Optimal choice of characteristics for a nonexcludable good.

The plan of the article is the following. The model and the basic properties of the optimal mechanism are presented in Section 2. We solve for the case of asymmetric information on the vertical dimension and the horizontal dimension in Sections 3 and 4, respectively. In Section 5, we characterize the optimal contract when agents at different locations also have different distributions of valuations. Moreover, we analyze situations in which the good can be located at different places over time. In Section 6, we study the mechanism when the principal maximizes welfare instead of revenue. Also, we determine the properties of the contract if one agent is also the producer of the good. Concluding remarks are collected in Section 7. All proofs can be found in the Appendix.

2. The model

* Basic ingredients. We consider two agents A and B indexed by i and j. Each agent ("he") is located at one extreme of a Hotelling line of measure N. Denoting by [y.sub.i] the location of agent i, we have [y.sub.A] = 0 and [y.sub.B] = N. An indivisible good can be produced and located somewhere on the line. (6) We denote by [[theta].sub.i] agent i's intrinsic valuation for the good (also referred to as "type") and we assume that [[theta].sub.i] [member of] [[theta].bar], [[bar.[theta]]]. Valuations are private information and they are independently drawn from a common knowledge distribution F([[theta].sub.i]) with continuous and strictly positive density f([[theta].sub.i]). It also satisfies the monotone hazard rate property: d[1-F([theta])/f([theta])]/d[theta] < 0. Agents care about the location x of the good. We assume that x can take a finite but arbitrarily large number of locations, and we order them from closest to agent A to closest to agent B: x [member of] {0, 1, ... , N - 1, N}. We denote by [[gamma].sub.i](= [absolute value of x-[y.sub.i]]) the distance between the location of the good and the location of agent i. The payoff of agent i as a function of his valuation and distance takes the following form:

v([[theta].sub.i],[[gamma].sub.i]) = [pi]([[theta].sub.i] - c[[gamma].sub.i], (1)

where, following the Hotelling terminology, e is a positive "transportation cost," [pi]' > 0, [pi]" < 0, and, for technical convenience, [pi]"' [greater than or equal to] 0. According to this formalization, the payoff is increasing in the valuation ([partial derivative]v/[partial derivative][[theta].sub.i] > 0) and decreasing in the distance ([partial derivative]v/[partial derivative][[gamma].sub.i] < 0). Moreover, valuation is relatively more important the bigger the distance between the location of the agent and the location of the good ([[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] > 0). In other words, high-type agents are relatively less sensitive to distance. Overall, agents are differentiated along two substitutable dimensions captured by two parameters, a vertical differentiation parameter (the valuation for the good) and a horizontal differentiation parameter (the distance between the good and the agent).

To be in the interesting case, the payoff of each agent when the good is produced is always greater than the payoff when it is not, which we normalize to 0 ([pi]([[theta].bar] - cN) > 0). Our setting is characterized by positive and type-dependent externalities. Each agent prefers to have the good produced and the payoff of agents increases with their valuation, independently of x.

We want to determine how the good is optimally located on the Hotelling line. We assume that the location decision is in the hands of a third party (from now on "principal" or "she"). Denote by e = [empty set] the event "the principal does not produce the good" and by e = x [member of] {0, ... , N} the event "the good is produced and located at x." In order to better concentrate on the inefficiencies of the allocation due to the asymmetry of information, we assume that producing the good is costless for the principal and generates no delay. Also, we concentrate in Section 3 on the case in which the principal maximizes revenue. This assumption is relaxed in Section 6.

[] Examples. The purpose of this subsection is to provide a few examples in which the ingredients of our theory are present and for which we believe our normative approach can be useful. (7)

Physical location of a nonexcludable private or public good. Agents A and B are two neighboring cities. The vertical differentiation parameter [[theta].sub.i] is the intrinsic demand for football of each city and the horizontal differentiation parameter is the distance between the city and the stadium. Also, c is simply a transportation cost. The payoff of each city when the stadium is built increases with its demand for football ([partial derivative]v/[partial derivative][[theta].sub.i] > 0) and decreases with the distance between the city and the stadium ([partial derivative]v/[partial derivative][[gamma].sub.i] < 0). Keeping c constant, inhabitants of a city supporting a football team are relatively more inclined to drive to attend an event ([[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] > 0). Also, each city prefers a stadium located far away rather than no stadium at all (positive externalities), and the utility of cities increases with their valuation, independently of the location (type-dependent externalities). The principal is an investor willing to build and manage a new stadium, and she maximizes revenue. Or, the principal is a local authority trying to make the two cities agree to finance a public stadium. The model can be applied to other decisions to locate a nonexcludable good such as a shopping mall or a hospital.

Creation of a private school. Agents A and B are two types of parents. The vertical differentiation parameter [[theta].sub.i] is the intrinsic willingness to pay for a new private school and the characteristics of the good is the emphasis of the school on languages versus sciences. Given our assumptions, the payoff of a group of parents increases with their valuation for private education. Parents disagree on the emphasis and the payoff decreases with the distance between the actual emphasis of the school and the desired emphasis of each type of parent. The parameter c captures how sensitive parents are to a departure from their preferred emphasis. Our model corresponds to the case where parents with a high valuation for the new school are relatively more willing to compromise on emphasis.

As a special case, the good may be French education, where [[theta].sub.A] is the valuation of French parents located in a foreign country for a new French school in that country (i.e., their willingness to pay to have the same education as in France (8)) and [[theta].sub.B] is the valuation of local citizens. The horizontal dimension captures, for instance, the emphasis on mathematics: French parents want to have the same curriculum as in France, however local citizens want part of the emphasis on mathematics replaced by local history and geography. Parents with high valuations are more likely to compromise on the curriculum because, for instance, there are few good alternatives to French education in the country considered. Also, the principal is an investor (9) or a parent willing to offer a personalized education to his own children and offering this new concept to other parents as well. (10) This special case is interesting because we observe that most French schools located in foreign countries do adapt the curriculum to the preferences of local citizens.

Services offered to club members. The principal is the administrator of a private golf or tennis club and maximizes revenue or welfare of club members. The club accepts families (agent A) who enjoy other activities besides sports (e.g., socializing, using a restaurant) and individuals (agent B) who come mainly to practice. Then, [[theta].sub.i] captures the intrinsic demand for the club in group i and the horizontal dimension is the quantity of activities beyond sports. Given our assumptions, club members with high valuation for the club are relatively more willing to compromise on the services offered.

Development of a new product. The principal is a monopolist deciding to develop a new product and maximizes profit. Agents A and B are two groups of consumers. The parameter [[theta].sub.i] is the demand for the new good in each group and [[gamma].sub.i] is the difference between the preferred and the actual characteristics of the good for group i. The model captures the fact that consumers with a high valuation for the good are relatively more willing to compromise on characteristics. Also, each group prefers to have the possibility to buy the good even if its main characteristic is not the preferred one.

Choice of the program of an opera season. The principal is the general director of the opera and maximizes either revenue or the welfare of attendants. Agents of type A represent the group of music lovers and agents of type B are casual attendants or tourists. The parameter [[theta].sub.i] represents the willingness to pay for tickets in each group and [[gamma].sub.i] is the difference between the preferred and the actual program offered at the opera. Each group is better off if the opera offers performances the coming year. However, they differ in their preferences over the program: type B agents prefer to attend well-known performances, whereas type A agents prefer rare productions. The latter group is also relatively more willing to compromise. The same logic extends to goods such as theater performances or temporary exhibits in art museums.