The Derivative and the Slope of a Curve

Elementary economics tells us that, given a total-cost function C = /(£?), where C denotes total cost and Q the output, the marginal cost (MC) is defined as the change in total cost resulting from a unit increase in output; that is. MC = AC/AQ. It is understood that A Q is an extremely small change. For the case of a product that has discrete units (integers only), a change of one unit is the smallest change possible; but for the case of a product whose quantity is a continuous variable, AQ can refer to an infinitesimal change. In this latter case, it is well known that the marginal cost can be measured by the slope of the total-cost curve. But the slope of the total-cost curve is nothing but the limit of the ratio AC/AQy when AQ approaches zero. Thus the concept of the slope of a curve is merely the geometric counterpart of the concept of the derivative. Both have to do with the "marginal" notion so extensively used in economics.

In Pig, 6.1, we have drawn a total-cost curve C which is the graph of the (primitive) function C — Suppose that we consider as the initial output level from which an increase in output is measured; then the relevant point on the cost curve is the point A. If output is to be raised to Qa + AQ = Qi, the total cost will be increased from Co to C0 + AC = C2; thus AC/AQ = {C2 - Q})/(Q2 ~ Qq). Geometrically, (his is the ratio of two line segments, EBjAE? or the slope of the line AB. This particular ratio measures an average rate of change—the average marginal cost for the particular AQ pictured- and represents a difference quotient. As such, it is a function of the initial value Qo and the amount of change A

What happens when we vary the magnitude of A£>? If a smaller output increment is contemplated (say, from Qo to Q\ only), then the average marginal cost will be measured by the slope of the line AD instead. Moreover, as we reduce the output increment further and further, flatter and flatter lines will result until, in the limit (as A Q ^ 0), we obtain the line KG (which is the tangent line to the cost curve at points) as the relevant line. The slope

FIGURE 6.1 C

FIGURE 6.1 C

of KG (— HG/K H) measures the slope of the total-cost curve at point A and represents the limit of AC/ AQ, as AQ 0, when initial output is at Q — Qci. Therefore, in terms of the derivative, the slope of the C = /(Q) curve at point A corresponds to the particular derivative value /'(£?o)-

What if the initial output level is changed from Qo to, say, Q2? In that case, point B on the curve will replace point A as the relevant point, and the slope of the curve at the new point B will give us the derivative value /'(&)■ Analogous results are obtainable for alternative initial output levels. In general, the derivative /'((?) a function of Q—will vary as Q changes.

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