Solving introductory microeconomics problems I have come across the following type of utility function: $$ f(K,L) = (\alpha K^{\frac{\sigma - 1}{\sigma}} + (1 - \alpha) L^{\frac{\sigma - 1}{\sigma}})^{\frac{\sigma}{\sigma - 1}} $$

I find it slightly reminiscent of the logarithm version of the Cobb-Douglas function, but clearly the exponents don't fit with that. So the question is: How would you interpret $\alpha$ and $\sigma$ in this case?. Is $\alpha$ still the relative fraction of capital and labour? How can I think about $\sigma$?

  • 1
    $\begingroup$ This does not seem to be a utility function, but a production function. $\endgroup$
    – Giskard
    Sep 5, 2018 at 9:12
  • $\begingroup$ What do you mean by "Is $\alpha$ still the relative fraction of capital and labour"? Even in the Cobb-Douglas function $\alpha \ln K + (1 - \alpha) \ln L$ the parameter does not show the ratio $K/L$. $\endgroup$
    – Giskard
    Sep 5, 2018 at 9:13
  • $\begingroup$ @denesp the question I have come across explicitly states that this is a utility function and that $\alpha$ and $\sigma$ should be interpreted from that point of view. However, apologies for the faulty description of "relative fraction". What I meant was $L/F$ and $K/F$. $\endgroup$
    – Jhonny
    Sep 5, 2018 at 10:02
  • $\begingroup$ @denesp So is Cobb-Douglas -but it is used also to model utility. $\endgroup$ Sep 7, 2018 at 0:00
  • $\begingroup$ @AlecosPapadopoulos My comment was on the notations $f$, $K$ and $L$. $\endgroup$
    – Giskard
    Sep 7, 2018 at 5:58

1 Answer 1


This is the CES production function, where CES stands for constant elasticity of substitution.

The parameter $\sigma$ captures the (constant) elasticity of substitution and $\alpha$ is the share parameter.

The Cobb-Douglas production function can be obtained as a special case of the CES class by taking $\sigma\to1$. For proof, I'd refer you to this post.


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