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
that describes preferences over n commodities, the expenditure function
![{\displaystyle e(p,u^{*}):{\textbf {R}}_{+}^{n}\times {\textbf {R}}\rightarrow {\textbf {R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/abe4a5b49efa4060b1b7855a1bdad21e4b19e2a0)
says what amount of money is needed to achieve a utility
if the n prices are given by the price vector
.
This function is defined by
![{\displaystyle e(p,u^{*})=\min _{x\in \geq (u^{*})}p\cdot x}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ff25b112d0bbd7ca0e4852db959e938bc558212b)
where
![{\displaystyle \geq (u^{*})=\{x\in {\textbf {R}}_{+}^{n}:u(x)\geq u^{*}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70881d7137ac5db3e11106ecb6844a514388840d)
is the set of all bundles that give utility at least as good as
.
Expressed equivalently, the individual minimizes expenditure
subject to the minimal utility constraint that
giving optimal quantities to consume of the various goods as
as function of
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}^{*}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef07e7b88db2567559ab4516362a8088b3f84501)
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 e(\lambda p,u)=\lambda e(p,u);}](https://wikimedia.org/api/rest_v1/media/math/render/svg/34765f5995cb6dfb8c441777886e8f93619d207b)
- 2. Continuous in
and ![{\displaystyle u;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f511ca1cadf3d2f1be744433d72cab2a82f1830)
- 3. Nondecreasing in
and strictly increasing in
provided ![{\displaystyle p\gg 0;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d7add2f30695a34aff14d59f140ecb491e398f92)
- 4. Concave in
![{\displaystyle p}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81eac1e205430d1f40810df36a0edffdc367af36)
- 5. If the utility function is strictly quasi-concave, there is the
Shephard's lemma
Proof
(1) As in the above proposition, note that
(2) Continue on the domain
:
(3) Let
and suppose
. Then
, and
. It follows immediately that
.
For the second statement , suppose to the contrary that for some
,
Than, for some
,
, which contradicts the "no excess utility" conclusion of the previous proposition
(4)Let
and suppose
. Then,
and
, so ![{\displaystyle e(tp+(1-t)p^{\prime },u)=(tp+(1-t)p^{\prime })\cdot x\geq }](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8d0fde374c58ea58e9e7aa41c687a35a90256e0)
.
(5)
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
and income level
:
[1]: 106
![{\displaystyle e(p,v(p,I))\equiv I}](https://wikimedia.org/api/rest_v1/media/math/render/svg/620147caa55d5e19f8acefb3b2d4ada0cab94637)
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.,
(2) For P, it is non-decreasing, i.e.,
;
(3)E(Pu) is a concave function. That is,
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
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}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d4d56a468e309b987aea07fb51797206082af24)
where
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
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b7afdbde211d2339d0147a95721a8eacd8e41cc3)
where
Then since
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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/53670857d39523a9a0511176eb1df172d7525bed)
Alternatively, the expenditure function can be found by solving the problem of minimizing
subject to the constraint
This yields conditional demand functions
and
and the expenditure function is then
![{\displaystyle e(p_{1},p_{2},u^{*})=p_{1}x_{1}^{*}+p_{2}x_{2}^{*}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4a143afebe2cdf1f4cd7834fb38035a216d4a949)
See also
References
Further reading