The following theorem and proof are taken from the Lecture Notes of Ariel Rubinstein.

Assume that the demand function is derived from maximizing a preference relation $\succeq$ which satisfies monotonicity. Then:


The hyperplane $H= \{ (p, w) | px(p^{\ast}, w^{\ast})=w \}$ is tangent to the $\succeq^{\ast}$-indifference curve at $(p^{\ast}, w^{\ast})$.


By the monotonicity of the preferences $(p^{\ast},w^{\ast}) \in H$. For any $(p,w)\in H$, the bundle $x(p^{\ast},w^{\ast}) \in B(p,w)$. Hence, $x(p,w) \succeq (p^{\ast},w^{\ast})$ and thus $(p,w) \succeq^{\ast} (p^{\ast},w^{\ast})$.

However, though I believe that I figured out each proposition in the proof separately, I couldn't grasp the proof itself. Can someone explain the reasoning here?


  • 1
    $\begingroup$ I suppose there is a typo. $x(p,w) \succeq (p^{\ast},w^{\ast})$ should be $x(p,w) \succeq x(p^{\ast},w^{\ast})$ $\endgroup$ Nov 29, 2022 at 18:08


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