Suppose we have the following equations for the MRS of a utility function. $$U(x, y)$$ Which of the following corresponds to a homothetic utility function?
$$MRS (x, y) = \frac{x^2+y^2}{xy}$$
$$ MRS(x, y) = 2(x + y)$$
The answer is 1. But why the second equation is not homothetic?
If a utility function is homothethic, $-\frac {\frac {\partial u}{\partial x} (tx)} {\frac {\partial u} {\partial y} (ty)} = -\frac {\frac {\partial u}{\partial x} (x)} {\frac {\partial u} {\partial y} (y)} $
In other words, the MRS must be homogenous of degree zero.
Hence, for $MRS(tx,ty)=2(tx+ty)=2t(x,y)=t MRS(x,y)$ is homogenous of degree 1
While $MRS(tx,ty)=\frac{(tx)^{2}+(ty)^{2}}{(tx)(ty)}=\frac{t^{2}(x+y)}{t^{2}(xy)}=\frac{x^{2}+y^{2}}{xy}=MRS(x,y)$ is homogenous of degree 0.
