Question

What is the relationship between the expenditure function and the Hicksian demand function?

Answer 1

This Hicksian Demand function is simply the solution to the cost/expenditure minimization problem subject to a target utility level. Hicksian demand function gives the optimal consumption bundle x*_i that not only achieves the target utility level but also minimizes the firm's expenditure. Once we obtain the optimal bundle, we can plug these back to our expenditure function, and this would give us the minimum expenditures required to obtain target utility given the prices and income.

Consider the expenditure minimization problem below: min E(x,y) = p_xx+p_yy subject to u(x,y) = u

Solving for this minimization problem will give us the hicksian demand function x*(p_x, p_y , u) and y*(p_x, p_y , u). Finally, when we substitute the hicksian demand function back to the expenditure function : E(x*(p_x, p_y , u),y*)(p_x, p_y , u) = p_xx*(p_x, p_y​ , u) + p_y y*(p_x, p_y​ , u), we will get our minmum expenditure required to obtain are target utility.