Sensitivity Analysis and Shadow Prices 721Shadow Prices
Let’s return to the original version of the computer firm’s problem.
Management is operating according to its optimal production plan: 100 stan-
dard models and 300 economy models per week, which together generate
$140,000 in contribution. At this solution, production uses 100 percent of hard-
disk capacity and all of the firm’s current labor supply. This prompts some nat-
ural questions for management to contemplate: How much would profits
increase by increasing hard-disk capacity? What about by increasing the labor
force? As we shall see, the notion of shadow prices for resources provides the
answers to these questions.
The shadow priceof a resource measures the change in the value of the
objective function associated with a unit change in the resource. To illustrate,
let’s compute the shadow price associated with hard-disk capacity. Suppose the
firm increases this capacity from 20,000 to 22,000. Figure 17.5 shows the capac-
ity increase as a rightward shift in the hard-disk constraint line. With the
increase in capacity, point C moves to the southeast. Nonetheless, (the newly
positioned) point C remains the optimal corner; that is, the firm should con-
tinue to utilize all of its disk capacity and labor. The hard-disk and labor con-
straints are 80S 40E 22,000 and 5S 5E 2,000, respectively. Solving
these as binding constraints, we find the optimal production plan to be S 150
and E 250. The firm’s new contribution is (500)(150) (300)(250)
$150,000. The 2000-unit increase in disk capacity has resulted in a $10,000
profit increase. Thus, the shadow price of an extra unit of capacity is
10,000/2000 $5.
To find the shadow price associated with additional hours of labor, we set
up an analogous calculation. Suppose the firm expands its labor force to 2,100
hours per week. The binding constraint equations are now 5S 5E 2,100
and 80S 40E 20,000. Thus, the optimal production plan is S 80 and
E 340. The new contribution is (500)(80) (300)(340) $142,000. The
addition of 100 labor-hours per week increases contribution by $2,000.
(Remember, the old contribution was $140,000.) Therefore, the shadow price
of labor (per hour) is 2,000/100 $20.
Each individual resource constraint has a shadow price. Each shadow price
measures the change in the objective function from a change in that resource
alone,that is, with the amounts of all other resources held constant. As is usual in
sensitivity analyses, we trace the impact of one effect at a time. Also, each shadow
price is constant as long as the same constraints are binding in the optimal solution.ThisCHECK
STATION 2How will the farmer’s mix of crops be affected if the price of wheat increases to $2.25?
If it falls to $.90? What if both crop prices fall by 15 percent? How high would the ratio
PW/PBhave to be to induce the farmer to produce only wheat? How low would the ratio
have to be for him to produce only barley?c17LinearProgramming.qxd 9/26/11 11:05 AM Page 721