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 be a firm's net supply function in terms of a certain good's price . Then:

for the profit function of the firm in terms of the good's price, assuming that 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 is given by the minimum of at some price, namely where holds. Thus, ; QED.

The proof is also a corollary of the envelope theorem.