Optimal choice of characteristics for a nonexcludable good.

Remark 3. The analysis shares some similarities with the optimal allocation of a public good studied in public economics (in the tradition of Clarke, 1971 and Groves, 1973) and provides an alternative and complementary perspective. In the standard setting, each agent's valuation is a function v([theta], q), where [theta] is his type and q, the quantity of public good. Given that all agents prefer more quantity to less, the main issue is to design a mechanism to prevent them from understating their type, getting away with a low payment while enjoying the public good (positive externality). In our setting, the valuation functions of agent A and B depend on the location x of the public good instead of the quantity provided. Here, agents have opposite preferences over locations. Also, given the positive externality, the incentives to underreport are present but the principal can use the location choice to mitigate them.

[] Optimal location when one agent is also the producer. Suppose now that agent A decides whether he produces the good and where he locates it. This captures the fact that the planner (e.g., a parent/a tennis player) can also have a private interest in the project (e.g., a private school/a tennis club). In order to better isolate the changes in the incentives of the new decision maker, we assume that B observes A's valuation [[theta].sub.A] for the good. Then, B does not have anything to infer from the mechanism proposed by A, and therefore A has no incentives to use the contract design to signal any information. (17)

Agent A offers a menu of contracts {[p.sub.x]([[??].sub.B]), [t.sub.B]([[??].sub.B])} such that, for each report [[??].sub.B], agent B pays a transfer [t.sub.B]([[??].sub.B]) and the good is located at x with probability [p.sub.x]([[??].sub.B]). Denote by [R.sub.A] the expected revenue of A (that is, the sum of his own valuation and the expected transfer raised from agent B) and by [u.sup.*.sub.B]([[theta].sub.B], [[theta].sub.B]) the utility of agent B with valuation [[theta].sub.B] and report [[theta].sub.B]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The objective of agent A is to solve

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([M.sub.B])

[p.sub.x]([[theta].sub.B]) [greater than or equal to] 0 [for all]x and [N.summation over x=0] [p.sub.x]([[theta].sub.B]) [less than or equal to] 1. ([F.sub.B])

In [P.sub.A], only [theta].sub.B] is private information. Because it is only required to grant informational rents to B, the objective function is the sum of the net surplus of agent A and the virtual surplus of agent B ([pi].sub.A]([[theta].sub.A], X) and [THETA]([[theta].sub.B], X), respectively). The monotonicity ([M.sub.B]) and feasibility ([F.sub.B]) constraints of agent B are the same as in Lemma 2, except that now the valuation [[theta].sub.A] is known. We denote by [x.sub.A] the location that maximizes the surplus from agent A's perspective:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

Given (2), (6), (13), [P.sub.A], and Proposition 2, we have the following.

Proposition 7. When agent A chooses the location, the optimal contract is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [r.sup.A.sub.B]([[theta].sub.A], [x.sup.A.sub.S]) is such that [[pi].sub.A]([[theta].sub.A], [x.sup.A.sub.S]) + [[THETA].sub.B]([r,sup.A.sub.B]([theta].sub.A], [x.sup.A.sub.S]), [x.sup.A.sub.S]) = 0. The location [x.sup.A.sub.S] is such that [alpha][x.sup.A.sub.S] / [alpha] [theta].sub.A] > 0, [alpha][x.sup.A.sub.S] / [alpha] [theta].sub.B]< 0, and [x.sup.A.sub.S] > max {[x.sub.S], [x.sub.F]} for all [[theta].sub.A] and [[theta].sub.B].

Again, depending on agent B's reported valuation, either the good is not produced (e = [empty set]) or it is situated at the location where the surplus is maximized (e = [x.sup.A.sub.S]). The novelty of this case is that agent A locates the good farther away from his own preferred location than in Proposition 2 ([x.sup.A.sub.S] > [X.sub.s]), and also farther away than under full information ([x.sup.A.sub.S] > [X.sub.F]). The idea is that, when agent A chooses the location, there is only one unknown parameter, B's valuation. In order to reduce B's rents, it is unambiguously better to bring the good closer to him. That same logic applies when we compare agent . A's optimal choice with the full information case.

