A Note on the Arrow-Lind Theorem

discount rate applied to expected returns. It would also argue in favor of state participation in private investments where this allows risks to be spread over a larger number of persons. A review of the explicit assumptions alone must cast doubt on the general validity of such applications. The assumption of independence is unrealistic for many investments, for example in infrastructure and "basic" industries whose returns are highly correlated with national income. The assumption that the share of the net benefits of an investment accruing to any person becomes negligible as population tends to infinity is unacceptable in at least three cases: for public goods, where the benefit is not "shared" but increases with the population; for projects whose scale must be adjusted roughly in proportion to the size of population (such as the construction of a grid system of electricity distribution); and for projects whose benefits accrue wholly or in part to a section of the population which is "small" in the sense of the theorem. The last reservation applies not merely to those projects which are specifically designed to benefit only a small part of the population, but also to those special benefits and costs from any project which happen to accrue unavoidably to limited groups. Arrow and Lind avoid this problem in their formal discussion by assuming that the government taxes all benefits and compensates all losses, although they acknowledge that this is unrealistic. Be that as it may, the present note accepts the Arrow-Lind approach more or less on its own terms, and considers more fully the role of certain implicit assumptions concerning the fiscal system and public expenditure. Specifically, it will be recalled that Arrow and Lind work with only two random variables, the disposable income of a typical individual and the income from distribution of project retums by the government. Although the latter is referred to in