Finally, shadow prices play a crucial role in evaluating new activities. To
illustrate, suppose the firm is contemplating the production and sale of a new
super-turbo computer (T). Each unit of this model has an expected contribu-
tion of $700, contains a 120-gigabyte hard disk and one DVD drive, and
requires 10 hours of labor. Should the firm produce this model? One way to
answer this question is to formulate the new, larger LP problem as follows:Maximize:
Subject to:.
Using an LP computer program to solve this problem, we find that S 100,
E 300, and T 0. Despite the higher unit contribution, no units of the turbo
model should be produced because its assembly would require a large quantity
of “expensive” labor.
We can reach the same conclusion much more quickly using the shadow
prices from our original problem. Suppose the firm considers producing one
turbo unit, T 1. The direct benefit is simply the unit’s contribution, $700.
What is the implicit cost of this unit? Because producing the unit uses the firm’s
limited resources, the firm will be able to produce fewer units of the other mod-
els, and total contribution from those models must fall. This loss in contribution
is an opportunity cost. Measuring this cost is straightforward. Producing a sin-
gle turbo unit uses 120 gigabytes of hard-disk capacity valued at $5 per unit (its
shadow price), one unit of DVD capacity valued at $0 (remember, its shadow
price is zero), and 10 hours of labor valued at $20 each. The total cost is.
Thus, if the firm produced this turbo unit, the change in its total contribution
would be 700 800 $100. Producing a single turbo unit (or indeed any
number of units) is a losing proposition. (If the firm produced 10 units, it
would generate 10 times the loss, $1,000.) Of course, if the unit contribution
were predicted to be $900, a comparison of benefit and opportunity cost would
show that the firm should introduce the turbo PC. This benefit-cost compari-
son would not indicate how many turbo units the firm should produce. The
precise, optimal value of T can be determined only by solving the new linear
program just illustrated.
Thus, we have the following general rule:A firm can profitably introduce a new activity if and only if the activity’s direct ben-
efit exceeds its opportunity cost, where opportunity cost is the sum of the resources
used, valued at their respective shadow prices.(120)(5)(1)(0)(10)(20)$800
5S5E10T2,000
80S40E120T20,000
ST 200
500S300E700T
724 Chapter 17 Linear Programmingc17LinearProgramming.qxd 9/26/11 11:05 AM Page 724