Discussion Papers In Economics
And Business

Relationship-specific Investment as a Barrier to Entry
Hiroshi Kitamura, Akira Miyaoka and Misato Sato

Discussion Paper 13-24

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRelationship-specific Investment as a Barrier to Entry
Hiroshi Kitamura, Akira Miyaoka and Misato Sato

Discussion Paper 13-24

September 2013

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANRelationship-specific Investment as a Barrier to Entry∗
Hiroshi Kitamura†

Akira Miyaoka‡

Misato Sato §September 16, 2013

Abstract
This study constructs a model of a relationship-specific investment in a dynamic
framework. Although such investment decreases operating costs and increases the current joint profits of firms in vertical relationships, its specificity reduces the ex-post flexibility to change a trading partner in the future. We demonstrate that whether the investment contract deters entry even in the absence of exclusionary terms depends on not only
the specificity but also the efficiency of the investment. We also show that an increase in
the investment efficiency does not necessarily improve the equilibrium social welfare.

JEL Classifications Code: L12, L41, L42.
Keywords: Vertical Relation; Entry Deterrence; Relationship-Specific Investment; Switching Costs∗
We would like to thank Katsuya Takii, Reiko Aoki, Koki Arai, Hiroaki Ino, Akira Ishii, Hideshi Itoh,
Masayuki Kanezaki, Keisuke Hattori, Noriaki Matsushima, Keizo Mizuno, Jun Nakabayashi, Hiroyuki Odagiri, Jun Oshiro, Daniel Rubinfeld, Daisuke Shimizu, Tetsuya Shinkai, Mitsuru Sunada, Noriyuki Yanagawa,Takenobu Yuki, conference participants at the Japanese Association for Applied Economics, Japanese Economic
Association, and Institutions and Economics International Conference and seminar participants at Hitotsubashi
University, Japan Fair Trade Commission, Kwansei Gakuin University, Osaka University, Osaka Prefecture
University, and Sapporo Gakuin University for helpful discussions and comments. We gratefully acknowledge
financial support from JSPS Grant-in-Aid for Research Activity start-up No. 22830075, for Young Scientists(B) No. 24730220, for Scientific Research (A) No. 22243022, and for JSPS Fellows No. 12J01593 and financial support from the Global COE program entitled “Human Behavior and Socioeconomic Dynamics” of Osaka
University. The usual disclaimer applies.
†
Corresponding Author: Faculty of Economics, Kyoto Sangyo University, Motoyama, Kamigamo, Kita-Ku,
Kyoto-City, Kyoto, 603-8555 Japan. E-mail: hiroshikitamura@cc.kyoto-su.ac.jp‡
Graduate School of Economics, Osaka University, 1-7 Machikaneyama, Toyonaka, Osaka, 560-0043,
Japan. Email: jge013ma@mail2.econ.osaka-u.ac.jp§
Department of Economics, The George Washington University, 2115 G street, NW Monroe Hall 340 Washington DC 20052, USA. Email: smisato@gwmail.gwu.edu

1 Introduction
In a number of vertical relationships, upstream firms make a relationship-specific investment
in downstream firms. For example, some firms develop original systems or instruments andintroduce them to downstream firms.1 Similarly, some firms provide special training programs to the downstream firm’s employees.2 These types of investments seem to decrease
the operating costs and increase the joint profits of firms in vertical relationships. In addition,
they seem to increase industry output and generate a socially efficient effect. From a static
viewpoint, these predictions are correct.
However, a relationship-specific investment usually generates specificity in vertical relationships. For example, through special training programs or the introduction of original
systems, employees in downstream firms can establish specific routines (called “habitual routines”) for their operations with a current trading partner. Owing to such specificity, whendownstream firms change the trading partner, employees must usually unlearn these old routines before commencing operations with a new trading partner (Postrel and Rumelt, 1992).
This generates switching costs such as temporary performance degradation or an increase
in retraining costs for the ex-post change of the trading partner. Therefore, a relationshipspecific investment may deter efficient future entry and lead to socially inefficient outcomesin a dynamic perspective.3
This study aims to theoretically explore the role of a relationship-specific investment as abarrier to efficient entry and its welfare implication based on a dynamic framework. It constructs a two-period model of entry deterrence with the relationship-specific investment. In
the first period, one upstream incumbent exists but in the second period, a more efficient firm
appears. The upstream incumbent offers the single downstream firm an investment contract
with some fixed payments such as an introduction campaign for new systems or instruments
for a limited time.
1

For example, Ticketmaster, the leading ticket sales and distribution company in the United States, develops
an original system for concert venues. See Whinston (2006). See also Holden and O’Toole’s (2004) study, which
explores the role of communication in determining the governance of manufacturer–retailer relationships.
2
Ticketmaster also trains a venue’s personnel in the use of its system.
3
Postrel and Rumelt (1992) refer to the difficulty experienced by several companies such as Wedgwood andNucor in changing employees’ old habits. Further, in the psychological literature, Shiffrin and Schneider (1977)
provide experimental evidence on this issue.

1

The key assumption in our model is that investment specificity generates switching costs
when the downstream firm changes its trading partner. When specificity is high, the acceptance of such investment restricts trading partner choices of the downstream firm because it
makes dealing with the entrant in the second period impossible. This exposes the downstream
firm to the intertemporal trade-off that once it accepts the specific investment, it becomes efficient in the current period but cannot transact with an efficient entrant in the future. Therefore,
for examining the existence of entry deterrence, we need to focus on the downstream firm’s
incentive to accept or reject the investment.
In this setting, we explore the existence of entry deterrence because of the relationshipspecific investment and show that not only the investment specificity but also its efficiency
is an important factor that makes entry deterrence possible. To understand the importance ofinvestment efficiency in triggering entry deterrence, we begin by analyzing the case where theinvestment is not efficient: that is, the investment does not reduce operating costs. We showthat the incumbent cannot then profitably design the investment contract to deter future entry
even when the investment is considerably specific. When the downstream firm rejects theinvestment, the entrant offers a better wholesale price than the incumbent does in the second
period. This leads to a higher level of rejection profits for the downstream firm, which theincumbent cannot profitably compensate under the investment contract.
In contrast, when the investment is efficient and reduces operating costs, the incumbent
can profitably design the investment contract to deter future entry if the investment is sufficiently specific. In this case, the acceptance of investment increases the joint profits of theincumbent and the downstream firm in the first period. We point out that this allows theincumbent to profitably compensate the downstream firm’s rejection profits.
Our results imply that an improvement in investment efficiency can trigger entry deterrence. This leads to an important welfare implication that an improvement in investment efficiency does not necessarily increase the equilibrium social welfare; that is, the relationshipbetween the investment efficiency and equilibrium social welfare becomes non-monotonic.
Hence, our results suggest that we should have a dynamic perspective when we discuss thewelfare effect of relationship-specific investments. In a static view, such investment decreases
the current operating costs of firms in vertical relationships and improves social welfare.
2However, in a dynamic view, the investment efficiency might reduce social welfare because
together with imposing switching costs, it can deter efficient future entry. Therefore, to evaluate the welfare effect of such investment, we need to consider these intertemporal effects.
Otherwise, we might obtain misleading predictions.
On interpreting our findings differently, our analysis predicts that entry deterrence owing
to an investment contract is more likely to be observed in an industry where downstream
firms are small companies. Small companies usually have fewer internal reserves and assets
because of difficulty in obtaining finance. Thus, upstream firms are more likely to need toincur investment costs to develop instruments and systems for downstream firms.
This study is related to the literature on entry deterrence with switching costs.4 Klemperer(1987) shows that entry deterrence is possible when the incumbent firm uses switching costs
in committing to a production capacity. In contrast, in the present study, entry deterrence
arises because switching costs restrict trading partner choices of the downstream firm and theinvestment can increase the present joint profits of the firms in the vertical relationship.
This study is also related to the literature on entry deterrence through exclusive contracts.
In the 1970s, the Chicago School (Posner, 1976; Bork, 1978) argued that if we consider
downstream buyers’ incentives to accept exclusive contracts, rational economic agents would
not engage in exclusive dealing for anticompetitive reasons, and thus, entry deterrence is
impossible by using exclusive contracts. In rebuttal of the Chicago School argument, postChicago economists show that entry deterrence is possible by using exclusive contracts in thepresence of scale economies5 (Rasmusen, Ramseyer, and Wiley, 1991; Segal and Whinston,
2000a) and downstream competition6 (Simpson and Wickelgren, 2007; Abito and Wright,
4

