Carole E. Scott is a Professor of Economics at the University of West Georgia and Editor-in-Chief of B>Quest.

Nothing, perhaps, characterizes both macro- and micro-economics as well as do functions intersecting at one, optimal point. In macroeconomics the most obvious example is the intersection of aggregate demand and aggregate supply curves at the full employment level of income. In microeconomics it is the profit-maximizing level of output is where the marginal revenue (MR) and marginal cost (MC) curves intersect: MR = MC.

Marginal revenue is the first derivative of the total revenue function, that is, it is the change in total revenue resulting from the sale of an additional unit of output. Diminishing returns dictates the shape of the marginal cost curve. Diminishing marginal utility determines the shape of the marginal revenue curve .

If real world marginal revenue curves are linear, businesses will find determining the optimum (profit maximizing) price to charge comparatively simple to identify so long as neither demand nor costs change, which, of course, they not infrequently do in the real world. Discovering the optimum price would be relatively easy because all other prices would produce either lower profits or losses; therefore, by a process analogous to the one used by an artillery battery: short, too far, on target, the profit maximizing price could be ascertained. If this is not true, even if costs do not change, maximizing profits in the real may be very difficult.

As has been true decade after decade, some of today's introductory economics textbooks only utilize linear individual and market demand curves. As a result, to be mathematically correct they must, and they do, employ linear marginal revenue curves to demonstrate how a firm can discover the profit maximizing price for its product(s). [1] Other textbooks only rarely display other than linear demand and marginal revenue curves.

Little mathematical sophistication is needed to see that if the individual demand curves for a product that are summed to produce the market demand curve are linear, the market demand curve must be convex. This is because, as price rises above the intercept of a given individual demand curve that has an intercept lower than some other individuals demand curves, the market demand curve at this and higher prices will not incorporate this demand curve; so the the slope of the market demand curve will be less elastic (steeper) beyond this price. Due to differences in income and taste, you can be confident that the intercept of different buyers' demand curves will not be the same.

The wider is the range of the intercepts of a given number of linear individual demand curves with given slopes, the more convex will be the market demand curve.Any student with even a slight knowledge of geometry will realize this. So, why for decades have the authors of economics textbooks continued to treat market demand curves as being linear?

Multiple Profit Equilibria

Because a convex demand curve may have a rising marginal revenue curve, there may be multiple profit equilibria, rather than the single one typically presented in economics textbooks.

As do some introductory macroeconomics textbooks, in his intermediate-level textbook, Hal Varian's initial display of the demand curve does show it as a curve convex to the origin. [Varian, 6] Subsequently, he shows a convex marginal revenue curve accompanying a convex demand (average revenue) curve. [Varian, 275] However, when he explains that profit maximization takes place where marginal revenue (MR) equals marginal cost (MC), he displays linear demand and marginal revenue curves. [Varian, 408, 411, 412, 414] Walter E. Nicholson also displays a marginal revenue curve that is convex to the origin when the demand curve has this shape. [Nicholson, 210] However, he, too, illustrates profit maximization with linear demand and marginal revenue curves. [Nicholson, 213, 298, 306, 320, 332]

If marginal cost curves were shown as being linear, the use of linear marginal revenue curves could be explained, but perhaps not be justified, as a way to make it easier for students to compute where MR = MC. However, marginal cost curves are displayed as curved lines. Undoubtedly, marginal cost curves are shown as a being non-linear in order to show the effect of a very important concept, diminishing returns.

Because the marginal cost curve is thought to decline to a minimum, from which point it steadily rises, a linear marginal revenue curve will cross it twice, the initial intersection being unprofitable and the second being the profit maximizing intersection. If the marginal revenue curve is not linear, additional points of equality (equilibrium) between marginal revenue and marginal cost could exist.

The singularity of a profitable equality of marginal cost and marginal revenue was questioned in 1982 in the April issue of Economic Inquiry by John P. Formby, Stephen Layson, and W. James Smith, who demonstrated the possibility of positively sloping marginal revenue curves whenever a convex demand curve is under consideration. Subsequently (1984) their argument was supported in the Southern Economic Journal by Peter J. Coughlin. More recently, in partnership with Steven R. Beckman, Smith has reasserted his belief in a positively sloping marginal revenue curve. Their views were later reinforced by Diane and Daniel Primont. [2]

"Our analysis," Formby, Layson, and Smith wrote, "lends theoretical support to Mrs. [Joan] Robinson's contention that demand curves with positively sloping MR may be highly relevant to real world markets and to A. A. Walter's suspicion that such demand curves may be pervasive." [Formby, 303]

Linear, Convex, Concave, or S-shaped?

