I have a function $g(x_1,x_2)=(x_1^{2}x_2)^3$. It was listed in my homework problem as being homogenous of a certain degree which I need to find out. I performed the following steps and getting an answer that is wrong. Can anyone point me in the right direction?
My attempt:
$g(tx_1,tx_2)=tg(x_1,x_2)$
$g(x_1,x_2)=(x_1^{2}x_2)^3$
$[(tx_1)^{2}(tx_2)]^3$
$[t^2x_1^{2}tx_2]^3$
$[t^{5}x_1^{5}t^{3}x^{3}_2]$
$t^{3}(t^{2}x^{5}_1x^{3}_2)$
So... the homework question wants me to find the degree of homogeneity. My problem is that, as I have shown above, it does not appear to be homogenous of any degree. I think it may be just a simple issue with my exponent(s) and the multiplying of them, but I am unsure. Again, if you have any pointers that would be appreciated.
A function $g: \mathbb{R}^{\ell} \to \mathbb{R}$ is said to be homogeneous of degree $k$ if for all $\alpha > 0$ and all $\mathbf{x} \in \mathbb{R}^{\ell}$, we have:
$$g(\alpha \mathbf{x}) = \alpha^{k} g(\mathbf{x}).$$
With that definition in mind, let's take a look at the function you're analyzing. Fix $\alpha > 0$ and $\mathbf{x} = (x_1, x_2) \in \mathbb{R}^{2}.$ We have:
\begin{align} g(\alpha x_1, \alpha x_2) &= ((\alpha x_1)^{2}\alpha x_2)^{3}\\ &= (\alpha^{2}x_1^{2} \alpha x_2)^{3}\\ &= \alpha^{9}(x_1^{2}x_2)^{3}\\ &= \alpha^{9}g(x_1, x_2). \end{align}
You seem to have made a mistake when computing powers of $\alpha$. It follows from the derivation above that $g$ is homogeneous of degree $9$.