Under asymmetric information, the good is always located closer to
the agent with lowest valuation than under full information. Moreover,
as long as [x.sub.F] and [x.sub.s] are interior, the distortion
increases as the difference in the valuations of the agents [absolute
value of [[theta].sub.A] - [[theta].sub.B]] increases. Also, if the
difference between valuations is sufficiently high ([absolute value of
[[theta].sub.A] - [[theta].sub.B]] > 1/2), then the agent with lowest
valuation enjoys the good at his favorite location.
[FIGURE 2 OMITTED]
Last, note that it is sometimes not possible to commit ex ante to
not supply the good. For example, the organizer of sports events may
have difficulties in shutting down the competition. In that case, we
have the following result.
Corollary 1. If the principal cannot commit to not supply the good,
then the allocation is ex post efficient and the good is located at
[x.sub.s].
In Proposition 2, the principal compares the surplus she extracts
at each possible location x and selects the location that provides the
highest payoff [x.sub.s] = arg [max.sub.x]
[[PHI].sub.A]([[theta].sub.A], x) + [[PHI].sub.B]([[theta].sub.B], x)
provided this payoff is positive. Then, the good is not produced when
[[PHI].sub.A] ([[theta].sub.A], [x.sub.s]) +
[[PHI].sub.B]([[theta].sub.B], [x.sub.s]) < 0. If she cannot commit
not to supply the good, it is located at [x.sub.s] even when
[[PHI].sub.A]([[theta].sub.A], [x.sub.s]) +
[[PHI].sub.B]([[theta].sub.B], [x.sub.s]) < 0. The inability to
commit does not affect the optimal location. The allocation is
suboptimal ex ante, but it is expost efficient.
Remark 2. The analysis extends to situations with more than two
agents. Suppose that there are m agents, indexed by k [member of] {1, 2,
..., m}. Agent k is located at [y.sub.k] [member of] [0, N], and let
[[pi].sub.k]([[theta].sub.k], x) [equivalent to] [pi]([[theta].sub.k] -
c[absolute value of x - [y.sub.k]) be the payoff of agent k when the
good is located at x. For each vector of valuations [theta] =
([[theta].sub.1], [[theta].sub.2] ..., [[theta].sub.m]), the optimal
location is [x.sub.M] = arg [max.sub.x] = [[summation].sup.m.sub.k=1]
[[PHI].sub.k]([[theta].sub.k], x) and the optimal contract is such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]([theta]) = l
[[summation].sup.m.sub.k=1] [[pi].sub.k]([[theta].sub.k], [x.sub.M])
> 0 and [p.sub.[phi]([[theta]) = 1 otherwise. Production occurs if
[theta] is above a given level. At location x, agent i faces the reserve
price [r.sub.i]([[theta].sub.i], x), where [[theta].sub.-i] is the
vector of valuations of other agents and [[PHI].sub.i] ([r.sub.i]
([[theta].sub.-i], x) + [[summation].sub.k[not equal to]i]
[[PHI].sub.k]([[theta].sub.k], x) = 0. The seller finds it profitable to
produce at location x if the configuration of valuations is such that
[[theta].sub.i] > [r.sub.i]([[theta].sub.i], x) for all
[[theta].sub.i]. For each [theta] such that it is profitable to produce
at some x, she picks the location [x.sub.M] that gives her the highest
payoff. If agents are evenly distributed on the Hotelling line, then,
other things being equal, the good is located where agents have the
lowest valuations.
4. Optimal location with unknown transportation costs
* In some applications, the principal is not fully informed about
the preferences for characteristics of the agents. Once a niche in a
market is identified, it is sometimes relatively easy to determine the
intrinsic demand for the new good but less obvious to know the
preferences for characteristics. Suppose, for instance, that an
entrepreneur wants to build a child-care center. The intrinsic demand
[[theta].sub.i] represents the number of parents interested in the
service and the horizontal differentiation parameter captures their
preferences for extra care in the evening. It is difficult for the
entrepreneur to know how flexible each parent is when she designs her
service.
In this section, we analyze how the efficient location is affected
when agents possess private information on the horizontal dimension and
we assume that the transportation cost of each agent is unknown to the
principal. Formally, agent i's transportation cost is [c.sub.i]
[member of] [c, [bar.c]] with [c.bar][greater than or equal to] 0.
Transportation costs are independently drawn from a common knowledge
distribution H([c.sub.i]) with continuous and strictly positive density
h([c.sub.i]) and satisfying the monotone hazard rate property
d[[H.sub.(c)/h(c)]]/dc > 0. To concentrate on the effect of
asymmetric information on the horizontal dimension, we assume
[[theta].sub.A] = [[theta].sub.B] = [theta]. Agent i's payoff is
w([c.sub.i], [[gamma].sub.i]) = [pi]([theta] -
[c.sub.i][gamma].sub.i]).
