilibrium e ̸= ê and consider its welfare, denoted by W (δ|e). Then, since W (δ|ê) is constant over all δ ∈ (δ ∗ , 1) andW (δ|e) = W (δ|ê) for all δ ∈ (0, δ ∗ ) due to Proposition 1, the social welfare
in any ineﬃcient equilibrium W (δ|e) must be non–monotonic with respect
to the discount factor δ. This is because W (δ|e) must be strictly less thanW (δ|ê) for some higher discount factors than δ ∗ .
For example, we ﬁrst consider the stationary equilibrium identiﬁed in
Proposition 3. Let es = {x∗ , U b , U s } denote this unique stationary equilibrium. Then the social welfare in the stationary equilibrium is given by

W (δ|es ) ≡ αx∗ S(as ) + (1 − αx∗ )δαx∗ S(as )1 − δ(1 − αx∗ )where note that trade will occur and hence the total surplus S(as ) will be
realized only when matched sellers choose the static equilibrium investment
as which occurs with probability x∗ ∈ [0, 1]. By Proposition 1 we knowthat x∗ = 1 when δ ∈ [0, δ ∗ ]. However, by Proposition 3 we have x∗ < 1when
δ > δ ∗ . Note that in such case x∗ is decreasing in δ. Also, sinceδαx∗ S(as )ss∗s1−δ(1−αx∗ ) = v(a ) − c(a )/β for all δ ∈ (δ , 1), we can rewrite W (δ|e ) as
W (δ|es ) = αx∗ S(as ) + (1 − αx∗ )[v(as ) − c(as )/β]for δ ∈ (δ ∗ , 1). Thus, as shown in Figure 1, the social welfare W (δ|es ) in thestationary equilibrium is non–monotonic with respect to the discount factor
δ: Smaller market friction reduces the social eﬃciency.
Second, we consider the non–stationary equilibria characterized in Proposition 4: We have the non–stationary equilibrium with period K cycle, denoted ens , for δ ∈ (δ K , δ K ). Then the social welfare in the non–stationary18

equilibrium ens , denoted W (δ|ens ), must be strictly less than that in theconstrained eﬃcient equilibrium W (δ|ê). This is becausebs
W (δ|ens ) = δ K−1 {UK+ UK}≤ δ K−1 {v(as ) − c(as )/β}< v(as ) − c(as )/β< αS(as ) + (1 − α){v(as ) − c(as )/β}= W (δ|ê)
where the ﬁrst inequality follows from the fact that all matched sellers choose
a = 0 and trades are not realized until period K in which v(as ) − c(as )/β ≥b + U s holds so that the matched sellers choose as , the second inequality
UK
K
from δ < 1 and K ≥ 2 and the ﬁnal inequality from S(as ) > v(as ) − c(as )/β
respectively. Thus, as shown in Figure 2, non–stationary equilibrium ens
may arise for δ ∈ (δ K , δ K ) even when we have the constrained eﬃcientequilibrium ê for all other discount factors. In such case the social welfare
is not monotonic with respect to the discount factor δ. Again, small searchcost has the adverse eﬀect on the market eﬃciency.
Next we will extend the basic model in the several directions.

5

Extensions

5.1

Heterogeneous AgentsIn this subsection we will allow buyers and sellers to be heterogeneous. Letθti ∈ Θ and ψtj ∈ Ψ be the random variables, called “types,” which aﬀect
the valuation of buyer i and investment cost of seller j in period t. We
denote by v(a, θ) the trade value which depends on the buyer’s current typeθ. We also denote by c(a, ψ) the seller’s investment cost which depends on
the seller’s current type ψ. At the beginning of each period every buyer andseller do not know their own types in that period. After matching, the typesof matched buyer and seller, θt and ψt , are realized according to some probability distributions and are then known to them. We assume that any typepair (θ, ψ) has positive measure. The realized types are common knowledge
for matched players but the types of future players will be changed. We also
assume that the probability distributions from which θt and ψt are drawn
are independent of time index t and that any type of any player is replaced
by a new entrant with the same type when he or she leaves the market.
These assumptions can ensure that the distributions of types can be kept
constant over time.
Suppose that in period t buyer i and seller j matched and observed theirrealized types as θ and ψ respectively. Given these types, in period t the19seller will obtain the following payoﬀ:
β max{v(a, θ) − Utb − Uts , 0} + Uts − c(a, ψ)(13)where the sum of the reservation values Utb and Uts are given bybs
Utb + Uts = δ{αEθ,ψ [v(at+1 , θ) − c(at+1 , ψ)] + (1 − α)(Ut+1+ Ut+1)}. (14)

Here Eθ,ψ [·] denotes expectation operator with respect to the types of buyers
and sellers (θ, ψ).
Matched seller of type ψ chooses his investment level at to maximize his
expected payoﬀ (13), given the reservation values Utb and Uts . By using asimilar argument to Lemma 1, we can show that the optimal investment
choice by each matched seller becomes a = 0 or a = as (θ, ψ) whereas (θ, ψ) ≡ arg max βv(a, θ) − c(a, ψ).
a∈A

Also, we can show that trade will occur when the seller chooses as (θ, ψ)again. Then let xt (θ, ψ) ∈ [0, 1] denote the equilibrium probability that sellerof type ψ chooses as (θ, ψ) when he matches with buyer of type θ in period t.
xt (θ, ψ) also means the probability of trade in period t when buyer of type θand seller of type s match with each other. Let xt ≡ (xt (θ, ψ))(θ,ψ)∈Θ×Ψ bea collection of possible trade probabilities in period t. Then we can deﬁne
an equilibrium as a path e ≡ {xt , Utb , Uts }∞t=0 as before.
For a given equilibrium e, we deﬁne the following set:L̃(e) ≡ {t = 0, 1, 2... | xt (θ, ψ) < 1 for some (θ, ψ) ∈ Θ × Ψ}.
L̃(e) is the set of the periods in which trades do not occur with some positive
probabilities according to equilibrium e. Then we can extend our ineﬃciency
result (Proposition 2) to the case of heterogeneous agents:
Proposition 6. There exists some δ ∈ (0, 1) such that for all δ ∈ (δ, 1) wehave supL̃(e) = +∞ in any equilibrium e.
Proof. See Appendix.

5.2

Ex Ante Contracts

We have so far assumed that matched parties cannot write ex ante contracts
at all before investment choice because trade decision is assumed to be notcontractible ex ante. In this subsection we will assume that it is ex anteveriﬁable whether or not trade is realized. Thus matched parties can write
ex ante contract contingent on trade outcome before sellers make speciﬁc
investments.

