Roy's identity

Roy's identity (debate as French roi, [ ʁwa - ] ) is an important sentence within the microeconomics. He notes an important relationship between the Marshallian demand and the indirect utility function. It was named after the French economist René Roy.

Representation and meaning

Despite the existence in many ways analogy between the concept of the indirect utility function and the output function demjeniger there is at first sight no direct analogy to Shephard's Lemma, after the corresponding derivative of the output function of the price of the corresponding Hicksian demand function. However, still a minor modification provides some comparability. The relationship is called Roy's identity.

Roy's identity: Be continuous and strictly monotonically increasing. Let further differentiable at one point and. Then for all ():

With regard to the two requirements of the theorem of the utility function, it would presuppose even suffice here that is continuous and the underlying preference - indifference relation satisfies the property of the local non-saturation and is convex. Note: It refers to a preference order as not locally saturated if for any and every environment exists a, with the (also see order of preference ). However, the more common practice stronger assumption of differentiability of the indirect utility function also allows a simpler proof of the relationship.

Evidence

The Lagrangian of the utility maximization problem (see the article Marshallian demand function ) is. The envelope theorem implies it applied that

As well as

( 2) used in (1) then instantly Roy's identity.