# Interpretation of Interesting Utility Function

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 $\sigma$?

• This does not seem to be a utility function, but a production function. – Giskard Sep 5 '18 at 9:12
• 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 $K/L$. – Giskard Sep 5 '18 at 9:13
• @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 $K/F$. – Jhonny Sep 5 '18 at 10:02
• @denesp So is Cobb-Douglas -but it is used also to model utility. – Alecos Papadopoulos Sep 7 '18 at 0:00
• @AlecosPapadopoulos My comment was on the notations $f$, $K$ and $L$. – Giskard Sep 7 '18 at 5:58

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.