Third degree price discrimination in linear-demand markets: effects on number of markets served and social welfare.

• Author: Kaftal, Victor

1. Introduction

   Does monopolistic third degree price discrimination reduce social welfare? The question has continued to intrigue economists and policy makers for more than half a century. The seminal work of Robinson (1933) shows that if a monopolist with a constant marginal cost sells in two distinct and independent markets having linear demands, then welfare falls with third degree price discrimination. Almost half a century later, Schmalensee (1981) reexamines Robinson's result and demonstrates that for any number of markets with linear demands, third degree price discrimination lowers welfare. Schmalensee (1981) also establishes, more generally, that price discrimination increases welfare only if it increases aggregate output. Subsequently, using ingenious duality approaches, Varian (1985) and Schwartz (1990) generalize the results for correlated demands and nonlinear marginal costs. (1)

   It is, therefore, well known that for linear market demands and constant marginal costs, monopolistic third degree price discrimination indeed lowers welfare. It is important to note, however, that the welfare reducing effect of third degree price discrimination is derived under a crucial assumption. It is assumed that if the monopolist is forced to charge the same price in all markets (uniform pricing), the monopolist would sell a positive quantity in each market. In a recent article, Cowan (2007) considers various forms of nonlinear demands and presents the necessary and sufficient and/or sufficient conditions for price discrimination to reduce welfare.

   Cowan (2007), however, also assumes that all markets are served under uniform pricing. Yet, when this assumption does not hold, the welfare effects of monopolistic third degree price discrimination turn out to be significantly more intricate.

   Observe that when all markets are not served under uniform pricing, price discrimination may increase welfare because price discrimination may serve markets that would not be served under uniform pricing. Thus, relative to uniform pricing, price discrimination has two countervailing effects on welfare. Aggregate welfare goes down in those markets that are originally served under uniform pricing; whereas, welfare goes up in those markets that are not served under uniform pricing. Therefore, the additional welfare from the "new markets" under price discrimination may offset any loss of welfare in the markets that are originally served under uniform pricing.

   The point is strongly emphasized in Schmalensee's (1981) seminal work. In the context of prescription drugs, an identical sentiment is recently echoed in Varian (2000). In the context of two markets, the possibility of price discrimination opening up "new markets" is first analyzed (graphically) by Battalio and Ekelund (1972). Layson (1994) also considers two markets but includes nonlinear demands and presents conditions for "opening of new markets" based on endogenous features, such as elasticities. In the context of patent policy, "opening of new markets" by price discrimination is explicitly incorporated by Hausman and MacKie-Mason (1988). Using two markets, they show that price discrimination by a patent holder is socially desirable in many cases because it allows the patent holder to "open new markets."

   There is a related literature in spatial economics, beginning with Greenhut and Ohta (1972) and Holahan (1975), that deals with spatial price discrimination and market opening with linear demand. The issue is whether all consumers should face a mill price plus transportation costs (no discrimination), or the firm should be allowed to set delivered pricing that is discriminatory but serves more consumers. This strand of literature bears some resemblance to our analysis with a continuous distribution of markets (see section 5 in this paper), though our analysis is more general because we allow both market size and demand intercept to vary.

   Given the abundance of examples where all markets are not served by a single price monopolist and the interests that researchers have placed on analyzing this issue, it is somewhat surprising that no necessary and sufficient conditions for the direction of welfare change have been established. The objective of the present paper is to fill this gap.

   We consider several markets with linear demands and characterize each demand by two exogenous parameters: the price intercept of the demand curve and the size of the market as measured by the area under the demand curve. Based on these exogenous parameters, we establish the necessary and sufficient conditions to determine the number of markets to be served under uniform pricing and whether welfare goes up or down under third degree price discrimination. We also establish sufficient conditions to determine the direction of welfare change based on either demand intercepts or market sizes. In particular, we demonstrate that even if some markets are ignored under uniform pricing, price discrimination increases welfare only in limited scenarios. It increases welfare only if both the price intercept of the demand and the market size are significantly lower in some markets.

   Formulation of these conditions based on the exogenous demand parameters complements the existing literature on price discrimination because the seminal work by Schmalensee (1981), Varian (1985), Schwartz (1990), and Malueg (1993) mostly establishes the necessary and/or sufficient conditions based on aggregate quantity, which is an endogenous variable. Conditions presented in Layson (1994) are based on features such as elasticities, which are also endogenous in nature. (2)

   To the best of our knowledge, Malueg and Schwartz (1994) are the first to investigate the necessary and/or sufficient conditions based on exogenous parameters. The focus of Malueg and Schwartz (1994), however, is very different from ours. They consider linear demands that are rotating or are parallel, and more importantly, they assume that the demand intercepts are uniformly distributed over [1 - x, 1 + x], where the parameter x measures demand dispersion. These special cases permit them to calculate exactly the levels of welfare under uniform pricing and under third degree price discrimination and to compute their ratios in terms of x. Uniform distribution being quite restrictive, Malueg and Schwartz also consider a special form of skewed distribution for which they can compute the welfare ratios in terms of x. However, they state, "Given the highly stylized nature of our model, we are hesitant to lean on it too heavily for predicting that global welfare would be higher under complete discrimination than under uniform pricing." They explain that "We cannot estimate the key parameters x and [t.sup.*] from actual distributions of per capita income." We, on the other hand, focus on general markets with general demand intercept distributions (which thus can describe actual markets) and establish necessary and sufficient conditions for welfare to increase. Therefore, our approach and our primary results are disjoint and complementary to Malueg and Schwartz (1994).

   In a concurrent article, He and Sun (2006) independently investigate the welfare effects of third degree price discrimination. They derive the necessary and sufficient conditions for a two market case and present two numerical examples for the three market case, one in which welfare increases and one in which welfare decreases, while we present the precise necessary and sufficient conditions for welfare to increase for any number of markets. In fact, He and Sun (2006) write "Yet, more general results regarding the condition under which discrimination will be beneficial when some markets are excluded under uniform price still remain unexplored." This is precisely the task undertaken in our paper.

   The paper is organized as follows. Section 2 describes the model. Section 3 introduces the concept of virtual markets as the natural tool for determining the profit maximizing uniform price and determines the number of markets being served under uniform pricing. Section 4 uses the virtual markets to determine the welfare impact of third degree price discrimination. Section 5 presents the continuous counterpart of our discrete framework, and section 6 concludes the paper. All proofs are in the Appendix.

2. Model

   A monopolist produces and sells a homogenous product in n distinct markets. The monopolist has a constant marginal cost of production, which is normalized to zero without loss of generality. Trade...