7. Supply and Demand

We are, of course, still very far from being able to give our equations hitherto formulated a practical application, or from being able to test them in this way. The bare number of equations required makes this impossible. To be able actually to formulate these equations, it would be necessary to know exactly the plans of every single consumer in regard to each of the different commodities and the size of the existing individual supplies, which is, of course, impossible.

Secondly, it was assumed in the foregoing that all the commodities to be exchanged are optionally divisible and that their consumption, in relation to a certain period of consumption, represents, even within the individual economy, a continuously variable magnitude.

Neither the one nor the other holds good in reality without qualification. In the case of several commodities, only a limited number of separate specimens can be used at a time in individual consumption. But even if the commodities are themselves optionally divisible, the consumption will in most cases only be able to vary by discontinuous steps; which renders a mathematical treatment of the above kind more difficult still, or makes it impossible.

But the case is different if we speak of the total sum of commodities which are exchanged on the market or consumed within the economic territory concerned. Firstly, the quantities of goods in question could then, as a rule, be much better determined statistically. Secondly—and this is nearly as important for an exact treatment—their total consumption, by virtue of the law of great numbers, will almost always be able to be regarded as a magnitude continuously varying, even if the individual consumption only changes by discontinuous steps. Jevons was therefore perfectly right in trying to unite the exchanging persons into groups or ‘trading bodies’; only, as we have seen, not much can be done with the concept of marginal utility of such a group. But we attain our end if, as Walras did, we conceive the prices or proportions of exchange of the commodities as variable and, what is more, as the only independent variables of the problem, or—which comes to the same thing—if we consider the exchange procedure from the point of view of supply and demand.

Let us first of all return to the exchange of two commodities.

If we solve all equations (4) in relation to x₁, y₁, x₂, y₂, . . .x'₁, y'₁ etc., every x and y and every x' and y' can be regarded as functions of p, where p is conceived as variable, if we leave out of account for the time being the equations (5) and (6). In other words, whenever both commodities are exchanged in the proportion of 1 : p, which in one way or another has been fixed in advance, then from every single possessor A_r of the commodity (A) comes a certain supplyx_r of this commodity and with it also a certain demandy_r for the commodity (B), where x_r and y_r each by itself, are functions of p, which must always stand to each other in the simple relation

In the same way, from every possessor B_q of the commodity (B) comes a certain supplyy'_q of the commodity (B) and a certain demandx'_q for the commodity (A). y'_q and x'_q, too, are functions of p, and stand in the same relation to each other as above. This we express better by

since all the x' express here supply and all the y' demand. p therefore denotes the price of the commodity (A) expressed in terms of (B); consequently or π denotes the price of the commodity (B) expressed in terms of (A).¹

If now we add together all the x’s and call the resulting sum X, then this sum expresses the total supply of the commodity (A). In the same way we obtain by addition of all the y’s the total demand Y for the commodity (B).

In the same way Y', the sum of all the y', expresses the total supply of (B), and X', the sum of all the x', the total demand for (A).

All these magnitudes become, therefore, functions of p or of π, the prices of the commodities reciprocal to each other, and, what is more, generally constant functions, even if the individual supplies and demands only vary by steps. If p rises a little and π consequently falls, the magnitudes X, Y, X' and Y' will, as we know from experience, rise by a very small amount, and fall respectively; and vice versa, if p falls and π rises. X, therefore, is transformed into X + dX (where dp and dX can also be negative) or into etc., if p changes into p + dp.

But this generally does not happen in such a way that with every shift of prices the possessors of (A) now increase or decrease their consumption of (B) by, perhaps, one hundredth each—which might not even be possible, according to the nature of the commodity (B). Most of them are probably not in the least disposed to increase or restrict their consumption of the goods concerned by the change in prices which has taken place. But some of them, while the price was still p, were presumably just about to consume the commodity (B) not yet used, or, on the contrary, to give up partly or completely their consumption of (B). For these, the rise or reduction in price dp is, as it were, the drop which causes the vessel to overflow. These alter their consumption and, what is more, not by an infinitely small, but by a relatively considerable amount, which, however, will be very small in comparison with the consumption of the majority of the consumers. This on the whole will be unchanged.

Let us now consider the equations (5) and (6). The former reduces itself to

X = X'

(10)

and simply expresses the fact that supply and demand of the commodity (A) must be equal in the case of equilibrium of the prices. By this the equation (6), or

Y = Y'

(11)

is also fulfilled, since Y is obviously = pX and Y' = pX'. Equality of supply and demand of the one commodity causes the same relation in respect of the other commodity. Using either of these equations, p can now be determined, if we have found out the forms of the functions X and X' or Y and Y'.

However, a more detailed examination shows that equality of supply and demand is indeed a necessary, but, at least from the theoretical point of view, not a sufficient condition for the equilibrium of the market, supposing the latter to be stable— if, that is to say, the proportion of exchange would automatically return to (approximately) the same position after an accidental shifting.

