In
microeconomics
Microeconomics is a branch of mainstream economics that studies the behavior of individuals and firms in making decisions regarding the allocation of scarce resources and the interactions among these individuals and firms. Microeconomics fo ...
, the expenditure function gives the minimum amount of money an individual needs to spend to achieve some level of
utility
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
, given a
utility function
As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...
and the prices of the available goods.
Formally, if there is a utility function
that describes preferences over ''n '' commodities, the expenditure function
:
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
:
where
:
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
:
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
>0,
:2. Continuous in
and
:3. Nondecreasing in
and strictly increasing in
provided
:4. Concave in
:5. If the utility function is strictly quasi-concave, there is the
Shephard's lemma
Shephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point ...
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
.
(5)
Expenditure and indirect utility
The expenditure function is the inverse of the
indirect utility
__NOTOC__
In economics, a consumer's indirect utility function
v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w. It reflects both the consumer's preferences and market co ...
function when the prices are kept constant. I.e, for every price vector
and income level
:
:
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
Example
Suppose the utility function is the
Cobb-Douglas function which generates the demand functions
[, pp. 111, has the general formula. ]
:
where
is the consumer's income. One way to find the expenditure function is to first find the
indirect utility function
__NOTOC__
In economics, a consumer's indirect utility function
v(p, w) gives the consumer's maximal attainable utility when faced with a vector p of goods prices and an amount of income w. It reflects both the consumer's preferences and market con ...
and then invert it. The indirect utility function
is found by replacing the quantities in the utility function with the demand functions thus:
:
where
Then since
when the consumer optimizes, we can invert the indirect utility function to find the expenditure function:
:
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
:
See also
*
Expenditure minimization problem
In microeconomics, the expenditure minimization problem is the dual of the utility maximization problem: "how much money do I need to reach a certain level of happiness?". This question comes in two parts. Given a consumer's utility function, pr ...
*
Hicksian demand function
In microeconomics, a consumer's Hicksian demand function or compensated demand function for a good is his quantity demanded as part of the solution to minimizing his expenditure on all goods while delivering a fixed level of utility. Essentia ...
*
Slutsky equation The Slutsky equation (or Slutsky identity) in economics, named after Eugen Slutsky, relates changes in Marshallian (uncompensated) demand to changes in Hicksian (compensated) demand, which is known as such since it compensates to maintain a fixed ...
*
Utility maximization problem
Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my uti ...
*
Budget constraint
In economics, a budget constraint represents all the combinations of goods and services that a consumer may purchase given current prices within his or her given income. Consumer theory uses the concepts of a budget constraint and a preference ...
*
Consumption set
The theory of consumer choice is the branch of microeconomics that relates Preference (economics), preferences to consumption expenditures and to supply and demand, consumer demand curves. It analyzes how consumers maximize the desirability of t ...
*
Shephard's lemma
Shephard's lemma is a major result in microeconomics having applications in the theory of the firm and in consumer choice. The lemma states that if indifference curves of the expenditure or cost function are convex, then the cost minimizing point ...
References
*
*
*
{{DEFAULTSORT:Expenditure Function
Consumer theory
Expenditure