In this model, a principal decides whether to produce one
indivisible good and which characteristics it contains. Agents are
differentiated along two substitutable dimensions: a vertical parameter
that captures their valuation for the good, and a horizontal parameter
that captures their disutility when the characteristics are distant from
their preferred ones. When valuations are private information, the
principal produces a good with characteristics more on the lines of the
preferences of the agent with the lowest valuation. Under asymmetric
information on the horizontal dimension, the principal biases the
decision in favor of the agent who incurs the highest disutility.
1. Introduction
* Consider the following problem. A principal needs to decide
whether to produce one indivisible good and, if she does, which
characteristics it contains. Production affects positively the utility
of two agents who are differentiated along two dimensions: a vertical
parameter captures their valuations for the good and a horizontal
parameter captures their differences in preferences for the
characteristics of the good. For example, a privately interested
investor is deciding whether to construct a football stadium and where
to locate it. Two neighboring cities are interested in the project. The
vertical differentiation parameter is the cities' demand for
football. The horizontal differentiation is the physical location of the
stadium. Each city prefers to have the stadium located between the two
cities or even in the neighboring city rather than no stadium at all.
Naturally, horizontal differentiation can also account for differences
in tastes.
It seems natural to conclude that the stadium should be built in
the city that values it the most. Interestingly, this is not always the
case in practice. In this article, we determine the conditions under
which this can occur. More generally, we characterize the optimal
contract between the principal and the agents when there is asymmetric
information either on the vertical dimension or on the horizontal
dimension. We assume in the first case that the intrinsic valuations for
the good are unknown. In the second, we suppose that agents face
privately known "transportation costs" for each unit of
distance between their preferred characteristics and the actual
characteristics of the good. Optimal contracting with asymmetric
information and positive externalities has already been studied in the
literature. Yet, previous studies suppose that the characteristics of
the good are given prior to the contracting stage. (1) To the best of
our knowledge, the endogenous choice of characteristics has been
overlooked.
The literature studying contracting problems with positive
externalities can be divided into two branches. The first branch
analyzes situations where the principal contracts separately with
several agents and the contract between the principal and one agent
generates positive externalities on other agents. This problem has been
studied in a general setting in Segal (1999) and Segal and Whinston
(2003). (2) It has also been investigated in specific settings by
Cornelli (1996) and Lockwood (2000), among others. (3) In these
articles, the item to be contracted upon is generally excludable and the
principal can contract only with a subset of agents if she finds it
optimal. In our model, the principal cannot produce one good for each
agent (i.e., she cannot serve both agents separately, specify a
different output level for each of them, or exclude one of them).
Instead, she must determine the characteristics of the good. One could
reinterpret "producing the good most preferred by one agent"
as "selling the good to that agent." Thus, our setting
resembles an auction, where the good can be allocated to one of the
agents but the other one still enjoys a positive utility when this
happens. However, unlike in the auction of an indivisible good, the
principal is not forced to produce a good with the characteristics most
liked by one agent. Instead, she chooses from a wide array of
combinations, ranging from most preferred by one agent to equally liked
by both.
The second branch studies the optimal contract between a principal
and several agents when the item that is contracted upon affects the
payoff of all agents. The literature on the provision of public goods in
the tradition of Clarke (1971), Groves (1973), and D'Aspremont and
Gerard-Varet (1979) discusses mechanisms to implement the socially
optimal level of public good with or without budget balance for the
government. (4) As in the present article, the good is nonexcludable and
all agents benefit from its provision. However, our focus departs in two
respects. First, the situations we have in mind are not necessarily
decisions to produce public goods and we do not impose budget balance.
Instead, we want to ensure that all parties participate and contribute.
Second and more importantly, in these analyses, the principal decides
over the quantity to be produced, and more quantity is always preferred
by agents. By contrast, in our model, the principal decides over an
attribute on which agents disagree, because the best characteristic for
one agent is also the worst for the other. This generates a new tradeoff
for the contract designer.
Last, even though the focus is different from the current analysis,
we should mention the literature on quality. For instance, Che (1993)
studies competition in a procurement environment where agents bid on the
price of a product and its quality. Quality is distorted downward under
asymmetric information in order to diminish the rent left to agents. The
author examines how two-dimensional auctions in which bids (on price and
quality) are evaluated by a scoring rule perform to implement this
second best. Incentives to provide quality have been studied also in
Lewis and Sappington (1988 and 1991).5 The present analysis focuses on
horizontal differentiation instead of quality. As discussed above, the
crucial difference is that agents disagree on the characteristics on the
horizontal dimension although they concur on quality. Also, there are no
externalities between agents in Che (1993), whereas it is a main
ingredient in our analysis. These two differences make the contracting
problem of a very different nature.
