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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, $x_1^*=x_2^*$.
