Tacit Collusion in a One‐Shot Game of Price Competition with Soft Capacity Constraints

• DOI: http://doi.org/10.1111/jems.12049
• Author: Marie‐Laure Cabon‐Dhersin,Nicolas Drouhin
• Published date: 01 June 2014
• Date: 01 June 2014

Tacit Collusion in a One-Shot Game of Price

Competition with Soft Capacity Constraints

MARIE-LAURE CABON-DHERSIN

Universit´

e de Rouen, CREAM

3 avenue Pasteur, 76100 Rouen, France

marie-laure.cabon-dhersin@univ-rouen.fr

NICOLAS DROUHIN

Ecole Normale Sup´

erieure de Cachan and CES-Universit´

e de Paris 1-Panth´

eon Sorbonne Ens Cachan,

61 avenue du Pr´

esident Wilson, 94230 Cachan, France

drouhin@ecogest.ens-cachan.fr

This paper analyzes price competition in the case of two firms operating under constant returns

to scale with more than one production factor. Factors are chosen sequentially in a two-stage

game generating a soft capacity constraint and implying a convex short-term cost function in

the second stage of the game. We show that tacit collusion is the only predictable result of the

whole game, that is, the unique payoff-dominant pure strategy Nash equilibrium. Technically,

this paper bridges the capacity constraint literature on price competition and that of the convex

cost function.

1. Introduction

The literature on Industrial Organization emphasizes the role of threats and retaliations

in a dynamic game framework to explain tacit collusion (Friedman, 1971; Abreu, 1986;

Benoit and Krishna, 1987; Feuerstein, 2005). This paper gives an example of a market

in which the collusive outcome arises as a predictable result of a nonrepeated price

competition duopoly with two stages. This is consistent with the experimental findings

of a remarkable degree of coordination around a collusive price by two firms (Abbink

and Brandts, 2008). Following Ivaldi et al. (2003), tacit collusion needs not to involve any

collusion in the legal sense, and, in particular, no communication between the parties.

“It is referred to as tacit collusion only because the outcome (in terms of prices set or

quantities produced, for example) may well resemble that of explicit collusion or even

of an official cartel.”

We start from the Bertrand competition model initiated by Dastidar (1995)1and

extended recently by Baye and Morgan(2002), Novshek and Chowdhury (2003), Hoernig

(2007), and Bagh (2010). In this setting, firms face convex costs and are committed

to satisfying the full demand. By lowering its price, a firm increases its revenue by

higher sales. But costs being convex, they will increase even more, making this deviation

nonprofitable. For this reason, a continuum of prices above the competitive price can

be sustained as Nash equilibria in pure strategies,2inducing a coordination problem in

We thank for their helpful comments and suggestions, the editors, two anonymous referees,various seminar

participants, and Simon Anderson, Francis Bloch, Raymond Deneckere, Joseph Harrigton, Edi Karni, Bertrand

Munier, and Andr´

edePalma.

1. See Vives (1999), section 5.1, pp. 117–123.

2. Other extensions with mixed-strategy equilibria and positive profit levels are provided by Baye and

Morgan (1999), Kaplan and Wettstein(2000), and Hoernig (2002).

C2014 Wiley Periodicals, Inc.

Journal of Economics & Management Strategy, Volume23, Number 2, Summer 2014, 427–442

428 Journal of Economics & Management Strategy

a nonrepeated framework. To claim the possibility of tacit collusion in this framework,

two questions still need to be answered: (1) Is the collusive outcome within the set of

the Nash equilibria? (2) Is there a plausible selection procedure to achieve the collusive

outcome as the unique solution? Dastidar (2001) answers the first question, but only

in the symmetric case. As far as we know, the second question has not been treated

explicitly.

According to Harsanyi and Selten (1988), two criteria can be used to solve the

coordination issue: The payoff dominance criterion, which appears to be the natural cri-

terion when it is common knowledge that both players are fully rational,3and the risk

dominance criterion that can be invoked when payoff dominance is insufficient to provide

uniqueness. Unfortunately,risk dominance may not be applied in the context of Bertrand

competition with convex costs, because of the infinity of equilibria.

In this paper, we will add a sequential choice of production factors into Dastidar’s

(1995) approach of Bertrand competition. As we will demonstrate, this plays a key role

in the coordination mechanism that leads to tacit collusion. Our model starts from a

constant returns to scale production function with two substitutable production factors

chosen sequentially. In the first stage, the firms invest, that is, they choose the quantity

of the fixed factors, quantity that will be invariable over the second stage. In the second

stage, firms compete on price and determine the quantity of variable factors needed

to satisfy the demand they will face. This implies that, when the first factor is fixed,

the short run marginal cost is convex in the second stage.4The sequential choice of

the production factors is certainly the central hypothesis of our approach. On the one

hand, it is a standard hypothesis in economic analysis, a textbook case renewing the

Marshallian tradition of distinguishing between short and long-term cost functions. On

the other hand, as far as we know, it appears that it has not been used in the recent

literature about price competition. However, we want to point out that this assumption

is a natural generalization of the notion of capacity constraints initiated by Edgeworth

(1925). There is a long tradition in Industrial Organization of considering firms that

are capacity constrained (Vives, 1980; Kreps and Scheinkman, 1983; Allen and Hellwig,

1986; Davidson and Deneckere, 1986; Allen et al., 2000). In those models, the constraint

is drastic (i.e., it is impossible to produce above the capacity). In our model, the choice

of the fixed factor corresponds to the choice of the production capacity. But the usual

way to model capacity constraints is equivalent, in our setting, with an assumption of

perfect complementarity of fixed and variable factors. In this case, when the capacity of

production is binding in the second stage, it is impossible to produce more, whatever the

quantity of variable factor.Our hypothesis of substitutability between fixed and variable

factors introduces a less drastic (soft) notion of capacity constraints,5and in many cases

a more realistic one. For example, we can take, as an illustration, the retailing sector.

The fixed factor refers to the surface needed to sell and the length of the shelves used

to display the products, whereas the variable factor can be interpreted as the number of

employees needed to fill up the shelves. For a given surface of the store, there exists an

3. In the sense of Aumann (1976), cf. Harsanyi and Selten (1988, p. 359). There is a huge literature on

equilibrium selection in games, see Cooper et al. (1990) for an introduction.

4. For simplicity, we will use a Cobb–Douglas production function, but what determines our results is

the convexity of the short-term marginal cost, which depends on the decreasing marginal productivity of the

variable factor, a universal assumption in economics. Our results do not depend on the specification of the

production function.

5. Our model provides foundations for the notion of nonrigid capacity constraint introduced by

Chowdhury (2009) directly in the cost function.