تبعیض قیمت غیر خطی با تعداد محدودی از مصرف کنندگان و قرارداد مجدد محدود

In the monopoly nonlinear pricing problem with unobservable consumer types, we study mechanisms in which the firm makes the set of consumer options conditional on the aggregate reports of consumer types it receives. Previous mechanisms that exploit knowledge of the true type distribution often have multiple equilibria or use noncredible contracts off the equilibrium path. When the monopolist can replace contracts after initial reports subject to the constraint that truthful consumers are not made worse off, the outcome is essentially the same as when the monopolist has full information. This holds whether or not the monopolist can make offers to consumers who reject all original contract offers. When the monopolist must guarantee nonnegative surplus to all truthful consumers in all contingencies, the equilibrium outcome has undistorted contracts but lower profits for the monopolist.

The best modern interpretation of second-degree price discrimination (or declining-block pricing) is that it arises as the solution to an asymmetric information problem. The monopolist knows the distribution of consumer types but cannot identify demand curves of individual consumers. While the monopolist takes the distribution of consumer types into consideration in choosing the optimal schedules, it does not use this information about market aggregates as aggressively as possible. In particular, the conventional solution has the monopolist choosing the nonlinear pricing schedule subject to incentive compatibility constraints (so that each consumer would truthfully reveal his type to the monopolist), but the contracts offered do not depend on the number of consumers who request each of the contract choices. One justification for this approach is that the monopolist is assumed to be selling to a large (infinite) number of consumers.1 An alternative assumption that we explore in this paper is that the monopolist has a small (finite) set of potential customers and knows the exact distribution of types. With precise information on the number of each type of consumer in the market, the monopolist can detect that some consumers seek to pay a lower average price per unit by reporting that they prefer a smaller quantity than they actually do. Using this information significantly changes the nature of the monopoly's optimal pricing policies. Each of these specifications is most appropriate in different contexts. The standard model may best fit a final good monopolist selling to a large number of individuals. Even though this number is still finite, uncertainty about the exact distribution of types can smooth things out, making a model with a continuum of consumers appropriate. On the other hand, an intermediate good monopolist may sell to a small number of oligopolists and, through access to general market research, may have reasonably precise information about the distribution of types to whom it is selling. Previous models have exploited the seller's information about consumer aggregates in a variety of ways. In a perfect information setting, Levine and Pesendorfer (1995) allow a monopolist to precommit to a pricing strategy in which no sales take place if any consumer chooses to purchase less than the quantity which yields zero surplus to the consumer. They also show that such an equilibrium is not robust to the introduction of noise in players' actions. Bagnoli et al. (1989) consider a durable-good monopoly and find that the monopolist can extract all surplus by making a sequence of price offers which depend on the history of purchases. This equilibrium is subgame perfect, but not unique.2Bagnoli et al. (1995) study a similar mechanism in a model of quality differences and find the monopolist can extract all surplus. We discuss the differences between their mechanism and ours in the conclusion. With asymmetric information, the idea that correlation between consumer types may help a monopolist extract additional surplus from consumers has been studied elsewhere, especially in auctions. In fact, Crémer and McLean, 1985 and Crémer and McLean, 1988; Brusco (1998), and Spiegel and Wilkie (2000) develop mechanisms by which the seller can extract the full surplus under certain conditions. Fudenberg and Tirole (1991, pp. 294–295) discuss the fact that these mechanisms may require large payments by consumers in certain outcomes of the game. Our correlation among types is a special case of the distributions studied by Crémer and McLean because the aggregate distribution is common knowledge among consumers and the firm. However, our mechanism bounds the losses to agents in outcomes off the equilibrium path and some of our assumptions are more general. We discuss differences between these models and ours more fully in the final section. The asymmetric information monopoly problem bears a strong analogy to the optimal income tax problem with a finite number of consumers.3Piketty (1993) shows that the government can implement any allocation on the full-information Pareto frontier as a dominance-solvable equilibrium by allowing the government to condition individual tax schedules on the aggregate reports from all workers. Thus, in contrast to the models of Mirrlees (1971) with a continuum of ability levels and Stiglitz (1982) with two ability levels and a continuum of workers of each type, no worker faces a positive marginal tax rate. Piketty calls such mechanisms “generalized tax schedules.” In effect, whenever too many taxpayers claim to be of low ability, the bundle given to these taxpayers is worse than the bundle intended for low ability taxpayers when everyone reports abilities truthfully. While researchers prior to Piketty were aware that mechanisms using aggregate information could sustain efficient allocations that violated the conventional incentive compatibility conditions, they generally disregarded such solutions since they viewed them as being plagued by problems of multiple equilibria. They thought the outcome when even one high-ability taxpayer claimed to be of low ability had to be set as very bad for low-ability types. In that case, a low-ability type would be better off to claim to be of high ability to avoid the punishment outcome.4 Although Piketty circumvents this difficulty and obtains a unique equilibrium, his approach has other unsatisfactory properties. First, the bundles offered taxpayers off the equilibrium path (after at least one taxpayer has