Shephard's lemma is a result in microeconomics having applications in the theory of the firm and in consumer choice.
[1] The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost-minimizing point of a given good (
The idea is that a consumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market.
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.
The equivalent result in the context of consumer theory was first derived by Lionel W. McKenzie in 1957.
[2] It states that the partial derivatives of the expenditure function with respect to the prices of goods equal the Hicksian demand functions for the relevant goods.
Similar results had already been derived by John Hicks (1939) and Paul Samuelson (1947).
In consumer theory, Shephard's lemma states that the demand for a particular good
, equals the derivative of the expenditure function with respect to the price of the relevant good: where
is the Hicksian demand for good
is the expenditure function, and both functions are in terms of prices (a vector
Likewise, in the theory of the firm, the lemma gives a similar formulation for the conditional factor demand for each input factor: the derivative of the cost function
with respect to the factor price: where
is the conditional factor demand for input
Both functions are in terms of factor prices (a vector
Although Shephard's original proof used the distance formula, modern proofs of Shephard's lemma use the envelope theorem.
[3] The proof is stated for the two good cases for ease of notation.
is the value function of the constrained optimization problem characterized by the following Lagrangian: By the envelope theorem the derivatives of the value function
is the minimizer (i.e. the Hicksian demand function for good 1).
Shephard's lemma gives a relationship between expenditure (or cost) functions and Hicksian demand.
The lemma can be re-expressed as Roy's identity, which gives a relationship between an indirect utility function and a corresponding Marshallian demand function.