Optimal choice of characteristics for a nonexcludable good.

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.