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

<p>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?</p>

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

<p>Thanks, </p>

CC BY-SA 3.0
2016-11-07T18:25:55.213Z
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 $C$.
optimal control
CC BY-SA 3.0
2016-11-07T18:45:22.060Z
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
CC BY-SA 3.0
2016-11-07T20:46:56.557Z
Are you talking about risk aversion over $c_i$ or over $C_t$? Is there any restriction on the value of $\eta$?
Herr K.
CC BY-SA 3.0
2016-11-08T00:44:57.563Z
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.
econovan
<dynamic-optimization><elasticity-of-substitution>
CES utility in dynamic setting
econovan
# CES utility in dynamic setting

(Asked by: username_0 on 2016-11-07)

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,

**Tags:** dynamic-optimization, elasticity-of-substitution

> **Comments:**
> **username_1**: 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 $C$.  
> **username_2**: I don't think this is CES utility. Wouldn't CES utility be: $$ U(c_1, c_2) = (c_1^{\eta} +c_2^{\eta})^{\frac{1}{\eta}}$$  
> **username_3**: Are you talking about risk aversion over $c_i$ or over $C_t$? Is there any restriction on the value of $\eta$?  
> **username_0**: 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.  

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## 0 answers

