- Shephard's lemma
Shephard's lemma is a major result in
microeconomics having applications inconsumer choice and thetheory of the firm . The lemma states that ifindifference curves of the expenditure orcost function are convex, then the cost minimizing point of a given good (i) withprice p_i is unique. The idea is that aconsumer will buy a unique ideal amount of each item to minimize the price for obtaining a certain level ofutility given the price of goods in themarket . It was named afterRonald Shephard who gave a proof using the distance formula in a paper published in1953 , although it was already used byJohn Hicks (1939 ) andPaul Samuelson (1947 ).Definition
The lemma give a precise formulation for the
demand of each good in the market with respect to that level of utility and those prices: the derivative of theexpenditure function (e (p, u) ) with respect to that price::h_i(u, p) = frac{partial e (p, u)}{ partial p_i}
where h_i(u, p) is the
Hicksian demand for good i, e (p, u) is the expenditure function, and both functions are in terms of prices (a vector p) and utility u.Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the
envelope theorem .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 anindirect utility function and a correspondingMarshallian demand function .ee also
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Convex preferences
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