Stochastic intertemporal utility maximization and the consumption function

The consumer's stochastic intertemporal utility maximization problem is studied in the case where all state variables are given exogenously as markov ito processes. It is shown that the feedback response function for optimal consumption can be conveniently expressed implicitly by writing consumption and wealth as functions of the marginal utility of money. These functions are shown to be related by a linear partial differential operator under the assumption of the existence of a feasible risk free portfolio. This linearity permits the development of a wide class of flexible functional forms for a consumption function which is consistent with stochastic intertemporal utility maximization (a).