The Lagrangian multiplier interpretation
Motivation. Consider the problem of utility maximization under the budget constraint Suppose the budget changes a little bit and let us see how the solution of the problem depends on changes in the budget. The maximized value of the utility will depend on , so let us reflect this dependence in the notation where, for each , is the maximizing bundle. The value of the Lagrangian multiplier in the FOC's , , will also depend on : The property that we will prove is
that is, the Lagrangian multiplier measures the sensitivity of the maximized utility function to changes in the budget.
The main problem is to maximize subject to . However, instead of a fixed constraint we consider a set of constraints perturbed by a constant : Here varies in a small neighborhood of zero (which corresponds to varying in a neighborhood of some in the motivating example). Now everything depends on , and differentiation with respect to will give the desired result.
Employing the constraint
As before, we assume the implicit function existence condition:
Changing the notation, if necessary, we can think that it is the last component of the gradient that is not zero: This condition applies to the perturbed constraint too and guarantees existence of the implicit function, which now depends also on : . Plugging it into the constraint we get , and differentiation yields
Employing the FOC's
Critical assumption. For each the perturbed maximization problem has a solution. At least in our motivating example, this assumption is satisfied.
The maximized objective function is denoted At each we have the right to use the FOC's for the Lagrangian. One of the FOC's is
To make use of (1), multiply this by :
It follows that
With we have , the Lagrange multiplier for the unperturbed problem, and as a result