- 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 () withprice 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 () with respect to that price::
where is the
Hicksian demand for good , is the expenditure function, and both functions are in terms of prices (a vector ) and utility .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
*
Convex preferences
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