Indirect utility function

An indirect utility function is a function used in microeconomics, which indicates the maximum level of utility that can reach a consumer goods at given prices and with a given budget. This differs from the ( direct ) utility function of a consumer, which is generally defined for certain quantities of goods.

Definition and meaning

The starting point for the derivation of the indirect utility function is the same as that used to derive the Marshallian demand. It consists in the utility maximization problem

( Details can be found in the article Marshallian demand function. ) One solution to this releasable by means of the Kuhn -Tucker method optimization problem is called the Marshallian demand, with the vector of the requested quantities of goods (), the associated price vector and the available consumption budget. In words, it is in such demand that is to say that quantity of goods - in Abgängigkeit of commodity prices - which is necessary in order to achieve the highest possible level of utility with a given budget. Substituting the Marshallian demand now back in the maximized function, as one calls the resulting function as an indirect utility function. It is therefore

While the Marshallian demand function gives the quantities of goods that are in demand in the utility maximum, the indirect utility function gives the utility level, which is reached at the maximum; In other words, the argument of the maximum, while the actual maximum yields.

Properties

It can be shown that under the usual conditions - continuous and strictly monotonically increasing - has among others the following properties:

• Steadily in and;
• Homogeneous of degree zero in, that is, for all and;
• Strictly increasing in and monotonically decreasing in ( for positive );
• Quasi- convex.

Under the above condition is also considered with respect to u (and already if u only the weaker assumption of local non-saturation is sufficient):

For the differentiability can be (u respect under the conditions described above ) refer to the following condition:

Classification

Related to the output function

Analogous to the relationship between shear and marshall Hicks'scher demands also exists between the - conceptually associated with the former - the indirect utility function and the - with the latter related - output function a close relationship. It is namely:

Relationship between expenditure and indirect utility function: Let the order of preference of consumers representable by a real-valued and continuous and strictly increasing utility function and represents. Then:

Roy's identity

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 ():

For the proof we refer to the article Roy's identity.

Example

For an example of the construction of an indirect utility function Marshallian demand function to the article being referenced.