Third degree price discrimination in lineardemand markets: effects on number of markets served and social welfare.
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 MacKieMason (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 among any of the markets is prohibited. Two distinct scenarios are considered: uniform pricing and price discrimination. Under uniform pricing, the monopolist must charge the same price in all markets and under price discrimination, the monopolist is allowed to charge a distinct price for each market. Using the notation [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], the demand D(p) = (2s/[[alpha].sup.2])[(a  p).sup.+] in any linear market can be characterized by using two independent exogenous parameters, [alpha] and S. The demand intercept a is the maximum willingness to pay; whereas, the size of the market S equals the area under the market demand and coincides with twice the maximum profit possible in that market. S is determined by several factors, such as the quantity demanded by each buyer at a given price and the number of buyers in the market. For example, in the context of demands for automobiles, a developed but small country such as Belgium would have a larger a but a smaller S relative to India, which has a lower per capita income but a larger population. The welfare in a linear market is W(p) = (S/ [[alpha].sup.2])[([[alpha].sup.2]  [p.sup.2]).sup.+]. For n markets, we assume that the intercepts are in increasing order 0 < [[alpha].sub.1] < [[alpha].sub.2] < ... < [[alpha].sub.n] and are all distinct because if there are two markets with the same intercepts, then we can combine them into a single market having the same as and a size S that equals the sum of the sizes of the two markets. Hence, for a uniform price p, the aggregate profit and aggregate welfare functions for the system of n markets are, respectively: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In other words, for a price [[alpha].sub.k1] [less than or equal to] p [less than or equal to] [[alpha].sub.k], [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and if p [greater than or equal to] [[alpha].sub.n], then [PI](p) = W(p) = 0. Because the function [PI](p) is continuous on [0, [[alpha].sub.n]] and vanishes outside (0, [[alpha].sub.n]), it must attain its maximum at a not necessarily unique 0 < [bar.p] < [[alpha].sub.n]. In the next section, we see how determining this maximum leads to the notion of virtual markets. For price discrimination, it is easy to verify that the maximal value of the aggregate profit function is [[PI].sub.d] = (1/2)[[summation].sup.n.sub.i=1]] [S.sub.i] and the corresponding welfare is [W.sub.d] = (3/4)[[summation].sup.n.sub.i=1]] [S.sub.i]. It is clear that (1/2)[[summation].sup.n.sub.i=1]][S.sub.i] [greater than or equal to] [PI]([bar.p]), and for n > 1, the inequality is strict. In section 4, we discuss conditions under which [W.sub.d] > W([bar.p]) and consider how many markets are served by the uniform price p = [bar.p]. [FIGURE 1 OMITTED] 3. Virtual Markets Before we can focus on comparing welfare under price discrimination and under uniform pricing, we need to examine more closely the algorithm for determining the profit maximizing price under uniform pricing. Mathematically, this requires determining the absolute maximum of the continuous (aggregate profit) function [PI](p) on the closed interval [0, [[alpha].sub.n]]. The function [PI](p) is piecewise defined. Its derivative, [PI]'(p), is only piecewise continuous, which complicates the procedure. Let us illustrate the procedure with the following example of three markets. The graph of the ensuing function [PI](p) is shown in Figure 1, and it is composed of parts of three parabolas. This figure shows that the absolute maximum of [PI](p) is the vertex of the second parabola. Intuition suggests that, in general, the absolute maximum can never occur where two of the parabolas intersect; that is, at a point [[alpha].sub.i], and hence must always be the vertex of one of the parabolas. In that case, the vertex must occur at a point [p.sub.k] belonging to an interval ([[alpha].sub.k1], [[alpha].sub.k]) (see by contrast, the vertex of the third parabola, which occurs in the second interval). As we show in the Appendix and state below in Proposition 1, this intuition is correct and the natural