The payoff is decreasing in the transportation cost ([partial
derivative]/[[partial derivative][c.sub.i] < 0) and high- cost agents
are relatively more sensitive to distance ([[partial
derivative].sup.2]w/[partial derivative][c.sub.i] [partial
derivative][[gamma].sub.i] <0). From the perspective of a
revenue-maximizing principal, the first-best location [X.sup.H.sub.F] is
such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (7)
Lemma 3. The optimal location is [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII]. It is decreasing in [c.sub.A] and increasing in
[c.sub.B].
Again, the principal always produces the good under complete
information. Also, she prefers to favor the agent with the highest
transportation cost, and the optimal location decreases with [c.sub.A]
and increases with [c.sub.B]. The increase in revenue extracted from the
agent with the highest transportation cost by moving the optimal
location closer to him offsets the loss of revenue from the agent with
the smallest transportation cost. Overall, [MATHEMATICAL EXPRESSION NOT
REPRODUCIBLE IN ASCII].
Under complete information, the same logic applies when the
transportation costs or the valuations are unknown. In both cases, the
agent with the lowest overall payoff (the lowest valuation
[[theta].sub.i] in Section 3 and the highest transportation cost
[c.sub.i] in the present section) is favored.
Under asymmetric information, the principal offers a menu of
contracts contingent on the reports [[??].sub.A] and [[??].sub.B]. Each
contract specifies probabilities [p.sub.x]([[??].sub.A], [[??].sub.B])
of producing the good at location x and transfers
[t.sub.i]([[??].sub.A], [[??].sub.B]) from agent i to the principal. As
in Section 3, the mechanism is optimal if it maximizes the revenue of
the principal under the incentive compatibility, the individual
rationality, and the feasibility constraints. For notational
convenience, let [[pi].sub.A]([c.sub.A], X) = [pi]([theta] - [c.sub.A]X)
and [[pi].sub.B]([c.sub.B], X) = [pi]([theta] - [c.sub.B]N - x)). The
counterpart of Lemma 2 is as follows.
Lemma 4. The optimal mechanism solves the following program
[P.sub.H]:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (M)
[p.sub.x]([c.sub.A], [c.sub.B])[greater than or equal to] 0 [for
all] x and [N.summation over (x=0)][p.sub.x]([c.sub.A], [c.sub.B])[less
than or equal to] 1 (F)
[[PSI].sub.A]([c.sub.A], x) = [[pi].sub.A]([c.sub.A], x) +
[[partial derivative][[pi].sub.B]([c.sub.B], x)/[partial
derivative][c.sub.B]/][H([c.sub.A]/[h.([c.sub.A]] (8)
[[PSI].sub.B]([c.sub.B], x) = [[pi].sub.B]([c.sub.B], x) + [partial
derivative][[pi].sub.B]([c.sub.B], x)/[partial
derivative][c.sub.B]][H(c.sub.B]/h([c.sub.B]. (9)
In the optimal contract, the expected utility of agent i = {A, B}
is
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
Under asymmetric information, low-cost agents can mimic high-cost
agents and the equilibrium utility must be decreasing in the
transportation cost. The main difference with Lemma 2 is that by
choosing a location, the seller implicitly chooses the degree of
asymmetric information with each agent. In particular, if the seller
decides to locate the good at x = 0, there is no asymmetric information
between the seller and agent A. In that case, the surplus of the agent
is [[pi].sub.A]([c.sub.A], 0) = [pi]([theta]); it is known and can be
fully extracted. More generally, the surplus that can be extracted by
the principal from agents A and B at location x are, respectively,
[PSI]([c.sub.A], x) and [psi]([c.sub.B], x), where the distortions due
to asymmetric information (second term in equations (8) and (9)) are
proportional to the distance (formally, [partial
derivative][[pi].sub.A]/[partial derivative][c.sub.A] = - x [pi]'
([theta] - [c.sub.A]x) and [partial derivative][[pi].sub.B]/[partial
derivative][[pi].sub.B]/[partial derivative][c.sub.B] = -(N
-x)[pi]' ([theta] - (N - x)[c.sub.B])). The two surpluses are
decreasing in the transportation cost ([partial
derivative][[PSI].sub.A]/[partial derivative][c.sub.A] < 0 and
[partial derivative][[PSI].sub.B]/[partial derivative][c.sub.B] < 0).
Let [x.sub.H.sub.S] be the optimal second-best location,
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (10)
Proposition 3. The optimal contract is such that
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],
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