## The Mathematical School

The Mathematical School

The Mathematical school is distinguished for its attachment to the study of exchange, from which it proposes to deduce the whole of political economy. Its method is based upon the fact that every ex­change may be represented as an equation, A = B, which expresses the relation between the quantities exchanged. Thus the first step plunges us into mathematics.

However true this may be, the application of the method must necessarily be very limited if it is always to be confined to exchange.

It is, however, a mistake to suppose that this is really the case, and one of the most ingenious and fruitful contributions made by the new school was to show how this circle could be gradually enlarged so as to include the whole of economic science.
Distribution, production, and even consumption are included within its ambit. Let us take distribution first and inquire what wages and rent are. In a word, what are revenues? A revenue is the price of certain services rendered by labour, capital, and land, the agents of production, and paid for by the entrepreneur as the result of an act of exchange.

And what is production? It is but the exchanging of one utility for another—a certain quantity of raw materials and of labour for a certain quantity of consumable goods. Even nature might be compared to a merchant exchanging products for labour, and Xenophon must have had a glimpse of this ingenious theory when he declared that "the gods sell us goods in return for our toil." The analogy might be pushed still further, and every act of exchange may be considered an act of production. Pantaleoni puts it elegantly when he says that "a partner to an exchange is very much like a field that needs tilling or a mine that requires exploiting."

And what are capitalization, investment, and loan but the exchange of present goods and immediate joys for the goods and enjoyments of the future?

It was a comparison instituted between the lending of money and an ordinary act of exchange that led Bohm-Bawerk to formulate his celebrated theory of interest. Bohm-Bawerk, however, is a repre­sentative of the Austrian rather than the Mathematical school.

Even consumption—that is, the employment of wealth—implies incessant exchanging, for if our resources are necessarily limited that must involve a choice between the object which we buy and that which with a sigh we are obliged to renounce. To give up an evening at the theatre in order to buy a book is to exchange one pleasure for another, and the law of exchange covers this case just as well as any other. It is the same everywhere. To pay taxes is to give up a portion of our goods in order to obtain security for all the rest. The rearing of children involves the sacrifice of one's own well-being and comfort in exchange for the joys of family life and the good opinion of our fellow men.

It is not impossible, then, to discover among economic facts certain relations which are expressible in algebraical formulae or even reducible to figures. The art of the Mathematical economist consists in the discovery of such relations and in putting them forth in the form of equations.

For example, we know that when the price of a commodity goes up the demand for it falls off. Here are two quantities, one of which is a function of the other. Let us see how the law of demand in its amended form would express this.

If along a horizontal line A B we take a number of fixed points equidistant from one another to represent prices—e.g., I, 2, 3, 4, 5 . . 10—and from each of these points we draw a vertical line to represent the quantity demanded at that price, and then join the summits of these vertical lines, which are known as the ordinates, we have a curve starting at a fairly high point—representing the lowest prices—and gradually descending as the prices rise until it becomes merged with the horizontal, at which point the demand becomes nil.

What is very interesting is that the curve is different for different products. In some cases the curve is gentle, in others abrupt, accord­ing as the demand, as Marshall puts it, has a greater or lesser degree of elasticity. Every commodity has, so to speak, its own characteristic curve, enabling us, at least theoretically, to recognize that product among a hundred.

We would naturally expect the supply curve to be just the inverse of the demand curve, rising with a rising price and descending with a falling one, so that by the time the price is zero supply is nil, whereas the demand is infinite.

But it is not quite correct to regard it as merely the inverse of the demand curve. A supply curve is really a much more complicated affair, because supply itself depends upon cost of production, and there are some kinds of production—agriculture, for example—where the cost of production increases much more rapidly than the quantity produced. In industry, on the other hand, the cost of production decreases as the quantity produced increases.

Mathematical political economy, not content with seeking relations of mutual dependence between isolated facts, claims to be able to embrace the whole field within its comprehensive formulae. Every­thing seems to be in a state of equilibrium, and any attempt to upset it is immediately corrected by a tendency to re-establish it. To deter­mine the conditions of equilibrium is the one object of pure economics.

The most remarkable attempt at systematization of this kind was made by Professor Walras, who endeavoured to bring every aspect of the economic world within his formula, a task almost as formidable as that attempted by Laplace in his Mecanique celeste.

