Discussion Paper No. 1040

MARKET EXPANSION
MAY HARM THE SUPPLIER
IN A BILATERAL MONOPOLY

Noriaki Matsushima
Laixun Zhao

August 2018

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Market expansion may harm the supplier in a bilateral monopoly∗

Noriaki Matsushima†
Institute of Social and Economic Research, Osaka University
Laixun Zhao‡
Research Institute of Economics and Business, Kobe University
August 6, 2018
Abstract
We consider a bilateral monopoly with a supplier and a buyer. Their trading terms are
determined through negotiations, but aﬀected by the buyer’s eﬀorts to search for outside
suppliers. We find surprisingly that a market expansion may harm the supplier.

Keywords: Bilateral monopoly, Nash bargaining, outside suppliers
JEL Classification Numbers: L14, L22∗

This paper is a part of the paper circulated under the title “Strategic Sourcing as a Driver for Free Revealingof Innovation” (2015) ISER Discussion Paper No. 936, Institute of Social and Economic Research, Osaka University. We divide the discussion paper into the two papers. This paper consists of Section 2 in the discussion
paper and a new introduction. Another one, Matsushima and Zhao (2018), consists of Sections 1, 3, 4, and 5 inthe discussion paper and a new explanation for the related real-world case. We are grateful to many researchersand seminar participants in APIOC2016 (Melbourne), Kobe University, MaCCI2015 (Mannheim), Osaka University, Tokyo Institute of Technology, and Universitat d’Alacant for their helpful comments. We would also
like to thank Cong Pan and Shohei Yoshida for research assistance. The first author thanks the warm hospitality at MOVE, Universitat Autònoma de Barcelona where part of this paper was written and a financial support
from the “Strategic Young Researcher Overseas Visits Program for Accelerating Brain Circulation” by JSPS. We
gratefully acknowledge financial support from JSPS Grant-in-Aid for Scientific Research (S) Nos. JP26220503,
JP15H05728, (A) No. JP17H00984, (B) Nos. JP24330079, JP15H03349, JP18H00847 and (C) No. JP24530248
and JP18K01593, and the program of the Joint Usage/Research Center for ‘Behavioral Economics’ at ISER,
Osaka University. The usual disclaimer applies.
†
Corresponding author: Noriaki Matsushima, Institute of Social and Economic Research, Osaka University,
Mihogaoka 6-1, Ibaraki, Osaka 567-0047, Japan. Phone: +81-6-6879-8571. E-mail: nmatsush@iser.osakau.ac.jp.
‡
Laixun Zhao, Research Institute of Economics and Business, Kobe University, Rokkodai 2-1, Kobe, Hyogo
657-8501, Japan. Phone: +81-78-803-7006. E-mail: zhao@rieb.kobe-u.ac.jp.

11 Introduction
Downstream firms in many industries often keep their outside supply sources to enhance thebargaining power over trading suppliers. In the context of the automobile industry, for instance,
it has been emphasized that finding alternative suppliers is a useful tactic to enlarge profitability(e.g., Wu and Choi, 2005).1 To capture this situation, we consider a bilateral monopoly wherea supplier sells its input to a buyer which is able to improve its disagreement payoﬀ througheﬀorts (e.g., searching for other suppliers, improving its own input facility) before the supplierand the buyer negotiate a two-part tariﬀ contract. The game considered here is quite simple: inthe first stage, the downstream firm makes a search eﬀort to improve its disagreement payoﬀ;
in the second stage, both firms bargain over the trading terms (a two-part tariﬀ). If bargaining
breaks down, the downstream firm obtains its disagreement payoﬀ (e.g., by ordering input fromthe second-best supplier it searched in the first stage, or using its own input facility (similar toInderst and Valletti, 2009)); in the third stage, the downstream firm sets the quantity of final
output.
We show that a market expansion may harm the supplier, opposite to what one might
conventionally expect. Intuitively, as the market size rises, the buyer raises its search eﬀort
to improve its disagreement payoﬀ, weakening the bargaining position of the supplier. As aconsequence, a market expansion while benefiting the downstream firm, can hurt the supplier.
Furthermore, we reverse the roles of the upstream and downstream firms, and find our basic
mechanism remains robust (this is available upon request).
Our simple model has a novel contribution to the literature of buyer-supplier relationships.
Many researchers have investigated how disagreement payoﬀs influence realized outcomesin buyer-supplier relations (e.g., Davidson, 1988; Horn and Wolinsky, 1988; Inderst, 2007;
Milliou and Petrakis, 2007; Iozzi and Valletti, 2014). Most of the papers discuss market structures in which at least one of the upstream and the downstream markets consists of more than1

