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

Formally, if there is a utility function <math>u</math> that describes preferences over L commodities, the expenditure function

<math>e(p, u^*) : \textbf R^L_+ \times \textbf R

\rightarrow \textbf R</math>

says what amount of money is needed to achieve a utility <math>u^*</math> if prices are set by <math>p</math>.
This function is defined by

<math>e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x</math>

where

<math>\geq(u^*) = \{x \in \textbf R^L_+ : u(x) \geq u^*\}</math>

is the set of all packages that give utility at least as good as <math>u^*</math>.

See also

• Expenditure minimization problem
• Hicksian demand function
• Utility maximization problem

The Personal Consumption Expenditure (PCE) is a price index, like the Consumer Price Index.

A measure of price changes in consumer goods and services.

It consists of the actual and imputed expenditures of households and includes data pertaining to durables, non-durables, and services. It is essentially a measure of goods and services targeted towards individuals and consumed by individuals.

Also referred to as “consumption”.

Similar to the consumer price index (CPI), PCE is a report (actually a part of the personal income report) put out by the Bureau of Economic Analysis of the Department of Commerce.

There are two broad indexes of consumer prices in the United States: the CPI and the chain price index for personal consumption expenditures (PCEPI). They are similar in many respects, but there are some important differences which can lead to large gaps between CPI and PCEPI inflation rates at times. The PCEPI uses a chain index which takes into account consumers’ changing consumption due to prices, while the CPI uses a fixed basket of goods with weightings that do not change over time.

The PCE is a fairly predictable report that has little impact on the markets.

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

Formally, if there is a utility function <math>u</math> that describes preferences over L commodities, the expenditure function

<math>e(p, u^*) : \textbf R^L_+ \times \textbf R

\rightarrow \textbf R</math>

says what amount of money is needed to achieve a utility <math>u^*</math> if prices are set by <math>p</math>.
This function is defined by

<math>e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x</math>

where

<math>\geq(u^*) = \{x \in \textbf R^L_+ : u(x) \geq u^*\}</math>

is the set of all packages that give utility at least as good as <math>u^*</math>.

See also

• Expenditure minimization problem
• Hicksian demand function
• Utility maximization problem

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In microeconomics, the expenditure minimization problem is the dual problem to the utility maximization problem: “how much money do I need to be happy?”. This question comes in two parts. Given a consumer’s utility function, prices, and a utility target,

• how much money would the consumer need? This is answered by the expenditure function.
• what could the consumer buy to meet this utility target while minimizing expenditure? This is answered by the Hicksian demand correspondence.

Expenditure function

Formally, the expenditure function is defined as follows. Suppose the consumer has a utility function <math>u</math> defined on <math>L</math> commodities. Then the consumer’s expenditure function gives the amount of money required to buy a package of commodities at given prices <math>p</math> that give utility greater than <math>u^*</math>,

<math>e(p, u^*) = \min_{x \in \geq{u^*}} p \cdot x</math>

where

<math>\geq{u^*} = \{x \in \textbf R^L_+ : u(x) \geq u^*\}</math>

is the set of all packages that give utility at least as good as <math>u^*</math>.

Hicksian demand correspondence

Secondly, the Hicksian demand correspondence <math>h(p, u^*)</math> is defined as the cheapest package that gives the desired utility. It can be defined in terms of the expenditure function with the Marshallian demand correspondence

<math>h(p, u^*) = x(p, e(p, u^*)).</math>

If the Marshallian demand correspondence <math>x(p, w)</math> is a function (i.e. always gives a unique answer), then <math>h(p, u^*)</math> is also called the Hicksian demand function.

See also

• Utility maximization problem

References

• Mas-Colell, Andreu; Whinston, Michael; & Green, Jerry (1995). Microeconomic Theory. Oxford: Oxford University Press. ISBN 0-19-507340-1

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

Formally, if there is a utility function <math>u</math> that describes preferences over L commodities, the expenditure function

<math>e(p, u^*) : \textbf R^L_+ \times \textbf R

\rightarrow \textbf R</math>

says what amount of money is needed to achieve a utility <math>u^*</math> if prices are set by <math>p</math>.
This function is defined by

<math>e(p, u^*) = \min_{x \in \geq(u^*)} p \cdot x</math>

where

<math>\geq(u^*) = \{x \in \textbf R^L_+ : u(x) \geq u^*\}</math>

is the set of all packages that give utility at least as good as <math>u^*</math>.

See also

• Expenditure minimization problem
• Hicksian demand function
• Utility maximization problem

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