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 relates the ordinary demand function to the derivatives of the indirect utility function.

[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.

The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income ( $m$ )in the indirect utility function ( $\psi\ (m, p)$ , 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 derivative of both sides of this equation with respect to the price of a single good $p i$ (with the utility level held constant) gives:

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

Rearranging gives the desired result:

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

[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.