Optimal choice of characteristics for a nonexcludable good.

where [r.sup.H.sub.B]([c.sub.A], [x.sup.H.sub.S]) is such that, [[PSI].sub.A]([c.sub.A], [x.sup.H.sub.S]) + [[PSI].sub.B]([r.sup.H.sub.B]([c.sub.A], [x.sup.H.sub.S]), [x.sup.H.sub.S] =. The reserve price [r.sup.H.sub.B]([c.sub.A], [x.sup.H.sub.S]) is such that [ar.sup.H.sub.B]/[partial derivative][c.sub.A] < 0. The location [x.sup.H.sub.S] is such that [partial derivative][x.sup.H.sub.S]/[partial derivative][c.sub.A] < 0, [[partial derivative][x.sup.H.sub.S]/[partial derivative][c.sub.A] < 0, [partial derivative][x.sup.H.sub.S]/[partial derivative][c.sub.B] > 0, and [x.sup.H.sub.S] [??] N/2 for all [c.sub.A] [less than or equal to] [c.sub.B]. Furthermore, for all [[c.sub.B], there exists [??] such that for all [c.sub.A] < [??], [x.sup.H.sub.S] > [x.sup.H.sub.F] and for all [c.sub.A], there exists [[??].sub.B] such that for all [c.sub.B] < [[??].sub.B], [x.sup.H.sub.S] < [x.sup.H.sub.F].

First, when [c.sub.A] = [c.sub.B], it is optimal to locate the good at half-distance. If ca increases, the surplus that can be extracted from A decreases and given agent A becomes relatively more sensitive to distance, it is profitable to move the location closer to A. Therefore, when [c.sub.A] > [c.sub.B], [x.sup.H.sub.S] < N/2. By the same token, when [c.sub.A] < [c.sub.B], then [x.sub.H.sub.S] > N/2.

Second, compared to the solution obtained under complete information, there are two effects going in opposite directions. Suppose [c.sub.A] < [c.sub.B], in which case [x.sup.H.sub.F] > N/2. Under asymmetric information and other things being equal, agent A must receive a higher informational rent. Therefore, the seller has an incentive to move the location closer to A to diminish the degree of asymmetric information with him. However, agent B is relatively more sensitive to distance. Then, it is also profitable to move the location closer to B. Overall, the bias obtained under complete information is decreased or increased, depending on which of the two effects dominates. This crucially depends on the shape of the utility function [pi](*) and the distribution h(*).

Third, as in the previous section, the good is not always produced: if the two costs are sufficiently high, the seller does not produce. Moreover, when ca increases, the surplus that can be extracted decreases and the seller requires agent B to have a low enough transportation cost to carry the project at each possible location x. In other words, [r.sub.B]([c.sub.A], x) decreases in [c.sub.A].

Last, the effects of asymmetric information in the present section can be contrasted with those obtained in Section 3. In both sections, the principal has incentives to increase the bias obtained under complete information. Given the agent who is a priori more valuable (the agent with the highest [theta] in Section 3 and the agent with the lowest c in this section) is also less sensitive to distance, moving the location away from him allows capturing extra rents from the other agent. However, it is now possible to increase or decrease the amount of asymmetric information with each agent by moving the location. This possibility generates a qualitative departure with respect to Section 3. As a result, the overall bias might not be exacerbated.

Example 2. Suppose that [pi]([theta] - [[gamma].sub.i]) = log ([theta] - [c.sub.i][[gamma].sub.i]). Under complete information, we have

[x.sup.H.sub.F] = N/2 + [theta]/2 ([c.sub.B] - [c.sub.A]/[c.sub.A][c.sub.B].

Under asymmetric information, [[PSI].sub.A]([c.sub.A], x) + [[PSI].sub.B] ([c.sub.B], x) is concave in x and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

If the distribution is uniform on [[bar.c],[c.bar]], then H(c)/h(c)[1/[c.sup.2]] is concave with a maximum at c* = 2[c.bar]. There two cases. When the support is small enough ([bar.c] < 2[c.bar]), we have [x.sup.H.sub.F] [??] [x.sup.H.sub.S] [??] [x.sup.H.sub.S] when [c.sub.A] [??] [c.sub.B]. Then, it is always optimal to move the good closer to the agent with the highest transportation cost and the bias is increased under asymmetric information. When the support is large ([bar.c] > 2[c.bar]), for all [c.sub.B], there exists [??] such that [H(c.sub.B]]/[h([c.sub.B])] = [1/[c.sup.2.sub.B]] = [H([[??].sub.B]/h([[??].sub.B])][1/[[??].sup.2.sub.B]. When [c.sub.A] [less than or equal to] min([c.sub.B], [[??].sub.B]), then [x.sup.H.sub.F] [less than or equal to] [x.sup.H.sub.S]. Then, the bias in favor of the agent with the highest transportation cost is increased. When [c.sub.A] [member of] (min([c.sub.B], [[??].sub.B]), max([c.sub.B], [[??].sub.B])), the principal finds it profitable to move the good closer to A and [x.sup.H.sub.F] > [x.sup.H.sub.S]. Last, when [c.sub.A] [greater than or equal to] max ([c.sub.B], [[??].sub.B], then [x.sup.H.sub.F] [less than or equal to] [x.sup.H.sub.S] and the bias is decreased.

