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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?

  1. $$MRS (x, y) = \frac{x^2+y^2}{xy}$$

  2. $$ MRS(x, y) = 2(x + y)$$

The answer is 1. But why the second equation is not homothetic?

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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.

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