4. Isolated Exchange

If for both the exchanging persons the marginal utility of one or other of the commodities in question, which we will call (A) and (B), is altered by the exchange, and consequently the price is not fixed in advance, then—supposing the exchange to be completely isolated, that is to say, supposing that other possibilities of obtaining the desired commodity do not exist—one cannot possibly speak of a fixed proportion of exchange which can be theoretically determined: the problem is indeterminate. Only this much is certain, that an exchange will take place wherever both contracting parties derive, or believe that they derive, advantage from it, and that it will continue as long as it promises a further gain of utility on both sides, be it ever so small. If we suppose in particular, as we also did in the previous cases, that it is a matter of continuous quantities, that is to say, of commodities which are optionally divisible and can also be consumed in optional quantities, it can be asserted that the exchange will cease only at the point at which the proportion of the marginal utility of the one commodity to that of the other is equal on both sides. If this condition is not yet fulfilled there will always exist on both sides a reason for continuing the exchange. If, after the exchange has taken place, in the estimation of the original possessor of (A) one unit of the commodity (B) is still equal in value to three units of the commodity (A), whilst the possessor of (B) estimates that this quantity is equal to only 2½ units of the commodity (A), then both believe that they will obtain an increase if the second of the contracting parties gives to the former another or several units of the commodity (B) against, for example, 2¾ units each of the commodity (A). But this tells us neither in what proportion the previous exchange took place nor how great the quantities were, nor consequently in what average proportion both commodities finally change their possessors.

The mathematical manner of treatment reflects this fact clearly. Let us suppose that one possessor has a units of the commodity (A), but as yet no units of (B); and that the other possessor has no units of (A), but b units of (B). Let us further assume that the function of marginal utility of the commodity (A) is F( ) for the former possessor and J( ) for the latter, and that the corresponding functions of the commodity (B) are f( ) and j( ) respectively. Then the exchange is continued up to the point where

(3)

x and y denote here the number of the exchanged units of (A) and (B) respectively.

But we have here only a single equation between two unknown quantities. The problem is consequently indeterminate; it has an infinite number of solutions. It could even appear as if, for each value of x, a y belonging to it could be found, and vice versa. This, however, is not so, because, as can easily be seen, the limiting condition must be added, that each of the exchanging persons ought to exchange with profit or at least without loss. The possible solutions consequently lie between two limits (margin pairs of x and y), in which cases the one or the other of the contracting parties has no profit at all (but also no loss). To determine these limits, when the functions of marginal utility are given on both sides, is a problem of the integral calculus. Let us think of the planned exchange as split up into an infinite number of partial exchanges, so that each time infinitesimal quantities, dx and dy, are exchanged against each other. If, then, the original possessor of commodity (A) gains nothing when he gives dx of (A) in exchange for dy of (B), the ratio of the marginal utilities to him of (A) and (B) must be the inverse of We therefore obtain each time

F(a − x).dx = f (y).dy

or, if we add up from zero to x and y on both sides,

in which case the upper limits must satisfy the integral of the equation (3).

Both these integrals, as can easily be seen, represent, for the possessor of (A), the total utility of the quantity of the commodity (A) given in exchange, and of the quantity of the commodity (B) taken in exchange, respectively. If, therefore, these functions of the total utility, now found by integration, are expressed by ϕ( ) and ψ( ) respectively, we get

ϕ(a) − ϕ(a − x) = ψ(y)

By this equation, in combination with equation (3), the values in question of x and y can be determined.

In the same way, if the analogous functions in respect of the possessor of (B) are expressed by χ and ω, the other limit of the possible proportions of exchange is given by

χ(x) = ω(b) − ω(b − y)

always in combination with (3). Between the limits thus determined, every proportion of exchange must be declared possible.

