Endogenous timing of moves in Bertrand–Edgeworth triopolies

• Date: 01 December 2016
• Published date: 01 December 2016
• DOI: http://doi.org/10.1111/ijet.12098

doi: 10.1111/ijet.12098

Endogenous timing of moves in Bertrand–Edgeworth

triopolies

Attila Tasn´

adi∗

We determine the endogenous order of movesin w hichthe fir ms set their prices in the frame-

work of a capacity-constrained Bertrand–Edgeworth triopoly. A three-period timing game that

determines the period in which the firms announce their prices precedes the price-setting stage.

We show that the firm with the largest capacity sets its price first, while the two other firms set

their prices later.Our result extends a finding by Deneckere and Kovenock from duopolies to tri-

opolies. This extension was made possible byHirata’s recent advancements on the mixed-strategy

equilibria of Bertrand–Edgeworth games.

Key wor ds Bertrand–Edgeworth, price leadership, oligopoly, timing games

JEL classification D43, L13

Accepted 13 October 2015

1 Introduction

A challenging question for oligopoly theory and its applications is the determination of the right

order of moves because this influences the market structure and its equilibrium behavior. In this

paper we address the timing problem for price-setting games, in particular, within the framework

of a homogeneous good capacity-constrained Bertrand–Edgeworth triopoly. The key feature of

these price-setting games is that the firms may serve less than the demand they are facing. The

main difficulty with these models is that one has to consider mixed-strategy equilibria since for the

interesting cases there is a lack of equilibrium in pure strategies.1

Deneckere and Kovenock (1992) determined the endogenous order of decisions in a homoge-

neous good Bertrand–Edgeworth duopoly game with capacity constraints. Their result was partially

generalized from duopolies to oligopolies byGangopadhyay (1993) who compared the simultaneous-

move case with the purely sequential-move case, which still did not determine the endogenous order

of moves for the oligopolistic case. The main difficulty in solving the timing problem lies in handling

the mixed-strategy equilibrium of the Bertrand–Edgeworth game. Beckmann (1965), Levitan and

Shubik (1972), Vives (1986), and Cheviakov and Hartwick (2005) determined the mixed-strategy

equilibrium solutions under quite restrictive assumptions such as linear demand or identical firms.

Under more general conditions the mixed-strategy equilibrium has been given in non-closed form

∗MTA-BCE “Lend¨

ulet” Strategic Interactions Research Group, Department of Mathematics, Corvinus University of

Budapest, Budapest, Hungary.Email: attila.tasnadi@uni-corv inus.hu

I am very grateful to an anonymous referee for comments and suggestions which helped to improveprevious versions of

this paper.Financial support from the Hungarian Scientific Research Fund (OTKA K-101224) is gratefully acknowledged.

1For more on Bertrand–Edgeworth games the reader is referred to Vives (1999).

International Journal of Economic Theory 12 (2016) 317–334 © IAET 317

International Journal of Economic Theory

Bertrand–Edgeworth triopolies Attila Tasn´

adi

by Kreps and Scheinkman (1983), Davidson and Deneckere (1986), Osborne and Pitchik (1986),

and Allen and Hellwig (1993) in the duopolistic setting.

The investigation of the triopolistic timing game has been made possible by recent character-

izations of the mixed-strategy equilibrium of capacity-constrained Bertrand–Edgeworth triopolies

by Hirata (2009), and independently by De Francesco and Salvadori (2010). Based on their charac-

terizations, we find that the large-capacity firm will emerge as the endogenous price leader, which

extends a result for the duopolistic case obtained by Deneckere and Kovenock (1992). Timing of

decisions within the price-setting framework usually results in known forms of price leadership, and

therefore we will also relate our results to the dominant firm model of price leadership.

In general, there is a growing literature on endogenous timing of decisions in oligopolies in both

the price-setting and the quantity-setting framework. The first contributions in this direction were

made by Gal-Or (1985), Dowrick (1986), and Boyer and Moreaux(1987) in works that compared the

outcomes of duopoly games with exogenously givenorderings of moves to find out whether the leader

or the follower has a more advantageous position. The recent literature aims to solve the conflict

concerning roles, and determines an endogenous order of moves under certain circumstances. Just

to mention some important works from the large number of contributions, we refer to Hamiltonand

Slutsky (1990), Deneckere and Kovenock(1992), van Damme and Hurkens (1999, 2004), Matsumura

(1999, 2002), Dastidar and Furth (2005), Yano and Komatsubara (2006, 2012), and von Stengel

(2010).

It is worth mentioning that another strand of the literature aims to generalize results from the

Bertrand–Edgeworth duopoly game with deterministic demand to the case of demand uncertainty.

For notable results in this direction we refer to Hviid (1991), Reynolds and Wilson(2000), de Frutos

and Fabra (2011), and Lepore (2012).

The remainder of this paper is organized as follows. Section 2 presents our framework and

an important lemma. Section 3 considers price-setting games with exogenously given orderings of

moves. Section 4 determines the solution of our three-period timing game. Section 5 contains some

concluding remarks.

2 Framework and preliminary result

Suppose that there are three firms in the market, where we shall denote the set of firms by N=

{1,2,3}. Weassume that the firms have zero unit costs2up to some positive capacity constraints. We

shall denote by kithe capacity constraint of firm iand by K=3

i=1kithe aggregate capacity of the

firms. Let the capacity constraints be ordered decreasingly, and so as to reduce the number of cases

we also assume that the two firms with the smallest capacities have identical capacities, k1>k

2=k3.

We summarize the assumptions imposed on the triopolists’ costfunctions as follows.

Assumption 1 There are three firms on the market with zero unit costs and capacity constraints k1>

k2=k3>0.

We refer to firm 1 as the large firm and to firms 2 and 3 as the small firms. The demand is given by

the function Don which we impose the following restrictions.

2Since we are considering the production-to-order version of the Bertrand–Edgeworth game (that is, production takes

place after the firms’ prices are revealed), the realassumption here is that the firms have identical unit costs.

318 International Journal of Economic Theory 12 (2016) 317–334 © IAET