Entry deterrence is analyzed in a number of studies wherein it arises owing to excess capacity (Spence,
1977; Dixit, 1980), quality uncertainty (Schmalensee, 1982), and cost uncertainty (Milgrom and Roberts, 1982).
5
Doganoglu and Wright (2010) explore exclusion in the presence of network externalities, which are an
example of scale economies. Fumagalli and Motta (2006) explore an extended model of Rasmusen, Ramseyer,
and Wiley (1991) and of Segal and Whinston (2000a), where buyers are not final consumers but competingfirms. They show that intense downstream competition reduces the possibility of exclusion. However, Abito
and Wright (2008) point out that this result depends on the assumption that buyers are undifferentiated Bertrand
competitors who need to incur epsilon participation fees to stay active and show that if buyers are differentiatedBertrand competitors, then intense downstream competition enhances exclusion. Wright (2009) also points outthat the equilibrium analysis of Fumagalli and Motta (2006) in the case of two-part tariffs contains some errors
and shows that exclusion arises when scale economies are sufficiently large or the entrant’s cost advantage is
not too big.
6
See also Wright (2008), Argenton (2010), Kitamura (2010, 2011), Johnson (2012), Gratz and Reisinger

3

2008). The present study has a similar motivation because we explore entry deterrence byconsidering downstream buyers’ incentive to accept an investment contract. In contrast to
earlier studies, however, this study shows that when a specific investment is made, the investment contract can deter efficient entry even in the absence of exclusionary terms, scaleeconomies, and downstream competition.
Exclusion with non-exclusive contracts has recently been analyzed by many economists(Elhauge, 2009; Elhauge and Wickelgren, 2011, 2012a, and 2012b; Semenov and Wright,
forthcoming). These studies consider the design of wholesale pricing to deter efficient entrants and show that exclusion is possible even in the absence of exclusionary terms. The
present study shows another way of exclusion with non-exclusive contracts by using the investment contract.
Finally, this study is related to the literature on relationship-specific investments and exclusive dealing (Marvel, 1982; Segal and Whinston, 2000b; De Meza and Selvaggi, 2007;
De Fontenay, Gans, and Groves, 2010). These studies examine the role of exclusive dealing
in enhancing the level of relationship-specific investment and show that exclusive dealing
may solve the hold-up problem and increase the level of investment in a specific market environment.7 In contrast, the present study focuses on the specificity of relationship-specificinvestments as a barrier to future entry.
The reminder of this paper is organized as follows. Section 2 sets up the model. Section
3 analyzes the existence of entry deterrence because of the relationship-specific investment.
Section 4 explores the welfare implications of the investment. Section 5 discusses the robustness of our results under linear wholesale pricing. Finally, Section 6 contains concluding
remarks. Proofs of the results are provided in the Appendix.
(forthcoming), and Kitamura, Sato, and Arai (forthcoming).
7
Recently, in an extension to Segal and Whinston (2000b), Fumagalli, Motta, and Rønde (2012) show that
disregarding the interaction between investment promotion and market foreclosure may lead to misleading results. They show that the investment promotion effect of exclusive dealing enhances anticompetitive exclusive
dealing.

4

2 The modelThis section sets up the model. We first characterize upstream and downstream markets in
2.1. Then, we introduce the timing of the game in 2.2.

2.1 Upstream and downstream markets
In the upstream market, two upstream firms exist, an incumbent (denoted by I) and an entrant(denoted by E). They produce an identical product but differ in cost efficiency. While theincumbent has a marginal cost of production cI , the entrant is more efficient and has a lower
marginal cost; that is, cE < cI .
In the downstream market, a monopolistic downstream firm exists (denoted by D).8 The
downstream firm purchases inputs from the upstream firm to produce a final product. We
assume that one unit of the final product requires one unit of the input and that the marginal
cost of transformation or resale is cD . We also assume that the demand for the final productQ is given by a simple linear function Q(p) = 1 − p where p is the final product’s price. We
also assume that cI + cD < 1 such that selling to final consumers is at least profitable.

2.2 The timing of the gameThe timing of the game follows that of Klemperer (1987) (see also Figure 1). The model
contains two periods (t = 1, 2). At Period 1, only the incumbent exists in the upstream market.
This may be because of a patent right, efficient marketing, or an industry protection policy.
Period 1 consists of three stages. At Period 1.1, the incumbent offers the downstream firm
an investment contract involving some fixed compensation x ≥ 0. This offer is interpreted
as an introduction campaign for new systems or instruments developed by the incumbent
for a limited time.9 The investment reduces the marginal cost of the downstream firm forthe operation with the incumbent by d ∈ [0, cD ). The level of d is interpreted as the degree
8

In the literature on anticompetitive exclusive dealing, in the absence of multiple downstream buyers, an
upstream incumbent cannot deter efficient entry. Therefore, our modeling strategy of assuming a downstream
monopoly clarifies the role of investment as an entry barrier.
9
As stated earlier, if downstream firms in an industry are small companies, they usually have limited internal
reserves and assets because they cannot obtain finance easily. Thus, upstream firms are more likely to incur
investment costs and offer a specific investment to downstream firms.

5

of investment efficiency. To simplify the analysis, we assume that the investment cost is
zero.10 The downstream firm decides to accept or reject this offer. We denote the decisionby θ ∈ {a, r}, where θ = a (r) indicates the acceptance (rejection). We assume that when
the downstream firm is indifferent between accepting and rejecting the investment contract,
it accepts the contract.11
At Period 1.2, the incumbent offers wholesale contracts consisting of two-part tariffs(wI|t=1 , F I|t=1 ) ∈ R2+ . In Section 5, we briefly discuss the case of linear wholesale pricing.12 At
Period 1.3, the downstream firm sells to final consumers. Note that because the incumbent
uses two-part tariffs, it optimally sets its per-unit wholesale price to equal its marginal costto remove the double marginalization problem; that is, wI|t=1 = cI . Therefore, if the downstream firm has accepted the investment contract at Period 1.1, then it operates at cI + cD − d.
However, if it has rejected the investment contract, then it operates at cI + cD (see also Table
1).
At Period 2, the entrant appears in the upstream market. Period 2 consists of four stages.
At Period 2.1, the entrant decides whether to enter the upstream market. We denote theentrant’s decision by λ ∈ {in, out}, where λ = in (out) indicates that the entrant enters (stays
out of) the market. We assume that the fixed cost of entry is negligible. Therefore, theentrant always enters the market if its post-entry profits (gross of entry costs) are positive. At
Period 2.2, all active upstream firms offer wholesale contracts consisting of two-part tariffs(wi|t=2 , Fi|t=2 ), where i ∈ {I, E}, to the downstream firm. As in Period 1, active upstream firms
optimally set the per-unit wholesale prices to equal their marginal costs; that is, wi|t=2 = ci for
i ∈ {I, E}.
At Period 2.3, given the wholesale contracts offered by the upstream firm(s), the downstream firm decides on a trading partner. If the downstream firm deals with the incumbent
and has accepted the investment contract at Period 1, then the downstream firm operates at
10Introducing a positive investment cost does not alter our main results qualitatively.
If we assume that the downstream firm rejects the investment when it is indifferent, some of the argumentsin this study must be modified slightly. However, the essence of our results is still valid. See footnotes 18 and
23 for details.
12
Two-part tariffs allow us to illustrate our main results in a simpler way. As discussed in Section 5, although
we find almost the same results under linear wholesale pricing, the welfare analysis becomes considerablycomplicated.
116

cI + cD − d. If the downstream firm deals with the incumbent but has rejected the investment contract at Period 1, then the incumbent makes the investment offer again without fixedcompensation x and the downstream firm accepts the investment offer.13 As a result, thedownstream firm operates at cI + cD − d. On the other hand, if the downstream firm decides todeal with the entrant, then the downstream firm operates at cE + cD .14 However, if the downstream firm has accepted the investment contract at Period 1, then it needs to incur switchingcosts K > 0. The level of switching costs is interpreted as the degree of investment specificity.
For example, in the context of “habitual routines,” these switching costs include a temporary
performance degradation or an increase in the costs of familiarizing employees with a newroutine.15 We adopt the tie-break rule that when the downstream firm is indifferent between
dealing with the incumbent or the entrant, it continues to deal with the incumbent. At Period
2.4, the downstream firm orders the product produced by upstream firms and sells to final
consumers.
The incumbent’s and the downstream firm’s operating profits at Period 1 when the downstream firm’s investment decision at Period 1 is θ ∈ {a, r} are denoted by ΠθI|t=1 and πθD|t=1 ,respectively. Similarly, the incumbent’s, the downstream firm’s, and the entrant’s operating
profits at Period 2 when the downstream firm’s decision at Period 1 is θ ∈ {a, r} and the enθ(λ)θ(λ)
trant’s decision at Period 2 is λ ∈ {in, out} are denoted by Πθ(λ)
I|t=2 , πD|t=2 , and ΠE|t=2 , respectively.