In order to make the foregoing a little clearer by an example, we may be allowed to make the simplifying assumption that for both the exchanging persons (which we will call A and B), the functions of marginal utility of the same commodity are identical, so that J( ) is identical with F( ) and j( ) with f( ), and their values depend only on the possessed or exchanged quantity of goods, not on the personal dispositions or other circumstances of A and B. Moreover, let us suppose that both functions of marginal utility can be replaced by approximating formulae of the first degree, α − 2βx and α' − 2β'y, and this over the whole sphere of the problem, which, of course, as already mentioned, can only be the case under special circumstances. The equation (3) then turns into

and if here numerator is added to numerator and denominator to denominator, each of these fractions becomes

The ratio of the marginal utilities of the two goods, when equilibrium has been attained, is therefore, under the above assumptions, constant, independently of the values of x and y concerned, and equal to the proportion of the average marginal utilities of the quantities possessed. In whatever proportion the commodities here change hands by a repeated exchange, the last exchange which leads to equilibrium will always take place in the same proportion.¹

Suppose that A has 10 oxen and B has 100 sheep, and that the marginal utility of oxen is expressed by 200 — 10x, and the marginal utility of sheep by 10 — 0·1y. That is to say, in B’s estimation, if he does not yet possess on ox, one ox is worth 200 (e.g. 200 Marks, if the value of 1 Mark is regarded as constant); for every ox which he takes in exchange, the value of an ox will seem to him 10 (10 Marks)) less, etc. The same is true for A, so that he, if he still possesses all the 10 oxen, estimates the value of 1 ox as 100 Marks only, but for every ox which he gives in exchange he will increase that value by 10 Marks, etc. In an analogous way the same is true of the marginal utility function of the sheep.¹ Properly speaking, we are dealing here with oxen in the same way as with sheep, namely as optionally divisible continuous quantities; so that it would be more correct to say that B estimates the first fraction, for example the first hundredth of an ox, as worth 2 Marks, the second hundredth as worth 1 Mark 90 Pfennig, etc.

We therefore have here

α = 200, 2β = 10, α' = 10, 2β' = 0· 1

When equilibrium has been attained, we necessarily get

or, written in a shorter way,

, consequently,

as follows by the addition of numerator to numerator and denominator to denominator. The last fraction expresses the constant and on both sides equal proportion of the marginal utilities in case of equilibrium, and consequently also the proportion in which both commodities are at last always exchanged.

The above equation finally reduces itself, as can easily be found, to

10x + 3y = 200

This equation must always be fulfilled after the exchange has taken place, but otherwise, within the above-mentioned limits, all possible proportions of exchange can occur. In order to determine these limits, we put, as we have already ascertained, supposing that A exchanges without any profit,

or

But if B exchanges without profit,

or

each time in conjunction with the equation

10x + 3y = 200

From these equations we obtain for the one limit

and for the other limit

The possible proportion of exchange will consequently be able to fluctuate between about 1 ox against 61 sheep and 3·4 oxen against only 55 sheep (or on an average 1 ox against about 16 sheep). In the first case B, and in the second A, will have exchanged without any profit (but also without loss).

As the proportion of marginal utility amounts in the end always to ‘1 ox worth 30 sheep,’ it could, for example, be supposed that both the contracting parties had from the beginning agreed to exchange in just this proportion. One would then have, beside the equation

10x + 3y = 200

which is always fulfilled, the equation

x = 30y

so that x = 2 and y = 60; that is to say, A gives 2 oxen to β and gets in return 60 sheep. It is easy to show that the gain of utility then becomes the same on both sides, namely 200 (Marks)¹

But if, for instance, β knows how to direct the proportions of exchange to his advantage, 3 oxen against only 56⅔ sheep (on an average 1 ox against 19 sheep) might be given by A. But A might perhaps not be inclined to do this in a single exchange, for although at first he values 1 ox as equivalent to 10 sheep, this proportion of marginal utility would have risen to ‘1 ox worth 30 sheep’ after the exchange, so that the transaction could appear to him as of doubtful use, though in reality it would bring him no loss according to our assumptions.

