Roy's identity

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Roy's identity (named for French economist Rene Roy) is a major result in microeconomics having applications in consumer choice and the theory of the firm. The lemma states that if indifference curves of the indirect utility function are convex in prices, then the cost minimizing point of a given good ( $i$ ), with price $p i$ , is unique. The idea is that a consumer will have an ideal amount of each item to minimize the price for obtaining a certain level of utility given the price of goods in the market.

[edit] 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 ( $i$ ) from some indirect utility function.

First, we obtain a trivial identity by substituting the expenditure function for wealth or income ( $m$ )in the indirect utility function ( $\psi\ (m, u)$ , at a utility of $u$ ):

$\psi\ ( e(p, u), p) = u$

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 $p$ ) is equal to that utility when evaluated at those prices.

Taking the partial derivative of both sides of this equation with respect to the price of a single good $p i$ and a constant utility level we have:

$\frac{ \partial \psi\ [e(u,p),p]}{\partial m} \frac{\partial e(u,p)}{\partial p_i} + \frac{\partial \psi\ [e(u,p),p]}{\partial p_i} = 0$ .

Rearranging, we obtain

$\frac{\partial e(u,p)}{\partial p_i}=-\frac{\frac{\partial \psi\ [e(u,p),p]}{\partial p_i}}{\frac{\partial \psi\ [e(u,p),p]}{\partial m}}=x_i(m,p)$

[edit] Application

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.