Optimal choice of characteristics for a nonexcludable good.

In these examples, the principal is not fully informed about the preferences of the agents. The demand for football, the private school, or the opera are generally unknown. Consequently, we assume that the principal does not observe [theta]. It also occurs that the principal does not observe how sensitive agents are to the choice of characteristics. We analyze this case in Section 4.

[] First-best. From now on in this section, we assume that the principal maximizes her expected revenue. To have a benchmark for comparison, we denote by [x.sub.F] the first-best location. It maximizes the payoff of the principal under full information. For any possible location x, the principal extracts all the surplus generated by the production of the good at that location. Formally, her total revenue is [pi]([[theta].sub.A] - cx) + [pi]([[theta].sub.B] - c(N - x)). Therefore, the good is located at [x.sub.F) such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

Lemma 1. The optimal location is [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. It is increasing in [[theta].sub.A] and decreasing in [[theta].sub.B].

The principal always produces the good. Given the substitutability of the vertical and horizontal differentiation parameters, she prefers to favor the agent with lowest valuation and to offer a good with characteristics more on the lines of his preferences. Formally, [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] when [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. The optimal location increases with [[theta].sub.A] (and becomes closer to B) and decreases with [[theta].sub.B] (and becomes closer to A). By doing so, the loss in the revenue extracted from the high-valuation agent is smaller than the gain in the revenue extracted from the low-valuation agent. In Section 3, we study how the presence of asymmetric information on the valuations modifies this allocation. In particular, we determine whether it exacerbates the bias or not.

3. Optimal location with unknown valuations

* Properties of the mechanism under asymmetric information. The contract must specify an allocation rule and payments when both agents accept the contract but also when at least one refuses it. Indeed, given the presence of externalities, the outside option of each agent is mechanism dependent. Note that the objective of the principal is to extract as much payment as possible from both agents. Therefore, she benefits from designing a mechanism in which the outside option is the smallest possible. Following the standard contracting literature, we assume that the principal can commit to any mechanism offered to the agents and each agent accepts the contract when the utility of accepting is at least equal to the outside option. Then, it is optimal for the seller to commit not to produce the good if at least one agent refuses to participate in the contract. The idea is simply that given the positive externalities, the worst possible scenario for any agent who refuses to participate is the one in which the good is never produced. Note that this threat is only credible if the principal can commit. On the other hand, it is costless for her, as it is only made off-the-equilibrium path.

From the revelation principle, we know that we can restrict the attention to a direct revelation mechanism. The principal offers a menu of contracts that depends on the pair of announced valuations ([[??].sub.A], [[??].sub.B]). The menu specifies a probability [p.sub.x]([[??].sub.A], [[??].sub.B]) of production at each possible location x, and a transfer [t.sub.i]([[??].sub.A], [[??].sub.B]) from each agent to the principal. We also denote by p[empty set]([[??].sub.A], [[??].sub.B]) the probability of not producing the good. For notational convenience, let [[pi].sub.i]([[theta].sub.i], x) [equivalent to] [pi]([[theta].sub.i] -c|x [y.sub.i]|) be agent i's payoff when the good is located at x. We have

[[pi].sub.A]([[theta].sub.A], x) = [pi]([[theta].sub.A] - cx) and [[pi].sub.B]([[theta].sub.B], x) = [pi]([[theta].sub.B] - c(N - x)), (3)

Also, let [u.sub.i]([[theta].sub.i], [[??].sub.i]) be the expected utility of agent i when his valuation is [[theta].sub.i], he announces [[??].sub.i], and the other agent discloses his true valuation [[theta].sub.j]. We denote by [u.sub.i]([[theta].sub.i]) [equivalent to] [u.sub.i]([[theta].sub.i], [[theta].sub.i]) his expected utility under truthful revelation. We have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

A mechanism {[p.sub.x](*), [t.sub.i](*), [t.sub.i](*)} is optimal if and only if it maximizes the expected revenue R:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and satisfies three kinds of constraints. First, incentive compatibility, which states that each agent must prefer to reveal his true valuation rather than any other one:

[u.sub.i]([[theta].sub.i]) [greater than or equal to] [u.sub.i]([[theta].sub.i], [[??].sub.i]) [[for all].sub.i], [[theta].sub.i], [[??].sub.i],.

