Distance function (microeconomics)

Fig. 1) I denotes the indifference curve to the utility level . ${\displaystyle {\overline {u}}}$

As a distance function (English: distance function ) is called in economics and there especially in microeconomics an implicit representation of the (direct) utility function . It can be interpreted in various ways; in geometric reading, it indicates for a given set vector and a given utility level how far one has to move on the set vector starting from the origin in order to arrive at the indifference curve on which this utility level is reached.

Classification and definition

The most obvious method to represent consumer preference orders in the form of a mathematical function is the (direct) utility function. It outputs a (real) value for a given combination of quantities of goods; By comparing the values ​​of different bundles of goods, it can then be concluded which bundle of goods is preferred to the other (or between which bundles of goods the household is indifferent). From the utility function, a set of bundles of goods (the so-called indifference set or curve) can in turn be formed, which contains all bundles of goods that generate a certain level of utility.

Definition via the utility function

The distance function is based on a consideration which - as can also be shown algebraically - is closely related to the idea of ​​utility maximization: One consider any arbitrary bundle of goods and a given utility level. At the same time one knows the utility function of the consumer. The value of the distance function is now precisely that scalar by which the bundle of goods has to be divided in order to optimally achieve the given level of utility. Formally: ${\displaystyle \delta }$

Definition I: Let be a tuple of sets of goods, a utility function and a certain utility level. Then the function is called ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ ${\displaystyle u:\mathbb {R} ^{n}\rightarrow \mathbb {R} _{+}}$ ${\displaystyle {\overline {u}}}$

${\displaystyle d(\mathbf {x} ,{\overline {u}})\equiv \max _{\delta }\left\{\delta \left|u\left({\frac {\mathbf {x} }{\delta }}\right)\geq {\overline {u}},\;\delta \in \mathbb {R} \right.\right\}}$

as a distance function.

Intuitive: if you consider any bundle of goods and any level of benefit . Then there are three possibilities: 1. The bundle of goods generates a higher benefit than . 2. The bundle of goods produces exactly the benefit . 3. The bundle of goods generates less benefit than . In case 1, the bundle of goods can be made smaller (that is, by dividing it), to the point that it only just provides the benefit ; in case 3, on the other hand, you have to enlarge it (i.e. by dividing it) in order to just achieve the benefit . Special case 2 is interesting. It can be seen that a bundle of goods creates utility if and only if the distance function has the value 1 there. This illustrates that no information from the utility function was lost in the construction of the distance function; the following applies: ${\displaystyle {\overline {u}}}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle \delta >1}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle \delta <1}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle \mathbf {x} }$ ${\displaystyle {\overline {u}}}$

${\displaystyle u(\mathbf {x} )={\overline {u}}\Leftrightarrow d(\mathbf {x} ,{\overline {u}})=1}$.

Definition via the expense function

Definition II: Let be a tuple of sets of goods at prices , a utility function and a certain utility level. Continue to be the expense function . Then the function is called ${\displaystyle \mathbf {x} =(x_{1},\ldots ,x_{n})}$ ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$ ${\displaystyle u:\mathbb {R} ^{n}\rightarrow \mathbb {R} _{+}}$ ${\displaystyle {\overline {u}}}$ ${\displaystyle e(\mathbf {p} ,{\overline {u}})}$

${\displaystyle d(\mathbf {x} ,{\overline {u}})\equiv \min _{\mathbf {p} }\left\{\mathbf {p} \cdot \mathbf {x} \left|e(\mathbf {p} ,{\overline {u}})=1\right.\right\}}$

as a distance function.

Theorem (Gorman 1976): Definitions I and II are equivalent.

properties

The distance function has the following properties, among others

• homogeneous of degree one in ; ${\displaystyle \mathbf {x} }$
• strictly monotonically increasing in and strictly monotonically decreasing in ; ${\displaystyle x}$ ${\displaystyle u}$
• concave in . ${\displaystyle \mathbf {x} }$

The Hessian matrix of the distance function is known in the literature as the Antonelli matrix .

See also

• Antonelli matrix (the Hesse matrix of the distance function)

literature

• Martin Browning: Dual Approaches to Utility. In: Salvador Barberà, Peter J. Hammond and Christian Seidl (eds.): Handbook of Utility Theory. Vol. 1. Kluwer Academic Publishers, Boston 1998, ISBN 0-7923-8174-2 , pp. 122-144.
• Richard Cornes: Duality and modern economics. Cambridge University Press, Cambridge u. a. 1992, ISBN 0-521-33601-5 .
• Angus Deaton: The Distance Function in Consumer Behavior with Applications to Index Numbers and Optimal Taxation. In: The Review of Economic Studies. 46, No. 3, 1979, pp. 391-405 ( JSTOR 2297009 ).
• Angus Deaton and John Muellbauer: Economics and consumer behavior. Cambridge University Press, Cambridge u. a. 1980, ISBN 0-521-22850-6 .
• William M. Gorman: Quasi-Separable Preferences, Costs, and Technologies. University of North Carolina, 1970, mimeo. [Comprehensively summarized in C. Blackorby and AF Shorrocks (eds.): Separability and Aggregation. The Collected Works of WM Gorman. Vol. 1. Oxford University Press, Cambridge a. a. 1996, ISBN 978-0-19-828521-2 (also online: doi : 10.1093 / 0198285213.003.0007 ).]

Individual evidence

1. ↑ See Cornes 1992, p. 76; Browning 1998, p. 132.
2. ↑ See Deaton 1979, p. 393.
3. ↑ See Cornes 1992, p. 76; Deaton / Muellbauer 1980, p. 55; Browning 1998, p. 132.
4. ^ William M. Gorman: Tricks with Utility Functions. In: Michael J. Artis and A. Robert Nobay (Eds.): Essays in economic analysis. Cambridge, Cambridge University Press 1976. See also Deaton 1979, p. 393.
5. ↑ See Cornes 1992, p. 79; Deaton / Muellbauer 1980, p. 55.