Expenditure function

In microeconomics, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of utility, given a utility function and the prices of the available goods.

Formally, if there is a utility function $u$ that describes preferences over n commodities, the expenditure function

e(p,u^{*}):{\textbf {R}}_{+}^{n}\times {\textbf {R}}\rightarrow {\textbf {R}}

says what amount of money is needed to achieve a utility $u^{*}$ if the n prices are given by the price vector $p$ . This function is defined by

e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x

where

\geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}

is the set of all bundles that give utility at least as good as $u^{*}$ .

Expressed equivalently, the individual minimizes expenditure $x_{1}p_{1}+\dots +x_{n}p_{n}$ subject to the minimal utility constraint that $u(x_{1},\dots ,x_{n})\geq u^{*},$ giving optimal quantities to consume of the various goods as $x_{1}^{*},\dots x_{n}^{*}$ as function of $u^{*}$ and the prices; then the expenditure function is

e(p_{1},\dots ,p_{n};u^{*})=p_{1}x_{1}^{*}+\dots +p_{n}x_{n}^{*}.

Expenditure and indirect utility

The expenditure function is the inverse of the indirect utility function when the prices are kept constant. I.e, for every price vector $p$ and income level $I$ :[1]^:106

e(p,v(p,I))\equiv I

Example

Suppose the utility function is the Cobb-Douglas function $u(x_{1},x_{2})=x_{1}^{.6}x_{2}^{.4},$ which generates the demand functions[2]

x_{1}(p_{1},p_{2},I)={\frac {.6I}{p_{1}}}\;\;\;\;{\rm {and}}\;\;\;x_{2}(p_{1},p_{2},I)={\frac {.4I}{p_{2}}},

where $I$ is the consumer's income. One way to find the expenditure function is to first find the indirect utility function and then invert it. The indirect utility function $v(p_{1},p_{2},I)$ is found by replacing the quantities in the utility function with the demand functions thus:

v(p_{1},p_{2},I)=u(x_{1}^{*},x_{2}^{*})=(x_{1}^{*})^{.6}(x_{2}^{*})^{.4}=\left({\frac {.6I}{p_{1}}}\right)^{.6}\left({\frac {.4I}{p_{2}}}\right)^{.4}=(.6^{.6}+.4^{.4})I^{.6+.4}p_{1}^{-.6}p_{2}^{-.4}=Kp_{1}^{-.6}p_{2}^{-.4}I,

where $K=(.6^{.6}+.4^{.4}).$ Then since $e(p_{1},p_{2},u)=e(p_{1},p_{2},v(p_{1},p_{2},I))=I$ when the consumer optimizes, we can invert the indirect utility function to find the expenditure function:

e(p_{1},p_{2},u)=(1/K)p_{1}^{.6}p_{2}^{.4}u,

Alternatively, the expenditure function can be found by solving the problem of minimizing $(p_{1}x_{1}+p_{2}x_{2})$ subject to the constraint $u(x_{1},x_{2})\geq u^{*}.$ This yields conditional demand functions $x_{1}^{*}(p_{1},p_{2},u^{*})$ and $x_{2}^{*}(p_{1},p_{2},u^{*})$ and the expenditure function is then

e(p_{1},p_{2},u^{*})=p_{1}x_{1}^{*}+p_{2}x_{2}^{*}

References

Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.
Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton., pp. 111, has the general formula.

Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (2007). Microeconomic Theory. pp. 59–60. ISBN 0-19-510268-1.
Mathis, Stephen A.; Koscianski, Janet (2002). Microeconomic Theory: An Integrated Approach. Upper Saddle River: Prentice Hall. pp. 132–133. ISBN 0-13-011418-9.
Varian, Hal R. (1984). Microeconomic Analysis (Second ed.). New York: W. W. Norton. pp. 121–123. ISBN 0-393-95282-7.

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[1] Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.

[2] Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton., pp. 111, has the general formula.

Expenditure function

Expenditure and indirect utility

Example

See also

References