## Finding Marginal Revenue Function from Ave rageRevenue Function

If we are given an average-revenue (AR) function in specific form,

the marginal-revenue (MR) function can be found by first multiplying AR by Q lo get the total-revenue (R) function:

and then differentiating R;

But if the AR function is given in the general form AR = /(0, then the total-revenue function will also be in a general form:

and therefore the "multiply out" approach will be to no avail. However, because R is a product of two functions of Q, namely, f(Q) and Q itself, the product rule can be put to work.

Thus we can differentiate R to get the MR function as follows:

However, can such a general result tell us anything significant about the MR? Indeed it can. Recalling that f(Q) denotes the AR function, let us rearrange (7.7) and write

This gives us an important relationship between MR and AR: namely, they will always differ by the amount Qf'(Q).

It remains to examine the expression Qf{Q). Its first component Q denotes output and is always nonncgativc. The other component, f\Q)„ represents the slope of the AR curve plotted against Q> Sincc "average revenue" and "price" are but different names for the same thing:

the AR curve can also be regarded as a curve relating price P to output Q: P = f{Q)-Viewed in this light, the AR curve is simply the inverse of the demand curve for the product of the firm, i.e., the demand curve plotted after the P and Q axes are reversed. Under pure competition, the AR curve is a horizontal straight line, so that f(Q) — 0 and, from (7.7'). MR — AR = 0 for all possible values of Q. Thus the MR curve and the AR curve must coincide. Under imperfect competition, on the other hand, the AR curve is normally downward-sloping, as in Fig. 7.2, so that f{Q) < 0 and, from {7.7'), MR - AR < 0 for all positive levels of output. In this case, the MR curve must lie below the AR curve.

The conclusion just stated is qualitative in nature; it concerns only the relative positions of the two curves. But (l.T) also furnishes the quantitative information that the MR curvo will fall short of the AR curve at any output level Q by precisely the amount Qf'(Q). Let us look at Pig. 7.2 again and consider the particular output level /VT For that output, the

FIGURE 7.2

FIGURE 7.2 expression Qf\Q) specifically becomes Nf'(N): if we can find the magnitude of Nf'(N) in the diagram, we shall know how far below the average-revenue point G the corresponding marginal-revenue point must lie.

The magnitude of Wis already specified. And f{N) is simply the slope of the AR curve at point G (where Q — AT), that is, the slope of the tangent line JM measured by the ratio of two distances OJ/OM.. However, wc see that OJ/OM — HJ/HG: besides, distance HG is precisely the amount of output under consideration. /V, Thus the distance jV/'{ AO, by which the MR curve must lie below7 the AR curve at output N, is

Accordingly, if we mark a vertical distance KG = HJ directly below point 6', then point K must be a point on the MR curve. (A simple way of accuratcly plotting KG is to draw a straight line passing through point//and parallel toVC; points is where that line intersects the vertical line t\G>)

The ¡same procedure can be used to locate other points on the MR curve. A11 wc must do. for any chosen point G' on the curve, is first to draw a tangent to the AR curve at G' that will meet the vertical axis at some point./'. Then draw a horizontal line from G' to the vertical axis, and label the intersection with the axis as H\ 11 we mark a vertical distance K'G' = H'J' directly below point G\ then the point Kf will be a point on the MR curve. This is the graphical wray of deriving an MR curve from a given AR curvc. Strictly speaking. the accurate drawing of a tangent line requires a knowledge of the value of the derivative at the relevant output that is, f!(N): hence the graphical method just outlined cannot quite exist by itself. An important exception is the case of a linear A R curve, where the tangent to any point on the curve is simply the given line itself so that there is in effect no need to draw any tangent at all. Then the graphical method wall apply in a straightforward way 