Roy's identity

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 $V(P,Y)$ is the indirect utility function, then the Marshallian demand function for good $i$ can be calculated as:

$x_{i}^{m}=-\frac{\frac{\partial V}{\partial p_{i}}}{\frac{\partial V}{\partial Y}}$

where $P$ is the price vector of goods and $Y$ is income.

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 $Y$ in the indirect utility function $V (P, Y)$, at a utility of $u$:

$V ( P, e(P, u)) = u$

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:

$\frac{ \partial V [P, e(P,u)]}{\partial Y} \frac{\partial e(P,u)}{\partial p_i} + \frac{\partial V [P, e(P,u)]}{\partial p_i} = 0$.

Rearranging gives the desired result:

$-\frac{\frac{\partial V [P, e(P,u)]}{\partial p_i}}{\frac{\partial V [P, e(P,u)]}{\partial Y}}=\frac{\partial e(P,u)}{\partial p_i}=h_i(P, u)=x_i(P, e(P,u))$

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 $V(p_{1},p_{2},Y)$ is the maximand of the constrained optimization problem characterized by the following Lagrangian:

$\mathcal{L}=U(x_{1},x_{2})+\lambda(Y-p_{1}x_{1}-p_{2}x_{2})$

By the envelope theorem, the derivatives of the maximand $V(p_{1},p_{2},Y)$ with respect to the parameters can be computed as such:

$\frac{\partial V}{\partial p_{1}}=-\lambda x_{1}^{m}$

$\frac{\partial V}{\partial Y}=\lambda$

where $x_{1}^{m}$ is the maximizer (i.e. the Marshallian demand function for good 1). Simple arithmetic then gives Roy's Identity:

$-\frac{\frac{\partial V}{\partial p_{1}}}{\frac{\partial V}{\partial Y}}=-\frac{-\lambda x_{1}^{m}}{\lambda}=x_{1}^{m}$

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.