20

Since ex ante contract is introduced, one might think that the ineﬃciency
identiﬁed in the previous sections can be reduced. However, we will showthat this is not always the case and rather the market outcome may be lesseﬃcient when ex ante contracts are possible than when they are not at all.
This implies that enlarging the set of contract space may not improve themarket eﬃciency when the market is more competitive in the sense thatsearch cost is so small.9
To keep consistency of all our assumptions, we also assume that eachmatched pair of a buyer and a seller can verify whether or not they actuallytraded the good in their current match but each of them cannot know thepast history of his or her matched partner such as whether he or she has
matched but has not traded the goods in past periods. 10
We also assume away the message games which use the messages sentby the parties and make the trade decision and payment contingent on
them. The message game is more general mechanism than the simple non–contingent contract which speciﬁes only ﬁxed payments. However, since ourmain purpose here is to show the possibility that ex ante contracting may
make the market eﬃciency worse, it suﬃces to focus on the simple case thatmessage games are ruled out: Our result shows at least that expanding theset of possible contracts does not always improve the market eﬃciency.
Now, since at the beginning of each period ex ante contracts are possible
contingent on trade outcome, in period t matched parties can write an ex
ante contract {Pt1 , Pt0 } where Pt1 (Pt0 ) denotes the payment made from thebuyer to the seller when the good is (not) traded.
Recall that the buyer’s valuation of the good is given by b(a) and theseller’s production cost by d(a). Here we assume that d(a) represents the
personal disutility seller incurs for delivering the good to buyer as well asc(a) is the personal disutility of investment. We also assume that both band d are increasing and positive for all a ∈ A. In the previous sections thetotal trade value v(a) = b(a) − d(a) matters for the equilibrium outcome.
On the other hand, once we handle the case of ex ante contracts, we needto make these notations b(a) and d(e) explicitly.
We make the following assumptions:
Assumption 2. (i) Each matched buyer has the full bargaining power tomake an oﬀer of ex ante contract to matched seller. (ii) Each seller has
9

We may be able to connect this result to the literature about price discrimination
among competing ﬁrms, which has discussed the eﬀects of enlarging the instruments ofﬁrms’ pricing policies on industry proﬁts, consumer surplus and social welfare. In somecases welfare may become lower when more instruments are available for ﬁrms. See Armstrong (2006) for the survey on related topics.
10
If matched parties can observe their past history, they can may make their current
strategies contingent on the observed past history of trade outcomes. This is not consistent
with the assumption of anonymous market.

21

no wealth. (iii) A = [0, a]. (iv) b, d and c are all continuous. (v) S(a) is
increasing in a ∈ [0, as ].
Assumption 2(i) says that ex ante contract is oﬀered by a matched buyer
to a matched seller in the take–it–or–leave–it fashion. One reason for why
buyer has full bargaining power ex ante but not ex post might be that, sinceseller’s investment is match speciﬁc and indispensable for trade, the seller has
some bargaining power ex post after investment was made. Assumption 2(ii)requires that any payment made from buyer to seller must be non–negative,
Pti ≥ 0 for i = 0, 1 and all t. We can readily show that any equilibrium
attains the ﬁrst best outcome when sellers are not wealth constrained (Seethe Remark below). Thus the limited liability of sellers will play the role ofmaking equilibrium comparison meaningful. Assumption 2(iii) and (iv) saythat seller chooses continuous investment which continuously aﬀects boththe buyer’s value b and seller’s disutility of delivery d. Finally Assumption2(v) is satisﬁed when S is a concave function because as is not higher thanthe ﬁrst best investment which maximizes S(a) over A.
Since contracts are incomplete and sellers are subject to limited liability,
by Assumption 2 we are concerned with the situations in which contract
enforcement is limited and severe as in emerging markets.
The timing of events in a period is changed as follows:
date 0 Matched buyer oﬀers an ex ante contract {Pt1 , Pt0 } to a matched seller
where Pti ≥ 0 for i = 0, 1. Then the seller decides whether or not to
accept it. If he rejects it, the parties obtain their reservation values
Utb and Uts .
date 1 The seller makes speciﬁc investment a ∈ A. This is observed to bothparties.
date 1.5 With probability β ∈ (0, 1), the seller makes a renegotiation contractoﬀer to the buyer while with probability 1 − β the buyer makes arenegotiation contract oﬀer to the seller. If the parties agree on renegotiation contract, this will be in force. Otherwise, the ex ante contract
will be in force.
date 2 Trade decision and payments are made according to binding contract(ex ante or renegotiated one).
Given an ex ante contract {Pt1 , Pt0 }, matched parties agree on trade of
the good if and only if the following voluntary trade conditions are satisﬁed(Hart and Moore (1988)):b(a) − Pt1 ≥ −Pt0 + Utb ,(15)

Pt1 − d(a) ≥ Pt0 + Uts(16)

for buyer and22

for seller respectively.
Inequality (15) says that the buyer’s payoﬀ from trade under the ex ante
contract (the left hand side) is not less than her payoﬀ from no–trade underthe ex ante contract (the right hand side). Inequality (16) has a similar
meaning for the seller.
If these conditions are satisﬁed, then the matched parties agree on tradeunder ex ante contract {Pt1 , Pt0 }. Thus they can avoid renegotiation insuch case. If these inequalities are not satisﬁed, no–trade outcome becomesstatus quo and hence they always renegotiate ex ante contract when ex post
surplus from trade is positive. However, in the presence of seller’s limited
liability, renegotiation may fail even when ex post surplus v(a) − Utb − Uts
is positive. The seller makes a renegotiation payment oﬀer R ≥ 0 to ensurethat the buyer accepts it, i.e., b(a) − R ≥ Pt0 + Utb . If b(a) − Pt0 − Utb < 0holds no such R ≥ 0 exists to satisfy the seller’s limited liability constraint.
Thus in that case trade will not occur and the buyer and seller will obtain
the payoﬀs −Pt0 + Utb and Pt0 + Uts respectively. On the other hand, when
the buyer makes a renegotiation oﬀer R ≥ 0, the limited liability constraintof seller is never binding because the seller’s acceptance condition requires
R − d(a) ≥ Pt0 + Uts which yields R ≥ d(a) + Pt0 + Uts ≥ 0.
Remark. The limited liability imposed on sellers plays the important role
here. Without the limited liability constraint, in any equilibrium the ﬁrst
best outcome can be attained under some mild conditions once we introduce
ex ante contracts (See Appendix).
Now the seller’s payoﬀ resulting from renegotiation (net of investment
disutility c(a)) can be written by β max{v(a) − Utb − Uts , 0} + Uts + Pt0 − c(a) if b(a) ≥ Pt0 + Utb ,sut (a) = s
Ut + Pt0 − c(a)otherwise(17)m
We deﬁne the cost minimizing investment, denoted a , as follows:am ≡ arg min d(a) + c(a),
a∈A

(18)

which minimizes the total disutility of seller, delivery and investment disutilities d(a) and c(a). We assume that am is unique and satisﬁes the following:
Assumption 3. S(am ) > v(as ) − c(as )/β.
Assumption 3 says that the seller’s bargaining power β is not so large
relative to the ex ante surplus S(am ) attained by the cost minimizing investment am .
23

We also deﬁne the ex ante contract, denoted C ∗ = {P 1 , P 0 }, which
implements the cost minimizing investment of seller am at the minimum
payment of buyer:
P 1 = d(am ) + c(am ),P 0 = 0.

