relates the ordinary (Marshallian) demand function to the derivatives of the indirect utility function. Specifically, denoting the indirect utility function
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
v
(
p
,
w
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
v
(
p
,
w
)
,
{\displaystyle v(p,w),}
the Marshallian demand function for good
i
{\displaystyle i}
can be calculated as
m
{\displaystyle {}^{m}}
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.
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
{\displaystyle 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
…