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## JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 102, 211-219 (1984) On the Arrow-Lind Theorem

### Abstract

This paper presents an extension of the Arrow-Lind theorem [ 1] on the asymptotic value of an uncertain public project when the benefits and costs of the project are shared among a large number of agents. The theorem has its origin in the following problem: how should such a project be valued in the absence of a complete set of contingent markets in the private sector? I adopt the Arrow-Lind approach in basing the value of the project on the sum of the individual agents ’ valuations. Arrow and Lind value a project by what I call the aggregate sale value. With their assumption that the IZ agents are identical, this is simply n times the representative agent’s sale value of his share of the project’s return. This concept of value is appropriate when all agents in the economy are identical since all agents are affected equally by the introduction of a public project. But in a world with different types of agents (income-preference pairs) some agents may be favorably affected while others may be adversely affected. In such a framework there are two natural concepts of value, the aggregate sale value and the aggregate purchase value, the latter being the sum of the individual agents ’ purchase values-the most that an agent will pay to purchase a given share of the project’s return. To extend the Arrow-Lind theorem we need to study the behaviour of both these values. In Section 2, I show that these two concepts of value are related to the Kaldor-Hicks criteria for a potential Pareto improvement. In Section 3, I show that when the share of the project held by an agent becomes arbitrarily small the individual purchase and sale values coincide and the idiosyncratic risk associated with the project goes to zero (Proposition 1). After introducing a concept of stochastic dependence for a pair of random variables which extends the concept of independence to a concept of positive or negative dependence (Definition l), I show that the individual sale value is less than (greater than) the expected value of the agent’s share if his incom

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