Definition
In consumer theory, Shephard's lemma states that the demand for a particular good i for a given level of utility u and given prices p, equals the derivative of the expenditure function with respect to the price of the relevant good:
where hi(p,u) is the Hicksian demand for good, e(p,u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility .
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 c(w,y) with respect to the factor price:
where xi(w,y) is the conditional factor demand for input, c(w,y) is the cost function, and both functions are in terms of factor prices (a vector w) and output .
Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the envelope theorem.
Read more about this topic: Shephard's Lemma
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