Indirect utility function

An indirect utility function is a function used in microeconomics that indicates the maximum level of utility that a consumer can achieve for a given price of goods and with a given budget. In this way it 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 for the derivation of the Marshallian demand . It consists in the utility maximization problem

${\displaystyle \max _{(x_{1},\ldots ,x_{n})\in \mathbb {R} _{+}^{n}}u(x_{1},\ldots ,x_{n})}$ under the secondary condition ${\displaystyle \sum _{i=1}^{n}p_{i}x_{i}\leq y}$

(Details can be found in the article Marshall's demand function . ) A solution to this optimization problem, which can be solved using the Kuhn-Tucker method , is called Marshall's demand , where the vector of the quantities of goods demanded ( ), the associated price vector and the available consumption budget are. In words, this demand is the quantity of goods - depending on the price of goods - that is required to achieve the highest possible level of utility with a given budget . If the Marshallian demand is now put back into the maximized function, the resulting function is called the indirect utility function . So it is ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$ ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ ${\displaystyle x_{i}\geq 0}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle y}$ ${\displaystyle y}$ ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$ ${\displaystyle v}$

${\displaystyle v(\mathbf {p} ,y)\equiv u\left[x_{1}(\mathbf {p} ,y),\ldots ,x_{n}(\mathbf {p} ,y)\right]{\overset {\mathrm {(vektoriell)} }{=}}u\left[\mathbf {x} (\mathbf {p} ,y)\right]}$.

While the Marshallian demand function supplies the quantities of goods that are demanded at the utility maximum, the indirect utility function delivers the utility level that is reached at the maximum; in other words is the argument of the maximum, while the actual gives maximum. ${\displaystyle \mathbf {x} (\mathbf {p} ,y)}$ ${\displaystyle v}$

properties

It can be shown that under the usual conditions - steadily and strictly monotonically increasing - has the following properties: ${\displaystyle v(\mathbf {p} ,y)}$ ${\displaystyle u(\cdot )}$

• steady in and ; ${\displaystyle \mathbf {p} }$ ${\displaystyle y}$
• homogeneous of degree zero in , ${\displaystyle (\mathbf {p} ,y)}$ i.e. H. for everyoneand; ${\displaystyle v(\alpha \mathbf {p} ,\alpha y)=v(\mathbf {p} ,y)}$ ${\displaystyle \mathbf {p} ,y}$ ${\displaystyle \alpha >0}$
• strictly monotonically increasing in ${\displaystyle y}$ and monotonically decreasing in ${\displaystyle \mathbf {p} }$ (for positive); ${\displaystyle y}$
• quasi-convex in . ${\displaystyle (\mathbf {p} ,y)}$

In addition, the following applies with regard to u (and even if u only satisfies the weaker assumption of local unsaturation):

(Jackson 1986): The following applies to all bundles of goods : Consider a utility level and one that are such that for all possible price vectors with strictly positive components. Then there is one with which for everyone . ${\displaystyle {\tilde {\mathbf {x} }}}$ ${\displaystyle {\overline {u}}\geq 0}$ ${\displaystyle \epsilon >0}$ ${\displaystyle v(\mathbf {p} ,\mathbf {p} \cdot {\tilde {\mathbf {x} }})\geq {\overline {u}}+\epsilon }$ ${\displaystyle \mathbf {p} }$ ${\displaystyle \delta \in (0,1)}$ ${\displaystyle v(\mathbf {p} ,\delta \cdot \mathbf {p} \cdot {\tilde {\mathbf {x} }})>{\overline {u}}}$ ${\displaystyle \mathbf {p} }$

The following condition can be referred to for the differentiability (under the stated conditions with regard to u ):

Be moreover continuously differentiable. If the utility maximization problem (see above) has a clear solution in an open environment around ( ), then the indirect utility function in this environment can be differentiated into . ${\displaystyle u(\mathbf {x} )}$ ${\displaystyle (\mathbf {p} _{0},y_{0})}$ ${\displaystyle y_{0}>0}$ ${\displaystyle v(\mathbf {p} ,y)}$ ${\displaystyle (\mathbf {p} ,y)}$

classification

Relation to the expenditure function

Analogous to the relationship between Marshallian and Hicksian inquiries, there is also a close relationship between the indirect utility function - which is conceptually linked to the former - and the expenditure function which is related to the latter . The following applies:

Relationship between expenditure and indirect utility function: Let the order of preferences of consumers be represented and represented by a real-valued utility function that increases continuously and strictly monotonically . Then: ${\displaystyle \mathbb {R} _{+}^{n}}$ ${\displaystyle u}$

1. ${\displaystyle e\left[\mathbf {p} ,v(\mathbf {p} ,y)\right]=y}$
2. ${\displaystyle v\left[\mathbf {p} ,e(\mathbf {p} ,{\overline {u}})\right]={\overline {u}}}$

Roy's identity

→ Main article : Roy's identity

Despite the in many ways analogy between the concept of the indirect utility function and that of the expenditure function, at first glance there is no direct analogy to Shephard's lemma , according to which the derivation of the expenditure function according to the price corresponds to the corresponding Hicksian demand function. However, a slight modification still provides a certain degree of comparability. The relationship is known as Roy's identity .

Roy's identity: Be steady and strictly monotonously increasing. Be further differentiable in one place and . Then applies to all ( ): ${\displaystyle u(\cdot )}$ ${\displaystyle v(\mathbf {p} ,y)}$ ${\displaystyle (\mathbf {p} _{0},y_{0})}$ ${\displaystyle \partial v({p}_{0},y_{0})/\partial y\neq 0}$ ${\displaystyle i}$ ${\displaystyle 1\leq i\leq n}$

${\displaystyle x_{i}(\mathbf {p} _{0},y_{0})=-{\frac {\partial v(\mathbf {p} _{0},y_{0})/\partial p_{i}}{\partial v(\mathbf {p} _{0},y_{0})/\partial y}}}$

See the article Roy's Identity for evidence .

example

For an example of the construction of an indirect utility function, see the article Marshall's demand function .

literature

• Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
• David M. Kreps: Microeconomic Foundations I. Choice and Competitive Markets Princeton University Press, Princeton 2012, ISBN 978-0-691-15583-8 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .

Remarks

1. ↑ Cf. largely Jehle / Reny 2011, pp. 29 ff. Some of the properties also follow the underlying preference-indifference relation even under the weaker assumption of local unsaturation . On this, Mas-Colell / Whinston / Green 1995, p. 59. (A preference order is referred to as locally unsaturated if for any and every environment around one exists, with which . See the article preference order . ) ${\displaystyle R}$ ${\displaystyle x_{a}\in X}$ ${\displaystyle \epsilon }$ ${\displaystyle U_{\epsilon }}$ ${\displaystyle x_{a}}$ ${\displaystyle z\in U_{\epsilon }}$ ${\displaystyle zPx_{a}}$
2. ↑ See Kreps 2012, p. 274.
3. ↑ Matthew O. Jackson: Continuous utility functions in consumer theory. A set of duality theorems. In: Journal of Mathematical Economics. 15, No. 1, 1986, pp. 63-77, doi : 10.1016 / 0304-4068 (86) 90024-8 .
4. ↑ See Kreps 2012, p. 262.
5. ↑ See Jehle / Reny 2011, p. 27 ff.
6. ↑ See Jehle / Reny 2011, p. 29; with slightly weaker assumptions Mas-Colell / Whinston / Green 1995, pp. 73 f.