The main features of the optimal contract are the following. We
assume that the vertical and horizontal dimensions are substitutable, in
the sense that the marginal importance attached by an agent to the
characteristics of the good decreases as his valuation increases.
Introducing a horizontal dimension generates a qualitative departure
only when this assumption is satisfied. In that case, the principal
always produces the good under full information. Besides, keeping the
transportation cost equal and constant for both agents, she prefers to
favor the agent with lowest valuation, that is, to offer a good with
characteristics more on the lines of his preferences than on the lines
of the preferences of the other agent. Given the substitutability of the
vertical and horizontal differentiation parameters, the loss in the
revenue extracted from the high-valuation agent under this strategy is
smaller than the gain in the revenue extracted from the low-valuation
agent. Alternatively, agents with high transportation costs are
relatively more sensitive to distance. Therefore, keeping the valuation
constant and equal for both agents, it is optimal under full information
to bias the decision in favor of the agent with the highest cost.
Asymmetric information on the vertical dimension induces two
distortions in the optimal contract, one for each agent. In fact,
because production of the good affects the utility of the two agents,
the optimal contract is such that the principal demands payments and
grants informational rents to both of them. Interestingly, under
incomplete information, the principal favors even more the agent with
lowest valuation than under full information. The idea is that the
principal distorts the characteristics of the good offered in order to
reduce the rents left to agents. Due to substitutability of
characteristics and valuation, marginal rents are greatest for the
lowest-valuation agent. Therefore, it is relatively more interesting to
reduce the rents of this agent, which is achieved by selecting
characteristics that are closer to his favorite ones. To sum up,
positive externalities together with the capacity to extract payments
from both agents induces the principal to select a convex combination of
characteristics, with a tendency to favor the agent with lowest
valuation. Asymmetric information exacerbates this bias.
When the principal does not fully observe the preferences on the
horizontal dimension, two opposite effects are at work. First, given
high-cost agents are relatively more sensitive to distance, it is
beneficial to bias the decision in favor of the agent with the highest
cost. Second, if the good possesses the preferred characteristics of one
agent, then that agent does not incur any cost. Then, the principal can
increase or decrease the amount of asymmetric information with each
agent by choosing the characteristics. Given a low-cost agent has
relatively fewer incentives to reveal truthfully his information and
must be granted higher rents, the principal can minimize them by
choosing a characteristic closer to the preferred characteristic of the
agent who turns out to have the lowest cost. Overall, the bias obtained
under complete information can be increased or decreased depending on
which of these two effects dominates.
The plan of the article is the following. The model and the basic
properties of the optimal mechanism are presented in Section 2. We solve
for the case of asymmetric information on the vertical dimension and the
horizontal dimension in Sections 3 and 4, respectively. In Section 5, we
characterize the optimal contract when agents at different locations
also have different distributions of valuations. Moreover, we analyze
situations in which the good can be located at different places over
time. In Section 6, we study the mechanism when the principal maximizes
welfare instead of revenue. Also, we determine the properties of the
contract if one agent is also the producer of the good. Concluding
remarks are collected in Section 7. All proofs can be found in the
Appendix.
2. The model
* Basic ingredients. We consider two agents A and B indexed by i
and j. Each agent ("he") is located at one extreme of a
Hotelling line of measure N. Denoting by [y.sub.i] the location of agent
i, we have [y.sub.A] = 0 and [y.sub.B] = N. An indivisible good can be
produced and located somewhere on the line. (6) We denote by
[[theta].sub.i] agent i's intrinsic valuation for the good (also
referred to as "type") and we assume that [[theta].sub.i]
[member of] [[theta].bar], [[bar.[theta]]]. Valuations are private
information and they are independently drawn from a common knowledge
distribution F([[theta].sub.i]) with continuous and strictly positive
density f([[theta].sub.i]). It also satisfies the monotone hazard rate
property: d[1-F([theta])/f([theta])]/d[theta] < 0. Agents care about
the location x of the good. We assume that x can take a finite but
arbitrarily large number of locations, and we order them from closest to
agent A to closest to agent B: x [member of] {0, 1, ... , N - 1, N}. We
denote by [[gamma].sub.i](= [absolute value of x-[y.sub.i]]) the
distance between the location of the good and the location of agent i.
