<p>There's a great Econometrica paper by Zhelobodko, Kokovin, Parenti and Thisse from 2012:</p>
<ul>
<li><a href="https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA9986" rel="nofollow noreferrer">&quot;Monopolistic Competition: Beyond the Constant Elasticity of Substitution&quot;</a></li>
</ul>
<p>They allow the elastisity of subsititution <span class="math-container">$\frac{u^{\prime}(x_i)}{x_iu^{\prime\prime}(x_i)}$</span>, constant in the CES case, to be a function of the average level of consumption <span class="math-container">$x$</span> and make the model Dixit-Stiglitz model of monopolistic competition shine with new colors.</p>
<p>Abstract:</p>
<blockquote>
<p>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.</p>
</blockquote>

CC BY-SA 4.0
2020-12-20T17:52:05.810Z
2020-12-20T18:03:16.400Z
2020-12-20T18:03:16.400Z
Konstantin
<p><a href="http://faculty.wcas.northwestern.edu/~yona/research/CalvoMarch162005.pdf" rel="nofollow">Eichenbaum and Fisher (2005)</a> 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.</p>

CC BY-SA 3.0
2015-04-17T14:21:57.713Z
2015-04-17T14:21:57.713Z
FooBar
<p>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. </p>

<p>CES utility functions with equal spending weights are similar to</p>

<p>$$ \left(\sum_i c_i^\frac{\epsilon-1}{\epsilon}\right)^\frac{\epsilon}{1-\epsilon}$$</p>

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

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

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

<p>Is there a common way of modeling these preferences? </p>

CC BY-SA 3.0
2015-04-12T06:54:35.163Z
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.
Rud Faden
CC BY-SA 3.0
2015-04-12T08:54:36.350Z
@RudFaden even that argument is new to me, given that preferences -- as I lied them down - appear symmetric. Would you have a reference?
FooBar
CC BY-SA 3.0
2015-04-12T09:28:52.760Z
Try this. Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities)
Rud Faden
<microeconomics><utility><preferences>
Non-Constant Elasticity of Substitution
FooBar
# Non-Constant Elasticity of Substitution

(Asked by: username_0 on 2015-04-11)

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?

**Tags:** microeconomics, utility, preferences
> **Comments:**
> **username_1**: 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.  
> **username_0**: @RudFaden even that argument is new to me, given that preferences -- as I lied them down - appear symmetric. Would you have a reference?  
> **username_1**: Try this. Will the Real Elasticity of Substitution Please Stand Up? (A Comparison of the Allen/Uzawa and Morishima Elasticities)  

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## 2 answers

### Answer by username_2 on 2020-12-20. Score: 3

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

* ["Monopolistic Competition: Beyond the Constant Elasticity of Substitution"](https://onlinelibrary.wiley.com/doi/abs/10.3982/ECTA9986)

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.


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### Answer by username_0 on 2015-04-17. Score: 2

[Eichenbaum and Fisher (2005)](http://faculty.wcas.northwestern.edu/~yona/research/CalvoMarch162005.pdf) 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.


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