6. Exchange of Several Goods. Indirect Exchange

If three or more commodities come to be exchanged on the market, not only do our formulae become, in a corresponding degree, more complex, but quite a new phenomenon appears, which is of the greatest importance from the economic point of view, namely the indirect exchange, which consists in the fact that a commodity is taken in exchange, not in order to be kept and consumed, but in order to be again given in exchange.

Suppose, for example, that three commodities (A), (B) and (C) are present on the market, which are to be simultaneously exchanged for one another. It could now seem as if each possessor of the commodity (A) would simply relinquish part of his possession of (A) against a certain quantity of (B) and another part of (A) against a certain quantity of (C), according to the law of the proportionality of the corresponding marginal utilities—and similarly with the possessors of (B) and (C)—so that the quantity of (A) given by the possessors of (A) to the possessors of (B) would constitute the remuneration for the quantity of (B) obtained, etc. This, however, win generally not be the case, for a general equilibrium on the market would thereby not yet be attained. Rather, the direct exchange is almost always followed by an indirect one, since at least some of the possessors of (A) derive their advantage by exchanging against each other certain quantities of (B), in order to exchange them afterwards for corresponding quantities of (C), or vice versa. An analogous operation can, of course, also be undertaken by the possessors of (B) or of (C), or simultaneously by the members of the different parties.

The same result can also be attained with the help of credit or money. The possessors of (A) then surrender certain quantities of (A) to the possessors of (B) without direct remuneration, or for money. On the other hand, they obtain from the possessors of (C) a corresponding quantity of (C) without direct remuneration, or for the money which they have just received from the possessors of (B). Finally, the possessors of (B) surrender a corresponding quantity of (B) to the possessors of (C) for just this sum of money, or against the claim which the possessors of (A) have on the possessors of (B) and which they have transferred to the possessors of (C); so that either the money finally returns to the starting-point or the claims are discharged. The result will be the same as in the case originally supposed, save that the quantities of (B), which previously went through the hands of the possessors of (A) as middlemen, are now transferred directly to the possessors of (C).

If credit and money transactions as well as wholesale trade are excluded for any reason, then the quantities of goods which are surrendered on both sides—one of one sort for one of another—must certainly be exchanged directly. But then the three proportions of exchange between (A) and (B), between (B) and (C) and between (C) and (A) will stand in no relation whatsoever; so that if, for instance, in the trade between the possessors of (A) and (B), two units of (A) are given for every unit of (B), and in the exchange between (B) and (C), three units of (B) are given for every unit of (C), then, in the exchange between (C) and (A), perhaps five, seven, or any number of units of (A) whatsoever, can be exchanged for each unit of (C), whilst in the case of free exchange, exactly six units of (A) would have to be given for every unit of (C).

Or vice versa. If we suppose that the proportions of exchange of the three commodities are dependent on each other, so that one of them is always determined, in the simple way indicated above, by the other two, then we cannot make the further stipulation that the quantities of goods finally sold should pay for each other, or should be directly exchanged against each other. The problem would then be overdeterminate.

We are here obviously confronted with one of the most important questions of the theory of exchange. The ‘exchange between three’ forms, so to speak, a connecting link, which leads from the state of primitive exchange to that of developed economy, where two producers or other possessors of commodities, as we know from experience, almost never exchange their goods directly. A will give his commodity to B, B will give the one he possesses to C, C his to D, etc., until the chain is completed, usually by way of various ramifications.

In order to simplify the mathematical treatment of this problem as far as possible, it is perhaps best to unite the different possessors of commodities not in several, but in one single group, each of whose members is already from the outset conceived as possessor of certain quantities of all these goods, and therefore, on the assumption of only three commodities, as the possessor of all three. Initially, one or two of these quantities can, of course, be zero.¹

Suppose the number of all the exchanging persons is n.

One of them has at the outset the quantities a_r, b_r and c_r of the commodities (A), (B) and (C) respectively, where r is an optional index number. After the completed exchange, he will possess the quantities a_r + x_r, b_r + y_r and c_r + z_r in which at least one of the magnitudes x_r, y_r, z_r must be negative and therefore expresses a quantity of goods given in exchange instead of a quantity of goods taken in exchange. But also two of these magnitudes could be negative, if the person concerned had originally possessed (at least) two of the three commodities, and had given away certain quantities of both for each quantity of the third commodity.

