Two-Part Pricing with Costly Arbitrage.

• Author: McManus, Brian

Brian McManus [*]

This paper considers the optimal two-part pricing strategy of a monopolist whose customers collude when they purchase the firm's product. In contrast to the sentiment in the existing price discrimination literature, I find that a monopolist's profit can actually increase when consumers share its good. When transaction costs for collusion are zero the firm can extract the full consumer surplus through two-part prices. When transaction costs are positive or there are a substantial number of consumers without access to resale, the firm may be hurt by arbitrage.

1. Introduction

   A standard assumption for models of price discrimination is that consumers are unable to engage in side transactions after they have purchased a firm's product. This assumption generally helps the firm screen its customers on the basis of their observed or unobserved characteristics. Restrictions on arbitrage are appropriate for many markets. [1] In Walter Oi's (1971) classic example of separate amusement park admission and ride prices, it is hard to imagine a situation in which many consumers can use tickets for rides when only one person is admitted to the park. But surely it is possible for two households to consider buying a lawnmower or theater subscription jointly to avoid the expense of each purchasing independently. Since Oi dissected the "Disneyland dilemma," it has been suggested (but never proven) that such cooperation among consumers would diminish the monopolist's ability to win high profit through nonlinear prices. Oi writes:

   A two-part tariff wherein the monopolist exacts a lump sum tax for the right to buy his product can surely increase profits. Yet, this type of pricing is rarely observed. That apparent oversight on the part of the greedy monopolist can partially be explained by the inability to prevent resale. If transaction costs were low, one customer could pay the lump sum tax and purchase large quantities for resale to other consumers. (1971, p. 88) [2]

   Similar arguments are presented in Phlips (1983), Tirole (1988), and Wilson (1993).

   In this paper I investigate firm profit and social welfare when consumers can engage in side transactions. A simple model is used to demonstrate that there are situations in which the profit of a monopolist that sets two-part prices can increase in the presence of post-sale arbitrage among consumers. The analysis below is divided into two scenarios. In the first, two consumers are able to engage in (costly) side transactions and they are the only individuals eligible to purchase from a monopoly firm. The firm is typically able to increase its profit relative to a model without arbitrage, but there are some (fairly limited) circumstances under which the firm is hurt by collusion between consumers. Profit only falls when transaction costs are neither too large nor too small and demand is appropriately behaved. In the second situation a pair of consumers can costlessly share a monopolist's product, but a number of independent consumers also demand the firm's good. Firm profit increases when the size of the popul ation that cannot engage in side transactions is not too large or when the firm would have sold only to high-demand consumers in a market without arbitrage.

   Previous research on price discrimination that has included side transactions or consumer coalitions has found that a monopolist's profit is reduced by the possibility of arbitrage. Alger (1999) submits a model in which a firm offers price/quantity bundles to consumers who can buy cooperatively and purchase multiple bundles. Alger finds that the introduction of multiple and joint purchasing increases the ability of consumers to retain surplus (relative to discriminatory pricing without arbitrage), but there are several differences between her model of cooperative purchase and the analysis below. First, Alger uses general nonlinear pricing (i.e., the firm offers a menu of discrete price/quantity bundles) rather than two-part prices. Second, coalition formation is costless when occurring among consumers with identical demand but prohibited otherwise. I specify below that the two consumers who are able to collude have different demand curves and that arbitrage may require the payment of a positive transaction co st. Third, Alger retains individual participation constraints for independent purchase. Any offered bundle that might be shared must leave a consumer with a nonnegative surplus if the bundle is purchased and consumed independently. In this paper I permit the monopolist to offer pricing arrangements that return nonnegative surplus only under joint purchase.

   Innes and Sexton (1993, 1994) study situations in which consumers consider forming a group that will replace an established monopolist in the production of a good. The monopolist uses price discrimination to prevent coalition formation among consumers with identical unit demand curves. The firm offers its product to some consumers at a discount so that the number of consumers who would benefit from a coalition is low enough to prevent a coalition from forming at all.