We will call this C ∗ the cost minimizing contract.
Then we show the following result.
Proposition 7. Suppose that matched parties can write ex ante contract
contingent on trade outcome after matching but before sellers’ investments
and that Assumption 1–3 hold. Then, for small search friction (large δ),
there exists a stationary equilibrium in which every matched buyer oﬀers
the cost minimizing contract C ∗ and every matched seller chooses the cost
minimizing investment am with certainty in every period.
Proof. See Appendix.
The intuition behind Proposition 7 is as follows: Suppose that the reservation values of buyer and seller are stationary and given byb

U =δαS(am )s, U = 0.
1 − δ(1 − α)Suppose also that every matched buyer oﬀers the cost minimizing contract
C ∗ to her matched seller. Then, if the seller accepts this contract andchooses the cost minimizing investment am , the buyer will obtain the payoﬀb(am ) − P 1 = S(am ) while the seller will obtain the payoﬀ equal to P 1 −d(am ) − c(am ) = 0. This can be a stationary equilibrium. Thus we haveto show two things for large δ ∈ (0, 1): First, every matched seller has the
incentive to choose am , given the contract C ∗ and the above reservationbsvalues U and U . Second, every matched buyer has no incentives to oﬀer
other contracts than C ∗ , given these reservation values.
First, given the contract C ∗ , if some matched seller deviates to choosea ̸= am , then he can be better oﬀ only when renegotiation of C ∗ will happen(because otherwise he will end up obtaining P 1 − d(a) − c(a) but this is lessthan the payoﬀ P 1 − d(am ) − c(am ) = 0 by choosing a = am ). However,bsunder renegotiation the seller’s payoﬀ is at most β max{v(a) − U − U , 0} +ss
U − c(a) which is less than U = 0 by choosing a = 0 because small frictionbs(large δ) makes the reservation values large so that U + U → S(am ) >v(as ) − c(as )/β due to Assumption 3. Thus when the search friction is smallenough every matched seller chooses am with certainty in order to avoid
renegotiation of the contract C ∗ .
Second, can some buyer improve her payoﬀ by oﬀering other contracts
than C ∗ ? Answer is No. This is because every matched buyer can implement
24

only the cost minimizing investment am by oﬀering ex ante contract: If
renegotiation occurs under ex ante contract {P 1 , P 0 }, this does not resultin higher payoﬀ of the buyer than S(am ). The reason for this is that whenrenegotiation happens the buyer will obtain the payoﬀ (1 − β) max{v(a) −bsbb
U − U , 0} + U which is however less than U because when δ is large
enough the seller will choose a = 0 under renegotiation as we have seen
above. Thus, if the buyer can be better oﬀ by ex ante contract, it must be
the case that the voluntary trade conditions hold and hence renegotiation
is avoided. For example suppose that the seller chooses a ̸= am . Then theseller’s payoﬀ becomes P 1 − d(a) − c(a) when renegotiation is avoided. We
can then show that the voluntary trade conditions must be satisﬁed withstrict inequalities due to the seller’s limited liability. If the voluntary tradecondition for the buyer b(a) − P 1 ≥ −P 0 + U b holds as equality, her payoﬀ is−P 0 + U b which is not higher than her reservation value U b due to P 0 ≥ 0.
If the voluntary trade condition for the seller P 1 − d(a) ≥ P 0 + U s holds asequality, then his payoﬀ is P 0 +U s −c(a) which is less than the payoﬀ attainedby slightly reducing the investment to a′ (a′ < a) and inducing renegotiation,
β max{v(a′ ) − U b − U s , 0} + U s + P 0 − c(a′ ) for a′ < a. Thus the voluntarytrade conditions must hold with strict inequalities both for the buyer andthe seller. However, then the seller can slightly change his investment froma to some a′′ towards am and obtain higher payoﬀ P 1 − d(a′′ ) − c(a′′ ) as longas a ̸= am . This argument shows that the buyer can implement the cost
minimizing investment am and hence obtain at most S(am ) by oﬀering ex
ante contract.
One important implication of Proposition 7 is that the equilibrium when
ex ante contracts are possible may be less eﬃcient than the equilibrium whenno ex ante contracts are possible. Next we will explore this issue.
First note that Proposition 5 still remains true even when we introduce
the limited liability constraint of sellers: Suppose that no ex ante contracts
are possible. Then the seller’s limited liability constraint results in failure of
renegotiation only if b(a) < Utb , even when the ex post surplus v(a)−Utb −Utsis positive. However, since in the∗ constrained
eﬃcient equilibrium we have
δαx S(as )b(as ) > v(as ) > U b + U s = 1−δ(1−αx
and
hence
b(as ) > Utb , the limited∗)
liability constraint is never binding in the constrained eﬃcient equilibrium
identiﬁed in Proposition 5.
Now we investigate the eﬀects of ex ante contracts on market eﬃciency.
We compare the social welfare attained in the equilibrium when ex ante
contracts are possible with that attained in the equilibrium when they arenot. In particular we consider the equilibrium shown in Proposition 7, which
we will call contract equilibrium, and the equilibrium shown in Lemma 3,
which we will call no–contract equilibrium. Recall here that the no–contractequilibrium is constrained eﬃcient among all possible equilibria when no25

ex ante contracts are written at all (Proposition 5). One might think that
our comparison between these equilibria is arbitrary because we will ruleout other equilibria. However, since our purpose is to demonstrate thepossibility that ineﬃciency of the markets with small friction becomes more
severe when ex ante contracts are introduced than when they are not, it
is suﬃcient to show that there exists at least one such equilibrium with exante contracts which is less eﬃcient than some equilibrium without ex ante
contracts.
By Proposition 7, the social welfare attained in contract equilibrium isgiven byαS(am )W c (α, δ) ≡,(19)
1 − δ(1 − α)when the discount factor δ is large because all matched sellers choose thecost minimizing investment am with certainty in all periods.
On the other hand, the social welfare in no–contract equilibrium wasdeﬁned as W (δ|ê) (see (12)). To make the dependence of W (δ|ê) on both δand α explicit, we can rewrite this as:
W n (α, δ) ≡ αS(as ) + (1 − α)∗δαx∗ S(as )1 − δ(1 − αx∗ )