If we further suppose that the equilibrium prices of the three commodities, measured according to an optional standard, are p_a, p_b and p_c, ¹ the principle of thrift (the principle of the greatest possible profit for everyone) demands that the possessor in question exchange up to the point at which, for him, the marginal utilities of the three commodities stand in the same proportion as their prices. We consequently have, if the marginal utilities of the three commodities for him are expressed by F_r( ), G_r( ) and H_r( )

F_r(a_r + x_r): G_r(b_r + y_r): H_r(c_r + z_r) = p_a: p_b: p_c

(7)

This amounts to two independent equations.

For each of the exchanging persons there exist two similar equations or, altogether, 2n equations.

We have now in addition to express the fact that for each possessor the amount realized by the goods taken in exchange is equal to the amount realized by the quantity of goods which he gave for them from his original stock of goods. We thus obtain, as can easily be seen, n equations of the type

x_rp_a + y_rp_b + z_rp_c = 0

(8)

But finally, three other equations must be considered here—to the effect that the algebraic sum of the (positive) quantities of each of the three commodities taken in exchange and the (negative) quantities given in exchange must be zero. We have therefore in addition

Of these equations, however, only two are independent, since the third can always be obtained from the others with the help of the n equations (8) (by their addition), as can easily be seen.

For the same number of unknowns, namely the 3n quantities x₁ . . . x_n, y₁ . . . y_n, z₁ . . . z_n and the two proportions of the three prices, we obtain therefore altogether 3n + 2 equations ; for instance

is then determined.

The absolute level of these prices themselves cannot, of course, be ascertained here, since they were reckoned according to an optional measure which cannot be exactly determined.

If, on the contrary, we had chosen one of the commodities, e.g. (A), as the standard of value, so that we had p_a = 1, p_b and p_c could, of course, be determined. They would then represent the price of (B) and of (C) respectively, expressed in terms of (A).

As we see, no difference is made here between the possessors of different commodities. It would be quite easy, however, to do this. We should then—assuming that, for instance, each person possesses at first only one commodity—have to divide the exchanging persons into three groups, in which case, according to our notation, all initial quantities b and c in the first group, the quantities c and a in the second group, and a and b in the third group would be zero. The other way of dealing with this problem would be exactly the same as above. But if one wanted to introduce here at the same time the condition that the sum of the y’s in the first group and the sum of the x’s in the second group, multiplied by p_b and p_a respectively, should be equal to one another (from which it follows directly that the sum of the z’s in the first group, multiplied by p_c and the sum of the x’s in the third group, multiplied by p_a must also be equal to one another as well as to the sum of the z’s in the second group and the sum of the y’s in the third group, multiplied by p_c and p_b respectively)—in other words, supposing that the transacting persons only obtain possession of the commodities by direct exchange—then the problem is overdeterminate and cannot be solved. We should then have not merely 3n + 2, but 3n + 3 equations, which would be independent of each other, whilst there are only 3n + 2 unknown magnitudes to be determined.

On the other hand one could easily introduce the condition of direct exchange, if one conceived the three proportions of exchange between (A) and (B), between (A) and (C) and finally between (B) and (C) as three magnitudes which are independent of each other.¹ The unknowns of the problem would then be increased by one, and would then amount to 3n + 3.

This is how Jevons treats the problem,² except that, as in the case of exchange between two commodities, he introduces the vague concept of the marginal utility of a ‘trading body,’ by which means he believes that he is able to reduce the number of equations to only 2 x 3 = 6.

But Jevons does not seem to have noticed that the state of equilibrium expressed by his equations excludes, in principle, the possibility of the wholesale trade as well as money and credit transactions, and that, if these are admitted, the equilibrium would immediately be disturbed afresh. He reminds us that the same pair of goods can only have one proportion of exchange in the same market, but he never mentions that in the case of a completely free exchange of three commodities there can only be two independent proportions of exchange (and generally in the case of n commodities only n − 1); indeed he treats these proportions of exchange as if all three would be independent.

Finally, so far as the question of the greatest possible profit is concerned, much the same applies here as in the case of exchange between two commodities only. Each party to the exchange attains, at the equilibrium prices fixed by free competition, the greatest possible profit attainable by him at just these prices. It is here specially to be remarked that, if initially only direct exchange is permitted, but subsequently the market is entirely freed, each of the exchanging persons will acquire a greater profit by the wholesale trade or stock-exchange operations which then take place, and in this way the total profit also can become greater. But the state of equilibrium thus attained will generally be different from that which would occur if trade were entirely free from the very beginning. For this reason it cannot be asserted that in the case of entirely free trade a greater total profit can invariably be obtained than if, for instance, only direct exchange were allowed. It can, however, easily be seen that this must on the whole be the case, and the more so, the more the division of labour is already carried through—which means that fewer direct exchange transactions can occur at all.