   The remainder of this paper proceeds as follows. In section 2 I present the model of consumer demand, firm pricing, and side transactions. The next section contains a review of optimal pricing by a monopolist when arbitrage is prohibited. In section 4 I consider the case of two colluding consumers and variable transaction costs. The analysis in section 5 covers a firm's pricing strategy when only a subset of the consumer population may engage in arbitrage. Conclusions and extensions to this research are discussed in section 6.

2. The Model

   Consider a market in which a single firm produces a good at a constant marginal cost, c. The firm may charge a separate per-unit price (p) and a fixed tariff (F) to capture profit. The timing of pricing and purchasing behavior is as follows. Aware of the composition of the consumer population and the prospects for arbitrage within it, the firm announces a nonlinear price schedule. Any consumer (or group of consumers) that is present in the market is then able to approach the firm and purchase at the posted prices. The firm cannot revise its price schedule between when prices are first announced and when all eligible consumers have had the opportunity to purchase.

   All consumers have demand functions that fall into one of two categories. N + 1 consumers have demand given by [D.sub.1](p), and N + 1 consumers have the demand curve [D.sub.2](p). The demand functions of the two types of consumer satisfy the following set of assumptions:

   ASSUMPTION 0 (A0). Demand has the properties:

   (i) [D.sub.1] and [D.sub.2] are continuous and twice differentiable;

   (ii) [D.sub.2](p) [greater than] [D.sub.1](p) [for all]p;

   (iii) [D'.sub.i](p) [less than]0 for i = 1, 2;

   (iv) -[D.sub.i](p) + (p - c)[k.sub.1][D".sub.1](p) + [k.sub.2][D'.sub.2](p) + 2[[k.sub.1][D'.sub.1](p) + [k.sub.2][D'.sub.2](p)] [less than or equal to] 0 for i = l, 2, for any [k.sub.1] [k.sub.2] [greater than or equal to] 0, and for any p [greater than or equal to] c;

   (v) Income effects are negligible; and

   (vi) If there is a finite p that solves [D.sub.1](p) = 0, it is greater than c.

   Part (ii) of A0 implies that the demand curves do not cross and part (iv) ensures that the firm's profit maximization problem is concave in p for all of the selling strategies discussed below. [3] For convenience, define [S.sub.i] as the surplus to a consumer of type i from purchasing the efficient quantity at a price p = c, i.e., [S.sub.i] [equivalent] [[[integral].sup.[infinity]].sub.c] [D.sub.i](p) dp. If a consumer does not purchase from the monopolist, she receives zero surplus.

   A single pair of consumers called "consumer 1" and "consumer 2" have the ability to purchase cooperatively. Consumers 1 and 2 belong to the low- and high-demand groups, respectively. This pair of potential buyers pays a nonnegative transaction cost, T, if 1 and 2 cooperate to avoid paying a fixed fee, F, once. [4] The negotiation process between the consumers regarding the payment of positive transaction costs is not explicitly modeled. I assume that consumers only incur the transaction cost if they are able to reach an agreement that increases their joint welfare (measured through consumer surplus) and makes neither consumer worse off. Once a welfare-improving agreement is identified, the consumers are able to buy from the firm and share the purchased good (and its expense) in a way that does not disturb the agreement. Although both the firm and the consumers would bear some expense to change T in certain situations described below, I assume that T is determined exogenously throughout this analysis. Followin g the terminology of Oi (1971) quoted in the introduction of this paper, side transactions between consumers are sometimes referred to as "resale" of the firm's product.

   Social welfare is measured without regard for its distribution among the consumers and the monopolist, that is, welfare is simply the unweighted sum of consumer surplus and firm profit. All comparisons of levels of welfare made in this paper are between two-part pricing schemes with and without resale. Transaction costs are considered to be equivalent to deadweight loss, as T is described as paid Out of consumer surplus.

   The above assumptions concerning the composition of the consumer population are certainly quite restrictive. One can imagine that a more general model with, say, uncertainty over the demand intensities of consumer coalition members would better represent the decision problem of the firm. Despite this, the results discussed below include a wide range of outcomes.

3. When Arbitrage Is Prohibited

   The purpose of this section is to review the firm's problem when arbitrage among consumers is prohibited. Profit maximization leads the firm to choose between two strategies: selling to all consumers (low and high demand) and selling only to consumers with the...