Let us imagine the whole of society included within one single room, say the London Stock Exchange, which is full of the tumult of those who have come to buy and sell, and who keep shouting their prices. In the centre, occupying the place usually taken up by the market, sits the entrepreneur, a merchant or manufacturer or an agri­culturist, as the case may be, who performs a double function.

On the one hand he buys from producers, whether rural or urban, landlords, capitalists, or workers, what Walras calls their "productive services," that is, the fertility of their lands, the productivity of their capital or their labour force, and by paying them the price fixed by the laws of exchange he determines the revenue of each; to the proprietor he pays a rent, to the capitalist interest, to the workman wages. But how is that price determined? Just as at the Exchange all values what­soever are determined by the law of demand and supply, so the entre­preneur demands so many services at such and such a price and the capitalist or workman offers him so many at that price, and the price will rise or fall until the quantity of services offered is equal to the quantity demanded.

The entrepreneur on his side disposes of the manufactured goods fashioned in his factory or the agricultural products grown on his farm to those very same persons, who have merely changed their clothes and become consumers. As a matter of fact the proprietors, capitalists, and workers who formerly figured as the vendors of services now reappear as the buyers of goods. And who else did we expect the buyers to be? Who else could they be?

And in this market the prices of products are determined in just the same fashion as we have outlined above.
All at once, however, a newer and a grander aspect of the equili­brium comes to view. Is it not quite evident that the total value of the productive services on the one hand and the total value of the products on the other must be mathematically equal? The entrepreneur cannot possibly receive in payment for the goods which he has sold to the consumers more than he gave to the same persons, who were just now producers, in return for their services. For where could they possibly get more money? It is a closed circuit, the quantity that comes out through one outlet re-enters through another.

With the important difference that it keeps much closer to facts, the ex­planation bears a striking resemblance to Quesnay's Tableau e'conomique.

We have two markets in juxtaposition, the one for services and the other for products, and in each of them prices are determined by the same laws, which are three in number:

(a) On the same market there can be only one price for the same class of goods.
(b) This price must be such that the quantity offered and the quantity demanded shall exactly coincide.
(c) The price must be such as will give maximum satisfaction to the maximum number of buyers and sellers.

All these laws are mathematical in character and involve problems of equilibrium.
In some such way would the new school reduce the science of economics to a sort of mechanism of exchange, basing its justification upon the contention that the Hedonistic principle of obtaining the maximum of satisfaction at the minimum of discomfort is a purely mechanical principle, which in other connexions is known as the principle of least resistance or the law of conservation of energy. Every individual is regarded simply as the slave of self-interest, just as the billiard-ball is of the cue. It is the delight of every economist as of every good billiard-player to study the complicated figures which result from the collision of the balls with one another or with the cushion.

Another problem of equilibrium is to discover the exact proportion in which the different elements combine in production. Jevons com­pares production to the infernal mixture which was boiled in their cauldron by the witches in Macbeth. But the ingredients are not mixed haphazard, and Pareto thinks that they conform to a law analo­gous to the law known in chemistry as the law of definite proportions, which determines that molecules shall combine in certain proportions only. The combination of the productive factors is perhaps not quite so rigidly fixed as is the proportion of hydrogen and oxygen which goes to form water. Similar results, for example, may be obtained by employing more hand labour and less capital, or more capital and less hand labour. But there must be some certain proportion which will yield a maximum utility, and this maximum is obtainable in precisely the same way as in other cases of equilibrium—that is, by varying the 'doses' of capital and labour until the final utility in the case both of capital and labour becomes equal. Generally speaking, this is the law that puts a limit to the indefinite expansion of industry, for whenever one element runs short, be it land or capital, labour or managing ability or markets, all the others are directly affected adversely and the undertaking as a whole becomes more difficult and less effective. Pareto rightly enough attaches the greatest importance to this law, and we have only to remember that it is the direct antithesis of the famous lav/ of accumulation of capital to realize its full significance.

There are several other cases of interdependence to which the new school has drawn attention, as, for example, that of certain comple­mentary goods whose values cannot vary independently. What is the use of one glove or one stocking without another, of a motor-car without petrol, of a table service without glasses? Not only is this true of consumption goods; it also applies to production goods. The value of coke is necessarily connected with the value of gas, for you cannot produce the one without the other, and this applies to all by-products. The possibility of utilizing a by-product always lowers the price of the main commodity.