2
$\begingroup$

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:

Theorem:

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

Proof:

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?

Thanks.

$\endgroup$
1
  • 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

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Browse other questions tagged or ask your own question.