(20)

s

δαx S(a )ss∗where x∗ ∈ [0, 1] satisﬁes 1−δ(1−αx∗ ) = v(a ) − c(a )/β for δ ∈ (δ , 1) and∗∗x = 1 for all δ ∈ (0, δ ).
We make the following assumption:

Assumption 4. as > am .
Then the direct comparison between W c (α, δ) and W n (α, δ) yields thefollowing result:
Proposition 8. Suppose that Assumption 1–4 hold. Suppose also that the
search friction measured by both α and δ is small, i.e., α and δ are close
to one. Then no–contract equilibrium can attain higher social welfare thancontract equilibrium, i.e., W n (α, δ) > W c (α, δ).
Proof. When δ tends to be close to one, we have W c (α, δ) → S(am ) while
W n (α, δ) → αS(as ) + (1 − α)[v(as ) − c(as )/β]which is close to S(as ) when α → 1.
By Assumption 2(v) S is increasing over [0, as ] and hence we have S(as ) >S(am ) due to Assumption 4. Q.E.D.
For Proposition 8 to be meaningful, Assumption 4 must be consistent
with Assumption 3. We can give an example to ensure this as follows:26

Example. Assume that A = [0, 1], b(a) = 1 + (1 − ρ)a, d(a) = 1 − ρa andc(a) = (1/2)a2 where ρ ∈ (0, β). Thus v(a) = a. In this quadratic example
we can show that as = β, am = ρ, S(am ) = ρ(1 − (1/2)ρ), v(as ) − c(as )/β =(1/2)β and S(as ) = β(1 − (1/2)β). Thus we have as = β > am = ρ which
satisﬁes Assumption 4. Also, when 1/2 > β, we have S(am ) > v(as ) −c(as )/β for all ρ ∈ (0, β) which satisﬁes Assumption 3.
Proposition 8 shows that introduction of ex ante contracts does not mitigate the holdup ineﬃciency caused by small market friction but rather exaggerates such ineﬃciency. In fact enlarging the set of available contracts may
make the market eﬃciency worse than the case of limited contracts. Thus
this might imply that it is not optimal policy to make markets competitivein developing countries with limited contract enforcement.

5.3

Two–Sided Investment

In this subsection we will extend the basic model to allow both matchedbuyers and sellers to make speciﬁc investments. One might think that, if
matched buyers also make non–contractible investment as well as sellers do,it becomes more diﬃcult to improve eﬃciency. However, this argument is nottrue in the dynamic search market model we are considering here. Surprisingly, we show that in completely symmetric case the two–sided investment
may help recover eﬃciency when the ineﬃciency identiﬁed in Proposition 2
arises in the case of one–sided investment.
To see this, consider the following example:
Example 1. Consider the symmetric and two–sided investment case: Each
matched buyer chooses match speciﬁc investment aB ∈ {0, 1} as well as eachmatched seller does so aS ∈ {0, 1}. Party i (i = B, S) incurs the investment
cost cai where c > 0. Also each party has the same ex post bargaining
power β = 1/2. Let v(aB , aS ) be the trade value of the good and assume
that v(·, ·) is increasing andS ∗ ≡ v(1, 1) − 2c > v(1, 0) = v(0, 1).
This also implies that it is the ﬁrst best optimal to implement high investments aB = aS = 1 from both parties because we have v(1, 1) − 2c >v(0, 1) − c and v(1, 1) − 2c > v(0, 0). Also assume S ∗ > 0.
Then we can show that there exists an equilibrium which implements theﬁrst best eﬃciency when the search friction tends to be small (δ → 1), eventhough this is not the case when only one party of matched pair invests.
To see this result, ﬁrst consider the case of one–sided investment. Thensuppose that δ is so high that there exist no equilibria in which every

27

matched investing party chooses high investment with probability one asshown by Proposition 2.
Now we consider the two–sided investment case that both parties of eachmatch choose investments, ai ∈ {0, 1}, i = B, S. Then we will show that
there exists a stationary equilibrium in which every matched party chooses
high investment aB = aS = 1 with probability one.
To see this, suppose that the reservation values UB∗ and US∗ are given as
follows:
δαS ∗UB∗ + US∗ =.
1 − δ(1 − α)Then consider the investment incentive of one party of a matched pair, sayi, given the other party choosing high investment aj = 1. If i chooses ai = 1
as well, then he obtains the payoﬀ:
1{v(1, 1) − UB∗ − US∗ } + Ui∗ − c.
2Here the ex post surplus v(1, 1)−UB∗ −US∗ is always positive for all δ ∈ (0, 1).
On the other hand, if i chooses ai = 0, then he can obtain the following
payoﬀ:
1
max{v(0, 1) − UB∗ − US∗ , 0} + Ui∗ .
2Then, since v(1, 1)−2c > v(0, 1) and deﬁnition of UB∗ +US∗ , the term v(0, 1)−
UB∗ − US∗ converges to v(0, 1) − [v(1, 1) − 2c] < 0 as δ → 1. Thus ex post
surplus is negative, which shows that i’s payoﬀ becomes Ui∗ by choosing
ai = 0. In particular, there exists some δ ∈ (0, 1) such that for all δ ∈ (δ, 1)
we have v(0, 1) − UB∗ − US∗ < 0. Thus party i chooses ai = 1 for all δ ∈ (δ, 1),
if the following inequality is satisﬁed
1{v(1, 1) − UB∗ − US∗ } + Ui∗ − c ≥ Ui∗ , i = B, S.
2
In fact we derive1{v(1, 1) − UB∗ − US∗ } − c2{}1δα[v(1, 1) − 2c]=v(1, 1) −−c
2
1 − δ(1 − α)1
> {v(1, 1) − [v(1, 1) − 2c]} − c21= 2c − c2= 0.
Thus there exists a stationary equilibrium in which all matched parties
choose the ﬁrst best investment aB = aS = 1, given their reservation values
28

∗

δαS
.
UB∗ and US∗ where UB∗ + US∗ = 1−δ(1−α)

