Price discrimination and efficient distribution.

• Author: Beard, T. Randolph

1. Introduction

   Economists have long warned of the danger that, on occasion, antitrust enforcement might be used perversely to protect competitors at the expense of competition. (1) Moreover, one of the most likely suspects for this charge has traditionally been Section 2 of the Clayton Act, as amended by the Robinson-Patman Act--namely the law on price discrimination. (2) Specifically, while the language of the Act proscribes discriminatory prices only "... where the effect ... may be substantially to lessen competition or tend to create a monopoly," judicial interpretation of the Act, as well as some other language contained in it, have resulted in a substantial weakening of that requirement. (3) As a result, there have been a number of cases in which the law appears to have been used to protect firms from aggressive but selective price cutting by either input suppliers or rival producers. Accordingly, many antitrust scholars have been particularly critical of both the law and enforcement practices in this area. (4)

   We believe that an important reason for this flawed public policy is the paucity of applicable economic theories involving price discrimination by intermediate product suppliers. As Katz (1987, pp. 154-5) points out, the bulk of the theoretical work on price discrimination has focused upon final product markets, while the vast majority of antitrust cases have involved intermediate goods. In fact, to our knowledge, only a handful of formal models have been introduced that treat discriminatory pricing by an input supplier, most notably those of Perry (1978), Katz (1987), and DeGraba (1990). These models either incorporate incentives for vertical integration or find price discrimination in favor of inefficient downstream firms, both of which tend to limit their relevancy in many real-word cases. (5)

   Also, while some economists instinctively believe that some (perhaps most) price discrimination is motivated by procompetitive efficiency considerations, none of the existing theories provide an unambiguous efficiency-based explanation for the practice. All models introduced to date--that apply to either intermediate or final product markets--indicate either a negative or an a priori indeterminate welfare effect. As a result, policymakers and the courts are left with an uneasy suspicion of the practice. And that suspicion is magnified by the lack of a better overall understanding of the observed behavior. As both Coase (1972) and Williamson (1979) note, incomplete understanding of a business practice often results in antagonism toward that practice in antitrust circles. (6)

   Here, we offer a theory of upstream price discrimination that potentially yields relatively strong conclusions regarding the likely impact on efficiency at the downstream (distribution) stage. Specifically, under the assumed conditions, price discrimination is shown to be both profitable upstream and efficiency improving downstream. Thus, such discrimination has an unambiguously positive effect on social welfare. Moreover, our theory appears to be consistent with the facts in several important secondary-line cases in which price discrimination has been alleged against an input supplier and, we believe, incorrectly judged to have significant anticompetitive effects.

   The article is organized as follows. Section 2 lays out the model and establishes the central welfare results. Section 3 briefly discusses our results and draws some policy implications. Section 4 provides a brief review of some antitrust cases to which our model seems to apply, and section 5 concludes.

2. The Model

   The basic economic process we seek to model here is the following. An upstream firm (manufacturer or wholesaler) with some, perhaps limited, market power sells its output to a set of downstream firms (distributors or retailers) that compete in the final product market. (7) Production at the downstream stage involves a fixed, one-to-one input-output ratio. That is, the downstream firms merely distribute or resell the upstream firm's product. (8)

   Two types of firms are assumed to be present in the downstream market: relatively efficient and relatively inefficient firms. We will generally use a subscript of "e" to denote the efficient firms and a subscript of "i" to denote the inefficient firms. By "efficient" we simply mean that the firms so labeled experience a comparatively lower cost of transforming the upstream, intermediate product into the final good. As a result of these cost differences, there exist corresponding final output price differences at the downstream stage. These downstream prices will be denoted by [P.sub.e] and [P.sub.i].

   We imagine a downstream retail sector in which some degree of price variation between efficient and inefficient retailers is possible because search or switching by consumers is costly. At this stage, we will leave unspecified the effects of wholesale prices ([w.sub.e] and [w.sub.i]) on the degree of consumer search or switching and on the (equilibrium) numbers and frequencies of efficient versus inefficient firms. These issues are important, and we will have more to say about them later. For now, we assume that some fraction ([pi]) of buyers patronizes the efficient retailers, and the remaining fraction of buyers (1--[pi]) patronizes the inefficient retailers.

