Shephard's lemma
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Shephard's lemma is a major result in
The lemma is named after Ronald Shephard who gave a proof using the distance formula in his book Theory of Cost and Production Functions (Princeton University Press, 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).
Definition
In consumer theory, Shephard's lemma states that the demand for a particular good for a given level of utility and given prices , equals the derivative of the expenditure function with respect to the price of the relevant good:
where is the
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 , is the cost function, and both functions are in terms of factor prices (a vector ) and output .
Although Shephard's original proof used the distance formula, modern proofs of Shephard's lemma use the envelope theorem.[3]
Proof for the differentiable case
The proof is stated for the two-good case for ease of notation. The expenditure function is the value function of the constrained optimization problem characterized by the following Lagrangian:
By the envelope theorem the derivatives of the value function with respect to the parameter are:
where is the minimizer (i.e. the Hicksian demand function for good 1). This completes the proof.
Application
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.
See also
References
- ISBN 0-393-95735-7.
- JSTOR 2296067.
- ISBN 0-07-057453-7.
Further reading
- Beavis, Brian; Dobbs, Ian M. (1990). "An Introduction to Duality Theory". Optimization and Stability Theory for Economic Analysis. New York: Cambridge University Press. pp. 117–133. ISBN 0-521-33605-8.