However, this result depends on the assumption that both buyers andsellers have the same investment cost as well as the same bargaining power.
In fact we can show that, when buyers and sellers are heterogeneous withrespect to their investment costs and bargaining powers, the ineﬃciency
result shown in Proposition 2 still appears. More precisely, we show that
there generically exist no equilibria in which both matched buyer and sellerchoose high investments (aB = as = 1) with probability one.
To see this, consider the following example:
Example 2. We modify Example 1 as follows: The buyer’s cost of choosing
high investment cB diﬀers from the seller’s cost cS (cB ̸= cS ).
Then the following incentive compatibility constraints must be satisﬁed
if there exists a stationary equilibrium in which every matched pair chooses
high investment aB = aS = 1 (ﬁrst best investment) with probability one:β{v(1, 1) − UB∗ − US∗ } + US∗ − cS ≥ US∗
and(1 − β){v(1, 1) − UB∗ − US∗ } + UB∗ − cB ≥ UB∗

where Ui∗ denotes the equilibrium reservation value of party i = B, S. The
reason why the above inequalities must hold is that each matched party canchoose low investment (ai = 0) and ensure at least the reservation value Ui∗
by himself.
Also the equilibrium reservation values UB∗ and US∗ must satisfy
UB∗ + US∗ =δα[v(1, 1) − cB − cS ].
1 − δ(1 − α)Let δ → 1. Then the above inequalities can be written byβ(cB + cS ) ≥ cS
and(1 − β)(cB + cS ) ≥ cB .
Since these inequalities must hold as equalities, we must have
1−β
cB
=
.
β
cS
However, a small perturbation of the parameter values violates this equality,
which shows that there generically exist no equilibria in which both matchedbuyer and seller choose high investments (the ﬁrst best investment) withprobability one.

29

Example 2 shows that the ineﬃciency result shown by Proposition 2
appears again even when both parties of matched pair make speciﬁc investments, if a slight perturbation of the parameters breaks exact symmetry
between buyers and sellers. In this sense our ineﬃciency result shown in
Proposition 2 is generically robust even when both parties of matched pairmake speciﬁc investments.

6

Conclusion

In this paper we have investigated the holdup problem in the search market
in which many buyers and sellers search for trading partners and speciﬁc investments are made before trade but after match. Then we have shown thatthe holdup problem imposes more serious ineﬃciency when search friction
becomes smaller: When search friction tends to be small, equilibrium investment is dropped down to zero and trade is delayed with positive probability
for inﬁnitely many times. Thus equilibrium delay of trade must occur with
non–trivial probability even in the complete information setting.

7

Appendix

7.1

Proof of Lemma 1. Every matched seller’s equilibrium payoﬀ ust in period t is given by∫ust ≡ {β max{v(a) − Uts − Utb , 0} + Uts − c(a)}dft .
A

Let uS (a) be the function of a:uS (a) ≡ β max{v(a) − Uts − Utb , 0} + Uts − c(a).
Then uS (a) can be written bys − U b } + U s − c(a) for a such that v(a) > U s + U b , β{v(a) − Ut+1
t
t
t+1
t+1
uS (a) = s
Ut − c(a)otherwise.
Now suppose that ft (A \ {0, as }) > 0. In particular take any a′′ ∈ A \ {0, as }and suppose that uS (a′′ ) ≥ max{uS (0), uS (as )}. Then, since any seller canchoose a = 0 and guarantee at least the payoﬀ Uts , we must have uS (a′′ ) ≥Uts , which can be written by
β max{v(a′′ ) − Utb − Uts , 0} ≥ c(a′′ ).
Then, since a′′ > 0 and c(a′′ ) > 0, v(a′′ ) > Utb + Uts must hold and hence theabove inequality yields v(a′′ ) − c(a′′ )/β ≥ Utb + Uts , which in turn impliesv(as ) − c(as )/β > v(a′′ ) − c(a′′ )/β ≥ Utb + Uts
30

by deﬁnition of as and Assumption 1. Thus v(as ) > Utb + Utb is satisﬁed andthen some matched seller will deviate to choose as with certainty and obtaina higher payoﬀ than uS (a′′ ). This is a contradiction. Thus ft (A\{0, as }) = 0
must hold in any period t. Q.E.D.

7.2

Proof of Lemma 2

First suppose that at = as was realized according to the equilibrium strategy
xt in period t. Then, since as was realized, xt > 0 must be satisﬁed. Since
any seller can choose a = 0 and guarantee at least the payoﬀ Uts , we musthave
β max{v(as ) − Utb − Uts , 0} ≥ c(as ),which in turn implies v(as ) > Utb + Uts due to c(as ) > 0. Since ex post
surplus v(as ) − Utb − Uts is positive, trade must occur after at = as was
realized.
Second suppose that at = 0 was realized. This implies xt < 1. Then,if v(0) ≥ Utb + Uts holds, we have v(as ) > Utb + Uts and hence the seller isbetter oﬀ by choosing as with certainty due to Assumption 1. Thus it must
be the case thatv(0) < Utb + Uts
which shows that ex post surplus v(0) − Utb − Uts is negative and thus trade
never occurs after at = 0 was realized. Q.E.D.

7.3

Proof of Proposition 1

By Lemma 1, we can identify the seller’s equilibrium strategy ft by theprobability xt ∈ [0, 1] putting on a = as and the probability 1 − xt on a = 0
respectively. Thus we will write the equilibrium strategy as xt instead of ft .
Take any equilibrium path {xt , Utb , Uts }∞t=0 . Then by Lemma 1 and 2,
the sum of the reservation values of a buyer and a seller can be written as
follows:bs
Utb + Uts = δ{αxt+1 S(as ) + (1 − αxt+1 )(Ut+1+ Ut+1)}(A1)

because every matched seller chooses the static equilibrium investment aswith probability xt ∈ [0, 1] in period t, only in which case trade occurs andthe trade value S(as ) is realized.
First we show that Utb + Uts < S(as ) for all t. Suppose that UTb + UTs ≥S(as ) for some period T . Then by (A1) we haveS(as ) ≤ UTb + UTs= δ{αxT +1 S(as ) + (1 − αxT +1 )(UTb +1 + UTs +1 )}≤ δαxT +1 S(as ) + (1 − δαxT +1 )(UTb +1 + UTs +1 )31

