Editing Expenditure function (section)

==Example==

Suppose the utility function is the [[Cobb-Douglas function]] <math>u(x_1, x_2) = x_1^{.6}x_2^{.4},</math> which generates the demand functions<ref>{{cite book |last=Varian |first=H. |year=1992 |title=Microeconomic Analysis |url=https://archive.org/details/microeconomicana00vari_0 |url-access=registration |edition=3rd |location=New York |publisher=W. W. Norton }}, pp. 111, has the general formula. </ref>   
:<math>  x_1(p_1, p_2, I) = \frac{ .6I}{p_1} \;\;\;\; {\rm and}\;\;\; x_2(p_1, p_2, I) = \frac{ .4I}{p_2},  </math>
where <math>I</math> 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 <math>v(p_1, p_2, I) </math> is found by replacing the quantities in the utility function with the demand functions thus: 

:<math> v(p_1, p_2,I)  =   u(x_1^*, x_2^*) = (x_1^*)^{.6}(x_2^*)^{.4}  =  \left( \frac{ .6I}{p_1}\right)^{.6}  \left( \frac{ .4I}{p_2}\right)^{.4}  = (.6^{.6} \times .4^{.4})I^{.6+.4}p_1^{-.6}  p_2^{-.4}  = K   p_1^{-.6}  p_2^{-.4}I, </math> 

where <math>K = (.6^{.6} \times .4^{.4}). </math>  Then since <math>e(p_1, p_2, u) = e(p_1, p_2, v(p_1, p_2, I)) =I</math>  when the consumer optimizes, we can invert the indirect utility function to find the expenditure function: 
:<math> e(p_1, p_2, u) =  (1/K)  p_1^{.6}  p_2^{.4}u, </math> 

Alternatively, the expenditure function can be found by solving the problem of minimizing <math>(p_1x_1+ p_2x_2)</math>  subject to the constraint  <math>u(x_1, x_2) \geq u^*.</math>  This yields conditional demand functions  <math>x_1^*(p_1, p_2, u^*)</math>  and <math>x_2^*(p_1, p_2, u^*)</math>  and the expenditure function is then 
: <math>e(p_1, p_2, u^*) = p_1x_1^*+ p_2x_2^*</math>