We assume no discounting between Period 1 and Period 2, for simplicity.16
Finally, in the analysis below, we assume that the following inequalities are satisfied:
13

This is because the investment (at least) does not harm the downstream firm’s profits at Period 2.
We do not explicitly consider the investment between the entrant and the downstream firm. If the entrant
could offer the investment contract at Period 2, the downstream firm that decides to deal with the entrant always
accepts the offer and this reduces the downstream firm’s operating costs to less than cE + cD . This does not alter
our main results qualitatively.
15
In this study, we assume that there is no switching cost when the downstream firm rejects the investment
contract at Period 1. Of course, there can be a positive switching cost to change the trading partner even when
the investment is not made. However, in the context of “habitual routines,” the ex-post change of trading partner
can be more costly when employees have established specific routines with the old partner than when they havenot. Therefore, our setting can be interpreted as one in which the general switching cost that does not depend
on whether the investment is made is normalized to zero; we only focus on the switching cost owing to theinvestment specificity.
16
Introducing a discount rate β ∈ (0, 1) between Period 1 and Period 2 does not change our results qualitatively.
14

7

Assumption 1.
0 ≤ d < cI − cE ≤ cD .

(1)

The second inequality guarantees that entry occurs at Period 2 as long as the investment
contract is rejected at Period 1. If the second inequality does not hold, the downstream firm
never deals with the entrant and entry never occurs at Period 2. The third inequality implies
that investment efficiency d is in fact bounded by cI − cE and not by cD . This assumption
simplifies the analysis below.

3 Relationship-specific investment and entry deterrence
In this section, we analyze the existence of entry deterrence owing to the relationship-specificinvestment. In 3.1, we describe the conditions to be satisfied so that the incumbent can deter
entry using an investment contract. In 3.2 and 3.3, we explore when these conditions are
actually satisfied. To understand easily the role of investment as an entry barrier, in 3.2, weanalyze the case where the investment is not efficient (d = 0) and in 3.3, we analyze the casewhere the investment is efficient (d > 0).

3.1 Conditions for entry deterrence with a relationship-specific investment
In this subsection, we describe the conditions to be satisfied for the incumbent to profitably
deter entry by using an investment contract. In brief, for the existence of entry deterrence
owing to a relationship-specific investment, the following two conditions must be satisfied
simultaneously:
Condition (i) Entry must be unprofitable for the entrant when the investment is undertaken
at Period 1.1,
Condition (ii) Entering into an investment contract at Period 1.1 must be profitable for boththe incumbent and the downstream firm.
In what follows, we describe these two conditions more formally.

83.1.1 Profitability of entry for the entrant
First, we examine Condition (i) and derive the condition that entry at Period 2.1 becomes
unprofitable for the entrant if the investment contract is made at Period 1.1. If the entrant
enters the market, the incumbent and the entrant compete for the downstream firm at Period
2.2. If the investment has been undertaken at Period 1.1, the downstream firm must incur
certain switching costs to change its trading partner. In this case, to attract the downstream
firm, the entrant must cover such switching costs in addition to guaranteeing the profits likely
to result for the downstream firm from the incumbent’s best offer. Therefore, we have thefollowing lemma that summarizes the condition under which entry at Period 2.1 becomes
unprofitable and the entrant does not enter the market:
Lemma 1. Suppose that the investment contract is made at Period 1.1. Then, the entrant
does not enter the market at Period 2.1 if and only if the investment is sufficiently specific or
efficient; that is, a pair of (K, d) satisfies K ≥ K̂(d) and d ≥ 0 simultaneously, whereK̂(d) =(cI − cE − d)((1 − (cI + cD − d)) + (1 − (cE + cD )))
,
4

(2)

and where K̂ 0 (d) < 0 and limd→cI −cE K̂(d) = 0.
Whether K ≥ K̂(d) holds depends on the degrees of investment specificity K, investment
efficiency d, and entrant’s efficiency cE . Note that while the left-hand side of K ≥ K̂(d) is
increasing in K, the right-hand side is decreasing in both d and cE . As the investment becomes
more specific (as K increases), the downstream firm must incur larger switching costs tochange its trading partner at Period 2. Then, the entrant must offer a better wholesale contract
to induce the downstream firm to change its trading partner. Similarly, as the investment
becomes more efficient (as d increases), the entrant needs to offer a better wholesale contract
to the downstream firm. In these cases, it tends to be unprofitable for the entrant to enter themarket at Period 2.1. By contrast, as the entrant becomes more efficient (as cE decreases), it
can yield higher profits since the joint profits of the entrant and the downstream firm increase.
Therefore, market entry at Period 2.1 is more likely to be profitable for the entrant. Hence,
when the investment is highly specific but efficient and the entrant is not so efficient, K ≥ K̂(d)is more likely to hold and the entrant tends to not enter the market at Period 2.1.
9

3.1.2 Investment contract design
Next, supposing that K ≥ K̂(d) is satisfied, we examine Condition (ii) and derive the condition that an investment contract deterring the efficient entrant needs to satisfy. Given theequilibrium entry behavior of the entrant at Period 2.1, the equilibrium profits of the incumbent, entrant, and downstream firm in the subgames after Period 1.1 are summarized in Table
2.17 For the investment contract to be accepted at Period 1.1, it needs to satisfy the following
two conditions simultaneously.
First, the investment contract must satisfy financial feasibility for the incumbent: theinvestment contract must enable the incumbent to yield higher profits, that is,rr
ΠaI|t=1 + Πa(out)I|t=2 − x ≥ ΠI|t=1 + ΠI|t=2 .

(3)Second, the investment contract must satisfy individual rationality for the downstream
firm: namely, the amount of x must induce the downstream firm to accept the investment
contract, that is,rr
πaD|t=1 + πa(out)D|t=2 + x ≥ πD|t=1 + πD|t=2 .

(4)

From inequalities (3) and (4), we have the following inequality:a(out)arrrr
ΠaI|t=1 + Πa(out)I|t=2 + πD|t=1 + πD|t=2 ≥ ΠI|t=1 + ΠI|t=2 + πD|t=1 + πD|t=2 .

(5)Inequality (5) implies that for the investment contract to be profitable for both the incumbent
and the downstream firm, this contract is required to at least increase the two-period joint
profits of contracting parties.

3.2 Benchmark: When investment is not efficient
From the discussion in 3.1, it is easy to see that the relationship-specific investment deters theentrant if and only if inequalities K ≥ K̂(d) and (5) hold simultaneously. Now, we explore
when these conditions are actually satisfied. To understand easily the role of investment as
an entry barrier, in this subsection, we analyze the case where the investment is not efficient(d = 0). In the next subsection, we analyze the case where the investment is efficient (d > 0).
17

For the detailed derivation of the equilibrium offer and profit of each firm, see Appendix A.

