The Existence of Capital Market Equilibrium and Asset Demand Correspondences

98 p.Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 1980.In the vast literature analyzing the structure and properties of capital market equilibrium which has appeared since the development of the Sharpe-Lintner-Mossin model, only one attempt appears to have been made to prove that in fact capital market equilibria exist. Furthermore, it appears that capital market theory has represented by the Sharpe-Lintner-Mossin model and its many variations does not include a well- defined theory of asset demand and asset demand correspondences.In this thesis, we examine in great detail the problem of the existence of general equilibrium in capital markets. Our overall approach to solving this problem is first to establish some of the basic elements of a rigorous theory of asset demand correspondences and then, using the properties of asset demand correspondences, to give simple direct proof of the existence of a capital market equilibrium.We begin by constructing a very general capital market model. In some respects the model we construct is similar to Mossin's 1966 model. In particular, we assume that unlimited short sales are possible and we focus on total asset returns rather than rates of return. Unlike Mossin, however, we assume that investorys probability beliefs are nonhomogeneous and that each investor probability beliefs are dependent on current asset prices. Most importantly, we replace the mean-variance portfolio selection problem with the more general expected utility portfolio problem.Given the usual assumptions concerning the utility function of the risk averse investor, we show that if probability beliefs depend on a continuous way on asset prices, then each investor's expected utility is a continuous function of both asset holdings and current asset prices. We also demonstrate that the investor's budget constraint can be viewed as a set-valued mapping continuously parameterized by current asset prices.Having established the basic properties of the investor's expected utility function and budget constraint set, we then use the theory of recession directions to derive the necessary and sufficient conditions for the existence of nonempty bounded optimal portfolio sets. Given the simplicity of these conditions, we are able to precisely describe the restrictions which must be imposed upon the basic portofolio problem in order to guarantee the existence of nonempty bounded optimal portfolio sets for the widest possible range of price vectors in the standard price simplex (DELTA). We conclude that if the investor is sufficiently risk averse, and if the investor assigns a positive probability to experiencing a loss by holding a portfolio with short positions, then for any vector of asset prices in the relative interior of the price simplex (i.e., in (DELTA)('0) ), the optimal portfolio set is nonempty and bounded.We next show that the optimal portfolio set is an upper semicontinuous function of the vector of relative asset prices on (DELTA)('0). Thus we essentially demonstrate that given the continuity properties of the investor's expected utility function and budget constraint sets, restrictions on investor risk aversion and probability beliefs which guarantee the existence of nonempty bounded optimal portfolio sets for any price vector in (DELTA)('0) , also guarantee the upper semicontinuity of asset demand correspondences on (DELTA) .Finally, we use the results on the existence and upper semicontinuity of investor demand correspondences to prove the existence of a capital market equilibrium. Thus, within the context of a capital market model allowing unlimited short sales and nonhomogeneous price-dependent probability beliefs, we show that if investors are sufficiently risk averse, and if each investor's probability beliefs are a continuous function of current price information, then a capital market equilibrium exists.U of I OnlyRestricted to the U of I community idenfinitely during batch ingest of legacy ETD