# How was CES utility function derived?

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.

• What do you mean by "derivation"? Finding the derivative? Or obtaining the CES function from some fundamental conditions? – Herr K. Oct 31 '20 at 4:48
• @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 Oct 31 '20 at 15:39
• 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 Nov 7 '20 at 11:22

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. In particular,

The CES aggregator is also sometimes called the Armington aggregator, which was discussed by Armington (1969).

Then, the CES utility function was popularized by Dixit and Stiglitz (1977) 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 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 is an insightful discussion by Ted Bergstrom.

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