See Matsushima and Zhao (2018) for a more detailed discussion of the literature.

2

one firm. Our paper shows that disagreement payoﬀs can influence firm profitabilities in an
unconventional way even under the simplest bilateral monopoly setting.2

2 Bilateral Monopoly
Consider a downstream firm and an upstream firm without any production cost for simplicity,
whose outputs are related in a one-to-one ratio. The downstream firm has an outside option:by incurring a fixed cost F, it can procure the input from a diﬀerent source at price w(e)(> 0) if
negotiation with the upstream firm breaks down, where e is the search eﬀort of the downstream
firm to improve the value of its outside option, at a cost of S (e). We assume w′ (e) < 0,w′′ (e) ≥ 0; and S ′ (e) > 0, S ′′ (e) > 0. In sum, the assumptions concerning outside options
follow those in Inderst and Valletti (2009) although here we endogenize the price w(e).
Let us investigate the simplest game structure: in the first stage, the downstream firm makes
a search eﬀort e to improve its outside option; in the second stage, both firms bargain over the
trading terms (a two-part tariﬀ, mq + T , where m is the wholesale price, q is quantity and T
is the fixed payment). If bargaining breaks down, the downstream firm executes its outside
option at the fixed cost F, with a marginal cost of w(e) (similar to Inderst and Valletti, 2009);
finally, in the third stage, the downstream firm sets the quantity of final output.3 The game issolved by backward induction.
To keep the model simple and clean, we assume the search cost to be sunk, in a way similar
to R&D investments in innovation models, where actual production and any price or profit
negotiation occurs afterwards.4 Let the inverse demand function in the downstream market
2

Recently, Inderst and Shaﬀer (2018) show the possibility that channel coordination in a vertical relationshipcan fail if some of the downstream firms have outside options.
3

In the next section, we show that the size of the fixed cost F does not aﬀect the main finding.

4

An alternative is to let the two parties bargain first and then the downstream firm search if bargaining breaks
down. In that case, however, because search begins after bargaining breaks down, delay of production occurs,
which is costly to the downstream firm. In order to avoid such delay cost, the downstream firm thus chooses tosearch before bargaining occurs. Furthermore, our model formulation endogenizes the outside opportunity of the3be p(q; a), where a is a positive parameter that can shift up the demand, such as market size,
income, etc.
In the last stage, the downstream firm chooses the final quantity q to maximize its profits(p(q; a) − m)q − T , resulting in the following first-order and second-order conditions:

F.O.C.

p(q; a) − m + pq (q; a)q = 0,

S.O.C.

2pq (q; a) + pqq (q; a)q < 0,where pq (q; a) = ∂p(q; a)/∂q and pqq (q; a) = ∂2 p(q; a)/∂q2 .
Let q(m; a) denote the equilibrium quantity in which the wholesale price is m. Then the
gross profits of the downstream and the upstream firms (excluding the search costs of thedownstream firm) in the second stage are given as respectively,
πD = [p(q(m; a); a) − m]q(m; a) − T ; πU = mq(m; a) + T,

The firms’ outside options are respectively,
πOD = [p(q(w(e); a); a) − w(e)]q(w(e); a) − F; πOU = 0.

During bargaining, they jointly maximize
G = {πD − πOD }{πU − 0}.
downstream firm along the lines of Inderst and Valletti (2009).

