Shephard's lemma, proved by Ronald Shephard in his 1953 work Theory of Cost and Production Functions, is a result in microeconomics stating that if the indifference curves implied by a firm's cost function or a consumer's expenditure function are convex, the quantity of a given good that minimizes cost at a given price is unique and equal to the derivative of the cost or expenditure function with respect to that good's price. Shephard's own proof used a distance-function argument; later economists, including Lionel McKenzie who derived the equivalent consumer-theory result in 1957, showed the same conclusion follows more directly from the envelope theorem. The lemma underlies standard treatments of cost minimization in the theory of the firm and expenditure minimization in consumer choice.
Facts
StatementIf the indifference curves implied by a firm's cost function or a consumer's expenditure function are convex, the cost minimizing quantity of a given good is unique and equals the derivative of the cost or expenditure function with respect to that good's price. 2 Classification
Statement Form Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Wikipedia: Shephard's lemma
WikipediaLead section, statement-form referenceQuote, Lead section, statement-form reference
The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost-minimizing point of a given good ( i ) with price p i } is unique.
2. Shephard's lemma, Wikipedia
Lede section
The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost-minimizing point of a given good (i) with price pi is unique.
History section
The lemma is named after Ronald Shephard, who proved it using the distance formula in his book Theory of Cost and Production Functions in 1953.
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