# Non-Constant Elasticity of Substitution

What's the literature on Non-Constant elasticities of substitution? Say, I'm interested in the elasticity between $c_1$ and $c_2$ increasing/decreasing in income/wealth.

CES utility functions with equal spending weights are similar to

$$\left(\sum_i c_i^\frac{\epsilon-1}{\epsilon}\right)^\frac{\epsilon}{1-\epsilon}$$

A simple way of getting non-constant elasticities would be to let $\epsilon = \epsilon(Y)$. Then the elasticity of substitution would vary with income:

$$\frac{d \log \frac{c_i}{c_j}}{d \log \frac{p_j}{p_i}} = \epsilon(Y)$$

But I feel that this mixture of preference-parameters and outcome variables is suboptimal.

Is there a common way of modeling these preferences?

• I don't think there is much written about this. Usually the challange is to show that there is CES. But in general CES only holds for 2 goods. With more than 2 goods there CES cannot exist between all of the goods. Commented Apr 12, 2015 at 6:54
• @RudFaden even that argument is new to me, given that preferences -- as I lied them down - appear symmetric. Would you have a reference? Commented Apr 12, 2015 at 8:54
• Try this. Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities) Commented Apr 12, 2015 at 9:28

They allow the elastisity of subsititution $$\frac{u^{\prime}(x_i)}{x_iu^{\prime\prime}(x_i)}$$, constant in the CES case, to be a function of the average level of consumption $$x$$ and make the model Dixit-Stiglitz model of monopolistic competition shine with new colors.