As we saw in the chapter, utility maximization—
Let’s demonstrate the equivalence of utility maximization and expenditure minimization using Antonio’s utility function from the Figure It Out in the chapter and the Lagrangian method (the first approach is identical to that for utility maximization except that you plug into the utility constraint instead of the budget constraint in the last step). We write out Antonio’s expenditure-
or
153
As before, solve for the first-
Then solve for λ in the first two conditions:
λ0.5B–0.5F0.5 = 5
λ = 10B0.5F–0.5
λ0.5B0.5F–0.5 = 2
λ = 4B–0.5F0.5
Set the two expressions for λ equal to each other and solve for F as a function of B:
λ = 4B–0.5F0.5 = 10B0.5F–0.5
4F0.5F0.5 = 10B0.5B0.5
F = 2.5B
Now substitute F as a function of B into the utility constraint:
100.5 = B0.5F0.5 = B0.5(2.5B)0.5 = (2.5)0.5B0.5B0.5
F* = 2.5B* = 2.5(2) = 5
This optimal bundle of goods costs Antonio
5B* + 2F* = 5(2) + 2(5) = $20
the minimum expenditure needed to achieve 100.5 units of utility.
Expenditure minimization is a good check of our cost-