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Sensitivity Analysis and Shadow Prices 723

capacity is increased beyond 24,000 or any higher amount. Any additions to

capacity beyond 24,000 go unused. What is the shadow price of each extra unit

of capacity beyond 24,000? Zero! The extra capacity has no effect on the fea-

sible region, the firm’s optimal plan, and its maximum profit. Because the

change in profit from the extra units is zero, the shadow price is zero as well.

Thus, we have demonstrated a third property concerning shadow prices:

Any resource that is not used fully in the optimal solution (i.e., has a nonbinding con-

straint) has a shadow price of zero.For example, in the original version of the prob-

lem (Figure 17.2), C is the optimal corner, where S 100 and E 300. Because

this production plan uses only 100 DVD drives, and 200 units of capacity are

available, the shadow price of DVD capacity is zero. Clearly, the firm would be

no worse off with less capacity (unless capacity were reduced below 100, in

which case the shortage of capacity would affect the optimal production plan).

To sum up, a constraint’s shadow price measures the improvement in the

objective that results from relaxing the constraint or, conversely, the decline

in the objective from tightening the constraint.

CHECK

STATION 3

For the farmer’s problem, compute the shadow prices of land and labor. How many

additional hours would the farmer have to expend before the shadow price of labor fell

to zero?

OPTIMAL DECISIONS AND SHADOW PRICES Shadow prices that emerge from

a linear program’s optimal solution measure implicit values for the firm’s lim-

ited resources. In the short run, these resources may be fixed. But in the longer

run, the firm frequently can expand or contract its resources, usually at some

cost. Shadow prices are essential for making these decisions. For instance, sup-

pose the computer producer can hire extra labor at a cost of $15 per hour

(wages plus fringe benefits). Should it do so? The answer is yes. The additional

contribution per labor-hour is $20 (simply the shadow price for labor found

earlier). Because the cost is only $15, the firm makes a net profit of 20   15 

$5 per labor-hour hired. It profitably can hire extra labor up to the point where

labor’s shadow price falls below $15. This occurs at a total labor supply of 2,500

labor-hours. At this point, the labor constraint becomes nonbinding (lies just

outside the DVD and hard-disk constraint lines); thus, its shadow price falls to

zero. Therefore, starting from 2,000 labor-hours, the firm could profit from

hiring as many as 500 extra hours.

Now, suppose the firm could engage a subcontractor to provide an extra

2,000 units of hard-disk capacity and100 hours of labor for a fixed fee of

$18,000. Should the firm accept this deal? Again, the answer is derived directly

from knowledge of the resource shadow prices. The total value to the firm of

the extra capacities is simply (2,000)(5) (100)(20) $12,000. Here, the val-

ues of the separate capacity increases (using the respective shadow prices) are

summed to arrive at the firm’s total benefit. Because this benefit is less than the

$18,000 cost, the firm should refuse the deal.

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