Optimal collusion with limited liability

• DOI: http://doi.org/10.1111/j.1742-7363.2013.12015.x
• Author: Etienne Billette de Villemeur,Bruno Versaevel,Laurent Flochel
• Published date: 01 September 2013
• Date: 01 September 2013

doi: 10.1111/j.1742-7363.2013.12015.x

Optimal collusion with limited liability

Etienne Billette de Villemeur,∗Laurent Flochel†and Bruno Versaevel‡

Collusion sustainability depends on firms’ ability to impose sufficiently severe punishments in

the event of deviation from the collusive rule. Weextend results from the literature on optimal

collusion by investigating the role of a limited liability constraint. We examine all situations in

which either structural conditions, financial considerations, or institutional circumstances set a

lower bound, possibly negative,to firms’ profits. For a large class of repeated games we show that,

when the limited liability constraint binds, there exists an infinity of multi-period punishment

paths that permit firms to implement the optimal collusive strategy. The usual front-loading

scheme is only a specific case and an optimal punishment profile can take the form of a price

asymmetric cycle. Wecharacterize the situations in which a longer punishment does not perform

as a perfect substitute for more immediate severity. In this case, the lowest discount factorthat

permits collusion is strictly higher than without the limited liability constraint, which hinders

collusion.

Key wor ds collusion, oligopoly, limited liability

JEL classification C72, D43, L13

Accepted 5 June2012

1 Introduction

In this paper, we characterizethe implementabilit y of a collusivestrategy by oligopolistic firms when

their ability to punish deviations over one or several periods is limited.

Firms in the same industry may increase profits by coordinating the prices they charge or the

quantities they sell. In a legal context in which collusive agreements cannot be overtly enforced, and

future profits are discounted,it is well known that an impatient firm may find it privately profitable to

deviate from a collusivestr ategy.This renders collusive agreements fundamentally unstable. However,

firms may design non-cooperative discipline mechanisms that help implement collusion.

Many papers have examined the structural conditions that facilitate the formation of cartels.

Most theoretical analyses rely on a class of dynamic models usually referred to as supergames. These

*University of Lille, EQUIPPE, France.Email: etienne.de-v illemeur@univ-lille1.fr

†Charles River Associates, Paris, France.

‡EMLYONBusiness School & GATE (UMR 5824 CNRS), Lyon,France.

This paper was completed while the first author was visiting the University of Montreal whose hospitality is gratefully

acknowledged. Several versionsbenefited from comments at the 2005 ETAPE seminar in industrial economics (University

of Paris 1), the 2005 Econometric Society WorldConference (University College London), the 2008 Dynamic Games in

Management Science conference (GERAD-HEC Montreal),and the 11th conference of the Society for the Advancement

of Economic Theory in 2011 (SAET). Weare grateful to Makoto Yano (editor) and an anonymousreferee for very helpful

suggestions. Special thanks are addressed to Nihat Aktas, LuisCorch ´

on, Jacques Cremer,Michel Le Breton, Onur ¨

Ozg¨

ur,

Richard Ruble, and Georges Zaccour for insightful comments or help in various other forms. All remaining errors are

ours.

International Journal of Economic Theory 9 (2013) 203–227 © IAET 203

Optimal collusion with limited liability Etienne Billette de Villemeur et al.

models feature a repeated market game in which firms maximize a flow of discounted individual

profits by non-cooperatively choosing a price or a quantity over an infinite number of periods.

When a deviation can be credibly and sufficiently “punished” via lower industry prices or larger

quantities in subsequent time periods, conditions on structural parameters can be derived which,

when satisfied, make collusion stable.

A majority of recent contributions to the literature have investigatedthe impact of various model

specifications on the sustainability of collusion with stick-and-carrot mechanisms in the style of

Abreu (1986, 1988). In this category of mechanisms, if a firm deviates from collusion, all firms

play a punishment strategy over one or several periods—the stick—which is more severe than Nash

reversion (i.e., it leads tolower instantaneous profits, possibly negative) before returning to a collusive

price or quantity. If a deviation occurs in a punishment period, the punishment phase restarts,

otherwise all firms resume the collusive behavior to earn supernormal profits—the carrot. More

specifically, Abreu (1986) exploits a single-period punishment mechanism for a class of repeated

quantity-setting oligopoly stage games with symmetric sellers of a homogeneous good, constant

positive marginal costs, and no fixed cost. For a given discount factor, the most severe punishment

strategy—following a deviation either from the collusive path or from a punishment rule—that

sustains collusion, is characterized. It results in the highest level of discounted collusive profits.

Our objective is to enrich the study of the circumstances that facilitatecollusion, or make it more

difficult to sustain.1This is done by investigating the exact role of an assumption, in the seminal

paper by Abreu(1986), according to which the price is strictly positive for all levels of industry output,

so that there is no floor for firms’ losses when the constant marginal cost is also specified above zero.

Indeed the quantity sold—and related costs—tend to infinity when firms charge below the marginal

cost and the price approaches zero.In that case, the single-period punishment that follows a deviation

can be made as severe as needed. Although the strategy set is assumed to be finite, the upper bound to

the available quantities is so high as to never be used as a punishment action that sustains collusion.

To our knowledge, most papers—if not all—that refer to Abreu (1986, 1988) actually overlook

this key assumption by introducing more structure. They typically borrow the same stick-and-carrot

mechanism with a single punishment period, although they either assume that demand is finite at all

prices, or that firms have limited production capacity.It follows that losses are bounded from below

in a punishment period, and collusion can be hindered. In that case, an extension of the punishment

phase to several periods appears as a natural substitute for more immediate severity. Fudenberg and

Tirole (1991, p. 165) emphasize that, when the severity of punishments is limited the punishment

phase should be longer, although “it is not obvious precisely which actions should be specified” in

the punishment phase. Our paper is novel in that it thoroughly examines this point. This is done in a

setup that encompasses the main assumptions in Abreu (1986). In our model, firms sell substitutable

goods (possibly differentiated), inversedemand functions are non-increasing (they can be finite at all

prices), the marginal cost is constant and non-negative (it can be zero), and therecan be a fixed cost.

In addition to standard incentive and participation constraints, a key specification that we introduce

is the limited liability constraint, which amounts to imposing a limitation on the lowest level of

profits a firm may earn. Whether the limited liability constraint binds or not impacts firms’ choices

of price or quantity in the punishment phase.

Interestingly,a limited liability constraint is not a technical sophistication that we add to standard

specifications. It is de facto present, or latent, in all models where demand or technological conditions

1The analysis of the connection between structural conditions and collusion stability with a stick-and-carrot mechanism

`alaAbreu has been extended to many aspects. The literature is briefly reviewed in a dedicated section that follows our

main results.

204 International Journal of Economic Theory 9 (2013) 203–227 © IAET