Optimal choice of characteristics for a nonexcludable good.

[] Optimal locations over time. In the previous sections, we have analyzed a static game. This assumption is strong for applications such as sports tournaments which take place at regular intervals. To address this question, we consider a simple extension of the model of Section 2. A sports tournament takes place at two periods [tau] = {1, 2}, and two cities A and B compete at date [tau] = 0 for hosting it at dates 1 and/or 2. The intrinsic demand for sports [[theta].sub.i] is private information of city i and it is the same across period. However, demand is increased by a positive and common knowledge shock [delta] the first time the game is organized in city i. This captures the fact that the event attracts consumers who would not ordinarily attend it. Or, organizing the tournament requires a one-time activity (e.g., repaving the streets) that increases the welfare of the city (e.g., it increases labor). Moreover, at each period, city i incurs a loss equal to - N when the tournament is held in city j. Overall, the instantaneous utility of city i is [tau]([[theta].sub.i] + [delta]) if the tournament takes place at city i for the first time, it is [tau]([[theta].sub.i]) if it takes place at city i for the second time, and it is [tau]([[theta].sub.i] - N) if it takes place at the competing city. Agents discount the future at rate [beta] [member of] [0, 1] and the organizers must organize the tournaments at both periods.

Let us denote by [x.sub.[tau]] {0, N} the location at date [tau]. The objective of the organizers is to design a mechanism specifying a probability [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] ([[??].sub.A], [[??].sub.B]) of locating the tournament at [x.sub.1] in the first period and [x.sub.2] in the second, conditional on the reported demands, as well as inter-temporal transfers [t.sub.A]([[??].sub.A], [[??].sub.B]) and [t.sub.B]([[??].sub.A], [[??].sub.B]) from city A and city B, respectively. Formally, the analysis extends Proposition 1, which corresponds to the case [beta] = 0 and [delta] = 0.

Proposition 5. The optimal contract is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[Z.bar].sub.A]([[theta].sub.B]) < [[theta].sub.B] and [[bar.Z].sub.A] ([[theta].sub.B]) > [[theta].sub.B] increase in [[theta].sub.B]. Besides, if [[theta].sub.B] = 1, it is optimal to alternate but the order does not matter. If [delta] = 0, it is not optimal to alternate.

The logic of Proposition 1 extends to the dynamic setting. Suppose that [[theta].sub.A] < [[theta].sub.B]. Then, it is always optimal to let city A organize the tournament at least once and the principal strictly prefers to organize the tournament in city A in the first period if [beta] < 1. Let us suppose this is the case; then the question is whether city A also organizes the second tournament.

First, it is optimal to organize both events in city A if the difference between the valuations is high enough so that the effect of the extra payoff is compensated. When the two valuations are relatively close, however, the principal might decide to organize the second event in city B.

Second, the region of valuations where alternating is optimal is affected by the size of the extra payoff& Alternating is more likely to occur when 3 is high and there is no incentive to do so when [delta] tends to zero. Indeed, when [delta] is sufficiently small, any small positive difference between the two valuations compensates for the extra payoff.

Third, when agents are fully impatient ([beta] = 0), only the first period matters and any location is optimal at date 2 from the perspective at date 0. On the other extreme, when agents are infinitely patient ([beta] = 1), the intertemporal payoff is the same when the tournament takes place first in city A or in city B. In that case, the principal requires to organize the tournament in city A at least once (but not necessarily in the first period). In that case, it is optimal to alternate when valuations are close enough but the order does not matter.

Fourth, private information does not modify these results qualitatively. The main difference is that the organizer must grant informational rents. Naturally, the incentives to reveal truthfully of each city are affected by the probabilities of hosting the event. Formally, suppose [[theta].sub.A] and [[theta].sub.B] are such that it is optimal under complete information to organize the event twice in city A. To disclose those valuations and achieve these locations, the organizer must grant the total second-period rent 1 - F[[theta].sub.A] / f [[theta].sub.A] [pi]' ([[theta].sub.A]) + 1 - F[[theta].sub.B] / f [[theta].sub.B] [pi]' ([[theta].sub.B] - N). However, if the seller decides to organize the event first at location 0 and then at location N, the second-period rent becomes 1 - F[[theta].sub.A] / f [[theta].sub.A] [pi]' ([[theta].sub.A] - N) 1 - F[[theta].sub.B] / f [[theta].sub.B] [pi]' ([[theta].sub.B] + [delta]). The organizer decides to alternate more (respectively less) often than under complete information if the second-period rent is smaller (respectively higher) under that scenario.

