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We derive bounds on the ratio of a monopolist's profit from third-degree price discrimination to that from uniform pricing. If the monopolist serves N independent markets, demand is continuous, and the cost function is superadditive, then the profit ratio is bounded by N. A linear-demand example is provided coming arbitrarily close to this bound. We provide examples showing the profit ratio can be unboundedly large when marginal cost is decreasing, demand is discontinuous, or fixed cost is positive. If the monopolist has access to certain demand-rationing strategies under uniform pricing, we can bound the profit ratio even for discontinuous demand functions and multiproduct cost functions.

How valuable is the ability to price discriminate? This question is central, for example, to the debate on parallel imports and international exhaustion of intellectual property rights. If allowing parallel imports eliminates price discrimination across countries, even consumers who benefit from a lower price may ultimately lose through a reduction in firms' ex ante investment incentives.1 The size of this incentive effect depends on the relative profitability of price discrimination. Recently, the issue of parallel imports has received particular attention in the international market for pharmaceuticals (see Danzon and Towse, 2003). In order to preserve low prices in poor countries, the European Union recently tightened restrictions on the re-importation on malaria, tuberculosis, and HIV/AIDS drugs from poor countries, enhancing pharmaceutical manufacturers' ability to engage in price discrimination across rich and poor countries. Recent U.S. policy moved in the opposite direction: by relaxing restrictions on the re-importation of pharmaceuticals from Canada, the United States reduced pharmaceutical manufacturers' ability to engage in price discrimination across the two countries. These policies were motivated by a desire to reduce prices in certain markets with little regard to what Danzon and Towse (2003) note may be potentially significant effects on firm's incentives to undertake pharmaceutical research and development.2 Our results on the relative profitability of price discrimination will provide bounds on the effect of such policies on firms' ex ante incentives to discover and develop pharmaceuticals as a function of the number of markets (countries) involved. To date, the economics literature has been silent on the profitability of price discrimination, focusing instead on the effect of price discrimination on static social welfare (defined as consumer surplus plus producer surplus). We address this gap in the literature by examining the effect of third-degree price discrimination on a monopolist's profit. Typically it is of little interest to ask whether third-degree price discrimination increases a monopolist's profit—because all price vectors feasible under uniform pricing are also feasible under discrimination, the monopolist's profit under price discrimination is weakly higher than under uniform pricing. 3 In this paper we ask how much price discrimination can increase a monopolist's profit. We show that if a monopolist faces N independent markets, demand is continuous, and the cost function is superadditive (related to diseconomies of scale), then the profit ratio is bounded by N. In cases where under price discrimination some markets are not served or are charged equal prices, we provide the tight bound that is strictly less than N. If the preceding conditions on demand and cost do not hold, the ratio of profit under third-degree price discrimination to profit under uniform pricing can be arbitrarily large, as we demonstrate in a series of examples. Our results relate to the public policy question of the effect of parallel importation of drugs and other goods on firms' ex ante investment incentives, as mentioned above. Our results have other practical implications for firm strategy and public policy. Firms' incentives to facilitate third-degree price discrimination by developing technologies to separate consumers into different markets and prevent arbitrage among them depend on the profitability of price discrimination. For example, Odlyzko (2004) notes the increasing complexity of price discrimination schemes in transportation, citing among other examples the evolution of canal tolls from simple uniform fees per boat for early canals in England to fees which varied not only by volume and weight of cargo but also by the type of commodity shipped and even the intended end use of the commodity for later canals in England and the United States. Shippers took various measures to evade the tolls including hiding high-toll commodities under low-toll ones; canal operators took various countermeasures including the use of books “listing canal boats, and the weight of cargo aboard as a function of how deeply in the water they lay” (Odlyzko, 2004, p. 331). Our results also add to economists' general theoretical understanding of price discrimination. To our knowledge, our paper is the first to derive bounds on the profitability of price discrimination relative to