claimed to be different from his true type) do not necessarily satisfy the government's budget constraint and hence may not be feasible. Second, ex post, the bundles given to consumers are not necessarily the optimal choices for the government given the information it has about taxpayers at that point. Hamilton and Slutsky (2003) find that imposing budget balance on and off the equilibrium path and requiring the allocations off the equilibrium path to satisfy certain optimality conditions limits the set of efficient allocations that the mechanism designer can achieve. The analogies between the nonlinear pricing and optimal income tax problems are significantly weakened by two differences. First, the firm does not face a budget constraint, in contrast to a redistributive government. Thus, budget balance off the equilibrium path does not affect the monopolist's problem. However, the problem remains of whether the bundles chosen off the equilibrium path are optimal given the monopolist's information at that point. A monopolist, having announced quantity–outlay pairs conditional on the number of consumers who request each of the pairs offered, may want to revise these offers after consumers make their choices if misrevelation has occurred. The off-equilibrium-path pairs were initially chosen to sustain the truthful outcome and not to maximize profits conditional on a false report. Second, a monopolist faces participation constraints that customers may choose to buy nothing if offered a very bad deal, in contrast to the fixed populations in the optimal income tax model.5 One modeling issue that arises in this context is whether the participation constraints apply only on the equilibrium path or off the equilibrium path as well. We consider both cases in our analysis. If the monopolist does not commit to fulfill contracts initially offered, consumers may be unwilling to reveal their types to the monopolist. The possibility we analyze in this paper is that the monopolist may be able to commit to fulfill the contracts offered, but it may not be able to commit not to revise those contracts with consent from consumers. Legal action by consumers may prevent the monopolist from making consumers who reported their types truthfully worse off than promised. However, even legally binding contracts can be altered if neither party is harmed. The monopolist is able to replace contracts, so long as the new contracts do not lower the utility of someone who has revealed her type truthfully.6 That is, a policy change which affects those who reveal themselves to be of a certain type could not lower the utility of someone of that type. However, the policy change could harm someone who has misrevealed, since an individual without “clean hands” would have his case thrown out of court. In this respect, our partial commitment differs in spirit from renegotiation in games (see, for example, Farrell and Maskin, 1989) and in principal-agent problems (see, for example, Fudenberg and Tirole, 1990), where renegotiation must lead to a Pareto improvement.7Baron and Besanko (1987) analyze a principal-agent model in which the principal can guarantee only that any truth-telling agent will earn nonnegative profit in the second period of a regulatory regime. Our partial commitment is similar in spirit, except that we allow the principal to guarantee any feasible utility level to truth-telling agents (and to condition the levels of these guarantees on the aggregate reports).8 We also require that the monopolist must induce a game among consumers which has a unique equilibrium. In the literature on implementation, several approaches exist to ensure uniqueness. When the outcome is only a Nash equilibrium, uniqueness does not often result, while requiring truth to be a dominant strategy for all consumers would essentially guarantee uniqueness. Mookerjee and Reichelstein (1992) consider when the requirement of having dominant strategies can be imposed with no loss. The monotonicity condition they find is not satisfied in our context. Thus, we take an alternative approach to ensuring uniqueness by requiring a weaker condition than existence of dominant strategies, namely that the game be dominance-solvable (solvable by iterated deletion of strictly dominated strategies). This is the approach used by Piketty (1993). It is also used in the literature on virtual implementation as in Abreu and Sen (1991), Abreu and Matsushima, 1992a and Abreu and Matsushima, 1992b, and Duggan (1997). These papers show that a planner can implement an allocation arbitrarily close to any outcome in many environments by use of a lottery in which the desired outcome is a prize with high probability. When the monopolist can commit to contracts, our model is consistent with the virtual implementation framework. However, we do not use lotteries, but implement an allocation close to the full-surplus extraction outcome with a relatively simple mechanism.9 In the simplest case, even though there are many information sets, the monopolist need only offer two different pairs of bundles. We move outside the implementation framework with our main results when the monopolist cannot commit and may revise contracts subject to partial commitment constraints. Now, the monopolist cannot be viewed as a “mechanism designer” operating outside the game, but must be viewed as a player in the game. As part of the overall game, the monopolist specifies a game among consumers by offering a menu of contracts and revising them after consumers reveal their types. We solve for the sequential equilibria of the nonlinear pricing games with this form of updating contracts. With contract revision subject to partial commitment, both consumer types purchase the efficient quantities, and the monopolist still extracts almost the entire surplus. This also holds if the monopolist can make offers to consumers who initially opted out. Restrictions on the minimum utility offered to truthful consumers both on and off the equilibrium path limit the monopolist's profit but outcomes are still efficient. Section 2 presents the basic model with two consumer types. Section 3 analyzes the contract revision game and Section 4 analyzes the game with offers to those who opt out in the first stage. Section 5 considers contract limitations. Section 6 considers some additional applications of our approach when the monopolist has less precise information. In Section 7, we present our conclusions and compare our results to other literature.