4

The first-order conditions ∂G/∂T = 0 and ∂G/∂m = 0 can be reexpressed as:
πD − πOD − πU = 0,
(πD − πOD )[q(m; a) + mqm (m; a)] − πU q(m; a)= (πD − πOD − πU )q(m; a) + (πD − πOD )mqm (m; a) = (πD − πOD )mqm (m; a) = 0,
where qm (m; a) = ∂q(m; a)/∂m. From the above, we obtainm = 0; T =p(q(0; a); a)q(0; a) − [p(q(w(e); a); a) − w(e)]q(w(e); a) + F.
2

Because bargaining is eﬃcient, the two firms set the wholesale price equal to the marginal costof the upstream firm, i.e., m = 0, to maximize their joint profits first; and then, they split theoverall profits through the lump-sum payment, T . As such, the “double marginalization” problem in a standard bilateral-monopoly situation is avoided. Expecting the bargained outcome,the profits of the two firms respectively becomeΠD (e) = [p(q(m); a) − m]q(m; a) − T − S (e)[p(q(0; a); a)]q(0; a) + [p(q(w(e); a); a) − w(e)]q(w(e); a) − F − 2S (e),2[p(q(0; a); a)]q(0; a) − [p(q(w(e); a); a) − w(e)]q(w(e); a) + F
ΠU (e) =.
2=

Then in the first stage, the downstream firm chooses search eﬀort, satisfying the following
first-order condition,w′ (e∗ )q(w(e∗ ); a) + 2S ′ (e∗ )∂ΠD (e)
=0⇔−
= 0.
∂e
2

5

(1)

3 Market size expansion
As promised, we now investigate how a marginal increase in parameter a (e.g., market size or
income) aﬀects the equilibrium outcome. Applying the envelop theorem yields
2∂ΠU (e∗ )de∗= pa (q(0; a); a)q(0; a) − pa (q(w(e∗ ); a); a)q(w(e∗ ); a) + w′ (e∗ )q(w(e∗ ); a)
. (2)
∂a
da
where pa ≡ ∂p(q; a)/∂a and e∗ is the equilibrium level of search eﬀort. Total diﬀerentiation of(1) givesde∗w′ (e∗ )qa (w(e∗ ); a)=,da−{w′′ (e∗ )q(w(e∗ ); a) + [w′ (e∗ )]2 qm (w(e∗ ); a) + 2S ′′ (e∗ )}where qm ≡ ∂q(m, a)/∂m. It is easy to show de∗ /da > 0, because the denominator is derived
from the second-order condition of (1).
Let us examine (2) in detail. If w(e∗ ) is suﬃciently small (e.g., approaching zero), the firstand second terms on the RHS cancel out, leaving only the third term and hence ∂ΠU (e∗ )/∂a <0. Hence,
Proposition 1 An increase in the parameter a decreases ΠU (e∗ ) if w(e∗ ) is suﬃciently small.
This Proposition implies that a market expansion actually harms the upstream firm, opposite to what one might conventionally expect. Intuitively, as the market size rises, the downstream firm raises its search eﬀort, which then weakens the bargaining position of the upstream
firm, leading to our result.
To summarize, positive shocks such as market size increases enlarge the pie and could
bring potential gains for all players. However, if some players can raise their outside options,they can take away more than the increase of the pie, leaving others worse oﬀ. Also note thatthe same logic applies to a setting with n outside options, where the firm would use the best
one of them and the rest of the options (n − 1) becomes irrelevant.