Textbooks often show linear marginal utility curves and linear demand and marginal revenue curves. However, it is not rare for them to be shown as being convex. However, R. D. G. Allen in his seminal work in mathematical economics believed individual demand curves might be s-shaped. He illustrated this possibility with the following example:

No sugar is demanded at prices above 1s. 3d per lb. As the price falls, the demand increases slowly at first, then rather rapidly and finally (for prices below 2d. per lb.) slowly again. The family never demands more than the "saturation" amount of 17 lbs. per adult per month. The demand curve cuts both axes and is first convex and then concave [s-shaped] to the price axis. The shape of such a family demand curve can vary, of course, over a wider range for different families and different goods. [Allen, 113]

It is clear that acquiring a second automobile for use over a given period of time has a lesser utility to a person than does the first automobile. The same can be said of the increase in total utility (marginal utility) experienced by eating a second egg for breakfast, that is, less additional utility is gained from eating the second egg than was gained by eating the first egg.

That an individual's demand curve ultimately becomes concave seems likely. For example, as a light bulb's price declines, a consumer may initially substantially increase his or her consumption of bulbs when price declines by a given amount, but as if falls further by this amount, because he or she is nearing his or her saturation point, they buy very few more. (Light bulbs are 20 cents cheaper; so let's buy a six-month supply. Oh, they have gone down another 20 cents. Shall we buy an equal number so that we will have a year's supply -- two years--three years!)

Then:

For Q_m to be linear, it must be the case that f(P) + g(P) = k, that is: (d/dP)[f(P) + g(P)] = 0, a seemingly unlikely event.

If an individual's marginal utility curve (MU) is linear, this means that, because the marginal utility of each unit is less than that of the previous unit's by a constant amount, marginal utility declines at an increasing rate, that is, the ratio MU_n+1/MU_n is declining. This is demonstrated below in Table One.

Table One

15/20 = 3/4 = 75 while 10/15 = 2/3 = 66.7

If marginal utility does not decline throughout its range at an increasing rate, it cannot be linear.

In economics textbooks students are told that the consumer should equate at the margin the ratios of utility to price for all products. If there is a change in the price of a can of soup that is not considered to be permanent, will this affect consumers' current purchases of things like television sets and automobile, or would it, instead, only affect the amount they purchase of one or more products with prices similar to the price of the soup?

The consumer wishing to buy more of an inexpensive product because its price has declined relative to the price of other products will obtain the money needed to buy more of it by reducing his or her purchases of another product or other products that are either highly divisible (gasoline rather than automobiles) or indivisible products whose price differs little from the product whose price has fallen.

Suppose that the price of a loaf of bread is reduced. Are consumers, as textbooks imply, going to reduce their purchases of everything else in order to purchase more bread? Certainly not! One reason why this is not what they will do is that the failure to buy even one big-ticket item would produce enough money to purchase a large amount of bread. While at the new price of bread at the margin a dollar gains more utility by being spent on a loaf of bread, the utility of the huge amount of bread which could be purchased with the money saved by not buying a big ticket item would likely be less than the utility lost by not purchasing the big ticket item like a refrigerator or sofa.

If a consumer does equate at the margin the ratio of marginal utility to price for each product to determine the amount of products A and Product X to demand, the consumer would need to solve the following simultaneous equations.

Where:

Equations:

(1): MUx = MUa or (k -2sQx)/Px = (c - 2tQa)/Pa

(2): PxQx + PaQa = B

Therefore: Q_x = [kP_a² + P_x(tB - P_ac]/(sP_a² + tP_x²)

As is shown in Table Two (below), these simultaneous equations reveal that the demand curve for X can be s-shaped. (The shape of the demand curve is revealed by the "change in quantity" column in this table. It shows how much quantity increases every time price declines by a dollar.) Assumed is that product X is infinitely divisible.

Table Two

Assumptions Table Two Is Based On:

k = $60	c = $60	Pa = $10
t = 1.5	s = 1.5	B = $200

Price	Quantity	Change in Quantity
$20	0.0	NA
$19	0.4	0.4
$18	0.9	0.5
$17	1.5	0.6
$16	2.2	0.7
$15	3.1	0.9
$14	4.1	1.0
$13	5.2	1.1
$12	6.6	1.4
$11	8.1	1.5
$10	10.0	1.9
$09	12.1	2.1
$08	14.6	2.5
$07	17.4	2.8
$06	20.6	3.2
$05	24.0	3.4
$04	27.6	3.6
$03	31.2	3.6
$02	34.6	3.4
$01	37.6	3.0

The above demand curve produces the revenue curves shown (below) in Table Three.

