Utility-expenditure duality, Microeconomics

Utility-Expenditure Duality:

Consider the minimisation of the  expenditures necessary to achieve a specified utility level. The solution for qi yields the compensated demand functions. If the solutions for qi are substituted in 491_Utility-Expenditure Duality.png one obtains the expenditure function E (p1,...,pn,U0), which gives the minimum expenditure necessary to achieve a given utility level. It is easy to show that E is homogeneous of degree one in prices and monotonically increasing in U0. It can also be shown that the expenditure function corresponding to a regular strictly quassi-concave utility function admitting no satiation is concave in prices. Finally, Shephard's lemma states  that the partial derivative of E with respect to the ith price is the ith compensated demand function. This can be shown as follows:   

1329_Utility-Expenditure Duality1.png

But the compensated demands are obtained by minimising expenditures for a given utility level U0; hence the change in total expenditures that is due to a small change in a price is zero. It follows that the second term above is zero and 

1028_Utility-Expenditure Duality2.png

The duality between utility and expenditure functions is formally identical to the duality between production and cost functions.

Posted Date: 10/26/2012 4:28:41 AM | Location : United States

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