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 ${\displaystyle u}$ that describes preferences over n commodities, the expenditure function

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

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

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

where

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

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

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

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

Features of Expenditure Functions

(Properties of the Expenditure Function) Suppose u is a continuous utility function representing a locally non-satiated preference relation º on Rn +. Then e(p, u) is

1.   Homogeneous of degree one in p: for all and ${\displaystyle \lambda >0}$, ${\displaystyle e(\lambda p,u)=\lambda e(p,u);}$

2.   Continuous in ${\displaystyle p}$ and ${\displaystyle u;}$

3.   Nondecreasing in ${\displaystyle p}$ and strictly increasing in ${\displaystyle u}$ provided ${\displaystyle p\gg 0;}$

4.   Concave in ${\displaystyle p}$

5. If the utility function is strictly quasi-concave, there is the Shephard's lemma

Proof

(1) As in the above proposition, note that

${\displaystyle e(\lambda p,u)=\min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}$ ${\displaystyle \lambda p\cdot x=\lambda \min _{x\in \mathbb {R} _{+}^{n}:u(x)\geq u}}$ ${\displaystyle p\cdot x=\lambda e(p,u)}$

(2) Continue on the domain ${\displaystyle e}$: ${\displaystyle {\textbf {R}}_{++}^{N}*{\textbf {R}}\rightarrow {\textbf {R}}}$

(3) Let ${\displaystyle p^{\prime }>p}$ and suppose ${\displaystyle x\in h(p^{\prime },u)}$. Then ${\displaystyle u(h)\geq u}$, and ${\displaystyle e(p^{\prime },u)=p^{\prime }\cdot x\geq p\cdot x}$ . It follows immediately that ${\displaystyle e(p,u)\leq e(p^{\prime },u)}$.

For the second statement , suppose to the contrary that for some ${\displaystyle u^{\prime }>u}$, ${\displaystyle e(p,u^{\prime })\leq e(p,u)}$ Than, for some ${\displaystyle x\in h(p,u)}$, ${\displaystyle u(x)=u^{\prime }>u}$, which contradicts the "no excess utility" conclusion of the previous proposition

(4)Let ${\displaystyle t\in (0,1)}$ and suppose ${\displaystyle x\in h(tp+(1-t)p^{\prime })}$. Then, ${\displaystyle p\cdot x\geq e(p,u)}$ and ${\displaystyle p^{\prime }\cdot x\geq e(p^{\prime },u)}$, so ${\displaystyle e(tp+(1-t)p^{\prime },u)=(tp+(1-t)p^{\prime })\cdot x\geq }$${\displaystyle te(p,u)+(1-t)e(p^{\prime },u)}$.

(5) ${\displaystyle {\frac {\delta (p^{0},u^{0})}{\delta p_{i}}}=x_{i}^{h}(p^{0},u^{0})}$

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 ${\displaystyle p}$ and income level ${\displaystyle I}$:^[1]: 106

${\displaystyle e(p,v(p,I))\equiv I}$

There is a duality relationship between expenditure function and utility function. If given a specific regular quasi-concave utility function, the corresponding price is homogeneous, and the utility is monotonically increasing expenditure function, conversely, the given price is homogeneous, and the utility is monotonically increasing expenditure function will generate the regular quasi-concave utility function. In addition to the property that prices are once homogeneous and utility is monotonically increasing, the expenditure function usually assumes

(1) is a non-negative function, i.e., ${\displaystyle E(P\cdot u)>O;}$

(2) For P, it is non-decreasing, i.e., ${\displaystyle E(p^{1}u)>E(p^{2}u),u>Op^{l}>p^{2}>O_{N}}$;

(3)E(Pu) is a concave function. That is, ${\displaystyle e(np^{l}+(1-n)p^{2})u)>\lambda E(p^{1}u)(1-n)E(p^{2}u)y>0}$ ${\displaystyle O<\lambda <1p^{l}\geq O_{N}p^{2}\geq O_{N}}$

Expenditure function is an important theoretical method to study consumer behavior. Expenditure function is very similar to cost function in production theory. Dual to the utility maximization problem is the cost minimization problem ^[2][3]

Example

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

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

where ${\displaystyle I}$ 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 ${\displaystyle v(p_{1},p_{2},I)}$ is found by replacing the quantities in the utility function with the demand functions thus:

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

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

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

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

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

See also

• Expenditure minimization problem
• Hicksian demand function
• Slutsky equation
• Utility maximization problem
• Budget constraint
• Consumption set
• Shephard's lemma

References

1. Varian, Hal (1992). Microeconomic Analysis (Third ed.). New York: Norton. ISBN 0-393-95735-7.
2. Jing ji xue da ci dian. Xiaomin Liang, 梁小民. (Di 1 ban ed.). Beijing Shi: Tuan jie chu ban she. 1994. ISBN 7-80061-954-0. OCLC 34287945.{{cite book}}: CS1 maint: others (link)
3. "CONSUMER CHOICE AND DUALITY" (PDF). 23 February 2024.
4. Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton., pp. 111, has the general formula.

Further reading

• Mas-Colell, Andreu; Whinston, Michael D.; Green, Jerry R. (2007). Microeconomic Theory. pp. 59–60. ISBN 978-0-19-510268-0.
• 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.