algorithm for determining the maximum of [PI](p) is to ignore the function [PI](p) itself and to compute the maxima of all the associated parabolas, and then to select the largest of these maxima. [FIGURE 2 OMITTED] Two natural questions arise: Do these associated parabolas have an economic interpretation of their own, and is their formal consideration useful also elsewhere, for example, for the comparison of welfare under price discrimination and uniform pricing? The answer to the first question is that these parabolas are the graphs of the profit functions of what we define below as "virtual markets." Section 4 provides an implicit answer to the second question: We found that studying the formal properties of the virtual markets is essential for our intuition and for deriving the results in this paper. DEFINITION 1. (Virtual Market): Given n markets with linear demands and using the notation of section 2, the kth virtual market (1 [less than or equal to] k [less than or equal to] n) is the market with linear demand [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The graph of the function [D.sup.(k)](p) is the tangent line to the aggregate demand function D(p) = [[summation].sup.n.sub.i=1]] [D.sub.i](P) at any point in the interval ([[alpha].sub.k1], [[alpha].sub.k]). In other words, we first consider the linear segment of the aggregate demand for p [member of] ([[alpha].sub.k1], [[alpha].sub.k]) and then extend it in both directions up to the axes. Figure 2 illustrates this construction by plotting the demand of the third virtual market for the case n = 5. The profit function of the kth virtual market is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], and it attains its maximum (1/2)[V.sub.k] at p = [p.sub.k], where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Note that [V.sub.k] is the size of the kth virtual market. PROPOSITION 1. Let [V.sub.k] = max {[V.sub.j]  1 [less than or equal to] j [less than or equal to] n}, then max [PI](p) = II([p.sub.k]) = (1/2)[V.sub.k]. REMARK 1. The optimal uniform price [bar.p] needs not be unique, but [bar.p] = [p.sub.[bar.k]] for some [bar.k]. If the maximum is achieved at more than one price, say [p.sub.k] and [p.sub.[bar.k]], it will be up to the monopolist to decide which price to choose; that is, whether to serve markets k to n or [bar.k] to n. Proposition 1 states that the uniform price [bar.p] that maximizes profit can be found by determining the virtual market that generates the largest profit. The advantage of doing so is that we trade the problem of finding the maximum of the aggregate profit, which is a continuous function with piecewise continuous derivatives, with the simpler problem of finding the maximum of (1/2)[V.sub.1], (1/2) [V.sub.2], ..., (1/2) [V.sub.n]. Further on, we consider n = 2 and n = 3 and present the necessary and sufficient conditions to determine how many markets will be served under uniform pricing. PROPOSITION 2. For n = 2, both markets are served under uniform pricing if and only if [S.sub.1]/[S.sub.2] + 2 [[alpha].sub.1]/[[alpha].sub.2] [greater than or equal to] 1. This proposition "interpolates" between the two intuitively obvious cases where both markets are served, namely where [[alpha].sub.1] [greater than or equal to] (1/2)[[alpha].sub.2] (the optimizing price for market 2 would automatically serve also market 1) and [S.sub.1] [greater than or equal to] [S.sub.2] (the profit possible in market 1 alone is larger than the profit in market 2). In other words, if [[alpha].sub.1] is sufficiently close to [[alpha].sub.2] ([[alpha].sub.1] [greater than or equal to] (1/2)[[alpha].sub.2]), no matter how small the size of market 1, it will be served (for example, consider the automobile markets in Luxemburg and Germany). Conversely, a monopoly drug manufacturer forced to charge the same price in a rich country and in a poor country (so [[alpha].sub.poor] [much less than] [[alpha].sub.rich]) will not ignore the poor one if it has sufficiently large population and hence market size (for example, consider EU and India). (3) The necessary and sufficient conditions to determine how many markets, are served get considerably more complex when we have more than two markets, and we see that new phenomena occur. Proposition 3 illustrates the result for n = 3. PROPOSITION 3. For n = 3, the number of markets being