Second, individual rationality, which implies that each agent must be willing to accept the contract offered by the principal (recall that in the case of nonacceptance of the contract, the good is never allocated, so the agent's reservation utility is zero) (11):

[u.sub.i]([[theta].sub.i]) [greater than or equal to] 0 [[for all].sub.i], [[theta].sub.i].

Last, the allocation rule must be feasible:

[p.sub.x] ([[theta].sub.A], [[theta].sub.B]) [greater than or equal to] 0 [[for all]x, [[theta].sub.i], [[theta].sub.j] and [n.summation over (x=0)] [p.sub.x]([[theta].sub.A], [[theta].sub.B]) [less than or equal to] 1 [[for all].sub.i], [[theta].sub.j].

Lemma 2. The optimal mechanism solves the following program P:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (M)

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

[[PHI].sub.A]([[theta].sub.A], x) = [[pi].sub.A]([[theta].sub.A], x) - [partial derivative][[pi].sub.A]([[theta].sub.A], x)/[partial derivative][[theta].sub.A] 1 - F([[theta].sub.A])/f([[theta].sub.A] (4)

[[PHI].sub.B]([[theta].sub.B], x) = [[pi].sub.B]([[theta].sub.B], x) - [partial derivative][[pi].sub.B]([[theta].sub.B], x)/[partial derivative][[theta].sub.B] 1 - F([[theta].sub.B])/f([[theta].sub.B] (5)

In the optimal mechanism, the expected utility of agent i = {A, B} is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The net surplus of agents A and B when the good is located at x are [[pi].sub.A]([[theta].sub.A], x) and [[pi].sub.B]([[theta].sub.B], x), respectively. Under complete information, this also corresponds to their willingness to pay and therefore to the maximum revenue that the principal can extract. Asymmetric information introduces a distortion in the agents' willingness to pay. This is the case because an agent with a high valuation can mimic an agent with a low valuation. To avoid mimicking, the seller must grant rents and the equilibrium utility of agents must increase in the valuation. Overall, the seller extracts only the virtual surplus [[PHI].sub.A]([[theta].sub.A], x) and [[PHI].sub.B]([[theta].sub.B], x) from agents A and B, respectively, when the good is located at x, that is, the net surplus adjusted for the informational rents. She chooses the location that maximizes the virtual surplus under the standard monotonicity (M) and feasibility (F) constraints. Given the concavity of [pi] and the monotone hazard rate property, the virtual surplus increases with the valuations of agents for all x: [partial derivative] [[PHI].sub.A]/[partial derivative] [[theta].sub.A] > 0 and [partial derivative] [[PHI].sub.B]/[partial derivative][[theta].sub.B] > 0. The analysis of the allocation mechanism considered here is an adaptation of the procedure introduced by Myerson (1981) in the context of an auction.

[] Optimal contract with two possible locations. In this subsection, we assume that the good can only be located at x = 0 or x = N. For instance, a sports tournament needs to be located in one of two cities and the physical location of the sport facilities already exists. The intrinsic demand for the game is unknown and the organizer must elicit this information to raise funds and locate the event optimally. In fact, this problem is formally identical to the optimal auction of an indivisible good with two bidders (A and B), private valuations, and positive type-dependent externalities. First, the principal may decide not to produce the good and both agents get utility 0. Second, she may produce the good and locate it at x = 0, then agent A gets utility [pi]([[theta].sub.A]) and agent B gets utility [pi]([[theta].sub.B] - cN). Third, she may produce the good and locate it at x = N, in which case agent A gets utility [pi]([[theta].sub.A] - cN) and agent B gets utility [pi]([[theta].sub.B]. Call [v.sub.i]([theta].sub.i] [equivalent to] [pi]([[theta].sub.i] and [[alpha].sub.i]([[theta].sub.i]) [equivalent to] [pi]([[theta].sub.i] - cN) (< [v.sub.i] for all [[theta].sub.i]). Locating the good at x = 0 and at x = N is formally equivalent to selling the good to agent A and to agent B, respectively: the agent who purchases it gets utility [v.sub.i] (increasing in his type [[theta].sub.i]) and the other one enjoys a positive externality [[alpha].sub.j] (also increasing in his type [[theta].sub.j]).

Using Lemma 2, equations (4) and (5), and ignoring for the moment constraint (M), it is immediate that in the optimal mechanism