<p>To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on <a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="noreferrer">constant elasticity of substitution</a>. In particular,</p>
<blockquote>
<p>The CES aggregator is also sometimes called the Armington aggregator, which was discussed by <a href="https://www.jstor.org/stable/3866403?seq=1" rel="noreferrer">Armington (1969)</a>.</p>
</blockquote>
<p>Then, the CES utility function was popularized by <a href="https://www.jstor.org/stable/1831401?seq=1" rel="noreferrer">Dixit and Stiglitz (1977)</a> in their study of optimal product diversity in a context of monopolistic competition.</p>
<p>If you want to understand how the CES utility function behaves when <span class="math-container">$\sigma=\infty$</span> or <span class="math-container">$\sigma=1$</span> <a href="https://www.uio.no/studier/emner/sv/oekonomi/ECON4415/h11/undervisningsmateriale/Lecture8revised3.pdf" rel="noreferrer">here</a> is a nice discussion of the basics of the CES utility function, which is widely used in trade.</p>
<p>If you want a deeper understanding of the elasticity of substitution and a discussion of the special class for which the elasticity of substitution is a constant <span class="math-container">$\sigma$</span>, <a href="https://econ.ucsb.edu/%7Etedb/Courses/GraduateTheoryUCSB/elasticity%20of%20substitutionrevised.tex.pdf" rel="noreferrer">here</a> is an insightful discussion by Ted Bergstrom.</p>

CC BY-SA 4.0
2020-10-31T09:36:35.480Z
2020-10-31T16:40:02.577Z
2020-10-31T16:40:02.577Z
emeryville
<p>The C.E.S functional has been introduced in Economics in the context of production theory, by</p>
<p><strong>Arrow, K. J., Chenery, H. B., Minhas, B. S., &amp; Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The review of Economics and Statistics, 225-250.</strong></p>
<p>There you can find a discussion of how it was derived.</p>
<p>A more pedagogic and detailed treatment can be found in</p>
<p><strong>Silberberg, E. (1990). The structure of economics; A mathematical analysis (International Edition), pp. 285-297</strong></p>

CC BY-SA 4.0
2020-10-31T22:50:58.187Z
2020-10-31T22:50:58.187Z
Alecos Papadopoulos
<p>Is there any book/papers that I can refer to the proof (derivation) of the CES utility function?</p>
<p>Or if anyone could help me with the derivation, I will be so much grateful to you.</p>

CC BY-SA 4.0
2020-10-31T04:48:38.433Z
What do you mean by "derivation"? Finding the derivative? Or obtaining the CES function from some fundamental conditions?
Herr K.
CC BY-SA 4.0
2020-10-31T15:39:05.553Z
@HerrK. I wanted to understand how people came up with the CES utility function. Not the derivatives, derivatives are very easy to find.
stochastic learner
CC BY-SA 4.0
2020-11-07T11:22:42.753Z
Did any of the excellent answers bellow happened to answer your Q? If so consider accepting one of them (ticking the check mark below up/downvote) - it improves site stats and gives you some points as well
1muflon1
<macroeconomics><microeconomics><utility><cobb-douglas><elasticity-of-substitution>
How was CES utility function derived?
stochastic learner
# How was CES utility function derived?

(Asked by: username_0 on 2020-10-31)

Is there any book/papers that I can refer to the proof (derivation) of the CES utility function?

Or if anyone could help me with the derivation, I will be so much grateful to you.

**Tags:** macroeconomics, microeconomics, utility, cobb-douglas, elasticity-of-substitution
> **Comments:**
> **username_1**: What do you mean by "derivation"? Finding the derivative? Or obtaining the CES function from some fundamental conditions?  
> **username_0**: @HerrK. I wanted to understand how people came up with the CES utility function. Not the derivatives, derivatives are very easy to find.  
> **username_2**: Did any of the excellent answers bellow happened to answer your Q? If so consider accepting one of them (ticking the check mark below up/downvote) - it improves site stats and gives you some points as well  

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## 2 answers

### Answer by username_3 on 2020-10-31. Score: 7

To understand the CES utility functions, which I guess is your question, a good starting point is the Wikipedia page on [constant elasticity of substitution](https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution). In particular,

> The CES aggregator is also sometimes called the Armington aggregator, which was discussed by [Armington (1969)](https://www.jstor.org/stable/3866403?seq=1).

Then, the CES utility function was popularized by [Dixit and Stiglitz (1977)](https://www.jstor.org/stable/1831401?seq=1) in their study of optimal product diversity in a context of monopolistic competition.

If you want to understand how the CES utility function behaves when $\sigma=\infty$ or $\sigma=1$ [here](https://www.uio.no/studier/emner/sv/oekonomi/ECON4415/h11/undervisningsmateriale/Lecture8revised3.pdf) is a nice discussion of the basics of the CES utility function, which is widely used in trade.

If you want a deeper understanding of the elasticity of substitution and a discussion of the special class for which the elasticity of substitution is a constant $\sigma$, [here](https://econ.ucsb.edu/%7Etedb/Courses/GraduateTheoryUCSB/elasticity%20of%20substitutionrevised.tex.pdf) is an insightful discussion by Ted Bergstrom.


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### Answer by username_4 on 2020-10-31. Score: 6

The C.E.S functional has been introduced in Economics in the context of production theory, by

**Arrow, K. J., Chenery, H. B., Minhas, B. S., & Solow, R. M. (1961). Capital-labor substitution and economic efficiency. The review of Economics and Statistics, 225-250.**

There you can find a discussion of how it was derived.

A more pedagogic and detailed treatment can be found in

**Silberberg, E. (1990). The structure of economics; A mathematical analysis (International Edition), pp. 285-297**


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