# Homogenous production function

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

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 $$1$$.