<p>This is the <a href="https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution" rel="noreferrer">CES production function</a>, where CES stands for constant elasticity of substitution. </p>

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

<p>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 <a href="https://economics.stackexchange.com/q/361/42">this post</a>.</p>

CC BY-SA 4.0
2018-09-05T13:55:21.340Z
2018-09-05T13:55:21.340Z
Herr K.
<p>Solving introductory microeconomics problems I have come across the following type of utility function:
$$ f(K,L) = (\alpha K^{\frac{\sigma - 1}{\sigma}} + (1 - \alpha) L^{\frac{\sigma - 1}{\sigma}})^{\frac{\sigma}{\sigma - 1}} $$</p>

<p>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: <strong>How would you interpret $\alpha$ and $\sigma$ in this case?</strong>. Is $\alpha$ still the relative fraction of capital and labour? How can I think about $\sigma$? </p>

CC BY-SA 4.0
2018-09-05T09:12:21.440Z
This does not seem to be a utility function, but a production function.
Giskard
CC BY-SA 4.0
2018-09-05T09:13:26.143Z
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
CC BY-SA 4.0
2018-09-05T10:02:32.500Z
@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
CC BY-SA 4.0
2018-09-07T00:00:21.660Z
@denesp So is Cobb-Douglas -but it is used also to model utility.
Alecos Papadopoulos
CC BY-SA 4.0
2018-09-07T05:58:53.933Z
@AlecosPapadopoulos My comment was on the notations $f$, $K$ and $L$.
Giskard
<microeconomics><utility><cobb-douglas>
Interpretation of Interesting Utility Function
Jhonny
# Interpretation of Interesting Utility Function

(Asked by: username_0 on 2018-09-05)

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 $\sigma$?

**Tags:** microeconomics, utility, cobb-douglas

> **Comments:**
> **username_1**: This does not seem to be a utility function, but a production function.  
> **username_1**: 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 $K/L$.  
> **username_0**: @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 $K/F$.  
> **username_2**: @denesp So is Cobb-Douglas -but it is used also to model utility.  
> **username_1**: @username_2 My comment was on the notations $f$, $K$ and $L$.  

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## 1 answers

### Answer by username_3 on 2018-09-05. Score: 5 | Accepted answer

This is the [CES production function](https://en.wikipedia.org/wiki/Constant_elasticity_of_substitution), 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](https://economics.stackexchange.com/q/361/42).


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