Collusion and heterogeneity of firms

• DOI: http://doi.org/10.1111/1756-2171.12174
• Published date: 01 March 2017
• Author: Federico Zincenko,Ichiro Obara
• Date: 01 March 2017

RAND Journal of Economics

Vol.48, No. 1, Spring 2017

pp. 230–249

Collusion and heterogeneity of firms

Ichiro Obara∗

and

Federico Zincenko∗∗

We examine the impact of heterogeneous discounting on collusion in dynamic Bertrand compe-

tition. We show exactly when collusion can be sustained and how collusion would be organized

efficiently with heterogeneous discounting. First, we show that collusion is possible if and only

if the average discount factor exceeds a certain threshold, with or without capacity constraints.

Next, we identify a dynamic pattern of market share that characterizes efficient collusion and

obtain the unique long-run prediction despite the presence of multiple equilibria. In the long run,

the most patient firm and the most impatient firm tend to dominate the market.

1. Introduction

To understand when and how collusion arises with heterogeneous discounting, we study a

dynamic Bertrand competition model, in which firms discount future profits at different discount

rates, and examine the impact of heterogeneous discounting on collusion. The model of dynamic

Bertrand/Cournot competition is the standard framework to analyze and understand collusion

(Tirole, 1988). The logic behind these models is simple: firms are willing to collude because

they fear a “price war” in the future. Clearly, the discount rate is the most critical parameter that

determines the effectiveness of such an intertemporal incentive scheme.

Almost all of the models of this kind, however, assume symmetric discounting. In fact, it is

often assumed that firms are completely symmetric in every aspect. Symmetric models are useful

for understanding a variety of issues associated with collusion due to its tractability.Nevertheless,

the symmetry assumption is unrealistic and, thus, limits the scope of applications of such models,

especially when symmetry is imposed on such a crucial parameter of the model.1This motivates

us to introduce heterogeneity of discount rates to the dynamic model of collusion.

∗University of California, Los Angeles; iobara@econ.ucla.edu.

∗∗University of Pittsburgh; zincenko@pitt.edu.

This article was developed from the second chapter (coauthored with Obara) of Zincenko’s doctoral dissertation at

UCLA. We wouldlike to thank the Editor and anonymous referees for their useful comments. We also like to thank the

participants at many seminars at various universities,2011 Asian Meeting of the Econometric Society at Korea University,

2012 Southwest Economic Theory Conference at UC San Diego, and 2013 North American Summer Meeting of the

Econometric Society at USC. All remaining errors are ours.

1Fershtman and Pakes (2000) emphasize the importance of heterogeneity among firms in the same market.

230 C2017, The RAND Corporation.

OBARA AND ZINCENKO / 231

There are at least two reasons to believethat future profit is discounted differently by different

firms. The first one is the cost of capital. If a firm faces a higher interest rate than do other firms for

any reason (e.g., asymmetric information), then the firm would value profits in the short run more

than would the other firm. The second reason is heterogeneity of managers’ discount rates. Even if

the cost of capital is the same across firms, the managers who run these firms may discount future

profits differently. According to Stein (1989), managers focus on the short term when their salaries

depend heavily on current stock prices. Shleifer and Vishny (1990) argue that short horizons of

investors lead to short horizons of managers who are averse to underpricing of their equity.

This type of aversion may occur when low equity prices increase the probability of replacement.

Narayanan (1985) emphasizes that managerial career concerns may result in a focus on short-term

horizons. Specifically,Narayanan (1985) states that, if the manager possesses private information

unavailable to investors and his ability is unknown, then he might choose quicker-return projects

to increase his wage. To the extent that these considerations (compensation scheme, investors’

horizon, managerial career concerns, manager’s private information) vary across firms, future

profit should be discounted differently by different firms.

Formally, our model is an infinitely repeated Bertrand (price-setting) game by firms with

heterogeneous discount rates and constant marginal costs. We assume that all of the firms produce

the same product and that the firm that charges the lowest price must serve the entire market,

as in the standard Bertrand game. When two or more firms charge the same (lowest) price, we

allow those firms to divide the market in a flexibleway through communication.2Thus, firms can

collude in two dimensions: price and market share.3

Our main findings consist of two parts. For the first part, we characterize exactly when

collusion is possible with heterogeneous discounting. We show that the average discount factor

is the key variable for determining the possibility of collusion. Specifically, we show that the

monopoly price (indeed, any profitable price strictly above the marginal cost) can be sustained

with nfirms if and only if the average discount factor exceeds n−1

n. If the average discount factor

falls strictly below this critical threshold, then the competitive outcome prevails in every period

in any equilibrium. Thus, heterogeneity in discounting does not discourage collusion, per se.

Because the allocation of market shares is flexible, a more patient firm is willing to give up more

market shares to more impatient firms, whose incentive constraints are then relaxed. Thus, the

distribution of discount rates matters, in general. In our Bertrand setting, it turns out that the first

moment of the distribution determines completely the possibility of collusion.

We can extend this result to the case where the firms face capacity constraints. In this case,

the critical level of the average discount factor depends on the price to be supported. The average

discount factor needs to be larger to support a collusive price that is closer to the monopoly

price. It may be the case that firms cannot collude at the monopoly price but can collude at a price

lower than the monopolyprice. As we focus on asymmetric discounting and keep every other part

of the model symmetric intentionally, we assume symmetric capacity constraints for the most

part. Nevertheless, we discuss asymmetric capacity constraints and illustrate the possibility that

asymmetry in capacity constraints facilitates collusion in the presence of asymmetric discounting.

For the second part, we examine how collusion should be organized. We provide an almost

complete characterization of all collusive equilibria that, from the firms’ perspective, are Pareto-

efficient. First, we show that the equilibrium prices are always set at the monopoly price in most

of the efficient collusive equilibria. If this is not the case, then the equilibrium prices must be

increasing monotonically and quickly until they reach the monopoly price and staythere forever.4

Thus, it is almost without loss of generality to focus on efficient collusive equilibria with the

2Genesove and Mullin (2001) document actual examples of communication for the sugar refining cartel, although

they do not appear to be related directly to the wayin which communication works in our model.

3Benoit and Krishna (1987) and Staiger and Wolak(1992) study a dynamic oligopoly model in which firms choose

capacities, then prices.

4Some efficient collusive equilibrium with very asymmetric payoffsmay require the market price to be lower than

the monopoly price initially (see the three-firm example in Section 4).

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