On the efficiency of price competition

Abstract

We study the efficiency of price competition among multi-product firms in differentiated oligopolies. Under a general affine demand model, we show that total surplus (sum of industry profit and consumers’ surplus) under competition is at least 75% of the maximum total surplus achievable by a centralized planner. We also show, in contrast to more stylized oligopoly models, that price collusion can increase total surplus and that competition does not, in general, yield a Pareto efficient trade-off between industry profit and consumers’ surplus. However, the maximum deviation of total surplus from Pareto optimality is less than 10%. These results have implications regarding the effectiveness of current anti-trust regulations.

Introduction

Price competition in oligopolies is the subject of extensive research in economics, operations management, and marketing. Of particular importance are questions relating to the effects of market structure on prices, producers’ surplus (industry profit), and total surplus (the sum of producers’ and consumers’ surplus), and the degree by which performance (as measured by total surplus) at equilibrium under competition deviates from optimal performance under centralized decision-making. Large deviations between equilibrium and optimal performance suggest opportunities for contractual, regulatory, or other measures that mitigate the loss of performance due to lack of coordination.

The vast majority of the current literature on price competition makes at least one of the following simplifying assumptions: (a) products are perfect substitutes, (b) only two firms compete, (c) firms are symmetric, and (d) each firm offers exactly one product. Clearly, each of these assumptions, widely invoked for tractability purposes, imposes significant loss of generality. In our model, the products are differentiated; they are gross, but not perfect, substitutes. The number of firms is arbitrary. Own and cross price effects and marginal costs are product-dependent. Multiple products can be offered by each firm. We retain, however, several common simplifying assumptions such as constant marginal costs and nonbinding capacity constraints. We employ an affine demand function where a unit price increase of a product leads to a fixed nonnegative change in the demand of other products as long as this product has positive demand. Affine demand is a common assumption for differentiated products (see, for example, [2], [8], [23]) and has been applied in several empirical studies (see, for example, [6], [10], [24]). Our model is summarized in Section 2 and the reader is referred to [11], where we provide a rigorous definition of affine demand that avoids negative values together with a proof of the existence and uniqueness of equilibrium. Several applications of differentiated oligopoly models in operations management are discussed in [2], [13].

Relaxing any one of assumptions (a)–(d) leads to qualitatively different insights into the nature of price competition. For example, when products are perfect substitutes, capacity is nonbinding, and marginal costs are constant and equal across products; it is well known that equilibrium prices are perfectly competitive, driving industry profits to zero and maximizing total surplus. By contrast, in a differentiated oligopoly setting, industry profits are positive and total surplus is suboptimal and does not necessarily increase with competition. However, we prove that the loss of total surplus due to competition cannot exceed 25%. In a differentiated oligopoly with symmetric firms, equilibrium prices are Pareto efficient in the sense that industry profit cannot increase without reducing consumers’ surplus. In contrast, we show that for asymmetric firms, equilibrium prices need not be Pareto efficient. Thus significant improvements in industry profit are possible without sacrificing consumers’ surplus. However, we show that the loss in total surplus relative to the Pareto efficient prices that guarantee the same level of consumers’ surplus never exceeds 10%. Our results are detailed in Section 3.

In an earlier paper, we study the loss of industry profits under the same setting analyzed here [12]. The main finding there is that the ratio of decentralized to centralized industry profit can be arbitrarily small. Upper and lower bounds are provided that depend on the demand sensitivity matrix but are independent of marginal costs. In the present paper, we focus instead on total surplus because it is the primary measure of overall welfare. The 25% constant factor efficiency loss presented here contrasts with the unbounded loss of industry profit.

We refer the reader to [25] for a survey of oligopoly models in the economics literature. We are unaware of any results in that literature that quantify the degree of performance loss in differentiated oligopolies due to price competition. The operations management literature has investigated the gap between centralized and decentralized solutions in serial systems consisting of a manufacturer and a retailer (e.g. [4], [18], [19]) as well as in other settings (e.g. [1], [3], [5], [7]).

Our work can also be related to the price of anarchy literature (see [17] for the seminal work and [21] for a survey). In particular our result that the maximum loss of efficiency due to price competition does not exceed 25% is reminiscent of the same bound obtained, in very different settings, in the context of traffic routing with congestion [22], in the context of network resource allocation with congestion [16], and in the context of a procurement game with option contracts [9]. It is an open question why these widely varying models yield the same worst case efficiency loss.