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## Economic Interpretation

The values have an important economic interpretation: If the right hand side of Constraint i is increased by , then the optimum objective value increases by approximately .

In particular, consider the problem

Maximize p(x)

subject to

g(x)=b,

where p(x) is a profit to maximize and b is a limited amount of resource. Then, the optimum Lagrange multiplier is the marginal value of the resource. Equivalently, if b were increased by , profit would increase by . This is an important result to remember. It will be used repeatedly in your Managerial Economics course.

Similarly, if

Minimize c(x)

subject to

d(x)=b,

represents the minimum cost c(x) of meeting some demand b, the optimum Lagrange multiplier is the marginal cost of meeting the demand.

In Example 1.1.2

Minimize subject to ,

if we change the right hand side from 1 to 1.05 (i.e. ), then the optimum objective function value goes from to roughly If instead the right hand side became 0.98, our estimate of the optimum objective function value would be      The first two constraints give , which leads to and cost of . The Hessian matrix is positive definite since a;SPMgt;0 and b;SPMgt;0. So this solution minimizes cost, given a,b,Q.

If Q increases by r%, then the RHS of the constraint increases by and the minimum cost increases by . That is, the minimum cost increases by 2r%. Since , the variance would increase by So the answer is 390+90=480.   Michael A. Trick
Mon Aug 24 14:26:21 EDT 1998