I want to prove that if a utility function that represents $\succsim$ has constant marginal elasticity of substitution (MRS) in proportional expansions along rays, that is: $$ MRS(x,y) = MRS(\alpha x, \alpha y), \ \forall \alpha > 0 $$ then $\succsim$ is homothetic.

The definition of homothetic preferences that I'm using is the one found in Mas-Colell et al.:

Definition 3.B.6: A monotone preference relation $\succsim$ on $X=\mathbb{R}^L_+$ is homothetic if all indifference sets are related by proportional expansion along rays; that is, if $x \sim y$ then $\alpha x \sim \alpha y$ for any $\alpha \ge 0$.

  • $\begingroup$ While this question is much better stated, the older version of this question now also has an answer to this one. Perhaps the two can be merged somehow. $\endgroup$ – Theoretical Economist Apr 18 '18 at 5:53