Stable cartels with a Cournot fringe.

• Author: Shaffer, Sherrill

1. Introduction

   The notion of an industry structure characterized by a small group of dominant firms plus a competitive fringe has a long tradition. More recent work explores conditions under which such a pattern constitutes an equilibrium, assuming collusive behavior among one group of firms and price-taking behavior within the fringe [2; 3; 4; 5; 6; 14; 17]. The alternative case of a Cournot fringe is analyzed briefly in Spulber [22, 471-73] and more extensively in Martin [14].

   Here we adopt the framework of this literature and build on it by analyzing the size and uniqueness of the stable cartel when the fringe is Cournot; endogeneity of Cournot versus Bertrand behavior within the fringe, given the stable cartel; possible endogeneity of the Stackelberg sequence of play between the cartel and the fringe; and effects of excludability from the cartel. Welfare effects are also briefly analyzed.

   Alternatively, cooperation within a cartel is equivalent to the outcome of horizontal mergers in the absence of synergies. As such, this paper presents a contrasting result to the analysis of exogenous Cournot mergers in Salant, Switzer, and Reynolds [18], endogenizes the merger decision, and demonstrates how a theory of mergers can be predicated on a Cournot fringe.

2. The Model

   Consider an industry of n identical firms facing a linear demand function. At first we take n as fixed and exogenous; a subsequent section considers the effects of entry. We shall explore the possibility that k of these firms will prefer to behave cooperatively to maximize their joint profits, while the n-k fringe firms choose their output levels noncooperatively. In the first stage each firm chooses to join either the cartel or the fringe. Consistent with the studies cited above, we do not address the question of which k out of n identical firms will join the cartel. Also consistent with most of those studies, we assume the existence of an enforcement mechanism for collusive behavior within the cartel.(1) Initially we assume (as in Spulber and Martin) that behavior within the fringe is Cournot and (as in all the cartel literature cited above) that the cartel behaves as a Stackelberg leader with respect to the fringe; subsections will explore the reasonableness of these assumptions.

   Define the market inverse demand function as p = a - bQ where Q = [Q.sub.c] + (n - k)[q.sub.f], the sum of the aggregate cartel quantity and the aggregate fringe quantity. In the simplest case, consider a linear cost function for each firm, [c.sub.i] = c[q.sub.i].(2) To ensure positive equilibrium output levels, we assume a [is greater than] c. Each member of the fringe will choose its output level to maximize its own profits, taking as given both the cartel output quantity and the output quantity of each other fringe firm. The first-order conditions for each fringe firm then yield a best-response function [q*.sub.f] = (p - c)/b, using notation similar to most of the literature cited.

   The cartel will choose its output level to maximize joint cartel profits, given the response function of the Cournot fringe. This is equivalent to having the cartel behave as a monopolist with respect to the residual demand curve given by [Q.sub.c] = [a + (n - k)c - (1 + n - k)p]/b, equivalent to an inverse demand function p = [a + (n - k)c - b[Q.sub.c]]/(1 + n - k).(3) Profit maximization for the cartel implies [Q.sub.*c] = (a - c)/2b and hence p* = [a + (1 + 2n - 2k)c]/[2(1 + n - k)]. The corresponding level of aggregate cartel profits is:

   [[Pi].sub.c](k) = [(a - c).sup.2]/[4b(1 + n - k)] (1)

   which is positive. Substituting p* into [q*.sub.f], we find [q*.sub.f] = (a - c)/[2b(1 + n - k)], implying a positive equilibrium profit level for each fringe firm equal to:

   [[Pi].sub.f](k) = [(a - c).sup.2]/[4b[(1 + n - k).sup.2]]. (2)

   The first contrast with the price-taking fringe is established by:

   PROPOSITION 1. [[Pi].sub.e]/k [is greater than] [[Pi].sub.f] if and only if k [is less than] (n + 1)/2.

   That is, for a sufficiently small cartel, each firm in the cartel earns higher profits than each fringe firm. Thus, the free-rider problem that characterizes the price-taking fringe in [3] does not necessarily arise with a Cournot fringe. Even though a fringe firm earns higher profits when behavior is Cournot than when it is price-taking, Cournot behavior of the fringe benefits the cartel members even more.(4) Indeed, the profit-maximizing output level of the cartel given above is the same with a Cournot fringe as if there were no fringe firms but merely a joint monopoly.

