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| '''Roy's identity''' (named for [[France|French]] [[economist]] [[René Roy]]) is a major result in [[microeconomics]] having applications in [[consumer]] choice and the [[theory of the firm]]. The lemma relates the ordinary (Marshallian) demand function to the derivatives of the [[indirect utility function]]. Specifically, where <math>V(P,Y)</math> is the indirect utility function, then the Marshallian demand function for good <math>i</math> can be calculated as:
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| :<math>x_{i}^{m}=-\frac{\frac{\partial V}{\partial p_{i}}}{\frac{\partial V}{\partial Y}}</math>
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| == Derivation of Roy's identity == | |
| Roy's identity reformulates [[Shephard's lemma]] in order to get a [[Marshallian demand function]] for an individual and a good (<math>i</math>) from some indirect utility function.
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| The first step is to consider the trivial identity obtained by substituting the [[expenditure function]] for [[wealth]] or [[income]] <math>Y</math> in the [[indirect utility function]] <math>V (Y, P)</math>, at a utility of <math>u</math>:
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| :<math>V ( e(u, P), P) = u </math>
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| This says that the indirect utility function evaluated in such a way that minimizes the cost for achieving a certain utility given a set of prices (a vector <math>p</math>) is equal to that utility when evaluated at those prices.
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| Taking the derivative of both sides of this equation with respect to the price of a single good <math>p_i</math> (with the utility level held constant) gives:
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| :<math>\frac{ \partial V [e(u,P),P]}{\partial Y} \frac{\partial e(u,P)}{\partial p_i} + \frac{\partial V [e(u,P),P]}{\partial p_i} = 0</math>.
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| Rearranging gives the desired result:
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| :<math>\frac{\partial e(u,P)}{\partial p_i}=-\frac{\frac{\partial V [e(u,P),P]}{\partial p_i}}{\frac{\partial V [e(u,P),P]}{\partial Y}}=x_i(Y,P)</math>
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| == Alternative Proof for the Differentiable Case ==
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| There is a simpler proof of Roy's Identity, stated for the two-good case for simplicity.
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| The indirect utility function <math>V(p_{1},p_{2},Y)</math> is the maximand of the constrained optimization problem characterized by the following Lagrangian:
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| :<math>\mathcal{L}=U(x_{1},x_{2})+\lambda(Y-p_{1}x_{1}-p_{2}x_{2}) </math>
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| By the envelope Theorem, the derivatives of the maximand <math>V(p_{1},p_{2},Y)</math> with respect to the parameters can be computed as such:
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| :<math>\frac{\partial V}{\partial p_{1}}=-\lambda x_{1}^{m} </math>
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| :<math>\frac{\partial V}{\partial Y}=\lambda </math> | |
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| where <math>x_{1}^{m}</math> is the maximizer (i.e. the Marshallian demand function for good 1). Simple arithmetic then gives Roy's Identity:
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| :<math>-\frac{\frac{\partial V}{\partial p_{1}}}{\frac{\partial V}{\partial Y}}=-\frac{-\lambda x_{1}^{m}}{\lambda}=x_{1}^{m} </math>
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| == Application ==
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| This gives a method of deriving the [[Marshallian demand function]] of a good for some consumer from the indirect utility function of that consumer. It is also fundamental in deriving the [[Slutsky equation]].
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| == References ==
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| *{{cite journal |last=Roy |first=René |year=1947 |title=La Distribution du Revenu Entre Les Divers Biens |journal=[[Econometrica]] |volume=15 |issue=3 |pages=205–225 |jstor=1905479 }}
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| [[Category:Microeconomics]]
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| [[Category:Underlying principles of microeconomic behavior]]
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| [[Category:Consumer theory]]
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