If, for instance, it is a matter of demand and supply of the commodity (A), it can generally be asserted that, if p [the price of (A) expressed in terms of (B)] increases, the demand for (A) always falls; if, on the contrary, p decreases, the demand for (A) will always increase.¹ If we could now be certain that, on the contrary, the supply of (A), at least near the equilibrium price found [i.e. the value of p, ascertained from (10) or (11)] would increase when the price rose, and would decrease when the price fell, then the stability of the equilibrium would obviously be secured ; for in the case of an accidental deviation of the price upwards the supply would be greater than the demand; in the case of a deviation downwards, the demand would, on the contrary, exceed the supply; in both cases the inequality of supply and demand would necessarily drive back the price to approximately the earlier position.

But we know in regard to the supply of (A) that this magnitude, multiplied by the price of (A), represents the demand for (B) (Y = pX).

If now the demand for (A) decreases when the price of (A), expressed in terms of (B), rises, then the demand for (B) must for the same reason diminish when the price of (B), expressed in terms of (A), rises, and consequently increase if the price of (A), expressed in terms of (B), rises. If therefore we put the demand for (A) or X' = ϕ(p) and the demand for (B) or Y = ϕ(p), then ϕ(p) is consequently a decreasing function (when p increases); ψ(p), on the other hand, is an increasing function of p. We therefore obtain for the supply of (A) or X the expression

which product, for different values of p, can under certain circumstances increase with increasing p, but also decrease. If ψ(p) increases more rapidly than p, this product increases; if ψ(p), on the other hand, increases less rapidly than p, it decreases.

When the price rises, therefore, not only the demand but also the supply of the commodity in question can decrease. If, now, the demand decreases more rapidly than the supply (and therefore, on the contrary, increases more rapidly when the price falls), the stability of the equilibrium is, as can easily be seen, even in these circumstances still secured. But there is nothing to prevent from decreasing or increasing even more rapidly than ϕ(p), near the value of p in question, since supply and demand of the same commodity proceed from different persons and are consequently totally independent of each other.¹

If this is the case, no real equilibrium of the price exists, but only a temporary equality of supply and demand; for as soon as the price moves even in the least degree upwards the demand will be greater than the supply and the price must consequently rise higher and higher, until the demand, decreasing, finally catches up with the decreasing supply once more. In the same way a small shift of the price downwards will cause the supply to exceed the demand, and leads therefore to lower and lower prices, until the demand, increasing, again catches up with the increasing supply.

In both cases equilibrium is finally reached, but the equilibrium price will in each case be a different one. Thus the further peculiarity arises, that not only one, but two different (stable) states of equilibrium of the market would theoretically be possible.

Walras, and Launhardt after him, have drawn supply and demand curves in hypothetical form. By this means the price is represented as abscissa of a right-angled system of coordinates, and the quantities of goods demanded or supplied as ordinates of the different curves. Mangoldt, by the way, in his Grundriss der Volkswirtschaftslehre, which was published in 1863, had already drawn similar curves, which, however, were eliminated by the editor of the later edition of his work.

I reproduce on the next page Launhardt’s diagram, in which, certainly, the peculiarity mentioned above does not appear.²Here, for the sake of greater clarity, two of these curves are drawn beneath the axis of the abscissae. If p is zero, i.e. if the commodity (A) is to be had for nothing, everybody, and consequently the possessors of (B) also, will provide themselves with it until saturation is reached, but they will not desire an infinite quantity of it. The demand curve therefore cuts the axis of ordinates at a certain distance from zero. If p increases, the demand for (A) on the part of the possessors of (B) decreases, and at a certain price this demand becomes zero.

The demand curve for (B) would now follow a similar course if the abscissae represented, instead of the price of (A) expressed in terms of (B), the price of (B) expressed in terms of (A)—that is to say, if w were chosen as abscissa. But in that case the demand for the commodity (B) will only begin at a value of p

different from zero. From then on the demand for (B) increases as p increases, but will never be able to exceed a certain magnitude, namely the quantity of (B) which would be desired if p were infinite and consequently were = 0, that is to say, if the commodity (B) could be had for nothing. The demand curve for (B) therefore approaches asymptotically a straight line which is drawn at this distance parallel to the axis of the abscissae. The two curves mentioned so far, by the way, are absolutely independent according to our assumptions.

Each of the other two curves, on the contrary, is totally determined by the form of each of the previous curves. If the demand for (A) is given by the function ϕ(p), the supply of (B), as we have seen, is necessarily represented by p. ϕ(p); in the same way expresses the supply of (A), if ψ(p) expresses the demand for (B).

A direct consequence of this is, that the point of intersection of the supply and demand curves of (A) must lie vertically above the point of intersection of the supply and demand curves of (B). Both points of intersection determine one and the same value of p, namely the equilibrium price.

As regards the supply curve of the commodity (A) in particular, this has, as can be seen, a highest point and approaches afterwards the axis of the abscissae asymptotically. But although it is quite independent of the form of the demand curve of the same commodity, its intersection point with the latter can lie just as well on the right side of the highest point as on its left side (as in the figure). These two positions of the intersection point correspond to our two above-mentioned cases of stable equilibrium of the price. But this does not prevent these two curves from being able to have more than one point, and if so at least three points of intersection in common, as, for example, is shown by the dotted line [representing the demand for (A)] drawn in our figure.¹ If this is the case, the two extreme intersection points, as we can easily convince ourselves, determine prices of stable equilibrium. The middle intersection point, on the contrary, shows no real equilibrium of prices, as was mentioned above, but only a temporary equality of supply and demand.