   The retail stage is assumed to be competitive, so that the final output prices of retailers are equal to marginal costs. We normalize the marginal cost of efficient retailers to zero and denote the marginal cost of inefficient retailers using "c." The downstream prices faced by consumers are [P.sub.e] = [w.sub.e] in the efficient markets and [P.sub.i] = [w.sub.i] + c in the inefficient markets. The consumer demand for the good, denoted by D(P), is generally assumed to depend only on the relevant final price. Hence, the consumer demand faced by the efficient retailers is [pi]D(Pe), and the demand faced by the inefficient retailers is (1--[pi])D([P.sub.i]).

   We will also make the simplifying assumption that the demand function is linear. The assumption of linearity will make the presentation very concise and will also allow us to relate some of the results to the notable findings of Robinson (1933), and later Schmalensee (1981), Varian (1985), and Layson (1988). (9) However, it should be noted that for small price changes, all smooth demand curves are well approximated by a linear function. The case of small efficiency differentials could, therefore, lean quite heavily on the analysis of linear functions. In order to ensure positive demand at key price levels in our analysis, we also generally assume that demand is still positive at the sum of the efficient monopoly price and the marginal cost of the inefficient retailers. (10)

   We begin the analysis with a benchmark case roughly corresponding to that described in Katz (1987). Suppose initially that [pi] (the fraction of buyers in the efficient market) is invariant to the retail prices. Normalizing the marginal costs of the upstream firm to zero, the optimal wholesale prices

   [w.sup.*.sub.e], [w.sup.*.sub.i] are defined by

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

   The first-order conditions and the linear demand curve immediately yield the following result:

   [w.sup.*.sub.i] = [w.sup.*.sub.i] - c/2 (2)

   In other words, the efficient retailer is charged more than the inefficient retailer; although, the difference must be less than the amount of the cost inefficiency (e). Another implication is that the retail price at inefficient stores ([P.sup.*.sub.i] = [w.sup.*.sub.i] + c) must exceed the price at the efficient vendors. This price discrimination in favor of the inefficient firm mirrors Katz (1987, Proposition 4) and DeGraba (1990) and generally corresponds to the conventional wisdom concerning price discriminating intermediate good monopolists.

   We next define the profit-maximizing uniform (non-discriminatory) wholesale price, [w.sup.*]:

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII](3)

   Combining the first-order conditions associated with Equations 1 and 3 together with the convex properties of linear functions will quickly lead one to the following characterization (11) of the optimal uniform wholesale price:

   LEMMA 1. [w.sup.*]([pi])=[pi][w.sup.*.sub.e] + (1- [pi])[w.sup.*.sub.i], for any [pi] [member of] [0, 1].

   Lemma 1 immediately yields the famous result attributed to Pigou (1920) that with linear demands, third-degree price discrimination produces no change in output compared to uniform monopoly pricing.(12) In turn, this result also illustrates that legal restrictions that impose uniform wholesale prices (for example, the Robinson-Patman Act) are welfare improving, given our current set of assumptions. To see this, define social welfare, SS([w.sub.e], [w.sub.i], [pi]), as

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

   THEOREM 1. SS([w.sup.*],[w.sup.*], [pi])> SS([w.sup.*],[w.sup.*],[pi]), for any [pi] [member of] (0, 1).

   Proof Combining Lemma 1 and the properties of a linear function, note that

   [pi]D([w.sup.*]) + (1 - [pi])D([w.sup.*] + c) = [pi]D([w.sup.*.sub.e]) + (1 -- [pi])D([w.sup.*sub.i] + c).

   Using this fact, SS([w.sup.*], [w.sup.*], [pi]) - SS([w.sup.*], [w.sup.*.sub.i], [pi]) equals

   [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

   Since the demand function is decreasing, both integrals must be positive. Hence, we have that SS([[w.sup.*],[w.sup.*],[pi]) - SS([w.sup.*.sub.e]...