Optimal choice of characteristics for a nonexcludable good.

if [[PHI].sub.A]([[theta].sub.A], 0) + [[PHI].sub.B]([[theta].sub.B], 0) > max{0, [[PHI].sub.A]([[theta].sub.A], N) + [[PHI].sub.B]([[theta].sub.B], N)}, then [p.sub.0]([[theta].sub.A], [[theta].sub.B]) = 1

if [[PHI].sub.A]([[theta].sub.A], N) + [[PHI].sub.B]([[theta].sub.B], N) > max{0, [[PHI].sub.A]([[theta].sub.A], 0) + [[PHI].sub.B]([[theta].sub.B], 0)}, then [p.sub.N]([[theta].sub.A], [[theta].sub.B]) = 1

Also, denote by [r.sub.i],([[theta].sub.j], x) the value of [[theta].sub,i] such that [[PHI].sub.i]([r.sub.i]([[theta].sub.j], x), x) + [[PHI].sub.j]([theta].sub.j], x = 0. (12)

Proposition 1. With two possible locations x [member of] {0, N}, the optimal contract is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

When the principal decides where to locate the good, she compares the virtual surplus at each location. Because externalities are positive and type dependent, the surplus depends on the valuations of both agents. First, the good will never be produced where the agent with highest valuation is located (formally, [[theta].sub.A] > [[theta].sub.B] [??] x [not equal to] 0 and [[theta].sub.B] > [[theta].sub.A] [??] x [not equal to] N). This is due to the type dependency of the externality and the substitutability between the vertical and horizontal dimensions ([[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] > 0). The key issue is that, at any location, the principal extracts payments from both agents. So, suppose that [[theta].sub.A] > [[theta].sub.B]. By definition, A is willing to pay more than B to have the good at his own location. However, by locating the good at x = N rather than at x = 0, the loss in the revenue extracted from agent A is smaller than the gain in the revenue extracted from agent B.

[FIGURE 1 OMITTED]

Second, the allocation is ex post inefficient as in the standard literature. Even if the auctioneer's utility of keeping the good is smaller than the bidders' lowest valuation, the good may not be sold in order to decrease the rents (Myerson, 1981). Under positive externalities, this inefficiency is diminished but persists: for some pairs ([[theta.sub.A], [[theta.sub.B]), the principal does not produce the good even though each agent has a positive utility under all locations.

Third, [r.sub.i] is the analog of a reserve price for bidder i in an auction mechanism. However, instead of being fixed, it depends negatively on the valuation of the other agent ([partial derivative][r.sub.i]/[partial derivative][[theta].sub.j] < 0). This new feature is due to the type dependency of the externality. As the valuation of one agent increases, his willingness to pay at any given location also increases. Therefore, the minimum valuation of the other agent above which the principal finds it optimal to produce decreases.

Last, note that ([partial derivative][pi]([[theta].sub.i] - N)/[partial derivative] N < 0 and ([partial derivative][pi]([[theta].sub.i] - c [[gamma].sub.i])/[partial derivative]c < 0. The size of the externality is inversely related to the length of the Hotelling line and to the transportation cost. We show that ([partial derivative][r.sub.i]([[theta].sub.j], x)/[partial derivative] N > 0. As the externality increases (i.e., as N decreases), the regulator can extract more payoff from the agents. Therefore, the event e = x [member of] {0, N} becomes relatively more profitable than the event e = [empty set] (i.e., [r.sub.i] decreases). Similarly, [partial derivative][r.sub.i]([[theta].sub.j], x)/[partial derivative]c > 0. The project is less profitable as the transportation cost increases and the principal prefers to decrease the probability of producing in order to diminish the rent. Of course, the allocation is more often ex post inefficient in that case. These results are depicted in Figure 1.

