374 MERTON where U is the instantaneous utility function, C is consumption, and e is the relative or constant absolute risk-aversion utility function, explicit solu- tions for the optimal consumption and portfolio rules were derived. The changes in these optimal rules with respect to shifts in various parameters such as expected return, interest rates, and risk were examined by the technique of comparative statics The present paper extends these results for more general utility functions, price behavior assumptions, and for income generated also from non capital gains sources. It is shown that if the"geometric Brownian motion hypothesis is accepted, then a general“ Separation'”or“ mutual fund theorem can be proved such that, in this model, the classical Tobin mean ariance rules hold without the objectionable assumptions of quadratic utility or of normality of distributions for prices. Hence, when asset prices are generated by a geometric Brownian motion, one can work with he two-asset case without loss of generality. If the further assumption is made that the utility function of the individual is a member of the family of utility functions called the"HARA"family, explicit solutions for the optimal consumption and portfolio rules are derived and a number of theorems proved. In the last parts of the paper, the effects on the consumption and portfolio rules of alternative asset price dynamics, in hich changes are neither stationary nor independent, are examined along with the effects of introducing wage income, uncertainty of life expectancy,and the possibility of default on(formerly)"risk-free 2. A DIGRESSION ON ITO PROCESSES To apply the dynamic programming technique in a continuous-time model, the state variable dynamics must be expressible as Marko tochastic processes defined over time intervals of length h, no matter how small h is. Such processes are referred to as infinitely divisible time. The two processes of this type? are: functions of Gauss-Wiener Brownian motions which are continuous in the"space"variables and functions of Poisson processes which are discrete in the space variables Because neither of these processes is differentiable in the usual sense, a more general type of differential equation must be developed to express the dynamics of such processes. a particular class of continuous-time 2 I ignore those infinitely divisible processes with infinite moments which include those members of the stable Paretian family other than the normal.374 MERTON where U is the instantaneous utility function, C is consumption, and E is the expectation operator. Under the additional assumption of a constant relative or constant absolute risk-aversion utility function, explicit solu￾tions for the optimal consumption and portfolio rules were derived. The changes in these optimal rules with respect to shifts in various parameters such as expected return, interest rates, and risk were examined by the technique of comparative statics. The present paper extends these results for more general utility functions, price behavior assumptions, and for income generated also from non￾capital gains sources. It is shown that if the “geometric Brownian motion” hypothesis is accepted, then a general ‘Separation” or “mutual fund” theorem can be proved such that, in this model, the classical Tobin mean￾variance rules hold without the objectionable assumptions of quadratic utility or of normality of distributions for prices. Hence, when asset prices are generated by a geometric Brownian motion, one can work with the two-asset case without loss of generality. If the further assumption is made that the utility function of the individual is a member of the family of utility functions called the “HARA” family, explicit solutions for the optimal consumption and portfolio rules are derived and a number of theorems proved. In the last parts of the paper, the effects on the consumption and portfolio rules of alternative asset price dynamics, in which changes are neither stationary nor independent, are examined along with the effects of introducing wage income, uncertainty of life expectancy, and the possibility of default on (formerly) “risk-free” assets. 2. A DIGRESSION ON 1~6 PROCESSES To apply the dynamic programming technique in a continuous-time model, the state variable dynamics must be expressible as Markov stochastic processes defined over time intervals of length h, no matter how small h is. Such processes are referred to as infinitely divisible in time. The two processes of this type2 are: functions of Gauss-Wiener Brownian motions which are continuous in the “space” variables and functions of Poisson processes which are discrete in the space variables. Because neither of these processes is differentiable in the usual sense, a more general type of differential equation must be developed to express the dynamics of such processes. A particular class of continuous-time 2 I ignore those infinitely divisible processes with infinite moments which include those members of the stable Paretian family other than the normal