which shows S(as ) ≤ UTb +1 + UTs +1 because 1 > δαxT +1 . Repeating thisargument, we have Ukb + Uks ≥ S(as ) for all k ≥ T . This then implies thatUkb + Uks ≥ S(as ) > v(as ) − c(as )/β and hence xk = 0 for all k ≥ T . Then,by Lemma 2 trade never occurs in all periods k ≥ T and thus every player
obtains his or her outside payoﬀ, zero, in each period from period T forever,irrespective of matching a partner. However, then Uki = 0 for all k ≥ T andi = b, s, which contradicts Ukb + Uks ≥ S(as ) > 0 for all k ≥ T . Thus wemust have S(as ) > Utb + Uts for all t.
Uti ≥ 0 must also hold for all t and i = b, s because any player can always
obtain his or her outside payoﬀ, zero, in any period by rejecting trade.
Then we derivebs
Utb + Uts = δ{αxt+1 S(as ) + (1 − αxt+1 )(Ut+1+ Ut+1)}bs≤ δ{αS(as ) + (1 − α)(Ut+1+ Ut+1
)}
{
}
T
−1
T
−1
(1 − α)s 1−δTbs= δ αS(a )+ δ (UT + UT )1 − δ(1 − α)where the inequality follows from the fact that S(as ) > Utb + Uts for all t inany equilibrium. Then, by taking large enough T (T → ∞) and noting thatUTb + UTs is bounded from above by S(as ) and from below by zero, for theabove inequality to hold for any T , we must have
Utb + Uts ≤δαS(as ), ∀ t.
1 − δ(1 − α)Thus, the matched seller’s payoﬀ function uS (a) deﬁned byuS (a) = β max{v(a) − Uts − Utb , 0} + Uts − c(a),can attain its maximum at a = as becausev(as ) − c(as )/β >δαS(as )≥ Utb + Uts
1 − δ(1 − α)for all δ ∈ (0, δ ∗ ). This shows that each matched seller chooses as withprobability one. Q.E.D.

7.4

Proof of Proposition 2

Now take an equilibrium path e = {xt , Utb , Uts }∞t=0 . Then we show that
supL(e) = +∞.
Suppose contrary to the claim that supL(e) < +∞ in some equilibrium
e. Then there must exist some period T such that we have xt = 1 for anyt ≥ T.
Consider period T + 1. Then we ﬁrst show thatUTb +1 + UTs +1 =32

δαS(as ).
1 − δ(1 − α)sδαS(a )To see this, ﬁrst suppose that UTb +1 + UTs +1 > 1−δ(1−α). Then we can showthat UTb +2 + UTs +2 > UTb +1 + UTs +1 because, if not, since xT +2 = 1 we have
UTb +1 + UTs +1 = δ{αS(as ) + (1 − α)(UTb +2 + UTs +2 )}≤ δ{αS(as ) + (1 − α)(UTb +1 + UTs +1 )}sδαS(a )
which shows UTb +1 +UTs +1 ≤ 1−δ(1−α), a contradiction. Thus UTb +2 +UTs +2 >b +U sUTb +1 +UTs +1 . Repeating this argument, we must have Ukb +Uks > Uk−1
k−1
for all k ≥ T +2. However, then there must exist some period K (K ≥ T +1)
b + U s > v(as ) − c(as )/β, which implies x
such that UK
K = 0 but thisKδαS(as )bcontradicts xt = 1 for all t > T . Thus we must have UT +1 +UTs +1 ≤ 1−δ(1−α).
sδαS(a )
Next suppose that UTb +1 + UTs +1 < 1−δ(1−α). Then, by a similar argument,bs
we can show that Ukb + Uks > Uk+1
+ Uk+1for all k ≥ T . However, thenbs
UM + UM < 0 for some period M > T , which contradicts to Uti ≥ 0 for allt and i = b, s. Thus we must have

UTb +1 + UTs +1 =δαS(as ).
1 − δ(1 − α)Then, given this value, each matched seller’s equilibrium payoﬀ in period
T + 1 is written by
usT +1 ≡ β{v(as ) − UTb +1 − UTs +1 } + UTs +1 − c(as )sδαS(a )
because xT +1 = 1. However, by substituting UTb +1 + UTs +1 = 1−δ(1−α)
into∗sthis expression, we can verify that for any δ ∈ (δ , 1) uT +1 must be lessthan UTs +1 which can be attained at least by choosing aT +1 = 0. This is acontradiction. Thus we must have supL(e) = +∞. Q.E.D.

7.5

Proof of Proposition 3

For a positive integer K ≥ 1, deﬁne the sequence {UtN }K
k=1 recursively as
follows:N
UkN = δUk+1, k = 1, 2, ..., K − 1,andNUK= δ{αS(as ) + (1 − α)U I }where
U I = δU1N .
Solving these values, we obtainUkN =δ K−k+1 αS(as ),1 − δ K+1 (1 − α)33

k = 1, 2, ..., K,

and
UI =

δ K+1 αS(as ).
1 − δ K+1 (1 − α)Since U1N > U I for all δ ∈ (0, 1) and both U1N and U I are increasing in δ,
we can ﬁnd some δ K and δ K such that for all δ ∈ (δ K , δ K ),U1N =

δ K αS(as )δ K+1 αS(as )> v(as ) − c(as )/β > U I =
.
K+1
1−δ
(1 − α)1 − δ K+1 (1 − α)N for all k = 2, 3, ..., K, we verify that U N > v(as ) −Then, since UkN > Uk−1ksc(a )/β for all k = 1, 2, ..., K.
Now consider the following strategies of matched sellers: In period t =τ K + τ − 1 (τ = 1, 2, ...) all matched sellers choose the static equilibrium
investment as with certainty. In all other periods all matched sellers choose
the least costly investment, zero, with certainty.
We also deﬁne the reservation values {Utb , Uts }∞t=0 as follows: I
for t = τ K + τ − 1 (τ = 1, 2, ...),
 Ubs
Ut + Ut = N
Uk for t = k − 1 (mod K + 1), k = 1, 2, ..., K.

Then, the above strategies become optimal, given these reservation values
because β{v(as ) − Utb − Uts } ≥ c(as ) for all t = τ K + τ − 1 (τ = 1, 2, ...),
which implies choosing at = as is optimal, while β{v(as ) − Utb − Uts } < c(as )for all other periods, which imply choosing at = 0 is optimal.
Finally, the above sellers’ strategies are consistent with the constructionof the reservation values. Q.E.D.

7.6

Proof of Proposition 5

By Proposition 1 we know that equilibrium is unique and same as the static
one when δ ∈ (0, δ ∗ ): In every period matched seller chooses the static
equilibrium investment as with probability one. Thus this equilibrium istrivially constrained eﬃcient because no other equilibria exist.
Next we will turn to the case that δ ∈ (δ ∗ , 1). In this case multiple
equilibria arise for some discount factors as we have shown in Proposition 3
and 4.
Consider the equilibrium ê shown in Lemma 3 and note that the social
welfare under ê can be written by
W (δ|ê) = αx̂0 S(as ) + (1 − αx̂0 ) + (1 − α)(Ûtb + Ûts )}= αS(as ) + (1 − α)[v(as ) − c(as )/β]for δ ∈ (δ ∗ , 1) becausev(as ) − c(as )/β =34

δαx∗ S(as ).
1 − δ(1 − αx∗ )andÛtb + Ûts =δαx∗ S(as )∀ t.
1 − δ(1 − αx∗ )Now take any other equilibrium, ẽ = {x̃t , Ũtb , Ũts }∞t=0 , than ê. Then we
have the social welfare under ẽ as
W (δ|ẽ) = αx̃0 S(as ) + (1 − αx̃0 )(Ũ0b + Ũ0s )
where
b
s
Ũtb + Ũts = δ{αx̃t+1 S(as ) + (1 − αx̃t+1 )(Ũt+1+ Ũt+1)}, t = 1, 2, ...

