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introduction to mathematical economics
ATC, MC MC
ATC
2.14 The relationship between average total cost and marginal cost.
FIGURE
0
Q1 Q 2 Q3
Q4 Q5
Q
At output level Q4 the slopes of the ray and tangent are identical (ATC = MC). Thus, at Q4 ATC is neither rising nor falling (i.e., dATC/dQ = 0). After Q4 the slope of the tangent not only becomes greater than the slope of the ray, but the slope of the ray changes direction and starts to increase. Thus, we see that at output level Q5, MC > ATC and ATC are rising. These relationships are illustrated in Figure 2.14. The situation depicted in Figure 2.14 illustrates a U-shaped average total cost curve in which the MC intersects ATC from below. The reader should visually verify that when MC < ATC, even when MC is rising, ATC is falling. Moreover, when MC > ATC, then ATC is rising. Finally, when MC = ATC, then ATC is neither rising nor falling (i.e., ATC is minimized). In some cases, the average curve is shaped not like U but like a hill: that is, the marginal curve intersects the average curve from above at its maximum point. An example of this would be the relationship between the average and marginal physical products of labor, which will be discussed in detail in Chapter 5.
PROFIT MAXIMIZATION: THE FIRST-ORDER CONDITION We are now in a position to use the rules for taking first derivatives to find the level of output Q that maximizes p, as illustrated in Table 2.3. Consider again the total revenue and total cost functions introduced earlier: TR(Q) = PQ; P = $18 TC (Q) = 6 + 33Q - 9Q 2 + Q 3 p = TR - TC = 18Q - (6 + 33Q - 9Q 2 + Q 3 ) p = -6 - 15Q + 9Q 2 - Q 3
(2.67)
