Economics theorem
Roy's identity (named after 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, denoting the indirect utility function as formula_1 the Marshallian demand function for good formula_2 can be calculated as
formula_3
where formula_4 is the price vector of goods and formula_5 is income, and where the superscript formula_6 indicates Marshallian demand. The result holds for continuous utility functions representing locally non-satiated and strictly convex preference relations on a convex consumption set, under the additional requirement that the indirect utility function is differentiable in all arguments.
Roy's identity is akin to the result that the price derivatives of the expenditure function give the Hicksian demand functions. The additional step of dividing by the wealth derivative of the indirect utility function in Roy's identity is necessary since the indirect utility function, unlike the expenditure function, has an ordinal interpretation: any strictly increasing transformation of the original utility function represents the same preferences.
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 (formula_2) from some indirect utility function.
The first step is to consider the trivial identity obtained by substituting the expenditure function for wealth or income formula_5 in the indirect utility function formula_9, at a utility of formula_10:
formula_11
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 formula_4) 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 formula_13 (with the utility level held constant) gives:
formula_14.
Rearranging gives the desired result:
formula_15
with the second-to-last equality following from Shephard's lemma and the last equality from a basic property of Hicksian demand.
Alternative proof using the envelope theorem.
For expositional ease, consider the two-goods case. The indirect utility function formula_16 is the value function of the constrained optimization problem characterized by the following Lagrangian:
formula_17
By the envelope theorem, the derivatives of the value function formula_16 with respect to the parameters are:
formula_19
formula_20
where formula_21 is the maximizer (i.e. the Marshallian demand function for good 1). Hence:
formula_22
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.
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