We will consider the following separate cases:
CASE 1: v(as ) − c(as )/β ≥ Ũ0b + Ũ0s .
In this case we have
W (δ|ẽ) = αx̃0 S(as ) + (1 − αx̃0 )(Ũ0b + Ũ0s )≤ αx̃0 S(as ) + (1 − αx̃0 )[v(as ) − c(as )/β]≤ αS(as ) + (1 − α)[v(as ) − c(as )/β]= W (δ|ê)
where the ﬁrst inequality follows from our supposition above and the second inequality from the fact that S(as ) = v(as ) − c(as ) > v(as ) − c(as )/β
respectively.
Thus the equilibrium ẽ never attains higher welfare than ê.
CASE 2: v(as ) − c(as )/β < Ũ0b + Ũ0s .
In this case we have x̃0 = 0. Let T > 0 be the period such that x̃t = 0 for
any t < T but x̃T > 0. 11
Then, by x̃T > 0, we must havev(as ) − c(as )/β ≥ ŨTb + ŨTs .
Under the equilibrium path ẽ, we then obtain
W (δ|ẽ) = δ T {αx̃T S(as ) + (1 − αx̃T )(ŨTb +1 + ŨTs +1 )}≤ δ T {αx̃T S(as ) + (1 − αx̃T )[v(as ) − c(as )/β]≤ δ{αx̃T S(as ) + (1 − αx̃T )[v(as ) − c(as )/β]}≤ δ{αS(as ) + (1 − α)[v(as ) − c(as )/β]}< αS(as ) + (1 − α)[v(as ) − c(as )/β]= W (δ|ê)11

If xt = 0 for all t, then by Lemma 2 no trades are realized in all periods and hence
W (δ|ẽ) = 0, which is less than W (δ|ê) > 0.

35

b + Ũ s , the second
where the ﬁrst inequality follows from v(as )−c(as )/β ≥ ŨK
Ksinequality from δ < 1, the third inequality from S(a ) > v(as ) − c(as )/β andthe last equality from the deﬁnition of ê respectively. Thus there exist noother equilibria which is more eﬃcient than ê in this case as well. Q.E.D.

7.7

Proof of Proposition 6

Suppose not, i.e., supL̃(e) < +∞ in some equilibrium e. This means that
there exists some ﬁnite time T ≥ 0 such that for all t ≥ T we have xt (θ, ψ) =1 for all (θ, ψ) ∈ Θ × Ψ. Thus, by using a smilar logic to the proof of
Proposition 2, we can show that the sum of the reservation values must be
stationary from period T onward and hence be constant at Ub + Us , i.e.,
U b + U s = δ{αEθ,ψ [S(as (θ, ψ), θ, ψ)] + (1 − α)(Ub + US )}where S(a, θ, ψ) ≡ v(a, θ) − c(a, ψ) is the ex ante total surplus.
Solving this, we have
Ub + Us =αδEθ,ψ [S(as (θ, ψ), θ, ψ)].
1 − δ(1 − α)Take some (θ′′ , ψ ′′ ) ∈ Θ×Ψ such that S(as (θ′′ , ψ ′′ ), θ′′ , ψ ′′ ) ≤ Eθ,ψ [S(as (θ, ψ), θ, ψ)].
Then we obtainv(as (θ′′ , ψ ′′ ), θ)−c(as (θ′′ , ψ ′′ ), ψ ′′ )/β < S(as (θ′′ , ψ ′′ ), θ′′ , ψ ′′ ) ≤ Eθ,ψ [S(as (θ, ψ), θ, ψ)]which shows that there exists some δ ∈ (0, 1) such that for all δ ∈ (δ, 1) wemust havev(as (θ′′ , ψ ′′ )) − c(as (θ′′ , ψ ′′ ))/β <δαEθ,ψ [S(as (θ, ψ), θ, ψ)].
1 − δ(1 − α)Thus seller of type ψ ′′ will never choose as (θ′′ , ψ ′′ ) when he matches withbuyer of type θ′′ in any period t ≥ T , which implies xt (θ′′ , ψ ′′ ) = 1 for allt ≥ T . However this is a contradiction. Q.E.D.

7.8

Proof of Proposition 7

Consider the following reservation values:b

Utb = U ≡δαS(am )s, Uts = U ≡ 0 ∀ t.
1 − δ(1 − α)Then we will show that, given these reservation values, each matched buyer
optimally oﬀers the cost minimizing contract C ∗ and each matched sellerchooses the cost minimizing investment am with certainty in any period.
bsLemma A1. Given C ∗ and (U , U ), every matched seller optimally chooses
36am with certainty when δ is large.
Proof. Suppose that matched seller chooses am . Then the voluntary tradeconditions under C ∗ are satisﬁed:bb(am ) − P 1 = S(am ) > P 0 + U =andδαS(am ),1 − δ(1 − α)s

P 1 − d(am ) = c(am ) > P 0 + U = 0.
Thus, if the seller chooses am , he obtains the payoﬀ P 1 − d(am ) − c(am ) = 0.
Now suppose that the seller chooses a ̸= am . Then, if the voluntarytrade conditions still hold, he obtains the payoﬀ P 1 − d(a) − c(a) which isless than zero by deﬁnition of am . If the voluntary trade conditions are notsatisﬁed, then renegotiation may happen and hence the seller can obtain at
most
s
b
s
us (a) = β max{v(a) − U − U , 0} + U − c(a).
bNote here that, if b(a) < U , the seller’s renegotiation payment oﬀer does
not satisfy his limited liability and hence renegotiation fails, in which cases
the seller obtains U − c(a). Thus the most proﬁtable case for the seller isbthat b(a) > U and thus renegotiation succeeds, in which case his payoﬀ isgiven as us (a) above. However, by Assumption 3, for large δ we havebs

