# Existence and uniqueness of demand, and symmetry implies equal demands given equal prices

Encountered the following problem during self study:

My take on the problem is that if we can show that the equation of the income expansion path is $$x_1=x_2$$ for all such $$U(x_1,x_2)$$ then we have proved the claim. I can see the result holding in examples for different utility functions, for example, $$U=x_1x_2$$, but I am unsure how to prove this more generally. Any help is appreciated

Proof by contradiction. Suppose $$(x_1^*, x_2^*)$$ solves the problem and $$x_1^* \neq x_2^*$$ is true. By symmetry of the utility and equal prices, $$(x_2^*, x_1^*)$$ which is not the same bundle as $$(x_1^*, x_2^*)$$ also solves the given problem contradicting that the solution is unique.

Direct Proof. Alternatively, you can also write the same argument directly. Given that $$(x_1^*, x_2^*)$$ solves the problem, by symmetry of the utility and equal prices, $$(x_2^*, x_1^*)$$ also solves the given problem. Since the solution is unique, $$(x_1^*, x_2^*)=(x_2^*, x_1^*)$$ must be true. Therefore, $$x_1^*=x_2^*$$.

• I feel stupid for not being able to see such a logical reason before. Thanks a lot. Apr 26, 2023 at 17:23