M 112 Short Course in Calculus
Chapter 2 – Rate of Change: The Derivative
Sections 2.5 – Marginal Cost and Revenue
V. J. Motto
Cost and Revenue
Management decisions within a particular firm or industry usually depend on the costs and
revenues involved. In this section we look at the cost and revenue functions.
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Example 1
If cost, C, and revenue, R, are given by the graph in Figure 2.47, for what
production quantities does the firm make a profit?
Figure 2.47:
Costs and revenues
Solution
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Marginal Analysis
Let C(q) be the function giving the cost of running q flights.
If the airline had originally planned to run 100 flights, its costs would be C(100).
With the additional flight, its costs would be C(101).
Therefore, Marginal cost = C(101) – C(100).
Now
and this quantity is the average rate of change of cost between 100 and 101 flights.
Figure 2.48: Marginal cost: Slope of one of these lines
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In Figure 2.48 the average rate of change is the slope of the secant line. If the graph of the
cost function is not curving too fast near the point, the slope of the secant line is close to
the slope of the tangent line there. Therefore, the average rate of change is close to the
instantaneous rate of change. Since these rates of change are not very different, many
economists choose to define marginal cost, MC, as the instantaneous rate of change of cost
with respect to quantity:
Figure 2.48: Marginal cost: Slope of one of these lines
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Sample Problem (page 120 # 12)
Cost and revenue functions for a charter bus company are shown in Figure
2.57. Should the company add a 50th bus? How about a 90th ? Explain your
answers using marginal revenue and marginal cost.
Solution
At q = 50, the slope of the revenue is larger
than the slope of the cost. Thus, at q = 50,
marginal revenue is greater than marginal
cost and the 50th bus should be added. At q
= 90 the slope of revenue is less than the
slope of cost. Thus, at q = 90 the marginal
revenue is less than marginal cost and the
90th bus should not be added.
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Figure 2.57
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