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'''Shephard's lemma''' is a major result in [[microeconomics]] having applications in the [[theory of the firm]] and in [[consumer]] choice.<ref>{{cite book |title=Microeconomic Analysis |edition=Third |authorlink=Hal Varian |first=Hal |last=Varian |year=1992 |publisher=Norton |location=New York |isbn=0393957357 }}</ref>


The [[lemma (mathematics)|lemma]] states that if [[indifference curves]] of the expenditure or [[Cost curve|cost function]] are [[convex function|convex]], then the cost minimizing point of a given good (<math>i</math>) with [[price]] <math>p_i</math> is unique. The idea is that a [[consumer]] will buy a unique ideal amount of each item to minimize the price for obtaining a certain level of [[utility]] given the price of goods in the [[market]].
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The lemma is named after [[Ronald Shephard]] who gave a [[Mathematical proof|proof]] using the distance formula in his book ''Theory of Cost and Production Functions'' (Princeton University Press, 1953).
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The equivalent result in the context of consumer theory was first derived by [[Lionel W. McKenzie]] in 1957. It states that the partial derivatives of the expenditure function with respect to the prices of goods equal the [[Hicksian demand function]]s for the relevant goods. Similar results had already been derived by [[John Hicks]] (1939) and [[Paul Samuelson]] (1947).
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==Definition==
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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:


:<math>h_i(\mathbf{p}, u) = \frac{\partial e (\mathbf{p}, u)}{ \partial p_i}</math>
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where ''h<sub>i</sub>('''p''',u)'' is the [[Hicksian demand]] for good <math>i</math>, ''e('''p''',u)'' is the [[expenditure function]], and both functions are in terms of prices (a [[Vector (geometric)|vector]] '''''p''''') and utility <math>u</math>.
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Likewise, in the [[theory of the firm]], the lemma gives a similar formulation for the [[Conditional factor demands|conditional factor demand]] for each input factor: the derivative of the cost function ''c('''w''',y)'' with respect to the factor price:
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:<math>x_i(\mathbf{w}, y) = \frac{\partial c (\mathbf{w}, y)}{ \partial w_i}</math>
 
where ''x<sub>i</sub>('''w''',y)'' is the [[conditional factor demands|conditional factor demand]] for input <math>i</math>, ''c('''w''',y)'' is the cost function, and both functions are in terms of factor prices (a [[Vector (geometric)|vector]] '''''w''''') and output <math>y</math>.
 
Although Shephard's original proof used the distance formula, modern proofs of the Shephard's lemma use the [[envelope theorem]].
 
==Proof for the Differentiable Case==
The proof is stated for the two-good case for ease of notation. The expenditure function <math>e(p_{1},p_{2},u)</math> is the minimand of the constrained optimization problem characterized by the following Lagrangian:
 
:<math>\mathcal{L}=p_{1}x_{1} + p_{2}x_{2} + \lambda(u-U(x_{1},x_{2})) </math>
 
By the [[envelope theorem]] the derivatives of the minimand <math>e(p_{1},p_{2},u)</math> with respect to the parameter <math>p_{1}</math> can be computed as such:
 
:<math>\frac{\partial e}{\partial p_{1}}=\frac{\partial \mathcal{L}}{\partial p_{1}}=x_{1}^{h} </math>
 
where <math>x_{1}^{h}</math> 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==
*[[Hotelling's lemma]]
*[[Convex preferences]]
 
==References==
{{reflist}}
 
{{DEFAULTSORT:Shephard's Lemma}}
[[Category:Underlying principles of microeconomic behavior]]
[[Category:Lemmas]]

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