10Assume that the relationship-specific investment is not efficient; that is, d = 0. If investment specificity K is sufficiently low such that K < K(0), then the downstream firm
can change the trading partner at Period 2 even when it accepts the investment at Period 1.
Then, the downstream firm tries to accept the investment at Period 1 and to deal with the entrant at Period 2. Therefore, the incumbent cannot deter the entrant by using the investment
contract.18
By contrast, if the investment is sufficiently specific such that K ≥ K̂(0) holds, then theacceptance of an investment contract induces the downstream firm to lose its best option of
accepting the investment contract at Period 1 but dealing with the entrant at Period 2. In sucha market environment, the downstream firm may strategically decide to reject the investment
offer so that it can deal with the entrant at Period 2. By considering the downstream firm’s
incentive to accept or reject the investment offer, the following proposition shows that when
the investment is not efficient, exclusion never occurs:
Proposition 1. Suppose that the relationship-specific investment is not efficient: d = 0. Then,the incumbent cannot deter the efficient entrant even when the investment is sufficiently specific (even when K ≥ K̂(0) holds).
The intuitive logic for the result under K ≥ K̂(0) is as follows (see also Figure 2). Whenthe downstream firm accepts the investment contract at Period 1, the entrant does not enter
the upstream market at Period 2 and only the incumbent offers wholesale contracts at both
Period 1 and Period 2. At each period, the joint profits of the incumbent and the downstream
firm coincide with monopoly profits for marginal costs cI + cD .
However, when the downstream firm rejects the investment contract at Period 1, the entrant enters the upstream market, and the incumbent and the entrant compete for the downstream firm at Period 2. In the equilibrium, the downstream firm buys from the entrant undera wholesale contract that leaves the downstream firm with slightly higher profits than themonopoly profits for marginal costs cI + cD . This implies that the acceptance of investment
18

In this case, the incumbent accepts the investment owing to the assumption that if the downstream firm is
indifferent between accepting and rejecting the investment contract, it accepts the contract. If instead we assume
that the downstream firm rejects the investment when it is indifferent, it is easy to see that the investment isrejected and the incumbent cannot deter the entrant.

11

reduces the two-period joint profits of the incumbent and the downstream firm; that is, inequality (5) does not hold. Therefore, the incumbent cannot profitably compensate the downstream firm for the rejection profits; that is, if the incumbent increases x to satisfy inequality(4), then inequality (3) does not hold.

3.3 When investment is efficient
We now assume that the relationship-specific investment is efficient; that is, d > 0. If investment specificity K is sufficiently low such that K < K̂(d), the downstream firm can choose
its best option of accepting the investment contract at Period 1 but dealing with the entrant at
Period 2. Therefore, while the investment is always accepted, the incumbent cannot deter theentrant with the investment contract.19
By contrast, if the investment is sufficiently specific such that K ≥ K̂(d), the incumbent
can deter the efficient entrant unlike in the case where d = 0:
Proposition 2. The incumbent can deter the efficient entrant if and only if the investment is
not only specific but also efficient. More precisely, entry deterrence is possible for a pair of(K, d) which satisfies K ≥ K̂(d) and d > 0 simultaneously.
Figure 3 summarizes Proposition 2. To understand easily the results here, we provide theintuitive logic by dividing the degree of investment specificity into two domains, K ≥ K̂(0)and K < K̂(0). First, Figure 3 implies that when the investment specificity is sufficiently highsuch that K ≥ K̂(0), entry deterrence is possible in the presence of even small investment
efficiency d > 0. In this case, the crucial difference from the previous subsection arisesin the subgame after the downstream firm accepts the investment contract at Period 1 (seealso Figure 4). Unlike the case where the investment has no efficiency (d = 0), here, if thedownstream firm accepts the investment contract at Period 1, then the investment increases
the joint profits of the incumbent and the downstream firm at Period 1 to the monopoly profits
for marginal costs cI + cD − d. An increase in the joint profits at Period 1 leads to an increase19

In this case, regardless of whether the investment contract is accepted at Period 1, the entrant always entersthe market at Period 2. This implies that while investing improves the joint profits of the incumbent and thedownstream firm at Period 1, it does not affect their joint profits at Period 2 (see Table 2). Therefore, thedownstream firm (strictly) prefers to accept the contract at Period 1 but to deal with the entrant at Period 2.

12

in the left-hand side of inequality (5) for all d > 0. This allows the incumbent to profitablycompensate the downstream firm for the rejection profits at Period 2.
Second, Figure 3 implies that when the investment specificity is sufficiently low such that
K < K̂(0), entry deterrence requires the higher investment efficiency d. In this case, theinvestment is always accepted regardless of the level of investment efficiency d. As discussed
in the interpretation of Lemma 1, an increase in investment efficiency d forces the entrant tooffer better wholesale prices, which reduces the entrant’s post-entry profit. Therefore, for thesufficiently efficient investment, entry is not profitable for the entrant when the investment
contract is accepted at Period 1. Since inequality (5) is satisfied for all d > 0, the incumbent
can deter the efficient entrant for the sufficiently higher investment efficiency d that satisfies
K ≥ K̂(d).
Combining the results of Propositions 1 and 2, we obtain important insights into entry
deterrence under relationship-specific investment. The incumbent cannot exclude rivals onlywith a specific investment: for exclusion, the investment needs to be not only specific but also
efficient.
Finally, note that our result in this section that entry deterrence can occur in the presenceof even small investment efficiency d > 0 is due largely to the assumption of two-part tariff
contracts. In Section 5, we show that entry deterrence requires sufficiently higher efficiency
under linear wholesale pricing.

4 Relationship-specific investment and social welfare
In this section, we explore the welfare implications of the relationship-specific investment.
In 4.1, we analyze when the relationship-specific investment should be made from a social
welfare perspective by comparing the level of social welfare in the cases when the investment
is and is not made. In 4.2, we explore the relationship between the equilibrium social welfare
and investment efficiency.

13

4.1 Social desirability of the investment
In the previous section, we observed that in the equilibrium, the investment contract is always
accepted at Period 1 except when (K, d) satisfies K ≥ K̂(0) and d = 0 simultaneously, as summarized in Figure 3. In this subsection, by comparing the level of social welfare in the cases
when the investment is and is not made, we explore the conditions under which making therelationship-specific investment is socially desirable (or undesirable).20 We analyze the two
types of investment separately, depending on whether the investment deters entry at Period 2.
In 4.1.1, we analyze the case where the investment deters entry. In 4.1.2, we analyze the casewhere the investment does not deter entry.
4.1.1 When investment deters entry
We first assume that the investment is sufficiently specific that it deters the efficient entrant
at Period 2 (i.e., K ≥ K̂(d) holds). In this case, the investment is accepted in the equilibrium
if and only if it is efficient. Therefore, here, we restrict our attention to the case of d >0. Let W INE denote the equilibrium level of social welfare (the superscript INE stands for“investment and no entry”), which is defined as follows:
W INE =

3[1 − (cI + cD − d)]2 3[1 − (cI + cD − d)]2
+
.
8
8

(6)

The first and second terms of the right-hand side of the above equation represent theequilibrium level of social welfare at Period 1 and Period 2, respectively. In the equilibrium,
investing at Period 1 reduces the downstream firm’s marginal cost by d. However, suchinvestment also leads to entry deterrence and the downstream firm deals with the incumbent
at Period 2.
We compare this with the case where the investment is not made at Period 1. In the latter
case, although the downstream firm’s marginal cost is not reduced at Period 1, the efficiententrant enters the upstream market and the downstream firm deals with the entrant at Period
20

In this subsection, we explore the investment’s social desirability under the assumption that trading partner
choices of the downstream firm at Period 2 are determined by the firms’ private incentives (i.e., whether K ≥K̂(d) holds). Instead, if we assume that a social planner or the government could force the downstream firm todeal with the entrant at Period 2, the evaluation of the investment’s social desirability could be different fromthat in this subsection. Details are available from the authors upon request.

14

2. Let W NIE denote the levels of social welfare in this case (the superscript NIE stands for“no investment and entry”), which is defined as follows:
W NIE =

3[1 − (cI + cD )]2 3[1 − (cE + cD )]2
+
.
8
8

(7)

By comparing equations (6) and (7), we obtain the following proposition:
Proposition 3. Suppose that the relationship-specific investment is sufficiently specific suchthat it leads to entry deterrence (K ≥ K̂(d)). Then, investing is socially desirable if and only
if this investment is sufficiently efficient; that is, W INE R W NIE if and only if d R d, wherer[1 − (cI + cD )]2 + [1 − (cE + cD )]2
d=− (1 − (cI + cD )) < cI − cE .
(8)
2
The relationship-specific investment generates two contrary effects on social welfare.
First, the investment improves the efficiency of the vertical relationship between the incumbent and the downstream firm. Because this effect increases social welfare at Period 1, it canbe interpreted as the current benefit on investing. From equations (6) and (7), the current
benefit CB can be derived as the difference between the first terms of the right-hand sides ofequations (6) and (7), that is:
CB(d|K ≥ K̂(d)) =3[1 − (cI + cD − d)]2 3[1 − (cI + cD )]2
−
.
8
8

(9)

