fulgurafrango [blog]
20 January 2017

This following question might come up if a student is particularly bright and observant, or if she just wants to make trouble:

Why are demand and supply diagrams in economics drawn they way they are, i.e., with quantity Q on the horizontal axis and price P on the vertical? (Figure 1.) After all, even introductory textbooks say that P is the independent and Q the dependent variable. Indeed, the demand function, for example, is written as

(1)                   Q = f(P),

also known as the “quantity-function form”. Just as y = f(x) is conventionally graphed with x on the abscissa (horizontal) and y on the ordinate (vertical), then properly speaking P should be measured on the horizontal and Q on the vertical resulting in the following diagram (Figure 2.) So why do we do use Figure 1 rather than Figure 2 when we graph functions like (1)?

The answer is to be found in the history of thought. The great populariser (though not the originator) of demand-and-supply diagrams was Alfred Marshall (1842-1924), and as it turns out, his version of a demand function frequently took the form of what has become known as the “price-function form” [Gordon 1982] (also called the “inverse demand function”):

(2)                   P = f(Q).

This was because Marshall built up his demand curve by asking, for each given quantity, what was the “demand price”, i.e., the maximum, a consumer would be willing to pay:

In such a market there is a demand price for each amount of the commodity, that is, a price at which each particular amount of the commodity can find purchasers in a day or week or year. The circumstances which govern this price for any given amount of the commodity vary in character from one problem to another; but in every case the more of a thing is offered for sale in a market the lower is the price at which it will find purchasers; or in other words, the demand price for each bushel or yard diminishes with every increase in the amount offered [Marshall 1920, V.III.12]. (Emphasis supplied.)

In this scheme, therefore, say for every given Q1, Q2,… Q­j, one correspondingly obtains P1 = f(Q1), P2 = f(Q2),…, and so on. Essentially, P in Marshall’s demand curve was the reservation price or bid that some consumer would make for the given quantity Q.

Analogously defined was the supply price:

As the price required to attract purchasers for any given amount of a commodity was called the demand price for that amount during a year or any other given time; so the price required to call forth the exertion necessary for producing any given amount of a commodity, may be called the supply price for that amount during the same time [Marshall 1920, IV.I. 11].

Again, the quantity is given, then the supply price is determined, so that P = g(Q).  Clearly, then, in Marshall’s system, Q was the independent variable and P the dependent, and he was well within his rights to draw the demand and supply curves the way he did, i.e., as in Figure 1.

In fact, here’s an original diagram that appears in Marshall’s book together with his explanation: To represent the equilibrium of demand and supply geometrically we may draw the demand and supply curves together as in Fig. 19. If then OR represents the rate at which production is being actually carried on, and Rd the demand price is greater than Rs the supply price, the production is exceptionally profitable, and will be increased. R, the amount-index, as we may call it, will move to the right. On the other hand, if Rd is less than Rs, R will move to the left. If Rd is equal to Rs, that is, if R is vertically under a point of intersection of the curves, demand and supply are in equilibrium [Marshall 1920, Book V, Ch. III, fn 19].

Marshall is describing how the difference between demand-price (Rd) and supply-price (Rs) for  a given quantity (OR) causes movement towards equilibrium, in what Samuelson would later call “Marshallian adjustment”.

Even antecedent to questions of equilibrium adjustment, however,  Gordon  says the reason for Marshalls preference for equation (2) had to do with his interest in investigating the idea of consumer’s surplus and producer’s surplus. Both are more easily explained and quantified in terms of a price-function form. In an analogous manner, diminishing marginal utility (which underlies consumer’s surplus) is also explained and graphed with quantity on the horizontal.

Marshall, a math whiz (“Second Wrangler”) at Cambridge, was only being faithful to his own system by drawing Q on the horizontal and P on the vertical axis: after all, his supply and demand functions were of the forms P = f(Q) and P = g(Q), respectively.

But if Figure 1 is from Marshall, how did we end up with Q = f(P) instead?

The quantity-form function (1)  is traceable  to Leon Walras (1834-1910). In this he followed the practice of a long line of Francophone economists beginning with the mathematician Antoine-Auguste Cournot (1801-1877), who was the true originator of the supply-and-demand diagram. Walras’s demand function was built up as it is done today—by quoting a price P and asking what the corresponding quantity demanded Q is at that price. An engineer by training (though uncompleted), Walras was no math slouch either and  knew perfectly well that a function such as (1) had to be drawn with Q on the vertical and P on the horizontal. Below is a diagram and quote from Walras’s major work, Elements of pure economics [1954 (1926)] that illustrates his orthodoxy: Let there be two co-ordinate axes, as drawn in Fig. 1 [above]: a horizontal price axis, Op, and a vertical demand axis, Od. On the price axis, starting at the origin, O, I lay off the lengths Op’a, Op”a, . . . representing various possible prices of oats in terms of wheat, or of (A) in terms of (B). On the other axis, beginning at the same origin O, I measure the length Oad1, representing the quantity of oats…which our holder of wheat will demand at the price zero. On lines drawn through the points p’ap”a , . . . parallel to the vertical demand axis I lay off the lengths p’aa’1,  p”a a”1, . . . representing the quantities of oats or (A) which will be demanded at the prices p’ap”a , . . . respectively [Walras 1954: 93-94]. (Original italics.)

Okay, so now we’ve shown that Marshall and Walras each had different systems, and that each used the proper graphical representation of his work.

What is anomalous however is that current practice uses “Marshall’s graphs and Walras’s equations” [Page 1980: 138].

The disconnect occurred as follows: Marshall first gained wide acceptance for his entire approach and apparatus — including Figure 1 — particularly in the Anglophone countries, where his textbook was dominant for more than three decades. Indeed such was Marshall’s unquestioned hegemony down to the time of Keynes and Joan Robinson that entire generations of economists took it on faith that “It’s all in Marshall.”  In turn, of course, the global dominance of Anglophone economics further amplified the reach of Marshall’s approach and diagrams.

Around the 1940s, however,  American and British economists themselves began increasingly to share Walras’s concern for the purely formal problem of general equilibrium, in which the quantity-function form was the rule. In this  shifting landscape, J.R. Hicks (Value and capital 1940) was an important transitional figure who redirected attention away from Marshall’s realistic but make-do approach to economics, towards the more abstract, axiomatic work of Continental writers.  Definitive results in general equilibrium, of course, would be forthcoming only later, culminating in the work of  K. Arrow and G. Debreu (themselves a symbolic personal partnership of Anglophone and Francophone economics).  The more rigorous and comprehensive results from general-equilibrium theory guaranteed that quantity-function forms would rule at the highest levels of the profession.

Alas, by that time, however, in a curious case of path-dependence, it was too late to wean teachers and students from Marshall’s diagrams.

The result is the present incongruity, where Walrasian logic is still illustrated using Marshallian pedagogical tools.

References

Gordon, S.  “Why did Marshall transpose the axes?”, Eastern Economic Journal 8(1): 31-45.

Marshall, A.  Principles of economics, 8th edition. London: Macmillan and Co.

Page, A.  Marshall’s graphs and Walras’s equations: a textbook anomaly” Economic Inquiry 18(1): 138-143.

Walras, L.  Elements of pure economics (W. Jaffe, trans.) New York: Allen and Unwin.