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I dont know how to demonstrate homogeneity degree of this function

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

Any idea?

Thanks.

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  • $\begingroup$ 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. $\endgroup$ – Regio Jun 3 '19 at 16:37
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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.

Kind regards, Pedro.

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