Inventories and Endogenous Stackelberg Leadership in Two‐Period Cournot Oligopoly

• Author: Sébastien Mitraille,Michel Moreaux
• Published date: 01 December 2013
• Date: 01 December 2013
• DOI: http://doi.org/10.1111/jems.12031

Inventories and Endogenous Stackelberg Leadership

in Two-Period Cournot Oligopoly

S´

EBASTIEN MITRAILLE

Universit´

e de Toulouse

ToulouseBusiness School

20, Bd Lascrosses - BP 7010 31068 ToulouseCedex 7, France

s.mitraille@esc-toulouse.fr

MICHEL MOREAUX

ToulouseSchool of Economics (IDEI and LERNA)

21, All´

ee de Brienne 31000 Toulouse, France

michel.moreaux@tse-fr.eu

Two-period Cournot competition between n identical firms producing at constant marginal cost

and able to store before selling has pure strategyNash-perfect equilibria, in which some firms store

to exert endogenously a leadership over rivals. The number of firms storing balances market share

gains, obtained by accumulating early the output, with losses in margin resulting from increased

sales and higher operation costs. This number and the industry inventories are nonmonotonic in

n. Concentration (HHI) and aggregate sales increase due to the strategic use of inventories.

1. Introduction

Inventory management is a key issue for many firms, be they large or small: well-known

management theories emphasize the need to stay lean, the gain to proceed with just-in-

time deliveries and zero inventories, or on the contrary the possibility to benefit from

economies of scale by purchasing or producing the economic-order quantity (Arrow

et al., 1951;Zipkin,2000), and hence storing. The importance of inventories in economic

analysis has also been recognized. For instance Arvan and Moses (1982) show that

economies of scale in the production of a storable good lead a monopolist to adjust its

output in the long run. Social losses then differ compared to the static case. Blanchard

(1983) investigates the behavior of finished good inventories in the U.S. automobile

industry.1Descriptivestatistics show that ”production smoothing is not the dominant element

of inventory behavior and that target inventory is probably a function of current sales” (p.374,

our emphasis), and econometric estimations confirm this finding. Using a sample of

U.S. firms from the good-producing manufacturing industries, Amihud and Mendelson

(1989) show that firms hold a larger level of inventories the greater their market power.

However an interesting theoretical question is to understand how a target level of

Wethank Daniel Spulber, an anonymous coeditor, and two referees for their useful comments. Thanks to Eric

Maskin and to participants of the Summer School on Game Theory and its Applications (Free University of

Bozen-Bolzano), to Gianni De Fraja, Ludovic Renou, to seminar participants at the University of Leicester

and at the University of Montpellier I (LAMETA), and to Joseph Harrington for their comments on an earlier

version. We also thank participants of the IIOC (Savannah), CEA (Montreal), ESEM (Vienna), and ASSET

(Lisbon) conferences.

1. Blanchard uses monthly data from January 1966 to December 1979 for 10 divisions of the main U.S.

manufacturers: five from GM, two from Ford, two from Chrysler, and one from American Motors.

C2013 Wiley Periodicals, Inc.

Journal of Economics & Management Strategy, Volume22, Number 4, Winter 2013, 852–874

Inventories and Endogenous Stackelberg Leadership 853

inventories is determined when firms are part of an oligopoly, as for example the U.S.

automobile industry,and more broadly to investigate how inventory levels vary with the

number of firms in competition. Is it possible to revert the causal relationship, and find

that a larger equilibrium level of inventories allows firms to enjoy larger equilibrium

market shares and consequently a greater market power on an oligopolistic market?

Moreover,how is the equilibrium level of inventories affected by changes in the number

of competitors, and what are the implications for consumers’ surplus? Answering these

questions is the object of this study.

Under constant returns-to-scale and in the absence of any capacity constraint, we

show that Cournot oligopolistic competition in which nfirms produce to store in a first

period, before producing again and selling in the market in second period, has a Nash-

perfect equilibrium in pure strategy no matter the number of firms in competition or the

cost of storage. Equilibria can be multiple and differ according to the number of firms

which store. The static Cournot–Nash outcome where no firm stores is the expected

unique equilibrium of the game when the cost of storage is large enough, but outside

this range it is not an equilibrium anymore. For intermediate or small values of the

cost of storage, equilibria are such that a subset of firms is storing a positive quantity

to exert endogenously some Stackelberg leadership over the other firms who behave

as followers, who do not store, and who produce and sell in second period only. The

number of firms storing at equilibrium results from the individual trade-off between a

larger market share,gained thanks to the endogenous commitment to be more aggressive

on the market by accumulating early the output, and a smaller margin, due to the increase

in aggregate sales and to the increase in the costs of operations, which are increased by

storage costs. As the number of competitors over the two periods increases, the number

of firms that can exert some leadership increases but the individual market share of each

leader decreases. Facing a margin reduction due to larger aggregate sales and to larger

operations costs, not storing may become more profitable than storing when the number

of competitors nis large enough, and the number of leaders, which increases when n

is small enough, may decrease when nis large enough. We confirm this intuition and

show that the number of firms storing and the aggregate inventories at equilibrium can

be nonmonotonic with the number of firms in competition. Finally as the equilibria are

asymmetric—except of course the Cournot one when it occurs—concentration increases

compared to what it would be at the static Cournot–Nash equilibrium, with identical

demand and costs. We show that the Herfindhal–Hirschman concentration index (HHI)

is strictly larger than the value it obtains in static Cournot competition, whereas the

price–cost margin is strictly lower due to the increase in aggregate sales caused by

inventories; consumers’ surplus is therefore larger than in static Cournot competition.

We illustrate these findings and study the effect of an entry of competitors on

the set of equilibria with an example. The number of firms storing and the aggregate

inventories are nonmonotonic with respect to the number of firms in competition. As

predicted, both the HHI and the aggregate sales reach levels strictly larger than what

they do in static Cournot competition when storing to gain market shares is a profitable

strategy: the market is more concentrated and aggregate sales are larger than what it

would be if storage were impossible. Togetherwith our theoretical findings, these results

suggest that observing a mass of firms storing is more likely on oligopolistic markets

than on market structures closer to the duopoly or to perfect competition. Moreover they

do also suggest that the HHI cannot be used as the sole measure of competitiveness on

markets where the finished product is storable and storage is used to exert some market

share leadership.