This interesting result of the theory, which was first noticed by Walras, is impugned in the well-known work by Auspitz and Lieben,² who assert that ‘the simultaneous validity of both demand curves [of the commodities (A) and (B)] is founded on assumptions which contradict each other.’ In this case, the authors go on to argue, one would have to assume firstly that ‘the prices or proportions of exchange of all other articles’ excluding the commodity (B) are constant against one another; and consequently, that the prices, on both sides, of all articles excluding the commodity (A), but including the commodity (B), are constant.

This objection seems to me to be unfounded. In Walras’s presentation as well as in our examination up to now, no account is taken in principle of the presence of other articles on the market; it is assumed that the demand for (A) comes exclusively from the possessors of the commodity (B) and that the demand for (B) comes exclusively from the possessors of the commodity (A), But we do not at all need to confine ourselves to this purely abstract assumption. If we put instead of the commodity (B) the sum total of all commodities on the market excluding the commodity (A), or what comes to about the same, if by one of the two commodities we understand money, then, at variable money prices of the commodity (A), the demand for (A) (which now comes from all other possessors of goods or consumers) as well as the supply of (A) [which is now determined by the demand on the part of the possessors of (A) for all other commodities] will on the whole have to follow the same course as in the case of only two commodities which we have considered.

The reciprocal proportions of exchange or the money prices of the other commodities exercise their influence, of course; but all these prices can be regarded, in otherwise unchanging circumstances, as dependent on the money price of the commodity (A). A demand and a supply curve of (A), as well as supply and demand curves of money dependent on them [those of the possessors of (A)], will consequently really exist; the supply curve of (A) will, if one draws the variable money price of (A) as abscissa, have a highest point, and from there it will approach the axis of the abscissae asymptotically, etc. The existence of differently characterized intersection points between these curves, as well as the possibility of several intersection points simultaneously, cannot therefore, at least a priori, be denied. The former result can even be regarded as a well-attested fact.

If, to be sure, one assumes, as Auspitz and Lieben do, that the valuation of money on the part of all exchanging persons is constant, then the supply curve of every single commodity must indeed always take a rising course, and we cannot then speak of several intersection points of the curves. This assumption can indeed be made in some cases, but by no means in all.

Let us take a few concrete examples. If, while the yield of the harvest and the size of supplies remain constant, the prices of corn for the year are for any reason higher than usual, and if importation of corn is excluded, one can by no means declare apriori that the supply of this commodity must now grow. It may be that the farmers—hitherto perhaps obliged to deny themselves much—desire to be better fed, now that their income has risen, or else to increase their own consumption of corn. The supply of corn will then, on the contrary, decrease. This, of course, supposes that the valuation of money on the part of the farmers has now decreased considerably; otherwise the raised price would induce them to increase their supplies and consequently to restrict their own consumption.

Or let us take as the commodity to be considered, the so-called ‘commodity labour.’ It is quite a common complaint amongst well-to-do people, that in times of relatively high wages people ‘do not want to work’; and this complaint is probably founded on fact. The worker allows himself more leisure than before if he is better paid, and the supply of labour decreases instead of increasing. At least, this is a possible consequence. But let it be repeated, this can happen only if we assume that the valuation of money on the part of the workers has decreased just because of their increased wages.

The descending part of the supply curve is consequently in both these cases cut by the demand curve (which always follows a descending course). If in these circumstances the two curves chance to run close together for a certain distance, the possibility of several intersection points, i.e. of several states of equilibrium of the same market at different prices, obviously exists.

In most cases, of course, only a very short segment of the theoretically possible supply and demand curves can in reality exist, since greater price fluctuations do not often occur because of other possibilities of purchasing and selling.

When the proportions of exchange of three or several (m) commodities are to be found, we obviously have to consider the total supply and the total demand of each commodity as functions of all proportions of exchange or prices of the commodities concerned. The equalization of the supply and demand of each separate commodity supplies m equations,¹ amongst which, however, only m — 1 are independent. The variable prices are here also m — 1 in number in that, for instance, one of the commodities itself is taken as the standard of value.²

A geometrical interpretation is, of course, excluded here. At best, if it is a question of only three commodities, we could speak of supply and demand surfaces, if the quantities of goods concerned, together with the two prices or proportions of exchange of the three commodities, are drawn as co-ordinates in three dimensions.

These indications may suffice to show that the conventional teaching of supply and demand, by means of the marginal utility theory, seems to be capable of considerable extension and deepening. It is true that one is very soon confronted thereby with an almost hopeless entanglement of interacting economic relationships; but if the exact mode of treatment can do nothing else, it will at least be able to distinguish sharply between that which we know or are able to penetrate, and that about which we know, or can know, really nothing at all; and this is, after all, the beginning of all true science.