The Demand for a Risky Asset

We investigate the effect on the demand for a risky asset when there are changes in either initial wealth or one of the asset return distributions in two asset expected utility portfolio problems. In the choice between a risky and a safe asset necessary and sufficient conditions are presented for first, second and third degree stochastic dominance shifts to yield increasing demand for the risky asset. The conditions may be illustrated graphically and used to demonstrate that many common utility functions fail to satisfy the increasing demand property. The results extend and generalize the analysis in Fishburn and Porter (Fishburn, P. C., R. B. Porter. 1976. Optimal portfolios with one safe and one risky asset: effects of changes in rate of return and risk. Management Sci.22 1064–1073.). In the two risky asset case sufficient conditions are presented for there to be an increase in the demand for one of the risky assets if either initial wealth is increased or this asset has a first degree stochastic dominance shift. The conditions, which build upon and extend earlier results of Arrow (Arrow, K. J. 1971. Essays in the Theory of Risk Bearing. Markham, Chicago, Ill.) and Cass and Stiglitz (Cass, D., J. E. Stiglitz. 1972. Risk aversion and wealth effects on portfolios with many assets. Rev. Econom. Studies39 331–354.), may be used to determine which utility functions display the increasing demand properties.