7. Concluding remarks

[] We have analyzed the optimal choice of a principal who decides whether to produce an indivisible good and which characteristics it contains. If the utility of agents is differentiated along two substitutable dimensions (an intrinsic willingness to pay for the good and a preference for characteristics), the principal offers a good with characteristics more on the lines of the preferences of the agent with lowest willingness to pay. Asymmetric information on the vertical dimension exacerbates this bias. If agents have different intensities of preferences for characteristics, it is optimal to bias the decision in favor of the agent who is the most sensitive to a deviation from his preferred characteristics. However, the inability to observe these intensities does not necessarily exacerbate the initial bias. The reason is that, when the intensities of preferences for characteristics are unknown, the principal can arbitrarily decrease the amount of asymmetric information with one agent by locating the good closer to him.

The analysis suggests that it is sometimes profitable to bias decisions against the preferences of the most interested parties. Coming back to the special case discussed in Section 2, according to our analysis, the reason why the French schools adapt the program to the tastes of local citizens is simply that, although French parents are a priori more willing to pay for French education, the school must offer something of value to local citizens in order to attract them (for instance, local citizens might have a different educational culture). Given French parents are ready to give up some features of French education as long as the main philosophy is preserved, the school maximizes its revenue by adopting that strategy. Examples of such biases can be found in other economic situations. For example, operas generally schedule an important number of well-known performances and only a few rare productions. This suggests that it is relatively easier to attract people who truly enjoy opera rather than people who attend it only on occasion.

The results rest on the assumption that individuals are differentiated along two dimensions. They assign an intrinsic valuation to the good but they have different preferences for its characteristics. Absent the second dimension, the principal takes a decision on the lines of the agent who values it most because it is the only way to generate a social value. In our setting, the principal generates a value also by choosing characteristics: the investor can locate the stadium in the city where there is already a high number of football supporters, but also she can locate it in a city in which residents go to football events only if they host them. Then, taking a decision on the lines of the agent who values the good most is not necessarily optimal. In other words, the optimal allocation of a nonexcludable good is affected crucially by the characteristics it contains and how they are perceived by economic agents.

(1) We mean by characteristic a feature of the good on which agents disagree because their tastes differ (in industrial organization jargon, a "horizontal differentiation" parameter).

(2) Segal (1999) studies the nature of inefficiencies depending on whether contracts are observable or not. Segal and Whinston (2003) consider a larger family of games of contracting where contracts between the principal and one agent are not observed by other agents. The article analyzes general properties of equilibrium outcomes that must be satisfied by all equilibria of all games considered.

(3) In Cornelli (1996), the firm has a high fixed cost of production. Positive externalities arise between consumers, because purchasing the good affects positively the probability that the firm finds it profitable to produce it. In Lockwood (2000), the agents' marginal cost of effort is private information and the output of an agent is affected positively by his effort and that of his coworkers.

(4) An important literature also discusses from a positive point of view how local public goods should be financed by residents and landowners. It addresses the issue of which type of tax should be used, taking into account how land prices affect location decisions as well as the size of the jurisdictions. See Scotchmer (2002) for a review.

(5) See also Laffont and Tirole (1993), Chapter 4 (and the literature therein) for a detailed analysis of the regulation of quality.

(6) It can be shown that, in equilibrium, the good will never be located outside [0, N].

(7) Other forces might also be at work in some of the examples. For instance, the principal might not have as much bargaining power in real life and parties might bargain instead of resorting to take-it-or-leave-it offers. Our theory provides an upper bound on the payoff the principal can obtain in that situation.

(8) These institutions aim at offering French education (and diplomas) to French citizens located abroad. Parents who plan to come back to France or expect to travel from country to country in the future value highly the fact that their children can get the same education at every location.

(9) Even though schools are public in France, most French Lycees in foreign countries are private institutions.