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. 10/14/20152

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? Solution Figure 2.47: Costs and revenues 10/14/20153

Figure 2.48: Marginal cost: Slope of one of these lines 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. 10/14/20154 Marginal Analysis

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 10/14/20155

Sample Problem (page 120 # 12) Cost and revenue functions for a charter bus company are shown in Figure Should the company add a 50 th bus? How about a 90 th ? Explain your answers using marginal revenue and marginal cost. Figure /14/20156 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.