# How can i determine the homogeneity degree of Stone Gaery function? [closed]

I dont know how to demonstrate homogeneity degree of this function

$$(X-\alpha)^{\beta}(Y)^{1-\beta}$$

Any idea?

Thanks.

• Probably you will only be able to put some bounds and state whether it shows decreasing/constant/increasing returns to scale. This type of function is sometimes called quasi-homothetic, so maybe you can find something on that. – Regio Jun 3 '19 at 16:37

If the function $$f(x,y) = (x-\alpha)^\beta \cdot y^{1-\beta}$$ is homogeneous of degree $$k$$, then it must be true that $$f(\lambda x, \lambda y) = \lambda^k \cdot f(x,y)$$ for $$\lambda \in \mathbb{R}$$ and some constant $$k$$. But see that:
$$f(\lambda x, \lambda y) = (\lambda x-\alpha)^\beta \cdot (\lambda y)^{1-\beta} \neq \lambda^k \cdot (x-\alpha)^\beta \cdot y^{1-\beta}$$
unless it should happen that $$\alpha = 0$$. In that case, the function will be homogeneous of degree one.