Second, on the contrary, the investment deters the efficient entrant and this decreasessocial welfare at Period 2. This effect can be regarded as the future loss on investing. The
future loss FL can be calculated as the difference between the second terms of the right-handsides of equations (6) and (7):FL(d|K ≥ K̂(d)) =3[1 − (cE + cD )]2 3[1 − (cI + cD − d)]2
−
.
8
8

(10)

By comparing equations (9) and (10), an increase in investment efficiency d leads to different effects. On the one hand, the current benefit is increasing in d because the efficientinvestment increases the social welfare at Period 1. On the other hand, the future loss isdecreasing in d because welfare loss at Period 2 decreases for the efficient investment. Therefore, if the investment is sufficiently efficient (d > d), then the current benefit dominates the
future loss and the investment becomes socially desirable.
154.1.2 When investment does not deter entry
We next assume that the investment is not so specific and that it allows entry by an efficiententrant at Period 2 (that is, K < K̂(d) holds). In the equilibrium for this market environment,
the investment contract with d ≥ 0 is always accepted at Period 1 but the efficient entrant
enters the upstream market and the downstream firm deals with the entrant at Period 2. Let
W IE denote the equilibrium levels of social welfare in this case (the superscript IE stands for“investment and entry”), which is defined as follows:"#3[1 − (cI + cD − d)]2
3[1 − (cE + cD )]2
IE
W =
+
−K .
8
8

(11)

One of the important features of W IE is represented in the brackets of the right-hand sideof equation (11). Although the entry occurs at Period 2, the downstream firm needs to incur
switching cost K, which constitutes the social cost, to deal with the entrant.21 This implies
that the investment may be socially undesirable even when it does not lead to entry deterrence
at Period 2.
As in 4.1.1, we compare this with the case where the investment is not made at Period 1.
In the latter case, although the downstream firm’s marginal cost is not reduced at Period 1, theentrant enters the upstream market and the downstream firm deals with the entrant withoutswitching costs at Period 2. Therefore, the level of social welfare in this case coincides withW NIE , represented by equation (7). By comparing equations (7) and (11), we obtain the following proposition, which shows when the investment that does not lead to entry deterrence
becomes socially desirable:
Proposition 4. Suppose that the relationship-specific investment is not so specific and that itallows entry at Period 2 (K < K̂(d)). Then, investing is socially desirable if and only if thisinvestment is not so specific or is sufficiently efficient. More precisely,
1. When the investment is sufficiently efficient (for d > d) or within an intermediate rangeof efficiency (for d < d ≤ d), investing is always socially desirable; that is, W IE > W NIE
21

In this model, switching cost K could be interpreted as damages for breach of contract. However, in thatcase, K is just a monetary transfer between firms and does not affect the social welfare. Alternatively, in thisstudy, we consider K as, for example, a temporary performance degradation, which harms social welfare.

16always holds, wherer3[1 − (cI + cD )]2 + 2[1 − (cE + cD )]2
d=− (1 − (cI + cD )),
5

(12)

and
2. When the investment is not too efficient (for d ≤ d), investing is socially desirable if
and only if the investment specificity is sufficiently low; that is, W NIE R W IE if and only
if K R K̃(d), where:K̃(d) =

3d((1 − (cI + cD − d)) + (1 − (cI + cD )))
.
8

(13)

As in the case of the investment that leads to entry deterrence in 4.1.1, the investment that
does not deter entry also has two contrary effects on social welfare: a current benefit and afuture loss. First, the investment improves the vertical relationship’s efficiency and increasessocial welfare at Period 1. From the first terms of the right-hand sides of equations (7) and(11), this current benefit on investing is the same as that in 4.1.1:
CB(d|K < K̂(d)) =3[1 − (cI + cD − d)]2 3[1 − (cI + cD )]2
−
.
8
8

(14)

Second, on the contrary, the investment generates a future loss that is different from thefuture loss in 4.1.1. In contrast to 4.1.1, here, investing does not lead to entry deterrence.
However, it requires the downstream firm to incur switching cost K to deal with the entrant
at Period 2. This becomes the future loss to make the investment that does not deter entry:
FL(d|K < K̂(d)) = K.

(15)

The comparison between equations (14) and (15) implies that the current benefit is increasing in investment efficiency d but the future loss is not. It also implies that the current
benefit does not depend on investment specificity K but that the future loss is increasing in
K. Therefore, if the investment efficiency is sufficiently high or the switching cost is not too
high, then the current benefit is predominant and such investing improves social welfare.
The results of Propositions 3 and 4 are summarized in Figure 5. Note that the curve
K = K̂(d), which we derived in Lemma 1, represents the critical values of K above which
17

Proposition 3 applies and below which Proposition 4 applies. From Figure 5, it can be seenthat overall, a relationship-specific investment is socially desirable when its efficiency is sufficiently high. This implies that a high-efficiency investment is more likely to improve social
welfare than a low-efficiency one. However, as we explain below, a close look at Figure 5
reveals that this argument does not always hold.
Figure 5 shows that for an intermediate level of investment specificity (i.e., for K ∈(K̂(d), K̂(d))), there is a non-monotonic relationship between the investment’s social desirability and its degree of efficiency. More precisely, as the investment becomes more efficient,
the welfare property of the investment shifts from socially undesirable to desirable, then backto undesirable, and, finally, to desirable.
This result reflects the property of the future loss on investing; the future loss discontinuously changes according to investment efficiency d. In contrast to the current benefit, thefuture loss differs in the cases when the investment deters and does not deter entry. In the former case, the future loss is derived from the failure of efficient entry owing to the existenceof a switching cost (equation (10)). By contrast, when the investment does not deter entry,
the future loss is derived from the switching cost itself (equation (15)).
Figure 6 illustrates the property of future loss for an intermediate level of specificity.
Figure 6 shows that if investment efficiency d increases above the threshold value d2 , thenthe future loss increases discontinuously. At this increase of d, the equilibrium outcome at
Period 2 moves from entry to entry deterrence. Because entry deterrence leads to higher
market prices and to a smaller level of consumer surplus, the change of equilibrium outcome
reduces the social welfare at Period 2. This leads to a discontinuous change in future loss.
By comparing the future loss and the current benefit in Figure 6, an increase in the investment efficiency (for d ∈ [d1 , d2 )) first makes the investment socially desirable as the current
benefit increases and dominates the future loss. However, a further increase in efficiency (ford ∈ [d2 , d)) enables the investment to deter entry at Period 2 and this raises the future loss
discontinuously, thereby making the investment socially undesirable. Finally, as the investment becomes sufficiently efficient (for d ≥ d), the current benefit on investing dominates the
future loss again. Therefore, investing becomes socially desirable.

184.2 Social welfare and investment efficiency
In this subsection, for a given degree of specificity K, we explore how a change in investment efficiency d ≥ 0 affects the equilibrium social welfare. Intuitively, an improvement ininvestment efficiency decreases the operating costs and thus increases the equilibrium social
welfare. This view seems correct because it is easy to see that W INE and W IE are increasing
in d from equations (6) and (11).22
However, the following proposition shows that the possibility of entry deterrence triggered by the investment efficiency leads to a non-monotonic relationship between the investment efficiency and equilibrium social welfare:
Proposition 5. For a given degree of specificity K, the equilibrium social welfare is increasing in investment efficiency d except where it discontinuously decreases by triggering entry
deterrence. More precisely, a slight improvement in investment efficiency d around d̂(K) reduces the equilibrium social welfare, where



0p
d̂(K) = 
 (1 − (cE + cD ))2 − 4K − (1 − (cI + cD ))f or K ≥ K̂(0),f or K < K̂(0).

