Nonlinear Pricing with Resale Isa E. HafaliryMay 8, 2007AbstractWe consider the problem of a monopolist— choosing an optimal nonlinearpricing scheme— facing two consumers who can resell some or all of the goodsto each other in a secondary market. We suppose that the valuations of theconsumers are drawn independently from a continuous distribution. We char-acterize the optimum direct mechanism and sh ow that the monopolist can bebetter o¤ or worse o¤ as compared to the without resale case , depending onthe speci…cs of the cost function of the monopolist and the utility functions ofthe cons ume rs.Keywords: Nonlinear pricing, Resale, Mechanism Design, Secondary Mar-ketsJEL Classi…cation Numbers: D42, D821 IntroductionConsider the adverse selection problem of a transaction between a seller (the monop-olist,) and buyers (consumers,) where the seller does not perfectly know how muchthe buyers are willing to pay for the goods. Suppose also that the seller sets theterms of the contract— i.e. a menu of quantities and prices. The problem of one prin-cipal facing one agent who has private information about his type was …rst analyzedby Mirrlees (1971). One monopolist and a consumer problem was then analyzed byMussa and Rosen (1978), and Maskin and Riley (1984), among others. The reader isreferred to Bolton and Dewatripont (2005), La¤ont and Martimort (2002) and Wilson(1993) for a detailed discussion of economics of adverse selection.All previous work on monopolist’s problem of optimal nonlinear pricing focusedeither on a single consumer or on multiple consumers who cannot resell the goodsto one another, except for Calzolari and Pavan (2006). Calzolari and Pavan (2006) PRELIMINARY AND INCOMPLETE, Comments are very welcome.yDepartment of Economics, Penn State University, Un iversity Park, PA 16802, USA. E-mail:[email protected] consider a monopolist who designs an allocation rule and a disclosure policythat optimally fashion the beliefs of the participants in the secondary market. Theyconsider two consumers with two types of valuations and show that it may be impos-sible to maximize the revenue with a deterministic selling procedure and disclosingonly the decision to trade. In their model, the monopolist has one unit of the goodat his hand. There is no question of how much to produce.In this paper, we also consider two consumers who can resell the good to eachother in the secondary market. However, we consider consumers drawing valuationsfrom a continuous distribution, rather than focusing on a …nite set of valuations. Inorder to be able to characterize optimal nonlinear pricing, we do not consider thedisclosure policy of the monopolist. Speci…cally, we suppose that the monopolistproposes a deterministic menu of o¤ers and no information is revealed after the tradebetween the monopolist and the consumers.We consider two di¤erent models of the nonlinear pricing problem. In the …rstmodel, the monopolist is assumed to have a production technology of constant mar-ginal cost, and the consumers are assumed to have concave utility functions. Thismodel was analyzed by Maskin and Riley (1984) for a model without resale, and itresults in quantity discounts, which is widely seen in practice. We …nd necessary andsu¢ cient conditions for the optimal nonlinear pricing schemes of the model with re-sale. Moreover, we show that the maximal revenue achievable in an environment withresale is less than the maximal revenue achievable in an environment without resale.This follows from the observation that in an environment without resale, since theutilities of the consumers are concave, the monopolist can o¤er a certainty equivalentof the transactions of the model with resale at a lower price.In the second model, the monopolist is assumed to have a convex cost function,and the consumers are assumed to have linear utility functions. For this model,the secondary market optimal behavior is easier to characterize. This is b ecauseconsumers have linear utility functions and hence announcing a unit price is theiroptimal selling procedure in the secondary market.1This model results in premiarather than quantity discounts. We characterize the optimal menu for this model andshow that the maximal revenue achievable in an environment with resale is more thanthe maximal revenue achievable in an environment without resale. This follows fromthe observation that in an environment with resale, by o¤ering the same quantitiesas in without resale optimal menu, the monopolist can demand higher amounts ofmonetary transfers from the consumers.We therefore conclude that the pro…tability of banning the resale (if the monop-olist has a power to do so) depends on the speci…cs of the environment. If the utilityfunctions of the consumers are concave, then the monopolist can gain by o¤eringa certainty equivalent of the transactions in the secondary market. Therefore, themodel without resale would give more revenue. If the cost function of the monopolist1Whereas in the …rst model, the consumers o¤er a nonlinear menu of quantities and prices toeach other in the secondary market.2is convex, then the monopolist can bene…t from selling to low value consumers andletting them to sell to high value consumers. Therefore, the model with resale wouldgive more revenue.2 Economic EnvironmentConsider a monopolist and two consumers, where the monopolist does not know howmuch the consumers are willing to pay for the good. Suppose that consumers’pref-erences depend on the preference characteristics : The characteristics  are privateinformation to the consumers. The monopolist and the other consumer only knowthe continuous cumulative distribution of ; F () on an interval;  — we supposethat F satis…es Myerson’s regularity condition.2The consumers’preferences are rep-resented by the utility functionu () = v (q)  T;where q is the number of units consumed, and T is the total amount paid. Themonopolist’s production cost is given by the function K (q) : The monopoly pro…tfrom selling q units against a sum of money T is then given by = T  K (q) :This paper considers the question of …nding the pro…t maximizing pair (T; q) thatthe monopolist will be able to induce the consumers to choose, in an environmentin which the consumers can resell some or all of the good that they have to eachother. We suppose that resale takes place in an imperfect information setting. Thatis,