I am struggling with with a problem regarding a function $F(x,y)$ that satisfies the following condition:
$F(\phi x,\phi^{\alpha+\beta-1}y)=\phi^\beta F(x,y)$, for all $\phi, \alpha, \beta > 0$, where $\alpha, \beta$ are constants between 0 and 1.
It is stated that this condition implies that:
$F(x,y)=x^\beta f(x^{1-\alpha-\beta}y)$ for some function $f$.
Is see that the latter expression satisfies the condition, but I fail to prove the implication. Can anyone help me out? Thanks in advance.
Notice that plugging in $x=1$ in the second equation suggests that we define $f(y) := F(1,y)$. Then, assuming the first equation holds, we have $$ x^\beta f(x^{1-\alpha-\beta}y) = x^\beta F(1,x^{1-\alpha-\beta}y) = F(x\cdot1,x^{\alpha+\beta-1}\cdot x^{1-\alpha-\beta}y) = F(x,y).$$