Roy's identity

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 w in the indirect utility function v ( p , w ) , at a utility of 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:

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 v ( p 1 , p 2 , w ) is the maximand of the constrained optimization problem characterized by the following Lagrangian:

By the envelope theorem, the derivatives of the maximand v ( p 1 , p 2 , w ) with respect to the parameters can be computed as such:

where x 1 m is the maximizer (i.e. the Marshallian demand function for good 1). Simple arithmetic then gives Roy's identity:

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.