served under uniform pricing is three iff [V.sub.1] [greater than or equal to] [V.sub.2] and [V.sub.1] [greater than or equal to] [V.sub.3]; two iff [V.sub.2] > [V.sub.1] and [V.sub.2] [greater than or equal to] [V.sub.3]; one iff [V.sub.3] > [V.sub.1] and [V.sub.3] > [V.sub.2]. Moreover, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Examining the previous three inequalities, it is possible to verify that all cases are possible; that is, one, two, or three markets are served even when [S.sub.1] > [S.sub.2] > [S.sub.3], and thus, [S.sub.1] [greater than or equal to] max{[S.sub.2], [S.sub.3]} is not sufficient to have market 1 served. The boundaries where equality is achieved (the transition curves) determine where small changes in the parameter may cause a change in the number of markets served. For instance, if [V.sub.3] > [V.sub.2] and [V.sub.3]  [V.sub.1] > 0 is small, only market 3 is served, but a small decrease in [S.sub.3] or [[alpha].sub.3], or a small increase in [S.sub.1] or [[alpha].sub.1], may cause all three markets to be served. Sufficient conditions for uniform pricing to serve all markets are presented in Proposition 4 below. PROPOSITION 4. If [S.sub.1] [greater than or equal to] [[summation].sup.n.sub.i=2][S.sub.i] and/or [[alpha].sub.1] [greater than or equal to] (1/2)[[alpha].sub.n], then all markets are served under uniform pricing. Also, the sufficient conditions presented here are minimally sufficient in the sense that if [S.sub.1] < [[summation].sup.n.sub.i=2][S.sub.i], then there exists ([S.sub.1], ..., [S.sub.n]) such that not all markets will be served under uniform pricing. Similarly, if [[alpha].sub.1] < (1/2)[[alpha].sub.n], then there exists ([S.sub.1], ..., [S.sub.n]) such that not all markets will be served under uniform pricing. In the next section, we shall see that virtual markets play an even more central role in comparing the welfare under uniform pricing with the one under price discrimination. 4. Impact of Price Discrimination on Welfare In this section, we discuss the impact of third degree price discrimination on welfare. It is well known since Robinson (1933) that if the demands are linear and all markets are served under uniform pricing, then welfare decreases with price discrimination. Thus, in the case of two markets, the comparison between uniform pricing and price discrimination is simple: Price discrimination decreases welfare when both markets are served and increases it if only one market is served. Already for three markets the situation is more complex: While price discrimination decreases welfare when all markets are served and increases it if only one market is served, if two markets are served both outcomes are possible. Proposition 5 presents in terms of virtual markets and for an arbitrary number of markets, n, necessary and sufficient conditions, for price discrimination to enhance welfare. PROPOSITION 5. If only markets k to n are served under uniform pricing, then the welfare arising from uniform pricing is [[summation].sup.n.sub.i=k][S.sub.i]  (1/4)[V.sub.k], and price discrimination increases welfare if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where for k = 1, the condition is 0 > (1/3){[[summation].sup.n.sub.i=1][S.sub.i]  [V.sub.1]}. Note that if k = 1, that is, all markets are served under uniform pricing, then welfare goes down with price discrimination by Proposition 5 and Lemma 3 in the Appendix. This confirms the well known result of Robinson (1933). On the other hand, if k = n, that is, only market n is served under uniform pricing, then welfare goes up with price discrimination by Proposition 5 and Lemma 1 in the Appendix, confirming an obvious fact. The intuition behind Proposition 5 is as follows: Relative to uniform pricing, price discrimination has two opposite effects on welfare. Welfare goes down for those markets that are served under uniform pricing (that is, markets k to n). On the other hand, welfare goes up in those markets that are not served under uniform pricing (that is, markets 1 to k  1). It can be verified that the loss in welfare for markets k to n is (1/4)([[summation].sup.n.sub.i=k][S.sub.i]  [V.sub.k]); whereas, the gain in welfare for markets 1 to k  1 is (314)[[summation].sup.k1.sub.i=1][S.sub.i], thus giving rise to Proposition 5. In a recent survey of price discrimination, Armstrong (2005) states "... when a weak marketa market with relatively few highvaluation consumersis also a relatively small market, then price discrimination is likely to help consumers in this market and improve overall welfare. But if the weak market is also a large market, then price discrimination is likely to harm consumers in the strong market, and harm welfare." Proposition 5 supports Armstrong's intuition when the weak markets give rise to a large aggregate market (that is, [[summation].sup.k1.sub.i=1][S.sub.i] is large). By extending the analysis to more than two markets, however, Proposition 5 shows that price discrimination also reduces welfare if the weak markets are too small because the increase in welfare in the weak markets is not enough to compensate for the loss in welfare in the strong markets that are served under uniform pricing. In other words, when only some markets are served by uniform pricing, then price discrimination enhances welfare if and only if the aggregate size of the weak markets is moderately large. To illustrate, let us revisit the case of three markets presented in Proposition 3. As already mentioned, price discrimination always lowers welfare if all markets are served (k = 1); that is, [V.sub.1] [greater than or equal to] [V.sub.2] and [V.sub.1] [greater than or equal to] [V.sub.3], and always increases it if only one market is served (k = 3); that is, [V.sub.3] > [V.sub.1] and [V.sub.3] > [V.sub.2]. If only markets 2 and 3 are served, then Proposition 5 states that price discrimination increases welfare if and only if [S.sub.1] > (1/3)([S.sub.2] + [S.sub.3]  [V.sub.2]); that is, if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The inequalities characterizing the cases by Proposition 3 are best illustrated by Figures 3 and 4, which illustrate the areas where welfare increases with price discrimination in terms of the variables [S.sub.1] and [[alpha].sub.1]. Notice that if the size [S.sub.1] is large (or if not, if [[alpha].sub.1] is sufficiently large to compensate), market 1 will be served (see Proposition 4). But in case it is not, price discrimination will benefit welfare only if [S.sub.1] (or [[alpha].sub.1]) are not too small. The policy makers often have less information about the market demands than the monopolist. Among the available information is the number of markets currently served by the monopolist and a reasonable estimate of the market sizes. This information alone may be sufficient to evaluate the impact of price discrimination as shown by the following example. [FIGURE 3 OMITTED] EXAMPLE 1. For n = 3, if only markets 2 and 3 are served, then welfare increases with price discrimination if [S.sub.1] > (1/3)min([S.sub.2], [S.sub.3]). [FIGURE 4 OMITTED] The sufficient condition in Example 1 generalizes to any number of markets. COROLLARY l. Suppose at least j markets are not served under uniform pricing. If j = n  1, price discrimination always increases welfare, and if j < n  1, price discrimination increases welfare if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. 5. Continuous Distribution of Linear Markets There are two reasons to consider a continuous distribution of markets instead of a finite number of markets as we have done so far. The first one is that continuous markets may arise naturally. For instance, as Cowan (2007) has pointed out, Coca Cola considered a marketing plan (never implemented) of pricing its vending machine products according to the local temperaturea continuous parameter. The second one is that a continuous distribution provides a simpler and, in the following sense, a less arbitrary way to represent a large number of markets. Indeed, as we have pointed out in section 2, if two markets have the same [alpha], they should be combined in our model into a single market. It is, therefore, arbitrary to distinguish markets with small differences between their as. Rather, it is more intuitive and more appropriate to combine all markets whose [alpha]s fall within the same (small) intervals. In this familiar Riemann sum limiting process, we can describe the system of markets via a size distribution function S([alpha]), that is, [[integral].sup.b.sub.a] S([alpha]) d[alpha] represents the total size of the markets with a [less than or equal to] [alpha] [less than or equal to] b. This is equivalent to considering a continuum