The payoff of agent i as a function of his valuation and distance takes
the following form:
v([[theta].sub.i],[[gamma].sub.i]) = [pi]([[theta].sub.i] -
c[[gamma].sub.i], (1)
where, following the Hotelling terminology, e is a positive
"transportation cost," [pi]' > 0, [pi]" < 0,
and, for technical convenience, [pi]"' [greater than or equal
to] 0. According to this formalization, the payoff is increasing in the
valuation ([partial derivative]v/[partial derivative][[theta].sub.i]
> 0) and decreasing in the distance ([partial derivative]v/[partial
derivative][[gamma].sub.i] < 0). Moreover, valuation is relatively
more important the bigger the distance between the location of the agent
and the location of the good ([[partial derivative].sup.2]v/[partial
derivative][[theta].sub.i][partial derivative][[gamma].sub.i] > 0).
In other words, high-type agents are relatively less sensitive to
distance. Overall, agents are differentiated along two substitutable
dimensions captured by two parameters, a vertical differentiation
parameter (the valuation for the good) and a horizontal differentiation
parameter (the distance between the good and the agent).
To be in the interesting case, the payoff of each agent when the
good is produced is always greater than the payoff when it is not, which
we normalize to 0 ([pi]([[theta].bar] - cN) > 0). Our setting is
characterized by positive and type-dependent externalities. Each agent
prefers to have the good produced and the payoff of agents increases
with their valuation, independently of x.
We want to determine how the good is optimally located on the
Hotelling line. We assume that the location decision is in the hands of
a third party (from now on "principal" or "she").
Denote by e = [empty set] the event "the principal does not produce
the good" and by e = x [member of] {0, ... , N} the event "the
good is produced and located at x." In order to better concentrate
on the inefficiencies of the allocation due to the asymmetry of
information, we assume that producing the good is costless for the
principal and generates no delay. Also, we concentrate in Section 3 on
the case in which the principal maximizes revenue. This assumption is
relaxed in Section 6.
[] Examples. The purpose of this subsection is to provide a few
examples in which the ingredients of our theory are present and for
which we believe our normative approach can be useful. (7)
Physical location of a nonexcludable private or public good. Agents
A and B are two neighboring cities. The vertical differentiation
parameter [[theta].sub.i] is the intrinsic demand for football of each
city and the horizontal differentiation parameter is the distance
between the city and the stadium. Also, c is simply a transportation
cost. The payoff of each city when the stadium is built increases with
its demand for football ([partial derivative]v/[partial
derivative][[theta].sub.i] > 0) and decreases with the distance
between the city and the stadium ([partial derivative]v/[partial
derivative][[gamma].sub.i] < 0). Keeping c constant, inhabitants of a
city supporting a football team are relatively more inclined to drive to
attend an event ([[partial derivative].sup.2]v/[partial
derivative][[theta].sub.i][partial derivative][[gamma].sub.i] > 0).
Also, each city prefers a stadium located far away rather than no
stadium at all (positive externalities), and the utility of cities
increases with their valuation, independently of the location
(type-dependent externalities). The principal is an investor willing to
build and manage a new stadium, and she maximizes revenue. Or, the
principal is a local authority trying to make the two cities agree to
finance a public stadium. The model can be applied to other decisions to
locate a nonexcludable good such as a shopping mall or a hospital.
Creation of a private school. Agents A and B are two types of
parents. The vertical differentiation parameter [[theta].sub.i] is the
intrinsic willingness to pay for a new private school and the
characteristics of the good is the emphasis of the school on languages
versus sciences. Given our assumptions, the payoff of a group of parents
increases with their valuation for private education. Parents disagree
on the emphasis and the payoff decreases with the distance between the
actual emphasis of the school and the desired emphasis of each type of
parent. The parameter c captures how sensitive parents are to a
departure from their preferred emphasis. Our model corresponds to the
case where parents with a high valuation for the new school are
relatively more willing to compromise on emphasis.
As a special case, the good may be French education, where
[[theta].sub.A] is the valuation of French parents located in a foreign
country for a new French school in that country (i.e., their willingness
to pay to have the same education as in France (8)) and [[theta].sub.B]
is the valuation of local citizens. The horizontal dimension captures,
for instance, the emphasis on mathematics: French parents want to have
the same curriculum as in France, however local citizens want part of
the emphasis on mathematics replaced by local history and geography.
Parents with high valuations are more likely to compromise on the
curriculum because, for instance, there are few good alternatives to
French education in the country considered. Also, the principal is an
investor (9) or a parent willing to offer a personalized education to
his own children and offering this new concept to other parents as well.
(10) This special case is interesting because we observe that most
French schools located in foreign countries do adapt the curriculum to
the preferences of local citizens.
Services offered to club members. The principal is the
administrator of a private golf or tennis club and maximizes revenue or
welfare of club members. The club accepts families (agent A) who enjoy
other activities besides sports (e.g., s