But supposing that he was first expected to exchange 1 ox for 13 sheep, then a second ox for 17⅔ sheep, then ½ ox for 11 sheep and finally another ½ ox for 15 sheep, then there would remain for him after each exchange respectively a proportion of exchange between sheep and oxen of more than 1 : 13, 1 : 17⅔, 1 : 22 and finally of just 1 : 30, so that each single exchange would have to seem to him undoubtedly profitable, although he has in fact finally exchanged just 3 oxen for not quite 57 sheep.

In the case of isolated exchange, too, of course, a kind of maximum problem is solved, for each of the exchanging persons strives after the greatest possible profit and is inclined to continue the exchange until he can derive no further profit from it. But since the whole problem is indeterminate, one can speak of a definite solution only when new conditions are added.

Such a condition would be, for instance, to determine the quantities of goods which are to be exchanged in such a way that the gain of utility attained by both the contracting parties together, in other words, approximately the ‘economic’ profit, becomes the greatest possible one. It is self-evident that, if this aim is attained by the exchange which has taken place, the proportion of marginal utility of both commodities on each side must be the same and that consequently the equation (3) must be fulfilled, for otherwise the exchange could always, as we have seen, be continued with a gain of utility on both sides, so that the gain of utility already attained could not possibly be the greatest possible one. But this does not mean that the solution of this problem belongs to the possible solutions mentioned above.

The mathematical treatment of this problem is very simple; one has only to express that the sum of the gains of utility on both sides, or, which is the same, the sum of the total utility attained on both sides

ϕ(a − x) + ψ(y) + χ(x) + ω(b − y)

is to be as great as possible. Since x and y are here independent of each other, one must consequently have at the same time

or, differently expressed,

F(a − x) = J(x)

and

By this the equation (3) is obviously exactly fulfilled; but whether the pair of values of x and y, so determined, really lies within the limits of the possible exchange, has still to be decided.

The matter becomes especially simple, if, as in our chosen example, the marginal utility functions are conceived as identical on both sides, F( ) with J( ) and f( ) with j( ). In this case the equations

F(a − x) = J(x) and f (b − y) = j(y)

are obviously fulfilled by and and in consequence of the general characteristics of the marginal utility functions, it is clear that they can have no other (real) solutions. In other words, the greatest possible total utility is attained under these assumptions if the existing supply is simply distributed in equal shares between both the exchanging persons. This, by the way, is evident.

In our example, therefore, A would give 5 oxen to β and would get 50 sheep for them. Thereby the conditioning equation

10 x + 3y = 200

is indeed fulfilled and the proportion of marginal utility turns out to be such that 1 ox is estimated on both sides as equal to 30 sheep, as was required by the theory. But this exchange lies far beyond the possible limits. Indeed, it would bring to A a loss instead of a profit, and is consequently excluded, if each of the exchanging persons pursues his own profit. (Compare, moreover, section 5.)

In what has gone before we set out from the hypothesis that the commodities which are to be exchanged cannot replace each other in any way, so that the marginal utility only depends on the possession of the commodity in question, but not on the possession of the other. In reality, however, this is not always, and perhaps never wholly, the case. In our example, therefore, it cannot in fact be without significance for the valuation of an ox, whether the possessor in question has or has not, besides a certain number of oxen, also sheep. Therefore it would correspond more to reality if, as Edgeworth¹ has done, one conceived the total utility for A of oxen and sheep together as a general function U of x and y, whereby the partial derivatives of U in relation to x and y (taken positively) obviously express the marginal utility for A of the oxen and sheep respectively. If V is the corresponding function for B, one obtains as a conditioning equation of the exchange (called ‘contract curve’ by Professor Edgeworth) the very elegant expression

which turns into the above equation (3), as soon as one is allowed to suppose that

U = ϕ(a − x) + ψ(y)

and

V = χ(x) + ω(b − y)

that is to say, when the utility (total utility as well as marginal utility) of each commodity only depends on the possessed quantity of this commodity.