Expense function

The expenditure function is a function used in microeconomics that indicates how much a consumer has to spend minimally in order to achieve a given level of benefit . The utility function and prices of the goods with which the utility can be achieved are specified.

Definition and meaning

The starting point for deriving the expense function is the same as that for deriving Hicks's demand . It consists in the problem of minimizing expenses

{\ displaystyle \ min _ {(x_ {1}, \ ldots, x_ {n}) \ in \ mathbb {R} _ {+} ^ {n}} \ sum _ {i = 1} ^ {n} p_ {i} x_ {i}}

under the constraints .

{\ displaystyle n}

{\ displaystyle u (x_ {1}, \ ldots, x_ {n}) \ geq {\ overline {u}}}

(Details on this can be found in the article Hicks's demand function . ) A solution to this optimization problem, which can be solved using the Kuhn-Tucker method , is called Hicks's demand , whereby the vector of the quantities of goods demanded, the associated price vector and an ex ante demanded (minimum ) is benefit level. In words, this demand is the amount of goods - depending on the prices of goods - that is required to achieve a given level of utility as cheaply as possible . If you put Hicks's demand back into the minimized function, the resulting function is called the output function . So it is ${\ displaystyle \ mathbf {x} ^ {h} (\ mathbf {p}, {\ overline {u}})}$ ${\ displaystyle \ mathbf {x} = (x_ {1}, \ ldots, x_ {n})}$ ${\ displaystyle \ mathbf {p} = (p_ {1}, \ ldots, p_ {n})}$ ${\ displaystyle {\ overline {u}}}$ ${\ displaystyle {\ overline {u}}}$ ${\ displaystyle \ mathbf {x} ^ {h} (\ mathbf {p}, {\ overline {u}})}$ ${\ displaystyle e}$

{\ displaystyle e \ equiv \ sum _ {i = 1} ^ {n} \ left [p_ {i} \ cdot x_ {i} ^ {h} (\ mathbf {p}, {\ overline {u}}) \ right] {\ overset {\ mathrm {(vectorial)}} {=}} \ mathbf {p} \ cdot \ mathbf {x} ^ {h} (\ mathbf {p}, {\ overline {u}}) }

.

While Hicks's demand function supplies the quantities of goods that are required in the minimum expenditure, the expenditure function supplies the amount of expenditure that is necessary for this; in other words, the argument is the minimum, while the actual minimum is, which is why one could have postulated directly from the above representation of the minimization problem that it is equal to the output function. ${\ displaystyle \ mathbf {x} ^ {h} (\ mathbf {p}, {\ overline {u}})}$ ${\ displaystyle e}$

properties

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

homogeneous of degree one in , ${\ displaystyle \ mathbf {p}}$ soand; ${\ displaystyle e (\ alpha \ mathbf {p}, \ alpha y) = \ alpha e (\ mathbf {p}, y) \; \ forall \ mathbf {p}, y}$ ${\ displaystyle \ alpha> 0}$
monotonically increasing in ; ${\ displaystyle \ mathbf {p}}$
strictly increasing in for ; ${\ displaystyle {\ overline {u}}}$ ${\ displaystyle \ mathbf {p}> \ mathbf {0}}$
concave in . ${\ displaystyle \ mathbf {p}}$

Relation to the indirect utility function

Derivation of the utility function

The expenditure function was constructed from the utility function above. Under certain conditions it is also possible to construct the utility function from the expenditure function.

Derivation of the utility function from the expenditure function: Let the utility function u be continuous, strictly monotonically increasing in and quasi-concave in . Be the expense function. Then the original utility function used for the construction of is given as follows: ${\ displaystyle \ mathbf {x}}$ ${\ displaystyle \ mathbf {x}}$ ${\ displaystyle e (\ mathbf {p}, {\ overline {u}})}$ ${\ displaystyle e (\ mathbf {p}, {\ overline {u}})}$

{\ displaystyle u (\ mathbf {x}) = \ max _ {\ overline {u}} \ left \ {{\ overline {u}} \ left | \ forall \ mathbf {p}: \; \ sum _ { i = 1} ^ {n} p_ {i} x_ {i} \ geq e (\ mathbf {p}, {\ overline {u}}) \ right. \ right \}}

literature

Friedrich Breyer: Microeconomics. An introduction. 5th edition. Springer, Heidelberg a. a. 2011, ISBN 978-3-642-22150-7 .
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 .
Geoffrey A. Jehle and Philip J. Reny: Advanced Microeconomic Theory. 3rd ed. Financial Times / Prentice Hall, Harlow 2011, ISBN 978-0-273-73191-7 .
Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-195-07340-1 .

Remarks

↑ For this, see largely Jehle / Reny 2011, pp. 32–39. Some of the properties already follow the underlying preference-indifference relation 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 for every environment around one exists, with which . See the article preference order .) ${\ displaystyle R}$ ${\ displaystyle x_ {a} \ in X}$ ${\ displaystyle \ epsilon}$ ${\ displaystyle U _ {\ epsilon}}$ ${\ displaystyle x_ {a}}$ ${\ displaystyle z \ in U _ {\ epsilon}}$ ${\ displaystyle zPx_ {a}}$
↑ For the proof cf. Browning 1999, p. 129 f.

[1] For this, see largely Jehle / Reny 2011, pp. 32–39. Some of the properties already follow the underlying preference-indifference relation 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 for 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] For the proof cf. Browning 1999, p. 129 f.