تبعیض قیمت کامل با آربیتراژ بدون هزینه

The ability of a monopoly seller to prevent resale is often presented as a necessary condition for first degree price discrimination. In this paper, we explore this claim and show that, even with costless arbitrage markets, price discrimination may continue to be both feasible and profit maximizing despite potential resale. With finite numbers of consumers, arbitrage markets may be ‘thin’, in the sense that there can be too few low-valuation consumers to supply high-valuation consumers. We examine both ex ante and ex post arbitrage markets and show how a monopoly can exploit potential ‘thinness’ to profitably price discriminate. In each case, we present sufficient conditions for equilibrium price discrimination. We note that the form of such discrimination depends on the nature of the arbitrage market and consider business strategies that a monopoly might adopt to exacerbate market thinness. Our results show how market depth and the effectiveness of arbitrage can be the key elements for price discrimination, rather than the per se prevention of reselling.

According to most treatments (e.g., Varian, 1989), price discrimination requires firms to (1) have some market power, (2) be able to sort consumers and (3) be able to prevent resale. When it comes to the benchmark case of first degree or perfect price discrimination the first two conditions become stronger in that the firm must also be able to make take it or leave it offers to consumers and possess perfect information regarding a consumer's type. This paper explores the possibility of first degree price discrimination in an environment where (3), the no resale condition, is not satisfied. It is commonly asserted that this change to the standard treatment undermines the possibility of price discrimination. Consider, for instance: It is clear that if the transaction (arbitrage) costs between two consumers are low, any attempt to sell a given good to two consumers at different prices runs into the problem that the low-price consumer buys the good to resell it to the high-price one. (Tirole, 1988, p.134) To our knowledge there have been no attempts to model the resale market to test whether it is in fact true that this possibility undermines price discrimination.2 To explore this, we completely relax (3) and assume the opposite: that resale can occur free of transaction or transportation costs. Moreover, we assume that any trader – the monopolist producer or arbitrageurs – can make take it or leave it offers to any consumer and knows that consumer's type. Despite the complete relaxation of the no resale condition, we demonstrate that price discrimination is still both feasible and potentially profitable for a monopolist seller. Our contribution highlights the importance of the number of low-valuation consumers in driving price discrimination. In a model with two consumer types and a monopolist with unlimited capacity, we model two variants of resale. In the first, following an initial round of sales by the monopolist, the monopolist and initial purchasers can act as sellers in an ex post market. We term this ex post arbitrage. In this situation, we demonstrate that there are conditions under which the monopolist sells to all consumers initially and charges high-valuation types a higher price than the price to low-valuation types. The possibility of ex post arbitrage constrains the monopolist's pricing to high-valuation consumers but does not prevent them from engaging in price discrimination. The reason is that, in equilibrium, the high-valuation consumers are not certain that low-valuation consumers will trade in the ex post market and are, therefore, willing to accept a higher up-front price. There is no discounting or risk aversion driving this result although the presence of either would strengthen it. In the second variant, we consider forward markets for the sale of the monopolist's product. In these markets, low-valuation consumers can sell forward contracts to high-valuation ones and settle these by purchasing the requisite stock from the monopolist later on. In this situation, it may be worthwhile for the monopolist to set a pricing policy equivalent to a perfect price discrimination outcome whereby each consumer is charged their willingness to pay. This is because the monopolist has an incentive to speculate on there being insufficient low types for trade to have occurred in the forward market rather than foreclosing on that possibility by setting a single high price. In each case, it is uncertainty regarding the resale market – specifically, the possibility that a resale market may be ‘thin’ – that drives the result. Put simply, when a monopolist can observe a consumer's type, the low-valuation consumers are those who can perform an arbitrage as well as a consumption function. The monopolist can choose to foreclose on these by not selling to those consumers (i.e., charging a high price) or, if there is a strong enough possibility that the numbers of such consumers may be low, sell to them and price discriminate. As such, the main contribution of this paper is to identify a new condition that permits price discrimination to arise – the expected number of low type consumers – and, in so doing, improve our understanding of the foundations of price discrimination. In addition, while our model is extreme in that a monopolist can perfectly observe a consumer's type and competition in the resale market is strong, it identifies strategies that enhance expectations that resale markets will be thin, such as volume purchase constraints, which may accompany practices of price discrimination. The paper proceeds as follows. Section 2 outlines the basic model structure and the textbook case of perfect price discrimination; to serve as a benchmark. Section 3 considers the case of ex post arbitrage while Section 4 analyses what happens if there is an ex ante or forward market. Finally Section 5 concludes.

We have demonstrated here that costless arbitrage does not necessarily constrain the practice of price discrimination. When consumers and the seller do not expect that the number of low-valuation consumers will be large, the seller finds it optimal to charge low and high value consumers different prices. Even when consumers can hedge against such risks in forward markets, the monopolist will still use observationally perfect price discrimination if it suspects that the number of potential arbitrageurs is low. This suggests that, at the very least, the textbook condition that arbitrage be impossible or prohibitively costly can be amended with an emphasis on the mass of consumers likely to be of low-valuation and to engage in arbitrage activities. Specifically, so long as there is a non-zero probability that there are L-types in the population, price discrimination is feasible. To our knowledge, this is the first attempt to consider in detail this particular assumption underlying the feasibility of price discrimination — either first or third degree. While the model is an extreme one in that there is perfect observability of types and very strong resale market competition, we believe that the insights from this model may be useful in studies of price discrimination. In particular, strategies that limit effective (as opposed to actual) arbitrage may be usefully employed by sellers wishing to charge different prices to different customers. These include strategies that obscure the nature of discounts being offered and who they are being offered to, their time sensitivity (as in the Gold Box of Amazon.com) and also the volume of purchases allowed at discounted prices. Each of these limit effective arbitrage.