Non-Constant Elasticity of Substitution

Question

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?

Answer 1

There's a great Econometrica paper by Zhelobodko, Kokovin, Parenti and Thisse from 2012:

• "Monopolistic Competition: Beyond the Constant Elasticity of Substitution"

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.

Abstract:

We propose a model of monopolistic competition with additive preferences andvariable marginal costs. Using the concept of “relative love for variety,” we providea full characterization of the free-entry equilibrium. When the relative love for varietyincreases with individual consumption, the market generates pro-competitive effects.When it decreases, the market mimics anti-competitive behavior. The constant elastic-ity of substitution is the only case in which all competitive effects are washed out. Wealso show that our results hold true when the economy involves several sectors, firmsare heterogeneous, and preferences are given by the quadratic utility and the translog.

Answer 2

Eichenbaum and Fisher (2005) follow Kimball (1995) in allowing for the possibility that the elasticity of demand is increasing in a firm’s price. Which is not exactly what I was looking for, but it's a start.