On the Relationship between Complete and Incomplete Financial Market Models

Abstract

A number of recent papers2 have dealt with models of firms under uncertainty in which the securities of the firms are traded in an incomplete security market. Although the securities market is incomplete in the sense that the number of independent securities is less than the number of states of nature, the results obtained are similar to those derived in an Arrow [1964]-Debreu [1959] complete market model. In these incomplete market models shareholders unanimously support or oppose any proposed production plan, and given a pricetaking assumption, all shareholders prefer that a firm maximize its market value. In spite of the similarity in results, the relationship between the complete and incomplete market models must be interpreted with care. Incomplete financial market models have been analyzed under two different classes of assumptions. Leland [1974b], Ekern and Wilson [1974], and Ekern [1973, 1974, 1975] have analyzed firm behavior under the assumption that investors hold optimal portfolios for the currently proposed production plans of firms. Then, if the vector of marginal returns for a firm is contained in the subspace spanned by the return vectors for all securities traded in the market, a proposed change in the production plan of a firm is unanimously supported or rejected by all shareholders. Leland [1973] has referred to this as ex post unanimity in the sense that the portfolios held by shareholders are optimal for the currently proposed production plans. Ex post unanimity, however, does not answer the question of how investors arrived at a position in which the optimal portfolios are held. In a manner analogous to an Arrow-Debreu model Radner [1974] considered the case in which shareholders do not hold optimal portfolios for the currently proposed production plans of firms. He then demonstrates that all ex ante or initial shareholders prefer the firm to maximize its value and also unanimously support or reject any proposed production plan when the consumption sets of