of linear markets with size a piecewise continuous function S([alpha]) [greater than or equal to] 0, for 0 [less than or equal to] [alpha] [less than or equal to] A. For example, S([alpha]) = (1/2)[alpha] for [alpha] [member of] [1  x, 1 + x] and S ([alpha]) = 0 otherwise, give rise to the rotating demands considered in Malueg and Schwartz (1994). Then all the results in sections 2, 3, and 4 and in the Appendix will go through essentially unchanged if we substitute sums with integrals. So the aggregate demand, profit, and welfare functions are for all 0 [less than or equal to] p [less than or equal to] A: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The virtual markets are the linear markets parametrized by 0 [less than or equal to] p [less than or equal to] A with demand function [D.sub.p](x) having its graph tangent to the graph of D(x) for x = p, namely, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. These linear markets have price intercept [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and size [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and hence attain their maximum profit (l/2)V(p) at the price x = [phi](p). PROPOSITION 6. The maximum of the profit function [PI](p) coincides with half of the maximum of the function V(p), and it occurs at (some) [bar.p] = [phi]([bar.p]). Welfare strictly increases with price discrimination if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. 6. Conclusion In this paper, we have analyzed the welfare impact of third degree price discrimination when all markets are not necessarily served under uniform pricing. We have considered n [greater than or equal to] 2 distinct markets with linear demands and have established the necessary and sufficient conditions for price discrimination to enhance welfare. Our approach and our primary results are disjoint and complementary to those in the existing literature. We have characterized each market demand by two exogenous parameters: the price intercept of the demand and the size of the market as measured by the area under the demand. Based on these two exogenous parameters, we have established the necessary and sufficient conditions to determine the number of markets to be served under uniform pricing and the direction of the welfare change under third degree price discrimination. To the best of our knowledge, this is the first extensive analysis to present the necessary and sufficient conditions for n [greater than or equal to] 3 markets. We have also demonstrated that the analysis with n [greater than or equal to] 3 markets offers additional insight to that with n = 2 markets. By extending the analysis to more than two markets, we have demonstrated that price discrimination may reduce welfare when the aggregate size of the weak markets is too small. Thus, when only some markets are served by uniform pricing, price discrimination enhances welfare if and only if the aggregate size of the weak markets is moderately large. A natural next step would be to extend our analysis to incorporate nonlinear market demands. When nonlinear demands can be reasonably approximated by linear demands, the results of the paper should qualitatively hold. When it cannot be done, however, further analysis is called for. Initial investigation indicates a few challenging tasks associated with nonlinear demands in general. A first step would be to extend the analysis to incorporate specific classes of nonlinear markets demands. It remains an agenda for future research. Appendix We leave to the readers the proofs that can be derived from the definitions via a direct computation or that are immediate consequences of the preceding results. Lemmas 18 are used to prove the remaining results. LEMMA 1. The nth virtual market coincides with the nth actual market; that is, [V.sub.n] = [S.sub.n] and [p.sub.n] = (1/2)[[alpha].sub.n]. In general, [p.sub.k] does not necessarily belong to the interval ([[alpha].sub.k1], [[alpha].sub.k]). LEMMA 2. The sequence {[p.sub.k]} is strictly increasing, and [p.sub.k] > (1/2)[[alpha].sub.k] for 1 [less than or equal to] k < n. PROOF. If 1 [less than or equal to] k < m [less than or equal to] n, then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, for 1 [less than or equal to] k < n [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED. LEMMA 3. (a) If 1 [less than or equal to] k < n, then [S.sub.k] < [V.sub.k] < [V.sub.k+1] + [S.sub.k]. (b) The sequence {[[summation].sup.n.sub.i=k][S.sub.i]  [V.sub.k]} is nonnegative and strictly decreasing. PROOF. Proof of Part (a). