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Sensitivity Analysis and Shadow Prices 725

In closing this section, it is worth making one further point about the rela-

tionship between marginal analysis and the optimal solutions of linear pro-

grams. Earlier we saw that a new activity is excluded (its quantity is set equal to

zero) if its unit benefit is less than its unit cost. What about activities that are

includedin the optimal solution? Recall that both standard and economy com-

puters are part of the PC firm’s optimal production mix. The marginal bene-

fit of producing an extra standard model is $500 (its contribution). Using the

resource shadow prices, its marginal cost is computed as

.

In the optimal solution, marginal benefit and marginal cost are identical.

Similarly, for the economy model, marginal benefit is $300 and marginal cost is

.

Again marginal benefit and marginal cost are identical. The following general

result holds for any linear program:

For any decision variable that is positive in the optimal solution, its marginal ben-

efit equals its marginal cost, where the latter is computed according to the resource

shadow prices.

Thus, once again we find that the relationship, MMB MC 0, holds at

the optimum solution.

(40)(5)(1)(0)(5)(20)$300

(80)(5)(1)(0)(5)(20)$500

CHECK

STATION 4

The farmer considers planting a third crop, soybeans. The price of soybeans is $1.75

per bushel. Growing 1,000 bushels of soybeans requires 2 acres of land and 4 hours of

labor per week. Is soybean production profitable? Explain.

Allocating HIV

Resources

In the fall of 1999, the Centers for Disease Control and Prevention called for

a panel of health scientists, economists, and policy experts to formulate a

framework and strategy for HIV prevention in the United States. A member

of that panel, Edward Kaplan of Yale University, has described the work of

the panel in analyzing the problem of HIV prevention.^7 The panel con-

cluded that a key goal was “to prevent as many new HIV infections as possi-

ble within the resources available for HIV prevention.” Although this broad

goal might seem obvious, current health-care measures often pursue other

ends. As Kaplan noted, formulating the problem as a constrained maxi-

mization problem was, at least at first, foreign to many on the panel who

were not economists.

(^7) E. Kaplan, “Allocating HIV Resources,” OR/MS Today(February 2001): 26–29.
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