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

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

edit
(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  ,  
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

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

edit

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 

 

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

edit

Suppose the utility function is the Cobb-Douglas function   which generates the demand functions[4]

 

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:

 

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

edit

References

edit
  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).
  4. ^ Varian, H. (1992). Microeconomic Analysis (3rd ed.). New York: W. W. Norton., pp. 111, has the general formula.

Further reading

edit