(16)

Note that d̂(K) for K < K̂(0) is the inverse function of K̂(d) (see equation (2)). WhenK ≥ K̂(0), the investment contract is rejected and entry occurs under d = 0. However, thepresence of (even small) investment efficiency enables the contract to be accepted and entry
deterrence occurs. Therefore, as the investment efficiency increases from zero to positive, theequilibrium social welfare changes from W NIE to W INE .
By contrast, when K < K̂(0), the investment contract is accepted for any d ≥ 0. However,whether entry deterrence occurs still depends on the investment efficiency. In this case, by
making entry less profitable for the entrant, the higher investment efficiency leads to entry
deterrence. Therefore, as the investment efficiency increases above d̂(K), the equilibriumsocial welfare changes from W IE to W INE .23 These discontinuous changes for both K ≥ K̂(0)22From equation (7), it is easy to see that W NIE is independent of investment efficiency d.
If we assume that the downstream firm rejects an investment contract when it is indifferent, the investment
is accepted only for d > 0 also under K < K̂(0). Therefore, the equilibrium social welfare under K < K̂(0)discontinuously changes not only around d = d̂(K) but also around d = 0. In the latter case, as the investment
efficiency slightly increases from zero to positive, the equilibrium social welfare under K < K̂(0) changes fromW NIE to W IE . Proposition 4 implies that this discontinuous change reduces the equilibrium social welfare.
23

19

and K < K̂(0) reduce the equilibrium social welfare.
The results in this section clarify the importance of a dynamic perspective for the discussion of relationship-specific investments. In a static view, the investment efficiency decreases
operating costs among vertical relations and has a welfare improving effect. However, ina dynamic view, the investment specificity generates a switching cost to change the trading
partner and has a welfare reducing effect regardless of whether the investment deters futureentry. In addition, from a dynamic perspective, an improvement in investment efficiency does
not necessarily increase social welfare, because it can trigger entry deterrence. Therefore, toconsider the welfare effect of the investment, we need to consider these intertemporal effectsof the investment. Otherwise, we may obtain misleading predictions.

5 Linear wholesale pricing
In this section, we briefly discuss the case when the upstream firms can use only linear wholesale pricing. For analysis, we make an additional assumption as follows:
Assumption 2.
cI − d <1 − cD + cE
.
2

(17)

Note that the right-hand side of this inequality represents a monopoly wholesale price by
the entrant. It is easy to see that this inequality is more likely to hold when cI is small or cE islarge. This assumption implies that the entrant’s cost advantage is so small that it cannot set
the monopoly price in wholesale competition with the incumbent.24 Combining Assumptions
1 and 2, under linear wholesale pricing, we restrict our attention to investment efficiency d,which satisfies the following inequality:)((cI − cE ) − (1 − (cD + cI )), 0 < d < cI − cE .
max
2

(18)

In the case of linear wholesale pricing, as the next proposition shows, we have qualitatively
the same results as under two-part tariffs but the investment efficiency becomes more important to trigger entry deterrence.
24

A similar assumption is often made in the exclusive dealing literature (e.g., Fumagalli and Motta, 2006;
Abito and Wright, 2008; Kitamura, 2010). Entry deterrence can still occur without this assumption, but theanalysis becomes more complicated.

20

Proposition 6. Suppose that the upstream firms can only use linear wholesale contracts.
Then, the incumbent can deter the efficient entrant through the investment contract if andonly if the investment is sufficiently specific and efficient; that is, a pair of (K, d) satisfies
K ≥ K̂(d) and d > d̃ simultaneously, where
√
6−2
d̃ =
(1 − (cD + cI )) > 0,2√
and where ( 6 − 2)/2 ≈ 0.2247.

(19)

Recall that under two-part tariffs, entry deterrence under the investment contract occurs
if and only if K ≥ K̂(d) and d > 0 (Proposition 2). Unlike under two-part tariffs, entry
deterrence under linear wholesale pricing requires a sufficiently higher efficiency; that is,d > d̃ > 0.25 Figure 7 illustrates the result of Proposition 6 when d̃ falls within the range ofinequality (18).26
In the case of linear wholesale pricing, the major difference from the case of two-parttariffs is the presence of a double marginalization problem. Under K ≥ K̂(d), when the investment contract is accepted, the incumbent maintains a monopoly throughout both periods
and the double marginalization problem reduces the joint profits of the incumbent and thedownstream firm in both periods. By contrast, when the investment contract is rejected, thatproblem reduces the joint profits of the incumbent and the downstream firm only at Period
1 because the wholesale price competition between the incumbent and the entrant eliminates
the problem at Period 2. This means that the left-hand side of inequality (5) decreases moresignificantly than the right-hand side. Therefore, under linear wholesale pricing, condition(5) becomes tighter and the investment needs to be still more efficient than the one under
two-part tariffs.
In relation to the welfare implications of the investment under linear wholesale pricing,various cases arise depending on the parameter values and the analysis becomes considerably√√d̃ falls within the range of inequality(18) if and only if 6/2 < (1 − (cD + cE ))/(1 − (cD + cI )) < 6. When√
(1 − (cD + cE ))/(1 − (cD + cI )) ≤ 6/2, we have d̃ ≥ cI − cE and the incumbent can never deter the entrant√ withthe investment contract that satisfies inequality (18). By contrast, when (1 − (cD + cE ))/(1 − (cD + cI )) ≥ 6, wehave d̃ ≤ [(cI − cE ) − (1 − (cD + cI ))]/2 and the incumbent can always deter the entrant under inequality (18).
26
As Figure 7 shows, under K < K̂(d), the investment contract is accepted but entry occurs regardless of thevalue of d. This is for the same reason as under two-part tariffs. See footnote 19.
2521

complicated. However, when d̃ falls within the range of inequality (18), we obtain results
similar to those under two-part tariffs: first, investing tends to be socially desirable when
the investment is not so specific but highly efficient; second, the equilibrium social welfare
discontinuously decreases as d exceeds the threshold value d̃.

6 Concluding remarks
This study explored the role of relationship-specific investment as a barrier to efficient entry.
The investment generates switching costs to change the trading partner, thereby reducing thedownstream firm’s ex-post flexibility to deal with the efficient entrant in the future. However,
the investment also increases the current joint profits of the incumbent and the downstream
firm. We showed that because of both the switching costs and an increase in current joint
profits, the incumbent could deter the efficient entrant in the future by using an investment
contract. We also showed that the investment’s welfare effect depends on both its specificity
and efficiency and that an increase in the investment efficiency can reduce the equilibriumsocial welfare.
These results provide an important policy implication. When we discuss the welfare effectof a relationship-specific investment, we need to have a dynamic perspective. From a static
perspective, the investment reduces current operating costs among vertical relationships andit seems to be efficient. From a dynamic perspective, however, the investment efficiency
itself enables the investment contract to be accepted even when it generates switching costs
to change a trading partner in the future. Therefore, the seemingly efficient investment canalso be the cause of inefficiency. To evaluate the relationship-specific investment correctly,
we need to pay attention to both of these opposing effects on social welfare.
By interpreting our results differently, we obtain an important policy implication that an
inefficient firm may survive for a long time because of relationship-specific investments. In
the international trade literature, a number of economists argue that vertical relationships
within Japanese companies, known as keiretsu, play the role of a structural trade barrier (e.g.,
Lawrence, 1991, 1993a, and 1993b; Spencer and Qiu, 2001).27 One important feature of27

In the Structural Impediments Initiative (SII) between Japan and the United States during 1989–1990,
particular attention was paid to the role of keiretsu. The U.S. government argued that keiretsu linkages made22

keiretsu is that relationship-specific investments are common among its members, for example, as in the auto industry. Our results might be one possible explanation of why foreign
firms cannot easily enter the Japanese market in the presence of relationship-specific investments within keiretsu groups.
Several outstanding issues require future research. First, there is a concern about the generality of our results. Although our analysis is in terms of a parametric example, the results
may extend to more general settings. Second, a concern about the role of strategic interaction between downstream firms exists. In our model, a downstream firm is a monopolist in
its final product market. However, the competition between downstream firms is an important research topic. We predict that, as discussed by Simpson and Wickelgren (2007) and
Abito and Wright (2008) in their studies on anticompetitive exclusive dealing, competition
between downstream firms enables the upstream incumbent to deter efficient entry under the
relationship-specific investment more easily.
Finally, a concern about the incumbent’s strategic behavior exists. According to our results, the incumbent may have an incentive to control the investment specificity. For example,
its investment may be too specific. We trust that this study will assist future researchers inaddressing these issues.

A Equilibria in the subgame following Period 1.1
In this Appendix, we analyze the equilibrium in each of the possible subgames after Period
1.1. Especially, we explore the wholesale contracts offered by upstream firms, realized profitof each firm, and entry behavior of the entrant in the equilibrium of each subgame.
foreign entry into the Japanese market difficult. At the end of the SII talks, the Japanese government agreed
to strengthen monitoring by its Fair Trade Commission of transactions among keiretsu firms and to take thenecessary steps toward eliminating any restraints on competition that might arise from their business practices(Lawrence, 1993b).

