5. Exchange in the Open Market

We have treated the individual exchange in such great detail merely in order to be able to demonstrate by means of a simple example the most important fundamental principles of the exact manner of treatment, not for the sake of its practical importance, for this is small. In modern economic life almost all proportions of exchange are determined by the open market or indirectly by its influence.

In the market, however, an element is added which causes the problem which we just now had to declare indeterminate, to appear relatively determinate. Jevons calls this the law of indifference, but it is in fact nothing other than competition, the mutual competition of buyers and sellers. Under the influence of competition, as we are accustomed to say, only one price can rule on the market and in its neighbourhood, so that all partial exchanges are carried out approximately in one and the same proportion of exchange.

It would, of course, be possible, and indeed it occurs quite often, that the one or the other party in the market attains in the first instance by an initial restraint a price higher than the one which later proves compatible with the general situation of the market; but then there is always the danger that some members of the party, cleverly using this good opportunity, might dispose of their whole stock at this artificially raised price, with the result that for the others the situation of the market would become so bad that in the end this procedure would bring them more loss than profit. It is just this latter circumstance that marks the principal difference between the market and the individual exchange. If one tries to avoid this danger by agreements in respect of the quantities of goods to be sold and bought, that is to say by cartels, etc., the conditions of the individual exchange are more or less repeated.

We simply suppose here as a fact that on the market one price or a proportion of exchange between every two commodities establishes itself within a short time for each commodity in which afterwards the bulk of the transactions are done. And supposing only two commodities are present on the market and are going to be exchanged against each other, let us set ourselves the task of finding out the proportion of exchange at which equilibrium is attained on the market. If this proportion is 1 : p so that p units of the commodity (B) are given against one unit of the commodity (A), each of the exchanging persons will exchange in just this proportion and he will, exactly as in the case of fixed prices treated above, exchange up to the point where for him the proportion of the marginal utility of the commodity (A) to that of the commodity (B) becomes p : 1. Let us suppose that there are m possessors of the commodity (A) and n possessors of the commodity (5), each of whom we suppose, for the sake of simplicity, to be originally provided with only one of the two commodities. If we then express the marginal utility function of the commodity (A) for the different possessors of this commodity and for those of the commodity (B) by F₁( ), F₂( ). . . F_m( ) and J₁( ), J₂( ). . . J_n( ) respectively, and the marginal utility function of the commodity (B) for those possessors by f₁( ), f₂( ). . .f_m( ) and j₁( ), j₂( ) . . . j_n( ) respectively, we get the system of equations:

in which a₁, a₂ . . . express quantities initially owned by the various possessors of the commodity (A), x₁, y₁, x₂, y₂, . . . express the quantity of (A) and (B) which each of them has given and taken in exchange respectively, and b₁, b₂ . . .; x'₁, y'₁, x'₂, y'₂ . . . have the same significance in relation to the original possessors of (B).¹

We have here, therefore, 2m + 2n equations. To these, two other equations have to be added, which tell us that the sum of the quantity of goods given in exchange and the quantity of goods taken in exchange must be equal for each of the two commodities ; consequently

x₁ + x₂ + . . . + x_m = x'₁ + x'₂ + . . . + x'_n (5)

and

y₁ + y₂ + . . . + y_m = y'₁ + y'₂ + . . . + y'_n (6)

Of the two latter equations, however, each can be derived from the other with the help of the equations (4).¹ We consequently obtain altogether 2(m + n) + 1 equations, which are independent of each other, or just as many as the number of the unknown magnitudes: x₁ . . . x_m, y₁ . . . y_m, x'₁ . . . x'_n, y'₁ . . . y'_n and p. Our problem is consequently theoretically solved. We will undertake the discussion of these equations and their discontinuities later on, when we deal with supply and demand.

It would simplify matters somewhat if we were permitted to suppose that the marginal utility function of one or the other commodity depended only on the quantity possessed, but not on the personal disposition of the exchanging persons, so that the functions F₁ . . . F_m, J₁ . . . J_n could approximately be replaced by one and the same function, perhaps F( ), just as the functions f₁ . . .f_m, j₁ . . . j_n can all be replaced by the function f( ). If, further, we suppose what seems more doubtful still, however, and can indeed apply only to one special case, namely that F( ) and f( ) can both be expressed sufficiently exactly for the whole field of this problem by one approximating function of the first degree, α − βx and γ − δy respectively, then we obtain by the addition of numerator to numerator and denominator to denominator in the equations (4) and with the help of (5) and (6)

provided that by A and B we express the size of the existing total supply of (A) and (B). The equilibrium price appears here, therefore, as about the proportion of the average marginal utilities of the commodities (A) and (B), or of those marginal utilities which would result if the existing supply were distributed equally amongst all exchanging persons. The equilibrium price depends only on the number of barterers and on the size of the total stock, but not on its original distribution. When p is already determined in this way, one obtains the other unknown magnitudes of the problem, x₁, x₂, etc., very simply by an equation of the first degree in each case.

