Economics - General

Shadow Price

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.

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