uniform pricing. Previous work has focused on conditions under which third-degree price discrimination decreases social welfare. Robinson (1933) showed that, if under uniform pricing a monopolist serves two independent markets with linear demands, allowing third-degree price discrimination leaves total output unchanged and therefore reduces welfare if discriminatory prices are different, because marginal benefits of consumption are not equalized across markets. Subsequent authors generalized this result. Schmalensee (1981) showed that for a monopolist with constant marginal cost facing independent demands, third-degree price discrimination raises social welfare only if it increases total output. Varian (1985) generalized Schmalensee's result to allow for interdependent demands and nondecreasing marginal cost, while Schwartz (1990) allowed for interdependent demands and any cost function that depends only on total output. Motivated by the prescriptions of U.S. antitrust law, which applies to price discrimination in the sale of intermediate goods, Katz (1987), DeGraba (1990), Yoshida (2000), and O'Brien (2003) studied discrimination by a monopoly producer of an input used by (possibly) competing downstream firms. Collectively, these authors showed that allowing discrimination has an ambiguous effect on input prices. Our paper is closest in spirit to Malueg (1993) and Armstrong (1999). Malueg (1993) provided quantitative bounds on the social welfare given particular restrictions on market demands. As emphasized above, the present paper differs because our bounds are on relative profit rather than relative social welfare. Armstrong (1999) bounded the profitability of simple two-part tariffs relative to optimal nonlinear tariffs charged by a multiproduct monopolist. He showed that the profit from simple contracts converges to that from optimal contracts as the number of goods grows. The present paper differs because we study third-degree price discrimination rather than second-degree. The nature of our results is quite different as well: profit from the suboptimal pricing scheme (uniform pricing) does not necessarily converge to profit from the optimal one (third-degree price discrimination); rather, we show the gap can grow without bound as the number of markets grows. The proof that our bound is tight for linear demands relies on some novel analytical arguments. The remainder of this paper is organized as follows. Section 2 bounds the relative profitability of price discrimination in the benchmark case, in which the monopolist is unable to ration demand. That is, the monopolist must serve all demand at price it sets. In Section 3, we argue that in many real-world markets the monopolist may easily be able to ration demand under uniform pricing. We thus turn to bounding the relative profitability of price discrimination under various plausible rationing assumptions. If the monopolist can ration demand by allowing itself to stock out, we show that the bounds provided in Section 2 hold even if demand is discontinuous. If the monopolist can ration demand by picking and choosing the markets in which it wishes to sell, then the bounds on the relative profitability of price discrimination can be extended to the case of a multiproduct monopolist whose costs are a function of the vector of market outputs rather than the sum of market outputs. This last extension has useful applications; for example, it covers the case in which a monopolist produces a homogeneous good in a central plant and then incurs different transportation costs for delivering the good to different markets. Section 4 concludes.

We have shown that if a monopolist facing N independent markets has a superadditive cost function, then profit under third-degree price discrimination cannot exceed N times the profit under uniform pricing. Indeed, we derived an even tighter bound: N⁎, the minimum number of distinct prices needed for the markets that are served under price discrimination. N⁎ is lower than N to the extent some of the N markets are not served, and to the extent discriminatory prices happen to be the same across several markets. We provided an example with linear demands showing the bound is tight. We provided further examples showing that the assumptions of superadditivity and continuity of demand are generally required for the bound. In Section 3, we showed that the bounds derived in Section 2 hold under a broader set of conditions if the monopolist is assumed to have some ability to ration demand. If the monopolist can ration demand by stocking out, that is, electing to sell only a fraction of the quantity demanded at the chosen uniform price, then the assumption on the continuity of demand is no longer needed to bound the relative profitability of price discrimination. If the monopolist can ration demand by bypassing markets, that is, being able to choose which markets it serves and which not, then the bound on the relative profitability of price discrimination can be extended to the case of a multiproduct cost function. The bound holds under a condition related to diseconomies of scope, slightly weaker than superadditivity of the cost function.