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

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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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