2. Different Uses of the Same Kind of Commodity

The simplest form of exchange is that in which the owner of a quantity of goods can and will make different uses of its different parts. The above-mentioned colonist, for example, will keep for himself, his poultry and parrots only a part of his stock of corn for food purposes; the rest he will convert into spirits. It is obvious, then, that he must proportion the two parts to each other in such a way that the marginal utility on both sides becomes the same—in such a way, that is to say, that the last quantity of the remaining corn gives him the same enjoyment as the last quantity of the corn converted into spirits.

Put into an analytical form, this would be expressed as follows: The smallest enjoyment of one unit—for example, one kilogram of corn (the marginal utility of corn)—is conceived as a diminishing function of the supply which still remains after converting part of it into spirits. If, for example, the original supply consisted of a kilograms of corn, and x kilograms of it have already been converted into spirits, so that a–x kilograms of corn are left, the marginal utility of corn, which was originally F(a), has now risen to F(a–x). In the same way the smallest enjoyment of one kilogram of corn converted into spirits (marginal utility of spirits or, more properly, of corn used for making spirits) is a diminishing function of the quantity of corn used in this way, and can consequently be expressed by f (x). Then the solution of this problem consists simply of equating these two functional values:

F(a – x) = f (x) (1)

Or one could conceive the marginal utility of spirits directly as a function of the quantity of spirits produced. If we suppose that from m kilograms of corn one obtains one litre of spirits, the supply of spirits produced amounts to litres. The enjoyment of the last litre of spirits produced must then be expressed by , where f₁ represents a new function. But now, when equilibrium has occurred, this enjoyment must be as great as the enjoyment of the last m units of the remaining corn, or, which is the same, the marginal utility of spirits (enjoyment of one litre of spirits) must be m-times as great as the marginal utility of corn (enjoyment of one kilogram of corn).¹ We therefore write

and the problem would be solved—if one knew the forms of the functions F( ) and f( ) or f₁( ), and could replace them by exact mathematical expressions. Then it would only remain to solve the first or the second of the above equations for x, which would be a purely mathematical task. Our colonist solves the same problem by the experimental method, without having heard anything of this theory. When he has produced too little spirits, he distils some more ; if he has produced too much, so that the remaining supply of corn is insufficient for his purposes, he will take particular note of this experience for the next year.

But even without knowing the exact forms of the functions, from these equations one can draw an important conclusion, which can, of course, also be easily arrived at without using any symbols. For one could also conceive the whole utility or value in use of the remaining supply of corn or of the quantity of corn converted into spirit as functions of the quantities in question—functions which, of course, grow with the variable quantities, but more slowly than these. If we express them by ϕ(a − x) and ϕ(x), the marginal utilities F(a − x) and f (x) are, as we have already shown, their differential coefficients, the former with respect to(a − x), the latter with respect to x.

If one sets oneself the task of determining x in such a way that

ϕ(a − x) + ψ(x)

becomes a maximum, this problem can, as is known, be solved by making the differential coefficient of the sum with respect to x equal to zero. One consequently has

or, since

and

F(a − x) = f (x)

which is the same equation as the one found at the beginning.

In other words, the solution of our original problem forms at the same time the solution of the problem of distributing the supply of corn between its two uses in such a way that the greatest possible total utility or total enjoyment arises from it.

This, however, is self-evident; for the purpose of the production of spirits was just to obtain from one part of the supply of corn a higher enjoyment than was obtainable by its direct consumption; and the production will be continued as long as a further gain of utility is obtainable, that is to say, until the greatest possible utility is attained.

Beyond this, almost nothing, as was said before, is known a priori about the behaviour of the functions ϕ( ) and ψ( ) or F( ) and f( ). At the outset it is only certain that ϕ( ) and ψ( ) grow with the variable quantities under the sign of the function, but more slowly than these, and when these disappear, they become zero themselves. From this it follows that their differential quotients F( ) and f( ) are diminishing functions. The simplest approximating formula which satisfies these conditions is the one in which z indicates any variable quantity:

ϕ(z) = αz − βz², ψ(z) = α'z − β'z²

consequently

F(z) = α − 2βz, f (z) = α' − 2β'z

where α and β, α' and β' respectively are positive constants, whose values must be determined for each case. If here, for example, β is very small compared with α, then at first ϕ(z) increases almost proportionally with z, but afterwards more and more slowly, reaching a maximum for ; after that it decreases, finally becoming zero and even negative. The same is true of ψ(z), if one replaces α and β by α' and β' respectively.

F(z) and f (z), on the contrary, have for small values of z almost the constant values α and α' ; if z increases, they always decrease ; they become zero where and respectively; and beyond that they become negative.

In this there is nothing which is inconsistent with experience, for the total utility as well as the marginal utility of a quantity of goods can finally become ‘negative,’ that is to say, can change into disutility, if the existing quantity becomes much too great. For example: water, manure, dross, sawdust, etc.

But what it does not show is whether so simple an approximating formula meets even one single case sufficiently exactly to be applicable. In most cases, this is even most improbable. Launhardt, however, has made the most extensive use in his work¹ of precisely this formula, without really examining even once how far it corresponds to the facts. It is at least doubtful, therefore, whether the fine results and conclusions which, by the help of this approximating formula, he has found and printed in italics, have anything to do with reality.

Nevertheless, it will be possible to assert, according to the analogy of physical events, that, if it is only a question of variations within certain narrower limits, such an approximating formula can be substituted within this sphere for the exact form of the functions, whatever the nature of the latter may otherwise be.

If, for instance, in our example above it is quite certain in advance that the value of x sought² must lie between two limits b and c, which are known to lie not too far apart, it will be possible within these limits to use without hesitation the approximating formulae; that is to say, instead of equation (1)

F(a − x) = f (x)

we write

α − 2β(a − x) = α' − 2β'x

In order to be able in this case to determine the constants α, β, α', β', it is necessary to know for at least two values of x which belong to this sphere, the corresponding four values of the functions of the marginal utilities F(a − x) and f (x).³ If we suppose that for x = b the marginal utility of corn is ν and the marginal utility of the corn converted into spirits ν', and that for x = c their values are w and w' respectively, α, β, α', β' can easily be expressed by ν, w, ν' and w', and we obtain

or

This expression is, as can be seen, homogeneous in relation to the magnitude of ν, ν' , w and w' and of degree zero. In other words, the value of x remains unchanged, irrespective of the measure according to which the marginal utility is estimated; only for both kinds of commodities or uses in question this measure must be one and the same. This, of course, cannot be otherwise. The utility of a commodity is something sui generis; it can be measured neither in metres nor in kilograms; it is comparable only with itself or with the utility of other goods.

The understanding of the whole matter is greatly facilitated if one conceives it geometrically according to the method employed by Gossen, Jevons and others. The successively diminished supply of corn and the marginal utilities belonging to it, both measured according to an optional unit, can be represented as abscissa and ordinate of a curve, whose area¹represents the total utility according to the principles of the integral calculus. In the same way the marginal utility of the quantity of corn converted into spirits can be expressed by the ordinate of another curve whose abscissa, which represents this quantity itself, is measured from point a towards the left.

The solution of this problem now consists simply in finding the point of intersection of these two curves. The use of the approximating formula simply tells us that both curves can be regarded as straight lines near the point of intersection (as is usual, if it is a question of short pieces).

The rest will then simply be interpreted geometrically.