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  one obtains the expenditure function E (p₁,...,p_n,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:   

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 

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