Remark 1. The reader might argue that sometimes high-type agents are also relatively more concerned about distance, that is, the payoff function satisfies the assumption [[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] < 0. Then, under both complete and asymmetric information, the good is located in 0 (respectively N) when [[theta].sub.A] [[theta].sub.B] (respectively [[theta].sub.A] < [[theta].sub.B]. The agent with the highest valuation is also the least willing to compromise on location, and the decision is biased toward that agent. Therefore, when [[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] < 0, horizontal differentiation affects only quantitatively the optimal location compared to a model with vertical differentiation only. In both cases, the good is located closer to the agent with the highest valuation. Also, observing that a good is located close to the interest of the party who enjoys it the least cannot be reconciled with the assumption [[partial derivative].sup.2]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i] < 0.

[] Optimal contract with several possible locations. We now analyze the case where the number of potential locations for the good is finite but arbitrarily large (x [member of] {0, 1, ..., N}). This formalization is more suitable to explain the optimal choice of characteristics of a new good or the optimal location of a new shopping mall to be built anywhere between two communities. Interestingly, this case cannot be reinterpreted as an auction of an indivisible good with externalities. Formally, it shares some features with the auction of a divisible good (Maskin and Riley, 1989): for example, locating the good at x = N/2 is similar to selling half of the good to one agent and half to the other one. However, there is a crucial difference between the two interpretations. In fact, not producing the good in our model (e = [empty set]) corresponds to not selling it in the auction case, and locating the good somewhere in the line (e = x) corresponds to splitting it entirely between the two bidders. Yet, in auctions of divisible goods there is a third possibility implicitly ruled out in our setting, which is to sell a fraction of the good and keep the rest. (13)

We denote by [x.sub.s] the optimal second-best location. It maximizes the sum of the virtual surplus, that is, the payoff of the principal given the asymmetry of information:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (6)

Proposition 2. When the set of locations is arbitrarily large, the optimal contract is such that: (14)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sub.B]([[theta].sub.A], [x.sub.s]) is such that [[PHI].sub.A]([[theta].sub.A], [x.sub.s]) + [[PHI].sub.B]([r.sub.B]([[theta].sub.A], [x.sub.s]), Xs) = 0. The reserve price [r.sub.B]([[theta].sub.A], [x.sub.s]) is such that [partial derivative][r.sub.B]/[partial derivative][[theta].sub.A] < 0. The location [x.sub.s] is such that > [[partial derivative][x.sub.s]/[partial derivative][[theta].sub.B] < 0, and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] N/2 for all [[theta].sub.A] [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

The location principle highlighted in Proposition 1 extends to the case of a large number of possible locations. The principal first determines which location [x.sub.s] maximizes the virtual surplus, and then compares this total payoff with the payoff under no production. If both agents have the same valuation, then the good is located halfway between the two. As before, when types are different, the good is located closer to the agent with lowest valuation, although it is not necessarily at his exact location ([MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]). Given informational rents, the principal may again decide not to produce the good (e = [empty set]). However, the ability to choose from a wider range of locations makes this event relatively less frequent than in Proposition 1. Moreover, the good is located more efficiently than in Proposition 1.

Asymmetric information induces the principal to increase the distance between the location of the good and that of the agent who values it the most, relative to the socially optimal level. In fact, the principal has to manage simultaneously two distortions, [[partial derivative][[pi].sub.A]/[partial derivative][[theta].sub.A]][1-F([[theta].sub.A]/f([[theta].sub.A] and [[partial derivative][[pi].sub.B]/[partial derivative][[theta].sub.B]][1-F([theta].sub.B]/f([[theta].sub.B]], and both increase with the distance between the agent and the good. As the valuation [[theta].sub.i] of an agent increases, the distortion becomes less sensitive to the distance [[gamma].sub.i]. To decrease the rents, it becomes relatively more interesting to bring the location of the good closer to the agent with lowest valuation. The properties of the optimal mechanism are represented in Figure 2.

Example 1. Let us assume c = 1, [pi]([[theta].sub.i] - [[gamma].sub.i]) = 4([[theta].sub.i] - [[gamma].sub.i]) - [([theta].sub.i] - [[[gamma].sub.i]).sup.2], and [[theta].sub.i] ~ U [1, 2]. We also let N = 1 and we assume that x [member of] [0, 1], so that [[theta].sub.i] - [[gamma].sub.i] [member of] [0, 2]. Using (2)-(3)-(4)-(5)-(6), the expressions for the optimal locations are

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].