6

Example

Assume that p = a − q, w(e) = w − e, and S (e) = γe2 /2, with a and γ being

positive constants. The second stage net profits of the two firms are then given asΠD (e) =

)
)
1( 2
1( 2
a + [a − w(e)]2 − 4F − S (e), ΠU (e) =a − [a − w(e)]2 + 4F
8
8

In the first stage, the first-order condition of the downstream firm is∂ΠD (e) a − w − (4γ − 1)ea−w== 0 → e∗ =.
∂e44γ − 1
Because the quantity under which bargaining breaks down, (a − w(e∗ ))/2, is smaller than whenagreement is reached, we require a < 4γw. Substituting e∗ into ΠU (e) givesΠU (e∗ ) =[(8γ − 1)a − 4γw](4γw − a) F
+ .
8(4γ − 1)2
2

The first term is positive if
4γw< a < 4γw.
8γ − 1
Diﬀerentiating ΠU (e∗ ) with respect to a yields∂ΠU (e∗ ) 16γ2 w − (8γ − 1)a=,∂a4(4γ − 1)2
that is negative if and only if
16γ2 w< a < 4γw,
8γ − 1
under which a rise in market size a decreases the profit of the upstream firm.

7

4 Conclusion
The present paper while simple, generates novel results that match the real practices of many
firms on vertically related production networks, such as automobile makers and parts suppliers. We have explicitly demonstrated conventionally counterintuitive situations when the
supplier may lose from a market expansion. This mechanism has wide applications underdiﬀerent circumstances, for instance, suppliers’ incentives in cost reduction, quality improvement, upstream collaboration and technology spillovers, and even worker training in the labormarket.
In a companion paper, Matsushima and Zhao (2018) generalize this paper’s setup to thecase of a bilateral duopoly, again with outside options of buyers, and incorporating costreducing investments (e.g., R&D) by suppliers which could strengthen their bargaining position. This extended setup is often seen in some Japanese manufacturers such as in the automobile industry. Matsushima and Zhao (2018) find surprisingly that each supplier has anincentive to unilaterally generate a technology spillover to its rival for free, if its own downstream buyer can find cheap alternative sourcing. Such a strategy causes a market-size shrink
within the vertical chain, which can hurt the downstream buyer but benefit the supplier, via thebargaining mechanism described above.

References[1] Davidson, C., 1988. Multiunit bargaining in oligopolistic industries. Journal of Labor
Economics 6(3), pp. 397–422.
[2] Horn, H. and Wolinsky, A., 1988. Bilateral monopolies and incentives for merger. RAND
Journal of Economics 19(3), pp. 408–419.

8[3] Inderst, R., 2007. Leveraging buyer power. International Journal of Industrial Organization 25(5), pp. 908–924.
[4] Inderst, R. and Valletti, T., 2009. Price discrimination in input markets. RAND Journal of
Economics 40(1), pp. 1–19.
[5] Inderst, T. and Shaﬀer, G., 2018. Managing channel profits when retailers have profitable
outside options. Management Science. Published online in Articles in Advance 21 May
2018. https://doi.org/10.1287/mnsc.2017.2953[6] Iozzi, A. and Valletti, T., 2014. Vertical bargaining and countervailing power. American
Economic Journal: Microeconomics 6(3), pp. 106–135.
[7] Matsushima, N. and Zhao, L., 2018. Technology spillovers and outside options in a bilateral duopoly. ISER Discussion Paper No. 1039, Osaka University.
[8] Milliou, C. and Petrakis, E., 2007. Upstream horizontal mergers, vertical contracts, andbargaining. International Journal of Industrial Organization 25(5), pp. 963–987.
[9] Wu, Z. and Choi, T.Y., 2005. Supplier-supplier relationships in the buyer-supplier triad:
Building theories from eight case studies. Journal of Operations Management 24(1), pp.
27–52.

