3. Exchange at Given Prices

If we now turn to exchange in its real sense, we can first deal with the simple case, where the proportion of exchange of two commodities—or, if we conceive one of them as the price commodity¹ and the other as the commodity, the price of the latter—is already fixed in advance, as, for example, is approximately the case in the retail trade. The buyer of the commodity then provides himself with so much of it and disposes of so much of the price commodity—in a proportion of exchange which has been fixed by the seller—so that finally the proportion of the marginal utilities of both commodities for the intended consumption period just equals the price.

Let us suppose, for example, that he has at the beginning the quantity b of the price commodity, or b units, but is still without the commodity, and that he must give for one unit of the commodity p units of the price commodity. If we then express the marginal utility of the commodity by F( ) and the marginal utility of the price commodity by f( ), we get

F(x) = p . f (b − y)

where x indicates the number of the acquired units of the commodity and y the number of units of the price commodity given in exchange. Moreover, we have here

y = p . x

so that the problem is solved as soon as the forms of the functions F( ) and f( ) are known. Very often it will happen that the function f( ) is a constant. If, for example, the price commodity is money, its marginal utility is determined by the income or even by the total wealth of the buyer, and these magnitudes do not as a rule vary noticeably by a single exchange. We have then quite simply

F(x) = p . ν

if by ν we express the constant utility of the unit of money (for the buyer), or what is usually called the ‘value of one £’ or of One florin,’ and if the price of one unit of the desired commodity amounts to p£ or florins at the moment. Within suitable limits, one could, of course, also use here an approximating formula of the first degree for F( ) [ and f( )] ; for instance, when x lies near a,

after which

F(x) = p . ν

becomes

p₁ expresses here the average price of the commodity, a the quantity of it which the buyer in question usually buys ; w and c are two constants, which, for the sake of symmetry, we have chosen in such a way that w shall express a magnitude of value or utility, and c a quantity of goods. The last equation, then, tells us that if the price demanded is a little under or over the average price, the buyer in question will purchase and consume more or less than usual of the commodity for the consumption period concerned in proportion to the difference of price.

That the price on the part of the seller is unalterably fixed, supposes, of course, that for him neither the marginal utility of the commodity nor that of the price commodity is altered by the exchange. This can happen either through the fact that his supply of the commodity concerned is very large in comparison with the quantity to be exchanged, or through the fact that he himself is only the connecting link between the real barterers, as in the wholesale trade. How in the last case the price is in fact determined, is a problem in itself, which we cannot deal with for a long time yet. It is clear, of course, that here also a maximum problem is solved. Suppose that for the buyer the total utility of the quantity of goods is expressed by ϕ(x) and the total utility of the price commodity by ψ(b − y). If he now wants to gain the greatest possible utility, that is to say, if ϕ(x) + ψ(b − y) is to be a maximum, we must get

But according to what has gone before,

and

We therefore obtain

F(x)dx = f (b – y)dy

dx and dy express here the small quantities of goods last exchanged against each other. Their proportion is consequently the constant price p. Or, which is the same, from the equation y = p . x we obtain dy = pdx. We have consequently

F(x) = p . f (b − y)

as above.