# Hotelling's lemma

Hotelling's lemma is a result in microeconomics that relates the supply of a good to the profit of the good's producer. It was first shown by Harold Hotelling, and is widely used in the theory of the firm. The lemma is very simple, and can be stated:

Let ${\displaystyle y(p)}$ be a firm's net supply function in terms of a certain good's price (${\displaystyle p}$). Then:

${\displaystyle y(p)={\frac {\partial \pi (p)}{\partial p}}}$

for ${\displaystyle \pi }$ the profit function of the firm in terms of the good's price, assuming that ${\displaystyle p>0}$ and that derivative exists.

The proof of the theorem stems from the fact that for a profit-maximizing firm, the maximum of the firm's profit at some output ${\displaystyle y^{*}(p)}$ is given by the minimum of ${\displaystyle \pi (p^{*})-p^{*}y^{*}(p)}$ at some price, ${\displaystyle p^{*}}$, namely where ${\displaystyle {\frac {\partial \pi (p)}{\partial p}}-y=0}$ holds. Thus, ${\displaystyle y(p)={\frac {\partial \pi (p)}{\partial p}}}$; QED.

The proof is also a corollary of the envelope theorem.