9Appendix (for reference only) Reversing the Roles of the Up& Down-stream Firms
In Appendix, we switch the roles of the upstream and the downstream firms in the basic model,
and prove a similar result will arise when the upstream (instead of the downstream) firm has
better outside options.
Again assume the two firms’ outputs are related by a one-to-one ratio, with the upstream
firm’s marginal cost being constant at c. As its outside option, the upstream firm can supply
a final product directly, of quality v(e), if negotiation with the downstream firm breaks down,
where e is its eﬀort to improve the value (v′ (e) > 0 and v′′ (e) < 0), at a cost of S U (e) with
S U′ (e) ≥ 0 and S U′′ (e) > 0.
The game structure is as follows: in the first stage, the upstream firm makes an eﬀort e
to improve its outside options; in the second stage, both firms bargain over the trading terms;
finally, in the third stage, the downstream firm sets the quantity of final output. The game issolved by backward induction as before.
As in the benchmark, when the upstream firm uses a two-part tariﬀ contract, it sets thewholesale price at its marginal cost, c. The fixed payment from the downstream firm to the
upstream firm is T d . In an abstract form, the gross profit of the downstream firm is πd (c), where
π′d (c) < 0 and π′′d (c) > 0.
On the other hand, its outside option is when the upstream firm directly enters the downstream market and supplies the final product, with a gross profit of πo (c, v), where ∂πo (c, v)/∂c <0 and ∂2 πo (c, v)/∂c2 > 0; that is, an increase in c diminishes the gross profit, at a decreasingrate. But the converse holds for quality, ∂πo (c, v)/∂v > 0 and ∂2 πo (c, v)/∂v2 > 0. The crosspartial derivative is ∂2 πo (c, v)/∂c∂v < 0.

10

Thus, T d is chosen to satisfy
πd (c) − T d = T d − πo (c, v).

Then the firms’ net profits in the second stage are respectively
πd (c) − πo (c, v(e)),
2πd (c) + πo (c, v(e))=− S U (e).
2

ΠD =
ΠU

And in the first stage, the upstream firm’s maximization problem leads to:∂πo (c, v(e)) ′· v (e) − 2S U′ (e) = 0.
∂v
Total diﬀerential of the above yields
[ 2
]
[
]
∂ πo (c, v(e))∂πo (c, v(e)) ′∂πo (c, v(e)) ′′′2′′· (v (e)) +· v (e) − 2S U (e) de +· v (e) dc = 0,
∂v2
∂v
∂c∂v
which gives de/dc < 0 because the terms in the first brackets are the second-order conditions
and negative, and the last term in the second brackets is also negative.
Finally, we examine how the upstream firm’s marginal cost aﬀects the downstream firm’s
profit.
∂πo (c, v(e)) ∂πo (c, v(e)) ′ de2dΠD
= π′d (c) −−· v (e)
dc
∂c
∂v
dc
de
∂π
(c,
v(e))
o
= π′d (c) −−2S U′ (e) .
|{z} | {z∂c } | {z
dc
}
(−)

(+)

(+)

If the third term is strong enough, the sign of dΠD /dc becomes positive. That is, the eﬃciency
improvement of the upstream firm can harm the downstream firm, even though it generates a11

bigger pie, analogous to the result in the basic model.

ExampleAssume that p = a − q, v(e) = v + e, and S (e) = γe2 /2, with a, v, and γ(> 1/4)

being positive constants. The second stage net profits of the two firms are then given asΠD (e) =))1(1((a − c)2 − [v(e) − c]2 + 4F , ΠU (e) =(a − c)2 + [v(e) − c]2 − 4F − S (e).
88In the first stage, the first-order condition of the upstream firm is∂ΠU (e) a − w − (4γ − 1)ev−c== 0 → e∗ =.
∂e44γ − 1
Because the quantity under which bargaining breaks down, (v(e∗ ) − c)/2, is smaller than whenagreement is reached, we require c > a − 4γ(a − v). Substituting e∗ into ΠD (e) givesΠD (e∗ ) =[(4γ − 1)a + 4γv − (8γ − 1)c]((4γ − 1)a − 4γv + c) F
+ .
8(4γ − 1)2
2

The first term is positive if−(4γ − 1)a + 4γv < c <(4γ − 1)a + 4γv.
8γ − 1

Diﬀerentiating ΠD (e∗ ) with respect to c yields∂ΠD (e∗ ) −(4γ − 1)2 a + 16γ2 v − (8γ − 1)c=,∂c4(4γ − 1)2
that is positive if and only if−(4γ − 1)a + 4γv < c <−(4γ − 1)2 a + 16γ2 v,
8γ − 1

under which a decrease in c decreases the profit of the downstream firm.
12