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Moreover, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, it follows from Lemma 2 that [p.sub.k+1] > [p.sub.k] > (1/2)[[alpha].sub.k], and hence, [(2[p.sub.k+1]  [[alpha].sub.k]).sup.2] [not equal to] 0. QED. Proof of Part (b). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, [[summation].sup.n.sub.i=n][S.sub.i]  [V.sub.n] = [S.sub.n]  [V.sub.n]  0, thus [[summation].sup.n.sub.1=n][S.sub.i]  [V.sub.k] [greater than or equal to] 0. QED. LEMMA 4. If 1 [less than or equal to] k < m [less than or equal to] n, then (a) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In particular, (b) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Also, (c) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. LEMMA 5. If 1 < m [less than or equal to] n, then [p.sub.m] [less than or equal to] [[alpha].sub.m1] implies that [V.sub.m] < [V.sub.m1]. If 1 [less than or equal to] m < n, then [p.sub.m] [greater than or equal to] [[alpha].sub.m] implies that [V.sub.m] < [V.sub.m+1]. LEMMA 6. If [V.sub.k] = max {[V.sub.j]  1 [less than or equal to] j [less than or equal to] n}, then [[alpha].sub.k1] < [p.sub.k] < [[alpha].sub.k]. PROOF. The cases where 1 < k < n follow immediately from Lemma 5. If k = 1 (respectively, k = n), then [p.sub.1] < [[alpha].sub.1] (respectively, [p.sub.n] > [[alpha].sub.n1]) still follows by Lemma 5, while 0 = [[alpha].sub.0] and [p.sub.1]  [p.sub.n] = (1/2)[[alpha].sub.n] < [[alpha].sub.n] are obvious. QED. LEMMA 7. (a) If [[alpha].sub.j1] [less than or equal to] p [less than or equal to] [[alpha].sub.j], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (b) If [[alpha].sub.j1] < [p.sub.j], then [PI]'([p.sub.j]) = 0 and [PI]([p.sub.j]) = (1/2)[V.sub.j]. PROOF OF PROPOSITION I. Let [PI]([bar.p]) = max [PI](p). Then [[alpha].sub.[bar.k]1] [less than or equal to] [bar.p] [less than or equal to] [[alpha].sub.k] for some [bar.k]. By Lemma 7(a), [PI]([V.sub.[bar.k]]) [less than or equal to] (1/2)[V.sub.k]. On the other hand, by Lemma 6, [[alpha].sub.k1] < [bar.p] < [[alpha].sub.k], and by Lemma 7(b), (1/2)[V.sub.k] = [PI]([p.sub.k]) [less than or equal to] [PI]([bar.p]). Hence max [PI](p) = [PI]([p.sub.k]) = (1/2)[V.sub.k]. QED. LEMMA 8. If only markets k to n are served under uniform pricing, then the welfare arising from uniform pricing is [[summation].sup.n.sub.i=k][S.sub.i]  (1/4)[V.sub.k]. PROOF OF PROPOSITION 2. Both markets are served iff [V.sub.1] [greater than or equal to] [V.sub.2]. By Lemma 4(b), [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the latter inequality follows from Lemma 1. QED. PROOF OF PROPOSITION 3. The conditions characterizing the number of markets served are a direct consequence of Proposition 1. The inequalities obtained are consequences of Lemma 4(a) and 4(b). QED. PROOF OF PROPOSITION 4. Suppose [S.sub.1] [greater than or equal to] [[summation].sup.n.sub.i=2][S.sub.i] From Lemma 3(a), [V.sub.1] > [S.sub.1], hence, [V.sub.1] > [[summation].sup.n.sub.i=2][S.sub.i] [greater than or equal to] [[summation].sup.n.sub.i=k][S.sub.i] for all k [greater than or equal to] 2. Also, from Lemma 3(b), [[summation].sup.n.sub.i=k][S.sub.i] [greater than or equal to] [V.sub.k] [mu] k [for all] 2; that is, [V.sub.1] is maximal. Now suppose, [[alpha].sub.1] [greater than or equal to] (1/2)[[alpha].sub.n]. Then for every 1 < m [less than or equal to] n, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], by Lemma 1 and 2. From Lemma 4, it follows that [V.sub.m]  [V.sub.1] < 0; that is, [V.sub.1] is maximal. In either case, [PI]([p.sub.1]) = max [PI](p) by Proposition 1, completing the proof. QED. PROOF OF PROPOSITION 5. As mentioned in section 2, the aggregate welfare with price discrimination is [W.sub.d] = (3/ 4)[[summation].sup.n.sub.i=1][S.sub.i]. Therefore, using Lemma 8, the gain (or loss) in welfare due to uniform pricing for 1 [less than or equal to] k [less than or equal to] n and [[alpha].sub.k] [less than or equal to] p [less than or equal to] [[alpha].sub.k] is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. In particular, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Therefore, welfare