Let us say you are a person trying to choose between buying bananas and oranges. What you are trying to do is maximize a utility function u(x,y) where x represents bananas and y represents oranges. You can not buy an infinite amount of each however. This is subject to a budget constraint. Thus, we have the following maximization problem.
- max u(x,y) s.t. p1x + p2y ≤ I
If we add functional form assumptions on the utility function we can form the following Lagrangian:
- L= ln(x) + αln(y) – λ[p1x + p2y – I]
Our first order conditions are:
- Lx: 1/x – λp1 =0
- Ly: α/y – λp2 =0
- Lλ: p1x + p2y – I =0
Our optimal level of bananas and oranges is:
- x* = I/[(1+α)p1]
- y* = Iα/[(1+α)p2]
- λ* = (1+α)/I
We solved for x* (bananas), y* (oranges), and λ*, but what the heck is λ? The term λ is the shadow price. The shadow price represents the following: assume that instead of I dollars of income, we had I+ε dollars. If we had the extra ε of income, the shadow price λ tells us by how much the objective function (utility function) would increase since we could buy a few more apples and bananas.