23

A.1 When the downstream firm accepts the investment contract at Period 1.1
First, suppose that the downstream firm accepts the relationship-specific investment at Period
1.1. At Period 1.2, to solve the double marginalization problem, the incumbent sets its perunit wholesale price to equal its marginal cost and extracts all of the downstream firm’s profit
through the fixed fee; that is, the incumbent offers the following wholesale contract:!
[1 − (cI + cD − d)]2
(wI|t=1 , F I|t=1 ) = cI ,
.
4

(20)

Therefore, the incumbent and the downstream firm respectively yield the following operating
profits at Period 1.3:
[1 − (cI + cD − d)]2,4
πaD|t=1 = 0.
ΠaI|t=1 =(21)

At Period 2.1, the entrant decides whether to enter the upstream market. If the entrant
does not enter the upstream market, the incumbent monopolizes the market and offers thesame wholesale contract as at Period 1. Therefore, the equilibrium operating profits for theincumbent, the entrant, and the downstream firm respectively become[1 − (cI + cD − d)]2,4
Πa(out)E|t=2 = 0,Πa(out)I|t=2 =(22)

πa(out)D|t=2 = 0.
By contrast, when the entrant enters the upstream market, the incumbent and the entrant compete for the downstream firm at Period 2.2. While the incumbent’s best offer (wI|t=2 , F I|t=2 ) =(cI , 0) leaves the downstream firm with [1−(cI +cD −d)]2 /4, the entrant’s best offer (wE|t=2 , F E|t=2 ) =(cE , 0) leaves the downstream firm with [1 − (cE + cD )]2 /4 − K.28 Therefore, which upstream
firm profitably captures the downstream firm depends on the specificity and efficiency of theinvestment. First, if [1 − (cI + cD − d)]2 /4 ≥ [1 − (cE + cD )]2 /4 − K, which is equivalent to28

Note that since the downstream firm has accepted the investment contract at Period 1.1, it must incur
switching costs to change the trading partner to the entrant at Period 2.

24

K ≥ K̂(d), the downstream firm’s switching costs are so high that the incumbent can profitably attract the downstream firm.29 Anticipating that the post-entry profit is zero, the entrant
does not enter the market in the equilibrium (owing to negligible entry costs). Therefore, inthis case, the equilibrium profit of each firm at Period 2 is given by equation (22).
Second, if [1 − (cI + cD − d)]2 /4 < [1 − (cE + cD )]2 /4 − K, which is equivalent to K < K̂(d),the level of switching costs is low enough for the entrant to profitably attract the downstream
firm. In this case, the incumbent and the entrant offer the following wholesale contracts:(wI|t=2 , F I|t=2 ) = (cI , 0),"#!
[1 − (cE + cD )]2[1 − (cI + cD − d)]2
(wE|t=2 , F E|t=2 ) = cE ,−+K+ε ,
4
4(23)

where ε is some infinitesimally small number.30 At Period 2.3, the downstream firm chooses
the entrant as the trading partner. At Period 2.4, each firm yields the following operating
profits:Πa(in)
I|t=2 = 0,#
[1 − (cI + cD − d)]2−
+K+ε ,
4
4[1 − (cI + cD − d)]2πa(in)
=
+ ε.
D|t=2
4

[1 − (cE + cD )]Πa(in)
E|t=2 =

2

"

(24)

Therefore, anticipating that the post-entry profit is strictly positive, the entrant enters themarket at Period 2.1 in the equilibrium.

A.2 When the downstream firm rejects the investment contract at Period 1.1
Next, suppose that the downstream firm rejects the relationship-specific investment at Period
1.1. At Period 1.2, the incumbent offers the following wholesale contract:!
[1 − (cI + cD )]2(wI|t=1 , F I|t=1 ) = cI ,
.
4

(25)

The incumbent can do this by offering wI|t=2 = cI and choosing the fixed fee thath makes the downstreami2firm indifferent between its offer and the entrant’s: F I|t=2 = [1 − (cI + cD − d)] /4 − [1 − (cE + cD )]2 /4 − K .
30
The entrant sets the per-unit wholesale price at the marginal cost level and chooses the fixed fee that leaves
the downstream firm with slightly more profit than the incumbent’s best offer.
29

25

Therefore, at Period 1.3, the incumbent and the downstream firm respectively yield:[1 − (cI + cD )]2,4
πrD|t=1 = 0.
ΠrI|t=1 =(26)

At Period 2.1, the entrant decides whether to enter the market. If the entrant enters the
upstream market, the incumbent and the entrant compete for the downstream firm at Period
2.2. While the incumbent’s best offer leaves the downstream firm with [1 − (cI + cD − d)]2 /4,
the entrant’s best offer leaves the downstream firm with [1 − (cE + cD )]2 /4.31 Therefore, theentrant can profitably capture the downstream firm. In the equilibrium, the incumbent andthe entrant respectively offer the following wholesale contracts:(wI|t=2 , F I|t=2 ) = (cI , 0) ,"#!
[1 − (cE + cD )]2[1 − (cI + cD − d)]2
(wE|t=2 , F E|t=2 ) = cE ,
−
+ε .
4
4

(27)

Then, the downstream firm decides to deal with the entrant and each firm yields the following
operating profits:
ΠrI|t=2 = 0,#
[1 − (cI + cD − d)]2
−
+ε ,
4
4
[1 − (cI + cD − d)]2
πrD|t=2 =+ ε.
4[1 − (cE + cD )]ΠrE|t=2 =

2

"

(28)

Therefore, anticipating the positive post-entry profit, the entrant enters the market at Period
2.1 in the equilibrium.

B Proofs
Proof of Lemma 1
Note that since we assume negligibly small entry costs, the entrant does not enter the market
if and only if its post-entry profit is less than or equal to zero. If the entrant enters the market,
the incumbent and the entrant compete for the downstream firm at Period 2.2. While the31

Since the downstream firm has rejected the investment contract at Period 1.1, there are no switching costs
for the downstream firm to change its trading partner.

26

incumbent’s best offer leaves the downstream firm with [1 − (cI + cD − d)]2 /4, the entrant’s
best offer leaves the downstream firm with [1 − (cE + cD )]2 /4 − K. Therefore, if[1 − (cI + cD − d)]2 [1 − (cE + cD )]2
≥
−K
(29)
4
4
holds, the incumbent profitably attracts the downstream firm and the entrant’s post-entry
profit becomes zero. By solving this inequality with respect to K, we have K ≥ K̂(d).
Q.E.D.

Proof of Proposition 1
See the proof of Proposition 2.
Q.E.D.

Proof of Proposition 2
Recall that the relationship-specific investment deters the entrant if and only if both of inequalities K ≥ K̂(d) and (5) hold. Substituting equilibrium profits (see Table 2 or Appendix
A) into inequality (5), it can be rewritten as follows:!
[1 − (cI + cD − d)]2
[1 − (cI + cD )]2 [1 − (cI + cD − d)]2
2
≥
+
+ ε.
(30)
4
4
4
It is easy to see that this inequality holds if and only if d > 0. Therefore, when d = 0,
inequality (30) does not hold and incumbent cannot profitably induce the downstream firm
to accept the investment contract, even if K ≥ K̂(d) under d = 0 is satisfied (Proposition 1).
On the other hand, when d > 0, inequality (30) always holds and the incumbent can deter theefficient entrant if and only if K ≥ K̂(d) is satisfied (Proposition 2).
Q.E.D.

Proof of Proposition 3
From equations (6) and (7), we have3[1 − (cI + cD − d)]2 3[1 − (cI + cD )]2 3[1 − (cE + cD )]2
R
+
4
8
8
r
[1 − (cI + cD )]2 + [1 − (cE + cD )]2⇔dR− (1 − (cI + cD )).
2

W INE R W NIE ⇔

27

(31)

Let d denote the right-hand side of last inequality. Then, we have W INE R W NIE if and only
if d R d.
Q.E.D.

Proof of Proposition 4
From equations (7) and (11), we have3[1 − (cI + cD )]2 3[1 − (cI + cD − d)]2R−K883d((1 − (cI + cD − d)) + (1 − (cI + cD )))⇔KR.
8W NIE R W IE ⇔(32)

Let K̃(d) denote the right-hand side of last inequality. It is easy to see that ∂K̃(d)/∂d > 0
and K̃(0) = 0. Recall that we now consider the case of K < K̂(d). From the properties ofK̂(d) and K̃(d), there exists d ∈ (0, cI − cE ) such that K̂(d) = K̃(d). Then, d can be obtained
as equation (12) and it is easy to see that d < d. When d > d, we have K < K̂(d) < K̃(d).
Therefore, W NIE < W IE always holds. On the other hand, when d ≤ d, we have K̂(d) ≥ K̃(d).
Therefore, we have W NIE R W IE if and only if K R K̃(d).
Q.E.D.

