Aggregative games and oligopoly theory: short‐run and long‐run analysis

• Published date: 01 June 2020
• Author: Simon P. Anderson,Daniel Piccinin,Nisvan Erkal
• Date: 01 June 2020
• DOI: http://doi.org/10.1111/1756-2171.12322

RAND Journal of Economics

Vol.51, No. 2, Summer 2020

pp. 470–495

Aggregative games and oligopoly theory:

short-run and long-run analysis

Simon P. Anderson∗

Nisvan Erkal∗∗

and

Daniel Piccinin∗∗∗

We compile an IO toolkit for aggregative games with positive and normative comparative statics

results for asymmetric oligopoly in the short and long run. We characterize the classof aggrega-

tive Bertrand and Cournot oligopoly games, and the subset forwhich the aggregate is a summary

statistic for consumer welfare. We close the model with a monopolistically competitive fringe for

long-run analysis. Remarkably, we show strong neutrality properties in the long run across a

wide range of market structures. The results elucidate aggregative games as a unifying principle

in the literature on merger analysis, privatization, Stackelberg leadership, and cost shocks.

1. Introduction

Many non-cooperative games in economics are aggregative games, where each player’s

payoff depends on its own action and an aggregate of all players’ actions. Examples abound in

industrial organization (oligopoly, contests, R&D races), public economics (public goods pro-

vision, tragedy of the commons), and political economy (political contests, conflict models), to

∗Department of Economics, University of Virginia; sa9w@virginia.edu.

∗∗Depar tment of Economics, Universityof Melbourne; n.erkal@unimelb.edu.au.

∗∗∗Brick Cour t Chambers; daniel.piccinin@brickcourt.co.uk.

An earlier version of this article was circulated as CEPR Discussion Paper No. 9511, with the title “AggregativeOligopoly

Games with Entry.” We thank David Myatt (the Editor) and three anonymous referees for their constructive comments.

We are also thankful to Suren Basov, David Byrne, Chris Edmond, Maxim Engers, Daniel Halbheer, Joe Harrington,

Simon Loertscher, Phil McCalman, Claudio Mezzetti, Volker Nocke, Martin Peitz, Frank Stähler, Jun Xiao, Jidong Zhou,

and especially the late Richard Cornes for comments and discussion. We also thank seminar participants at the Federal

Trade Commission, Johns Hopkins University, New YorkUniversity, National Universityof Singapore, and University of

Mannheim, and conference participants at the Monash IO Workshop (2018), Australian National University Workshop

in Honor of Richard Cornes (2016), North American Winter Meeting of the Econometric Society (2013), Australasian

Economic Theory Workshop (2011), EARIE (2010), and CORE Conference in Honor of Jacques Thisse (2010) for their

comments. Imogen Halstead, Jingong Huang, Boon Han Koh, Charles Murry, and Yiyi Zhou have provided excellent

research assistance. The first author thanks National Science Foundation for financial support and the Department of

Economics at the University of Melbourne for its hospitality. The second author gratefully acknowledgesfunding from

the Australian Research Council (DP0987070).

470 © 2020, The RAND Corporation.

ANDERSON, ERKAL AND PICCININ / 471

name a few.1In oligopoly theory, a prominent example is the homogeneous product Cournot

model. Commonly used differentiated product demand models like logit, CES, and linear dif-

ferentiated demand all fit in the class. These oligopoly models are widely used in disparate

fields. Outside of industrial organization, the CES model is central in theories of international

trade (e.g., Helpman and Krugman, 1987; Melitz, 2003), endogenous growth (e.g., Grossman

and Helpman, 1993), and new economic geography (e.g., Fujita, Krugman, and Venables, 2001;

Fujita and Thisse, 2002). The logit model forms the basis of the structural revolution in empirical

industrial organization.

One reason why models like logit and CES are so popular is uncovered through recogniz-

ing them as aggregative games. The oligopoly problem in broad is complex: each firm’s action

depends on the actions of all other firms. An aggregative game reduces the degree of complex-

ity drastically to a simple problem in two dimensions. Each firm’s action depends only on one

variable, the aggregate, yielding a clean characterization of equilibria with asymmetric firms

in oligopoly.

We study the positive and normative economics of aggregative games for asymmetric

oligopoly models. Our first aim in Sections 2 and 3 is to provide a toolkit for IO oligopoly

aggregative games. In Section 2, we develop the key properties of these games using the device

of the inclusive best reply (ibr) function, and relate our analysis to standard IO techniques using

best reply functions. In particular, we show how standard intuition from strategic substitutes or

complements carries over easily to the aggregative game approach.

In Section 3, we consider the demands and utility functions for which Bertrand and Cournot

differentiated product oligopoly games are aggregative so that the toolkit applies. Even though

payoffs are a function of the aggregate in these games, consumer welfare does not have to be.

Where it is, the aggregative structure of the game can be exploited to dramatically simplify the

consumer welfare analysis. Tracking the aggregate pins down the consumer welfare results. We

characterize the Bertrand and Cournot games where consumer welfare depends on the aggregate

variable only. In such cases, the toolkit analysis delivers positive as well as normative properties

of equilibria in asymmetric oligopoly models.

In Sections 4 and 5, we apply the toolkit to provide a compendium of comparative stat-

ics results for oligopoly models in the short and long run, respectively. In Section 4, we in-

troduce a general concept of ibr “aggression,” which we use to compile a ranking of firms’

actions (e.g., prices and quantities), profits, and market shares across a wide range of char-

acteristics and market events, such as ownership structure, technological changes, and tax or

regulatory advantages. Our analysis underscores the analytical tractability that comes with re-

ducing the problem to two dimensions, by providing a graphical analysis for asymmetric firm

types.

In Section 5, we consider aggregative oligopoly games with endogenous entry and inves-

tigate the long-run effects (both positive and normative) of alternative market structures and

events. We close the model with a monopolistically competitive fringe, which competes with an

exogenously determined set of “large” oligopolistic firms. This constitutes an interesting market

structure in its own right, following the pioneering work of Shimomura and Thisse (2012) and

Parenti (2018). By allowing for a continuum of marginal entrants, this device provides a clean

solution to free-entry equilibrium without needing to account for the integer issues that arise

under oligopoly.

Long-run analysis with explicitly aggregative games had not been explored in the literature

before Anderson, Erkal, and Piccinin (2013), who close the model with symmetric oligopolists

that make zero profits.2The current analysis complements that in Anderson, Erkal, and Pic-

cinin (2013) and shows that the results are qualitatively the same if the model is closed with a

1See Corchón (1994, Table 1) for a diverselist of applications where aggregative games emerge. See Cornes and

Hartley (2005, 2007a, and 2007b) specifically for examples of aggregative games in contests and public goods games.

2See Polo (2018) for a survey of the theoretical literature on entry games and free entry equilibria.

C

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