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 2*r*%.

Since , the variance would increase by

So the answer is 390+90=480.

Mon Aug 24 14:26:21 EDT 1998