9781118041581

price measures both the benefit of added capacity and the cost of reduced capac-

ity. For instance, we saw that a 2,000-unit increase in hard-disk capacity raised

profit by $10,000 (implying a shadow price of $5). By the same token, we can

confirm that having 3,000 extra units of capacity raises profit by $15,000, whereas

1,000 extra units raises profit by $5,000. In fact, a 2,000-unit drop in capacity (a

move from 20,000 to 18,000) causes a $10,000 fall in contribution (from $140,000

to $130,000). In each case, the shadow price per unit change in capacity is $5—

a constant.

Now consider what happens if we expand hard-disk capacity beyond a total

of 24,000 units—say, from 24,000 to 24,400 units. If we were to draw the new

constraint in Figure 17.5, we would find that it lies entirely outside (i.e., to the

right of) the downward-sloping labor supply constraint. Clearly, the hard-disk

constraint is no longer binding. Instead, the DVD and labor constraints are

binding, implying S 200 and 5E 5S 2,000. Thus, the optimal product mix

is S 200 and E 200. The optimal mix remains the same when hard-disk

722 Chapter 17 Linear Programming

600

500

300

400

200

100

Economy Models

0 100 200 300 400 500

Standard Models

A

B

C

D

FIGURE 17.5

The Shadow Price

of Hard Disks

If the firm can increase

its hard-disk capacity

by 2,000, it will operate

at point C and increase

its profit by $10,000.

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