Roy's identity (named for 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
Contents
where
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 (
The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income
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
Taking the derivative of both sides of this equation with respect to the price of a single good
Rearranging gives the desired result:
with the second-to-last equality following from Shephard's lemma and the last equality from a basic property of Hicksian demand.
Alternative proof for the differentiable case
There is a simpler proof of Roy's identity, stated for the two-good case for simplicity.
The indirect utility function
By the envelope theorem, the derivatives of the maximand
where
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.