Roy's identity

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.

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:

:$frac\{partial\; e(u,p)\}\{partial\; p\_i\}=-frac\{frac\{partial\; psi\; [e(u,p),p]\; \}\{partial\; p\_i\{frac\{partial\; psi\; [e(u,p),p]\; \}\{partial\; m=x\_i(m,p)$

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.

References

*Roy, René (1947). "La Distribution du Revenu Entre Les Divers Biens," Econometrica, 15, 205-225.