When is a Risky Asset "Urgently Needed"?

Abstract

The demand for commodities in standard applications typically is increasing in income, whereas the demand for the risk free asset in the classic portfolio problem often decreases with income. The latter is shown to occur if and only if the consumer is uncertainty preferences over assets satisfy the condition that the risk free asset is more readily substituted for the risky asset as the quantity of the risky asset increases. In this case, the risky asset is said to be "urgently needed" following the terminology of Johnson in his classic 1913 certainty analysis [19]. The asset and certainty settings differ in critical ways which result in a much greater likelihood for the urgently needed preference property to be satisfied in the portfolio problem. We provide several sufficient conditions for when the risky asset will be urgently needed and a surprisingly simple, complete characterization for widely popular members of the HARA (hyperbolic absolute risk aversion) class. For more general preferences, two examples are given where it is possible to fully describe the region of asset space in which the risky asset is urgently needed. Finally, using a standard representative agent model we show that the risky asset being urgently needed is equivalent to the equilibrium (relative) price of the risky asset increasing with its own supply.