Bounding the welfare effects of third-degree price discrimination
研究垄断企业实行三级价格歧视时，社会福利与统一定价下的比值范围，发现若弱市场需求函数为凹函数，该比值有明确上下界。
Under Pigouvian third-degree price discrimination, a profit-maximizing monopolist typically charges different groups of customers different prices (resale between groups is assumed to be impossible). These differences in price entail a loss of efficiency because marginal valuations of the output are not equal across buyers. The net welfare effect of allowing price discrimination is ambiguous though, because the total output under discrimination may exceed that under uniform pricing. Nevertheless, under quite general conditions, third-degree price discrimination by a monopolist can increase (static) welfare only if total output is greater under discriminatory pricing than under uniform pricing (Richard Schmalensee, 1981; Hal R. Varian, 1985; Marius Schwartz, 1990). Although there has been much analysis of whether price discrimination raises or lowers total output or welfare, the size of these effects has remained largely unexplored. Varian (1985) compares discriminatory pricing to uniform pricing for a monopolist and provides some bounds for the change in welfare in terms of market prices and outputs; but these relationships give little sense of the relative size of welfare changes resulting from third-degree price discrimination. I address this issue by asking what can be said about the ratio Wd/ Wu, where Wd and Wu denote welfare under discriminatory and uniform pricing, respectively (welfare is measured as the sum of producer surplus and Marshallian consumer surplus). Section I provides two examples showing that this ratio can range from zero to infinity. In these examples, all markets are served under uniform pricing, and demand in one of the markets is strictly convex. Considering a two-market model in which a monopolist with constant marginal cost serves two independent markets under uniform pricing, I show in Section III that if demands in both markets are concave, then Wd/ W, is bounded below by 2. (The assumptions of constant marginal cost and independent demands allow producer surplus and consumer surplus to be identified separately for each market. Separability across markets is central to the approach taken in this paper.) Following Joan Robinson (1933), I call a market (weak) if the discriminatory price in that market is at least as great as (no greater than) the profit-maximizing uniform price. Surprisingly, the key to bounding Wd/ WU is that the demand function in the weak market be concave. If one market is strong, the other is weak, and demand in the weak market is concave, then Wd/ W' can be bounded above by 2.5 (or even 1.75 or 1.6 with additional assumptions on the demand functions). These bounds arise for several reasons. First, the monopolist's profit under uniform pricing must be at least as large as the profit in the weak market under discrimination. Moreover, if at all prices demand in the strong market is as great as in the weak market, then profit under uniform pricing must be at least twice as great as profit in the weak market under discrimination (otherwise the monopolist could do better by charging the monopoly price for the weak market). Second, this relationship between profits in the two markets is useful in bounding welfare because welfare in each market can be related to profit in that market. Section II shows that if demand is con* Department of Economics and A. B. Freeman School of Business, Tulane University, New Orleans, LA 70118. I thank Marius Schwartz and two anonymous referees for their comments on earlier drafts of this paper.