U +U =

δαS(am )> v(as ) − c(as )/β.
1 − δ(1 − α)sThus maxa∈A us (a) = U = 0 when δ is high enough, which shows that theseller’s deviation payoﬀ is not higher than zero. Thus in either case the seller
has no incentives to deviate from choosing am . Q.E.D.
Lemma A2. Every matched buyer optimally oﬀers C ∗ when δ is large,bs
given (U , U ).
Proof. If a matched buyer oﬀers C ∗ , then she can obtain the payoﬀ S(am )by Lemma A1 when δ is large. Then we will show that the buyer cannot
obtain higher payoﬀ by oﬀering diﬀerent contracts.
To see this, suppose that some matched buyer oﬀers Ĉ = {P̂ 1 , P̂ 0 } ̸= C ∗and that her matched seller’s investment is realized as â according to his
mixed strategy, given this contract. We can suppose that â > 0 because, if
the seller’s mixed strategy puts all positive probabilities on a = 0, then thebuyer’s payoﬀ must be at most S(0), which is less than his reservation valueb
U when δ is large enough.12
For large δ, U b can be close to S(am ) which is higher than S(0) because S(am ) =b(a ) − (d(am ) + c(am )) > b(am ) − (d(0) + c(0)) > S(0) = b(0) − d(0) − c(0).
12

m

37

Suppose ﬁrst that the voluntary trade conditions are satisﬁed under Ĉ:bb(â) − P̂ 1 ≥ −P̂ 0 + U ,sP̂ 1 − d(â) ≥ P̂ 0 + U .

(A2)
(A3)

Thus the seller obtains the payoﬀ P̂ 1 − d(â) − c(â).
Note also that the buyer can guarantee the payoﬀ S(am ) by oﬀeringC ∗ because the seller accepts it and chooses am following no renegotiationb(due to Lemma A1). Then, since S(am ) > U , the buyer’s deviation payoﬀbmust be strictly higher than his reservation value U . From this, the aboveﬁrst voluntary trade condition (A2) must be satisﬁed with strict inequality
because otherwise the buyer’s payoﬀ is not higher than the reservation valuebb
U b , i.e., −P̂ 0 + U ≤ U due to P̂ 0 ≥ 0.
We will then show that â = am must hold. To see this, suppose â ̸= am
and consider two separate cases:s

CASE 1: P̂ 1 − d(â) = P̂ 0 + U .
In this case the seller obtains the payoﬀ P̂ 1 − d(â) − c(â). However, if
the seller slightly reduces investment from â > 0 and chooses a′ < â (this ispossible because A is closed interval and c is continuous), then renegotiation
may or may not occur. In either case the seller obtains at least the following
payoﬀ
U s + P̂ 0 − c(a′ ) = P̂ 1 − d(â) − c(a′ )which is higher than the supposed payoﬀ P̂ 1 −d(â)−c(â) due to c(â) > c(a′ ).
Thus the seller has the incentive to deviate from â.
s

CASE 2: P̂ 1 − d(â) > P̂ 0 + U .
In this case, if the seller can slightly change investment level from â towards
am and choose a′ , then the voluntary trade conditions can be still satisﬁed
because b and d are continuous and we already know that the voluntarytrade condition for the buyer (A2) is strictly satisﬁed under a = â. Thus
the seller can obtain the following payoﬀ
P̂ 1 − d(a′ ) − c(a′ )which is higher than the supposed payoﬀ P̂ 1 − d(â) − c(â) (by a′ → am ).
Thus the seller has the incentive to deviate from â.
The above CASE 1 and 2 show that â = am . However, for the purposebsof implementing a = am given the reservation values U and U , it is themost proﬁtable for the buyer to oﬀer the contract C ∗ because then she canextract the full surplus S(am ) for implementing am . This implies that eachmatched buyer has no incentives to oﬀer other contracts than C ∗ , given thebsreservation values (U , U ).
38

Thus the remaining case is only that renegotiation happens under thedeviation contract Ĉ. In that case the payoﬀs of the parties are determinedby the ex post bargaining. Then the seller will obtain the following payoﬀ
from the supposed investment level â:bss

us (â) = β max{v(â) − U − U , 0} + U + P̂0 − c(â).
bmsδαS(a )
However, since U + U = 1−δ(1−α)> v(as ) − c(as )/β for large δ under
Assumption 3, we havesus (â) ≤ U + P̂ 0 − c(â).
Thus the seller must have chosen â = 0 with certainty.
Then the buyer’s payoﬀ can be at mostbsb

(1 − β) max{v(0) − U − U , 0} + U − P̂ 0
b

b

s

m

δαS(a )
which is however not larger than U because we have U + U = 1−δ(1−α)>
ssmv(a ) − c(a )/β > v(0). Since the buyer can obtain S(a ) by oﬀering thebex ante contract C ∗ and S(am ) > U , she optimally oﬀers the contract C ∗ .
Q.E.D.

Lemma A1 and A2 have established the result that every matched buyer
oﬀers the cost minimizing contract C ∗ and every matched seller chooses ambsδαS(am )and U = 0. Also these reservation
with certainty, given U = 1−δ(1−α)
values are consistent with the contract oﬀer of buyers and investment choiceof sellers as well. Thus this can be a stationary equilibrium. Q.E.D.

7.9

Proof of the Remark

We make the mild assumption that d(a) + c(a) is increasing in a ∈ (am , a](this will hold when d + c is convex function). We deﬁne the ﬁrst best
investment as a∗ which maximizes ex ante surplus S(a). Then, ﬁx any pairof reservation values (Utb , Uts ) in period t and consider the following ex ante
contract {Pt1 , Pt0 }:b(a∗ ) − Utb = Pt1 − Pt0 ≥ d(a∗ ) + Uts ,which is oﬀered by a matched buyer. Note here that Utb + Uts < S(a∗ ) for
all t and hence such Pt1 and Pt0 exist for these inequalities to be satisﬁed.
Given this contract, the voluntary trade conditions hold if the sellerchooses a∗ . In that case the seller obtains Pt1 − d(a∗ ) − c(a∗ ). On the other
hand, if the seller chooses a < a∗ , then these conditions will not hold (sinceb(a) < b(a∗ ) for a < a∗ ) and hence renegotiation will occur. Then the seller
obtains the payoﬀ β max{v(a) − Utb − Uts , 0} + Uts + Pt0 − c(a). However, this39

is less than the payoﬀ Pt1 − d(a∗ ) − c(a∗ ) obtained by choosing a∗ . Since theseller has no incentives to choose a > a∗ ,13 the seller optimally chooses a∗ .
Then the buyer can extract the full surplus S(a∗ ) by setting Pt0 = Utb −S(a∗ ).
Q.E.D.

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13

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40

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42W (δ|es )δ∗0Figure 1:

Social Welfare in Stationary Equilibrium

43

1

δ

W (δ|ê)

W (δ|ens )

δ∗

0

δK
Figure 2:

Social Welfare in Non–Stationary Equilibrium

44

δK

1

δ