This observation, which is at any rate interesting, was made by Launhardt. It is open to doubt whether any practical importance can be attached to it. As we have already several times remarked, this rule can only be generally valid, i.e. valid for all forms of functions, if it is a question of very small deviations, that is to say, if all exchanging persons are from the outset or by previous exchange in possession of approximately equal quantities of the same commodity, so that the marginal utility of the commodity (A) as well as that of the commodity (B) is already nearly equal for all of them. This, however, will not often come about in reality; for even if the marginal utility function were identical throughout, the amounts of property would nevertheless be different. From this it follows that this function can indeed be replaced by a series of different approximating functions, but not by one and the same formula,¹as the validity of the rule requires.

The treatment of the problem of exchange given above derives from Walras. Jevons, who has also availed himself of the mathematical method, but in a less correct way, believed that he could summarize the solution in two equations by regarding all possessors of the one as well as of the other commodity as a trading body. According to Jevons, for each of these trading bodies, in respect of each of the commodities, a kind of collective marginal utility holds good, which can be regarded as a function of the possessed or acquired total supply. If A and β are the total supplies of the commodities (A) and (B), and X and Y the exchanged total quantities of these, and if the mentioned collective marginal utility is expressed by F( ),J( ),f() and j( ) respectively,¹ we obtain

In this case the proportion of exchange to be determined is of course given by

But Jevons never says clearly what is really meant by this collective marginal utility of a trading body, and it seems as if he himself had not formed a sufficiently clear idea of it. The marginal utility of a commodity for a trading body can scarcely be anything else but the average marginal utility, the arithmetical mean, or else any mean of the individual marginal utilities of its members. But neither is it clear how the proportion of exchange can depend on this average marginal utility in the way Jevons demands, nor can one understand how it could be conceived as a function of the size of the possessed total supply, since the average marginal utility in fact also depends on the distribution of this supply and, what is more, on the distribution after the exchange, which is still unknown.²

If the members of the party, instead of operating each for himself on the market, were to buy and sell on joint account, in other words, if they formed a real trading body instead of a trading body which was only feigned, then we could indeed speak of their collective marginal utility; but then the reciprocal competition would be excluded. We should still be in the sphere of isolated exchange and there would be no fixed equilibrium price.

Jevons’s solution is therefore insufficient, although he has correctly grasped the fundamental idea of the theory.

But if with Walras one takes, instead of the exchanged total quantities themselves, their proportion, namely the average proportion of exchange, as the independent variable, it is indeed possible, as we shall soon see, to unite the equations of the exchange in one single formula, which is then nothing other than the mathematical expression for the equality of supply and demand.

In the case of exchange in the open market also, as well as in the cases treated previously, a maximum problem is solved ; but only in the sense that each of the exchanging persons (and consequently all of them together) obtains the greatest possible gain of utility which can be attained by him (or them) at the price fixed on the market. On the other hand, this would obviously not be the case if a uniform price were fixed in advance in some other way, e.g. by governmental order. That being so, only one market party, the one not favoured, could exchange until saturation was reached; but at no time could all the members of the other party, or perhaps even a single member, sell such a great amount of their goods as would be profitable for them at this price. Equilibrium on the market would then be impossible, since the supply of the favoured commodity would always exceed the demand.¹

It can, however, not be asserted that the gain of utility attained by all the exchanging persons together is necessarily smaller in the latter case than in the case of entirely free competition.

Generally speaking, of course, this will prove true; for if the fixed price deviates very much from the equilibrium price, the exchanged quantities of goods become in the end so small that the gain of utility on both sides, too, lags behind the gain of utility attainable in the case of free competition. Up to a certain limit, however, the profit of the favoured party is increased with each such shifting of the price; and it cannot generally be proved that the profit of the other party decreases thereby in a corresponding degree.

Still less can it be asserted that the distribution of the com modities which is most favourable economically, that is to say, the greatest possible general satisfaction, arises from free competition. If this problem is conceived in the absolute sense, its solution, as can easily be seen, requires that the marginal utility of all exchanging persons should become the same in relation to each separate commodity.¹ But this situation will quite often lie beyond the limits of the possible exchange, as it would bring to some of the exchanging persons loss instead of profit. This, however, does not prevent the problem from being solved in the relative sense, that is to say, in so far as it is compatible with the fundamental condition of exchange. But this could obviously only happen if the individual transactions were carried out at different prices, instead of at the single joint price required by free competition.²

But after all, the question of the most suitable distribution of goods forms a problem which is entirely different from that of the theory of exchange. For it supposes that utility or satisfaction can also be compared for different persons, whilst the theory of exchange only proceeds from the possibility of comparing the utilities of different commodities for one and the same person; which is quite a different matter.