Arbitrage Pricing by choosing a consumption withdrawal plan, c, and an optimal portfolio choice x,so as to maximize the expected increment in V; i.e max edv n optimum, consumption will be withdrawn to the point where its marginal equals the marginal utility of wealth The individual portfolio choice will result from the optimization of a locally quadratic form exactly as in the static CAPM theory with the additional feature that covariances of the change in wealth, du, with the changes in state variables, ds, will now be influenced by portfolio choice and will, in general, alter the optimal portfolio. By solving this optimization problem and using the marginal utility condition, u(c)=ve, the individual equilibrium sets factor risk premia E-Eo=(R/c)(ac/as)03; (wvu)/Vu, the individual coefficient of relative risk aver a, is the local variance of (independent) factor S,.(The interested reader referred to Cox, Ingersoll and Ross [8]for details. Notice that the premia E Eo can be negative if consumption moves counter to the state variable. In this case portfolios which bear positive factor s risk hedge against adverse movements in consumption, but too much can be made of this, since by simply redefining s to be-g the sign can be reversed. The sign, therefore, is somewhat arbitrary and we will assume it is normalized to be positive. Aggregating over individuals yields (3) One special case of particular interest occurs when state dependencies can be ignored. In the log case, R=l, for example, or any case with a relative wealth criteria(see Ross [48])the risk premia take the special form E-Eo=R(∑xb)02 where x is the individual optimal portfolio. This form emphasizes the general relationship between b, and o,. Normalizing 2x,by to unity by scaling s, we E-Eo= Roj The risk premium of factor j is proportional to its variance and the constant of proportionality is a measure of relative risk aversion For other utility functions, individual consumption vectors can be expressed in erms of portfolios of returns and similar expressions can be obtained. In effect, nce the weighted state consumption elasticities for all individuals satisfy the aPt pricing relationships, they must all be proportional 2 Breeden [5] has developed the observation that homogenous beliefs about Es and bs imply perfect correlation between individual random consumption changes. His results depend on th assumption, made also by aPt, that k<N