6. Extensions

* Optimal location selected by a social planner. It is difficult to reconcile the revenue-maximizing assumption with some applications, such as the decision to locate a public good (a hospital or a public school) between two communities. Suppose that the principal is a benevolent utilitarian regulator. (15) She offers a menu that specifies, for every pair ([[??].sub.A], [??].sub.B]), a probability [p.sub.x]([[??].sub.A], [??].sub.B]) of locating the good at x together with a subsidy [s.sub.i]([[??].sub.A], [??].sub.B]) to agent i. Following the regulation literature, we assume that subsidies are socially costly: $1 transferred to an agent is raised through distortionary taxation and costs $ (1 +[lambda]) to taxpayers, with [lambda] > O. (16) Let [??]([[theta].sub.i], [??].sub.i]) be the expected utility of agent i when his valuation is [[theta].sub.i], his report is [[??].sub.i], and agent j' report is truthful, then.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The utilitarian regulator maximizes social welfare W. Given the shadow cost [lambda] of public funds, it is simply the payoff of the agents when the good is produced at x ([[pi].sub.A] and [[pi].sub.B]) minus the social costs of transferring [s.sub.A] and [s.sub.B]. Formally,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The regulator's optimization program is therefore

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Let us denote by [x.sup.W.sub.S]([lambda]) the optimal second-best location.

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (12)

Given (2), (6), (12), 79w, and Proposition 2, we have the following.

Proposition 6. When a regulator chooses the location, the optimal contract is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [r.sup.W.sub.B]([[theta].sub.A], [x.sup.W.sub.S]; [lambda]) is such that [[DELTA].sub.A]([[theta].sub.A], [x.sup.W.sub.S])+ [[LAMBDA].sub.B]([[??].sub.B]([[theta].sub.A], [x.sup.W.sub.S]; [lambda]), [x.sup.W.sub.S]) = 0. The location [x.sup.W.sub.S] is such that [alpha] [x.sup.W.sub.S] / [alpha][[theta].sub.A] > 0, [alpha] [x.sup.W.sub.S] / [alpha][[theta].sub.B] < 0, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Furthermore, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] for all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] = [X.sub.F], and [x.sup.W.sub.S] ([infinity]) = [x.sub.S].

The characteristics of the optimal contract offered by a benevolent regulator and a privately interested party are very similar: location of the good closer to the agent with lowest valuation, distortion due to asymmetric information, possibility of not producing the good, and so on. The main difference is that, in the regulation case, the relative weights of efficiency versus rent extraction in the objective function of the principal are entirely determined by [lambda].

When transferring funds is costless ([lambda] = 0), the regulator is interested exclusively in the efficiency of her action. Therefore, she takes the same decisions as under full information ([x.sup.W.sub.S] (0) = [x.sub.F] and [[??].sub.i]([[theta].sub.j], [x.sup.W.sub.S](0); 0) = [[theta].bar]), even if it comes at the expense of a substantial subsidy. If subsidies from taxpayers to agents are prohibitively costly [lambda] = [infinity]), then the regulator's objective is formally equivalent to maximize welfare under the constraint that agents can be taxed but not subsidized ([[s.sub.i] [less than or equal to] 0). This case is identical to the case of a privately interested principal, who trades off efficiency and rents but will never choose to subsidize agents. The optimal decision coincides with that of Proposition 2: [x.sup.W.sub.S]([infinity]) = [x.sub.S] and [[??].sub.i]([[theta].sub.j], [x.sup.W.sub.S]([infinity]); ([infinity]) = [r.sub.i]([[theta].sub.j], [x.sub.S]). When the cost of public funds is positive but finite [lambda] [member of] (0, [infinity]), the regulator is more concerned with increasing efficiency and less concerned with decreasing rents than a privately interested party. This is reflected in her choices: [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] all [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [[??].sub.i]([theta].sub.j], [x.sup.W.sub.S]([lambda]); [lambda]) [member of] ([[theta].bar], [r.sub.i]([[theta].sub.j], [x.sub.S])).