increases with price discrimination, that is, [DELTA]W([p.sub.k]) > 0 if and only if [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. QED. PROOF OF COROLLARY 1. If at least j markets are not served under uniform pricing, thus j [less than or equal to] k  1 where only k to n markets are served; that is, [V.sub.k] is maximal. The case where k = n is obvious, so assume that k < n. If [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], then [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the last inequality follows from Lemma 3(a). [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], where the equality follows from Lemma 1, and the last inequality follows from the assumption that market k is served, and hence [V.sub.k] [greater than or equal to] [V.sub.n]. In either case, the conclusion follows from Proposition 5. QED. We are grateful to Rick Harbaugh, Laura Razzolini, David Sappington, and two anonymous reviewers. Debashis Pal is also grateful to Taft Research Center for financial support. Received January 2006; accepted March 2007. References Armstrong, Mark. 2005. Recent developments in the economics of price discrimination. Working Paper, University College London. Battalio, Raymond C., and Robert B. Ekelund. 1972. Output change under third degree discrimination. Southern Economic Journal 39:285 90. Cowan, Simon. 2007. The welfare effects of third degree price discrimination with nonlinear demand functions. The RAND Journal of Economies 38:41928. Greenhut, M. L., and H. Ohta. 1972. Monopoly output under alternative spatial pricing techniques. American Economic Review 62:70513. Hausman, Jerry A., and Jeffrey K. MacKieMason. 1988. Price discrimination and patent policy. The RAND Journal of Economics 19:253 65. He, Yong, and Guang Zhen Sun. 2006. Intermarket income heterogeneity, intramarket income dispersion, indivisible consumption and price discrimination. Pacific Economic Review 11:59 74. Holahan, William L. 1975. The welfare effects of spatial price discrimination. American Economic Review 65:498503. Layson, Stephen K. 1994. Market opening under third degree price discrimination. Journal of Industrial Economics 42:33540. Malueg, David. 1993. Bounding the welfare effects of thirddegree price discrimination. American Economic Review 83:1011 21. Malueg, David, and Marius Schwartz. 1994. Parallel imports, demand dispersion, and international price discrimination. Journal of International Economics 37:16795. Malueg, David, and Christopher Snyder. 2006. Bounding the relative profitability of price discrimination. International Journal of Industrial Organization 24:9951011. Robinson, Joan. 1933. The economics of imperfect competition. Macmillan: London) Schmalensee, Richard. I981. Output and welfare implications of monopolistic thirddegree price discrimination. American Economic Review 71:242 47. Schwartz, Marius. 1990. Thirddegree price discrimination and output: generalizing a welfare result. American Economic' Review 80:125962. Varian, Hal R. 1985. Price discrimination and social welfare. American Economic Review 75:87075. Varian, Hal R. 2000. Economic scene; a big factor in prescription drug pricing: location, location, location. The New York Times. September 21, 2000. New York: The New York Times Company. (1) Varian (1985) also presents the upper and lower bounds of welfare change due to price discrimination. More recently, Malueg (1993) and Malueg and Snyder (2006) present further bounds for the changes in welfare and profits, respectively. (2) Although Cowan (2007) presents the necessary and sufficient and/or sufficient conditions based on the exogenous parameters, Cowan assumes that all markets are served under uniform pricing. (3) By applying Lemma 4 in the Appendix, Proposition 2 can be extended to two groups of markets by taking [S.sub.1] (resp., [[alpha.sub.1]) the size (resp., intercept) of the largest virtual market in group one, and similarly for [S.sub.2], [[alpha].sub.2]. Victor Kaftal * and Debashis Pal ([dagger]) * Department of Mathematical Sciences, 839 C Old Chemistry Building, University of Cincinnati, Cincinnati, OH 45221, USA; Email victor.kaftal@uc.edu. ([dagger]) Department of Economics, 1204 Crosley Tower, University of Cincinnati, Cincinnati, OH 45221; Email debashis.pal@uc.edu; corresponding author. 

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