# CES utility in dynamic setting

Suppose I have a multiperiod consumption-saving problem with two or multiple goods able to be consumed.

If the utility within a period is Constant Elasticity of Substitution, ie. $C = (c_1^\frac{\eta-1}{\eta} + c_2^\frac{\eta-1}{\eta})^\frac{\eta}{\eta-1}$ and utility across periods is $U = \sum_{t=0}^\infty C_t$, do these preferences correspond to risk neutral preferences?

Clearly if there was only one good and utility was $\sum C_t$ but I am just not sure with two goods.

Thanks,

• I think it is risk neutral as you you use a linear utility. After all, you aggregate the different consumption good and find a homogenous $C$. Nov 7 '16 at 18:25
• I don't think this is CES utility. Wouldn't CES utility be: $$U(c_1, c_2) = (c_1^{\eta} +c_2^{\eta})^{\frac{1}{\eta}}$$
– BKay
Nov 7 '16 at 18:45
• Are you talking about risk aversion over $c_i$ or over $C_t$? Is there any restriction on the value of $\eta$? Nov 7 '16 at 20:46
• I must have misplaced the outer exponent. The right hand side of the first equation should be raised to \eta/(eta-1). The value of \eta lies between zero and infinity. With standard CES, a value of zero corresponds to perfect complements and as \eta approaches infinity we get perfect substitutes. After thinking about this some more, I have almost convinced myself that it requires \eta approaches infinity. Nov 8 '16 at 0:44