<p>I'm answering if there is any production function which is homogeneous and CES too. </p>

<p>How about a CES production function only. </p>

<p><span class="math-container">$y=(x^\rho +y^\rho )^\frac{1}{\rho }$</span> for <span class="math-container">$0\neq \rho &lt;1$</span></p>

<p>This function has Elasticity of Substitution : <span class="math-container">$\sigma =\frac{1}{(1-\rho )}$</span> which is a constant. </p>

<p>Considering homogeneity,</p>

<p><span class="math-container">$ y=((\lambda x)^\rho +(\lambda y)^\rho )^\frac{1}{\rho }$</span></p>

<p><span class="math-container">$y=(\lambda^\rho (x^\rho +y^\rho ))^\frac{1}{\rho }$</span></p>

<p><span class="math-container">$y=(\lambda ^\rho )^\frac{1}{\rho }(x^{\rho }+y^\rho )^\frac{1}{\rho }$</span></p>

<p><span class="math-container">$y=\lambda (x^\rho +y^\rho )^\frac{1}{\rho }$</span> which is the initial production function. </p>

<p>Since, the degree of <span class="math-container">$\lambda $</span> is 1, so, it's homogeneous of degree <span class="math-container">$1$</span>.</p>

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2018-11-12T14:38:18.243Z
2018-11-12T14:38:18.243Z
Henam
<p>Could one define a production function which is homogenous as having constant elasticity of substitution. Just want clarification .
Thanks</p>

<production-function>
Homogenous production function
Rivaldo
# Homogenous production function

(Asked by: username_0 on 2018-11-10)

Could one define a production function which is homogenous as having constant elasticity of substitution. Just want clarification .
Thanks

**Tags:** production-function

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## 1 answers

### Answer by username_1 on 2018-11-12. Score: 2

I'm answering if there is any production function which is homogeneous and CES too.

How about a CES production function only.

$y=(x^\rho +y^\rho )^\frac{1}{\rho }$ for $0\neq \rho <1$

This function has Elasticity of Substitution : $\sigma =\frac{1}{(1-\rho )}$ which is a constant.

Considering homogeneity,

$ y=((\lambda x)^\rho +(\lambda y)^\rho )^\frac{1}{\rho }$

$y=(\lambda^\rho (x^\rho +y^\rho ))^\frac{1}{\rho }$

$y=(\lambda ^\rho )^\frac{1}{\rho }(x^{\rho }+y^\rho )^\frac{1}{\rho }$

$y=\lambda (x^\rho +y^\rho )^\frac{1}{\rho }$ which is the initial production function.

Since, the degree of $\lambda $ is 1, so, it's homogeneous of degree $1$.


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