As an economic application of the quotient rule, let us consider the rate of change of average cost when output varies.
Given a total-cost function C = C(Q), the average-cost (AC) function is a quotient of two functions of Q> since AC = C(Q)/Q, defined as long as Q > 0. Therefore, the rate of change of AC with respect to Q can be found by differentiating AC:
From this it follows that, for Q > 0, d C(Q)> dQ Q <
Since the derivative C( Q) represents the marginal-cost (MC) function, and C(Q)/Q represents the AC function, the economic meaning of (7.10) is: The slope of the AC
FIGURE 73
FIGURE 73
curve will be positive, zero, or negative if and only if the marginal-cost curve lies above, intersects, or lies below the AC curve. This is illustrated in Fig. 7.3, where the MC and AC functions plotted are based on the specific total-cost function
To the left of Q = 6, AC is declining, and thus MC lies below it; to the right, the opposite is true. At Q = 6, AC has a slope of zero, and MC and AC have the same value.*
The qualitative conclusion in (7.10) is stated explicitly in terms of cost functions. However, its validity remains unaffected if we interpret C{Q) as any other differentiate total function, with C(Q)/Q and C'(Q) as its corresponding average and marginal functions. Thus this result gives us a general marginal-average relationship. In particular, we may point out, the fact that MR lies below AR when AR is downward-sloping, as discussed in connection with Fig. 7.2, is nothing but a special case of the general result in (7.10).
f Note that (7.10) does not state that, when AC is negatively sloped, MC must also be negatively sloped; it merely says that AC must exceed MC in that circumstance. At Q = 5 in Fig. 7.3, for instance, AC is declining but MC is rising, so that their slopes will have opposite signs-
1. Given the total-cost function C - Q3 - 5 Q2 + 12 Q + 75, write out a variable-cost (VC) function. Find the derivative of the VC function, and interpret the economic meaning of that derivative.
2, Given the average-cost function AC = Q2 - 4Q + 174, find the MC function. Is the given function more appropriate as a long-run or a short-run function? Why?
3. Differentiate the following by using the product rule;
(a) (9x2-2)(3x~1} (c) xz(4x + 6) (<?) (2 - 3x}(1 + *)(* +2)
(b) (3x + 10)(6x2 - 7x) (d) (ax - b){cxl) (f) (x2 - 3)x 1
4. (a) Given AR = 60 - 3Q, plot the average-revenue curve, and then find the MR curve by the method used in Fig. 7.2. (fa) Find the total-revenue function and the marginal-revenue function mathematically from the given AR function.
(c) Does the graphically derived MR curve in {a) check with the mathematically derived MR function in (fa)?
(d) Comparing the AR and MR functions, what can you conclude about their relative slopes?
5. Provided mathematical proof for the general result that 9'vsn a linear average curve, the corresponding marginal curve must have the same vertical intercept but will be twice as steep as the average curve,
6. Prove the result in (7,6) by first treating gMftOO as a single function, g(x)h(x) = </>(x), and then applying the product rule (7+4),
7. Find the derivatives of:
8. Given the function f(x) -ox^bf find the derivatives of:
10. Find the marginal and average functions for the following total functions and graph the results. Total-cost function: (o) C-3Q2 + 7Q+12 Total-revenue function:
(b) R^XOQ-Q2 Total-product function:
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