Table Three

Price	Total Revenue	Marginal Revenue
$20	$0.00	NA
19	$8.24	$8.24
18	$16.98	8.74
17	$26.22	9.24
16	$35.96	9.74
15	46.15	10.19
14	56.76	10.61
13	67.66	10.90
12	78.69	11.03
11	89.59	10.90
10	100.00	10.41
9	109.39	9.39
8	117.07	7.68
7	122.15	5.08
6	123.53	1.38
5	120.00	-3.53
4	110.34	-9.66
3	93.58	-16.76
2	69.23	-24.35
1	37.62	-31.61

Observe that marginal revenue rises until price reaches $11.

The marginal utility curves for products A and X that the previous two tables were based upon were identical. In the following example (See Table Four below.), the intercept of X's marginal utility curve is increased, and the slope of both products' marginal utility curves is increased to 2.

Table Four

Assumptions Table Four Is Based On:

The increase in the utility provided by product X results in more of it being demanded at each price. It's demand curve remains s-shaped. Observe that, as Allen surmised, the demand curve turns concave at relatively low prices.

If the slope of the two curves is reduced to 1.5 (See Table 5 below.), the demand curve for X continues to be s-shaped.

Table 5

If we increase slope to 3, the resulting demand curve, shown in Table 6 (below), is also s-shaped. The steeper slope causes fewer units to be demanded.

Table 6

If the slope of MUx is 3, while MU2 is reduced to 2 (not shown), the demand curve for X remains s-shaped, but it is enormously flattened out.

Conclusions

The linear marginal revenue curves that almost exclusively "populate" economics textbooks may be rare in the real world,and marginal revenue curves with a rising segment at high prices may not be uncommon. Contrary to what is shown in economics textbooks, at low prices, demand curves may often be concave. If all this is true, the profit maximizing price is not as easily determined as students are led to believe. This does not benefit either students or the repute of professional economists.

Footnotes

The marginal revenue curve is the first derivative of the total utility curve. If it is a straight line, it takes the form of k - 2sQ, where k is the intercept, s the slope, and Q the quantity, while the total revenue function is kQ - sQ² .

1. O'Sullivan Arthur and Steven M. Sheffrin, Economics, Principles and Tools (Upper Saddle River, 1998) display only linear individual demand curves for goods and services and dollars. See pages 19, 59, 251, 236, 237, 61, 62, 68, 69, 70, 73, 74, 78, 80, 83, 87, 93, 102, 105, 108, 111, 112, 116, 121, 122, 125, 129, 136, 137, 138, 142, 143, 146, 148, 183, 184, 186, 187, 201, 202, 206, 207, 212, 213, 216, 217, 221, 231. 232, 234, 236, 237, 238, 249, 260, 274, 281, 282, 285, 293, 295, 300, 306, 314, 334, 336, 340, 341, 344, 347, 360, 363, 366, 489, 494, 662, 666, 667, 675.

2. In Smith's earlier article with Formby and Layson it was demonstrated that upward sloping marginal revenue schedules are easily produced because simple parallel shifts in strictly convex demand and/or rotations always transform demand schedules with downward sloping marginal revenue into ones with positively sloping schedules. In this paper he and Beckman utilize utility theory to produce upward sloping marginal revenue schedules. The Primonts extended the results of Beckman and Smith.

Sources

Allen, R. D. G., Mathematical Analysis for Economists (New York, 1964).

Beckman, Steven R. and W. James Smith, "Positively Sloping Marginal Revenue, CES Utility and Subsistence Requirements," Southern Economic Journal (October, 1993), 297-303.

Baumol, William J. and Alan S. Blinder, Economics (New York, 1964).

Coughlin, Peter J., "Changes in Marginal Revenue and Related Elasticities," Southern Economic Journal (October 1984), pp. 568-573.

Formby, John P., Stephen Layson, and W. James Smith, "The Law of Demand, Positive Sloping Marginal Revenue, and Multiple Profit Equilibria, Economic Inquiry (April, 1982), pp. 303-311.

Kamerschen, David R. and Lloyd M. Valentine, Intermediate Microeconomic Theory (Dallas, 1981).

Leftwich, Richard H., The Price System and Resource Allocation (Hinsdale, Ill., 1979).

Lipsey, Richard G., Peter O. Steiner, and Douglas D. Purvis, Economics (New York, 1984).

O'Sullivan Arthur and Steven M. Sheffrin, Economics, Principles and Tools (Upper Saddle River, 1998)

Primont, Diane F. and David Primont, "Further Evidence of Positively Sloping Marginal Revenue Curves," Southern Economic Journal (October, 1995), pp. 481-485.

Robinson, Joan, The Economics of Imperfect Competition (London, 1933).

Samuelson, Paul A., Economics (New York, 1970).

Walter E. Nicholson, Intermediate Microeconomics and Its Application (Fort Worth, 1994)

Walters, A. A., "Monopoly Equilibrium," Economic Journal (January 1980), p. 161-162.

Varian, Hal R., Intermediate Microeconomics, A Modern Approach (New York, 1996)