Hotelling's lemma

Hotelling's lemma states that derivatives of a firm's profit function with respect to output and input prices yield its supply and input demands.

Profit function: A function giving a firm's maximum profit as a function of market prices and other parameters. Hotelling's lemma extracts the firm's choices from this optimized value function.
Profit maximization: The choice of production plans that yields the greatest revenue minus cost under given conditions. The lemma begins with the profit-maximizing production plan.
Comparative statics: The analysis of how an economic equilibrium or choice changes when an underlying parameter changes. Price derivatives of profit provide direct comparative statics for a firm's choices.
Harold Hotelling: An American mathematician and economist whose work helped establish mathematical methods in economics. The lemma bears Hotelling's name and follows his work on economic value functions.
Output supply: The quantity of a good a firm chooses to sell at given prices and conditions. The derivative of profit with respect to the output price gives the firm's supply.
Production set: The set of feasible input-output combinations available to a firm. The firm's profit function maximizes the value of plans in this feasible set.
Supply function: A relationship mapping prices and other conditions to the quantity a producer offers. Hotelling's lemma recovers this function from the profit function.
The Economics of Exhaustible Resources: Harold Hotelling's 1931 article analyzing the efficient extraction and pricing of exhaustible resources. Hotelling's broader contributions to economics include this influential analysis of resource prices.
Input demand: The quantity of a production input a firm chooses at given prices and conditions. The negative derivative of profit with respect to an input price gives its demand.
Price-taking: An assumption that an economic agent treats market prices as fixed when making choices. Fixed prices let the firm optimize against a price vector and differentiate profit by its components.

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