The intuition is as follows. When the support is large enough, low values of the transportation cost are relatively less likely, but at the same time, a low cost-agent has more possibilities to mimic a high-cost agent. Then, the rents that must be granted to a low-cost agent are relatively higher when the support is large. It becomes relatively more profitable to use the location to "regulate" the amount of asymmetric information. When the difference between the transportation costs is substantial, the principal reduces the bias obtained under complete information by moving the good closer to the agent with the smallest cost.

5. Asymmetric preferences and dynamic choices

* Optimal location when agents' preferences are not symmetric. In this section, we want to determine how the efficient location is modified when observing the preferences of the agent on the horizontal dimension convey relevant information on his preferences on the vertical dimension. For instance, suppose an entrepreneur wants to offer private education with primary emphasis on science. Some parents think that mathematics should be an important ingredient of the curriculum (group A) while other parents believe that a large part of mathematics and some of science should be replaced by arts and languages (group B). If the public system emphasizes arts rather than mathematics and science, not only parents with a high valuation for education in science are expected to be more willing to compromise on the content in mathematics versus arts ([partial derivative]v/[partial derivative][[theta].sub.i][partial derivative][[gamma].sub.i]>0), but also parents in group B should be willing to pay less for private education (high values of [[theta].sub.B] are less likely). Then, the intensity of preferences for private education is correlated to the preferences for the courses taught.

To capture this idea, we extend the model of Section 2 and assume that the likelihood of a given valuation [theta] is [f.sub.A]([theta]) at location 0 and [f.sub.B]([theta]) at location N. The two distributions satisfy the monotone likelihood ratio property (MLRP)

d/dv [[f.sub.A]([theta])] / [f.sub.B]([theta])] > 0.

MLRP says that an agent in location 0 is relatively more likely to have a high type and relatively less likely to have a low type than an agent in location N. It implies that (1 - [F.sub.A] / ([f.sub.A]([theta])) > (1 - [F.sub.B]([theta])) / [f.sub.B]([theta]) and [F.sub.A]([theta]) < [F.sub.B]([theta]). The virtual surplus becomes

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

and the optimal second-best location is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (11)

Given (6), (11), and Proposition 2, we have the following.

Proposition 4. The optimal contract is such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

where [r.sup.C.sub.B]([[theta].sub.A], [x.sup.C.sub.S]) is such that [[THETA].sup.C.sub.A] ([[theta].sub.A], [X.sup.C.sub.S]), [X.sup.C.sub.S]) = 0. For all [[theta].sub.A] and [[theta].sub.B], we have [X.sup.C.sub.S] > [X.sub.S]. Moreover, for all [[theta].sub.B], there exists [[??].sub.A] ([[theta].sub.B]) < [[theta].sub.B] such that [X.sup.C.sub.S] = N/2. Last, for all [[theta].sub.B], there exists [[??].sub.A]([[theta].sub.B]) < [[theta].sub.B] such that [x.sup.C.sub.S] [??] [X.sub.F] for all [[theta].sub.A] [??][??].sub.A]([[theta].sub.B]).

The result [X.sup.C.sub.S] > [X.sub.S] is intuitive. Given that it is optimal to distort the location against high- valuation agents and, given that the likelihood of a high valuation is now higher at location 0, the location is moved closer to agent B. Therefore, the range of valuations of agent A for which the good is located closer to B is increased ([x.sub.C] > N/2 when [[theta].sub.A] > [[??].sub.A]([[theta].sub.B])). Also, compared to the first-best location, the distortion due to asymmetric information is not symmetric. Sometimes, the good is located closer to B, even if his realized valuation is higher than that of A.

In terms of the previous example, because the public system offers a poor education in science but a relatively better education in arts rather than mathematics, the principal attracts both types of parents by favoring the group with the best outside option, that is, parents who think that a good education in arts is relatively more important. Parents who value mathematics more than arts enrolel their children because this is still the best option to learn science.