We have studied the effects of allowing a monopolist to condition nonlinear pricing contracts on the number of consumers who choose each of the menus of contracts. In addition, we require that all contract options, on and off the equilibrium path, be robust to revisions which benefit both the monopolist and truthful consumers. The monopolist can extract all consumer surplus when there are no restrictions on the contracts which it can offer on and off the equilibrium path, even when it can only make partial commitments. In effect, the monopolist is able to implement its most preferred outcome from the set of undistorted outcomes. This contrasts with Hamilton and Slutsky's (2003) result for the optimal income tax problem where the budget constraint prevents the social planner from implementing some undistorted outcomes. When the monopolist can only offer contract menus which do not violate the individual rationality constraints on and off the equilibrium path, the dominance-solvable outcome has undistorted contracts, but these contracts are less profitable for the monopolist than the conventional distorted ones. Unlike the simplest mechanisms where the monopolist exploits his knowledge of the precise distribution of consumer types, our mechanism does not have a multiplicity of Nash equilibria. Nor unlike mechanisms analogous to Piketty's generalized tax schedules do our mechanisms require the monopolist to commit to inferior outcomes off the equilibrium path. However, our mechanisms do require that the monopolist can offer contracts with negative surplus to truthful consumers off the equilibrium path. When we require that truthful consumers are never made worse off by participating in the mechanism, we find that the monopolist can only implement less profitable outcomes, albeit ones which are efficient. The Bagnoli–Salant–Swierzbinski mechanisms (1989, 1995) also extract all or almost all consumer surplus by changing offers over time after some consumers accept the early high-price offers. While these solutions are subgame-perfect Nash equilibria, they are not unique. In particular, in the infinite horizon game, if some consumers with high valuations only buy when price falls near marginal cost, the equilibrium price will drop to marginal cost quickly and the outcome is closer to Coase's prediction. In contrast, our solutions are unique because of the dominance solvability property. Other mechanisms (Crémer and McLean, 1985, Crémer and McLean, 1988 and Spiegel and Wilkie, 2000) allow a monopolist to avoid paying information rents to consumers by exploiting correlation between consumer types. There are several differences between these environments and ours, as well as in the results. First, since everyone knows the precise number of each consumer type in our model, the correlation is quite special (perfectly negative correlation when there is one consumer of each type, for example). This special correlation is allowed by Crémer–McLean but is not required. Second, the Crémer–McLean mechanism requires consumer risk neutrality, while our mechanism does not impose this. Since our mechanism does work with risk-neutral consumers, it would appear that the contract revision is the crucial feature that prevents implementation by a dominant strategy mechanism. Third, Crémer–McLean find dominant strategy mechanisms in some cases and Bayesian Nash mechanisms in others, while we construct dominance-solvable mechanisms (which are intermediate in restrictiveness). Fourth, Spiegel and Wilkie decompose the payment portion of their mechanism into two parts—one which depends on one's own reported type and one which depends on the vector of others' reports. Such a decomposition is not possible in our mechanism since the surplus received depends on the complete vector of reports. Thus, there appears to be no nesting relationship of our model with theirs. One problematic property of the Crémer–McLean mechanism is that the transfers paid by agents may need to be quite large. In our model where the aggregate distribution of types is common knowledge, the vectors of conditional probabilities of one consumer's type given another's type are sufficiently different that that does not occur. Thus, while our mechanism does not incorporate very large payments by consumers in any state of the world, a Crémer–McLean mechanism would not require very large payments either in the circumstances we consider.