Shephard's lemma is a major 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 (
i
pi
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).
In consumer theory, Shephard's lemma states that the demand for a particular good
i
u
p
hi(p,u)=
| \partiale(p,u) | |
| \partialpi |
where
hi(p,u)
i
e(p,u)
p
u
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)
xi(w,y)=
| \partialc(w,y) | |
| \partialwi |
where
xi(w,y)
i
c(w,y)
w
y
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 case for ease of notation. The expenditure function
e(p1,p2,u)
l{L}=p1x1+p2x2+λ(u-U(x1,x2))
By the envelope theorem the derivatives of the value function
e(p1,p2,u)
p1
| \partiale | = | |
| \partialp1 |
| \partiall{L | |
where
| h | |
| x | |
| 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.