   Stability of Cartels

   We next analyze cartel stability with a Cournot fringe. Following D'Aspremont et al. [3], Daskin [4], Donsimoni, Economides, and Polemarchakis [6], and Martin [14], a cartel can be defined as internally stable if it is not profitable for a cartel member to defect to the fringe; apart from the degenerate case of k = 1, this property is equivalent to the condition:

   [[Pi].sub.c](k)/k [is greater than or equal to] [[Pi].sub.f](k - 1). (3)

   (If k = 1, internal stability is assured since defection from a "cartel" of one firm is not possible.) If equality holds in (3) then we shall call the cartel weakly internally stable.

   Likewise, a cartel is said to be externally stable if it is not profitable for a fringe firm to join the cartel; apart from the degenerate case of k = n, this property is equivalent to the condition:

   [[Pi].sub.f](k) [is greater than or equal to] [[Pi].sub.c](k + 1)/(k + 1). (4)

   (If k = n, external stability is assured since no fringe firm exists to join the cartel.) If equality holds in (4) then we shall call the cartel weakly externally stable. Note that any combination of fringe and cartel satisfying both (3) and (4) will constitute a subgame perfect equilibrium outcome, in that no precommitment mechanism would be needed to enforce the output levels implicit in each firm's initial choice of group (fringe or cartel). For the moment we shall work with these two conditions (3) and (4) only, although a third condition will be introduced in a subsequent section.

   As proved in the appendix, we then have:

   PROPOSITION 2. In the game defined above, internally stable cartels exist only for:

   (a) k [is an element of] {2, ..., n} for n [is an element of] {2, 3, 4}, excluding the degenerate case of k = 1;

   (b) k [is less than or equal to] n/2 + 1 for even n [is greater than] 4; and

   (c) k [is less than or equal to] (n + 1)/2 + 1 for odd n [is greater than] 4.

   Note that the cartel with k = n = 4 is only weakly internally stable, satisfying (3) as an equality; a cartel member here is indifferent between colluding and defecting, and if one defects it is then indifferent between remaining in the fringe and returning to the cartel. As we shall verify below, this implies that the cartel with k = 3 and n = 4 is only weakly externally stable.

   Note also that the range of n that allows internal stability is consistent with the range over which cartel firms earn higher profits than fringe firms, as well as with a portion of the range in which the reverse is true.

   Turning to external stability, and as proved in the appendix, we find:

   PROPOSITION 3. In the game defined above, no externally stable cartel exists for n [is an element of] {2, 3}, excluding the degenerate case of k = n. Externally stable cartels exist for:

   (a) k [is greater than or equal to ] n/2 + 1 for even n [is greater than or equal to] 4; and

   (b) k [is greater than or equal to] (n + 1)/2 + 1 for odd n [is greater than] 4.

   Following standard practice, we shall define a cartel to be stable if it is both internally stable and externally stable. Then an immediate corollary of Propositions 2 and 3 is:

   PROPOSITION 4. The game defined above with n [is greater than] 4 has a unique stable cartel at k = n/2 + 1 for even n, or k = (n + 1)/2 + 1 for odd n. For n [is an element of] {2, 3}, the unique stable cartel is joint monopoly. For n = 4, k = 3 gives a cartel that is internally stable and weakly externally stable, while k = 4 gives a cartel that is externally stable and weakly internally stable.

   The result for n [is an element of] {2, 3} follows since the criterion of external stability is vacuous when the fringe is empty.(5) The uniqueness (apart from the case of n = 4) of the stable cartel with a Cournot fringe facing linear cost and demand parallels earlier uniqueness results for a stable cartel with a price-taking fringe, linear demand, and quadratic costs [4; 6].

   However, the difference in the relative size of the cartels is striking. In the model of Donsimoni, Economides, and Polemarchakis, the relative size is a decreasing function of the size of the economy; and Daskin and Martin obtain stable cartels of only 3 to 4 firms [4; 14]. By contrast, a stable cartel facing a Cournot fringe contains just over...