Proof of Proposition 5
We show that a slight improvement in investment efficiency d around d̂(K) discontinuously
reduces the equilibrium social welfare. Since the reasons behind this result differ dependingon K ≥ K̂(0) and K < K̂(0), we explore these two cases separately. To indicate the dependence on investment efficiency d, we denote welfare levels when the investment is made at
Period 1 as W INE (d) and W IE (d).
First, we show that the equilibrium social welfare under K ≥ K̂(0) discontinuously dicreases at d = d̂(K) = 0. Note that the equilibrium social welfare in this case is equal to W NIE
for d = 0 and W INE (d) for d ∈ (0, cI − cE ). Since Proposition 3 shows W NIE > W INE (d) fora sufficiently small d > 0, it can be seen that a slight improvement in investment efficiency
from d = 0 reduces social welfare in the equilibrium.
28

Next, we show that the equilibrium social welfare under K < K̂(0) discontinuously dip
creases at d = d̂(K) = (1 − (cE + cD ))2 − 4K−(1−(cI +cD )). Note that the equilibrium social
welfare in this case is equal to W IE (d) for d ∈ [0, d̂(K)) and W INE (d) for d ∈ [d̂(K), cI − cE ).
From equations (6) and (11), we have3(1 − (cE + cD ))2 5− K,
42d→d̂(K)−023(1 − (cE + cD ))W INE (d̂(K)) =
− 3K,
4
lim

W IE (d) =(33)

and it is easy to see that limd→d̂(K)−0 W IE (d) > W INE (d̂(K)). Therefore, the equilibrium social
welfare discontinuously decreases when d exceeds or even equals d̂(K).
Q.E.D.

Proof of Proposition 6
As in the case of two-part tariffs, for the existence of entry deterrence under linear wholesalepricing, the two conditions mentioned in Subsection 3.1 must be satisfied simultaneously.
First, entry at Period 2.1 must be unprofitable for the entrant if the investment contract
is accepted at Period 1.1. Note that the entrant does not enter the market if and only if its
post-entry profit is less than or equal to zero. If the entrant enters the market, the incumbent
and the entrant compete for the downstream firm at Period 2.2. While the incumbent’s best
offer wI|t=2 = cI leaves the downstream firm with [1 − (cI + cD − d)]2 /4, the entrant’s best offer
wE|t=2 = cE leaves the downstream firm with [1 − (cE + cD )]2 /4 − K. Therefore, if[1 − (cI + cD − d)]2 [1 − (cE + cD )]2
≥
−K
4
4

(34)

holds, the incumbent profitably attracts the downstream firm and the entrant’s post-entry
profit becomes zero. By solving this inequality with respect to K, we have K ≥ K̂(d).
Next, supposing that K ≥ K̂(d) is satisfied, an investment contract which deters the efficient entrant must be profitable for both the incumbent and the downstream firm, that is,inequality (5) must be satisfied. Below, we derive the equilibrium profit of each firm in thepossible subgames after Period 1.1.
29

First, suppose that the downstream firm accepts the investment contract at Period 1.1. At
Period 1.2, as an upstream monopolist, the incumbent offers wI|t=1 = (1 − (cD − d) + cI )/2.
Therefore, the incumbent and the downstream firm respectively yield the following operating
profits at Period 1.3:
[1 − (cI + cD − d)]2,8
[1 − (cI + cD − d)]2a.
πD|t=1 =16
ΠaI|t=1 =(35)

At Period 2.1, assuming K ≥ K̂(d) holds, the entrant does not enter the upstream market.
In this case, the incumbent offers the same wholesale price as at Period 1, and the equilibrium
operating profits for the incumbent, the entrant, and the downstream firm respectively become[1 − (cI + cD − d)]2,8
Πa(out)E|t=2 = 0,Πa(out)I|t=2 =

πa(out)D|t=2 =

(36)

[1 − (cI + cD − d)].
162

Suppose next that the downstream firm rejects the investment contract at Period 1.1. At
Period 1.2, the incumbent offers wI|t=1 = (1 − cD + cI )/2. Therefore, at Period 1.3, theincumbent and the downstream firm respectively yield:[1 − (cI + cD )]2,
8[1 − (cI + cD )]2
πrD|t=1 =.
16
ΠrI|t=1 =(37)

At Period 2.1, the entrant makes the entry decision. When it enters the upstream market,
the incumbent and the entrant compete for the downstream firm at Period 2.2. While theincumbent’s best offer wI|t=2 = cI leaves the downstream firm with [1 − (cI + cD − d)]2 /4, theentrant’s best offer wE|t=2 = cE leaves the downstream firm with [1 − (cE + cD )]2 /4. From
Assumption 1, it is easy to see that the entrant can profitably capture the downstream firm.
Therefore, in equilibrium, the entrant always enters the market and the entrant’s wholesale
price becomes wE|t=2 = cI − d − ε, where ε is some infinitesimally small number, from30

Assumption 2. Then, each firm yields the following operating profits:
ΠrI|t=2 = 0,
(cI − cE − d)(1 − (cI + cD − d))ΠrE|t=2 =,2[1 − (cI + cD − d)]2
πrD|t=2 =
+ ε.
4

(38)

By substituting above equilibrium profits in each case into inequality (5), it can be rewritten as follows:!
3[1 − (cI + cD − d)]2
3[1 − (cI + cD )]2 [1 − (cI + cD − d)]2
2
≥
+
+ ε.
16
16
4

(39)

By rearranging this inequality, we have inequality (19). Therefore, under linear wholesalepricing, the incumbent can deter the efficient entrant if and only if inequalities K ≥ K̂(d) and(19) are both satisfied.
Q.E.D.

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34

When D accepts the investment at
Period 1.1
Period 1When D deals with I: 

Period 2

When D rejects the investment at
Period 1.1When D deals with E: 
(D must incur the switching costsWhen D deals with I: )When D deals with E: (There is no switching cost for D)

Table 1: Downstream firm’s total marginal cost at each period in each subgame

When D accepts the investment
at Period 1.1

When D rejects the investment
at Period 1.1

Period1
If

Period2
If, E does not enter the market:

E always enters the market:, E enters the market:

Table 2: Equilibrium profits and equilibrium entry behavior
in the subgames after Period 1.1
Note:  is some infinitesimally small number.

Period 1
1.1

1.2

1.3

I offers an investment
contract and D decides
to accept or reject it.

I offers wholesale contract.

D sells to consumers.

Period 2
2.1

2.2

2.3

2.4

E decides whether
to enter the
upstream market.

All active upstream
firms offer wholesale
contracts.

D decides on a
trading partner.

D sells to consumers.

Figure 1: TimelineWhen D accepts the investment
at period 1.1
Period 1pWhen D rejects the investment
at period 1.1
Period 1

p

Joint
profitsof I&D

Joint
profitsof I&D

Q
Period 2

p





Q
Period 2

p

Joint
profitsof I&D

Joint profits
of E&DQ

Figure 2: Joint profits of upstream and downstream firms
when investment is not efficientQEntry (without investment)
Entry Deterrence(with investment)
Entry
never
occursEntry(with investment)
Figure 3: Area of entry deterrenceWhen D accepts the investment
at period 1.1

When D rejects the investment
at period 1.1

Period 1

p

Joint
profits of
I&D

Period 1

p









Joint
profitsof I&D

Q

Q

Period 2p

Joint
Profits
profits of
II&D&D

Period 2pJoint profits
of E&DQ

Figure 4: Joint profits of upstream and downstream firms
when investment is efficientQInvestment is socially
undesirable

Investment
is sociallydesirable



Entry
never
occurs

Figure 5: Social desirability of the relationship-specific investment

current benefit
future loss

Entry
Deterrence

Entry

(

)

(

)

future loss under
entry deterrence

future loss under entry

current benefitFigure 6: Current benefit and future loss
for an intermediate level of the specificity∈( ( )( ))Entry(without
investment)

Entry Deterrence(with investment)
Entry
never
occursEntry(with investment)

Figure 7: Area of entry deterrence under linear wholesale pricing