The Collected Economics Articles of Harold Hotelling

Harold Hotelling was one of the most important of the twentieth-century pioneers of mathematical economics and mathematical statistics.¹ In both fields he is recognised for his powerful theoretical contributions² and he was a most effective and caring teacher. His influence, both as an economist and as a statistician, is felt not only through his publications (a large number of which are seminal³), but also through his students, amongst whom one can count many of the leading economists and statisticians of the next generation.⁴

In the older treatments of depreciation the cost, or “theoretical selling price” of the product of a machine, was conceived of as determined causally by the addition of a number of items of which depreciation is one. In other words, depreciation was first computed by some rather arbitrary formula not involving the theoretical selling price, which was then found by the addition of depreciation to operating costs and division by quantity of output. It will be shown in this paper that depreciation and theoretical selling price must be computed simultaneously from a pair of equations which are frequently a bit complicated. The differences in the results obtained from the arbitrary and mathematical formulae are often very large.

After the work of the late Professor F. Y. Edgeworth one may doubt that anything further can be said on the theory of competition among a small number of entrepreneurs. However, one important feature of actual business seems until recently to have escaped scrutiny. This is the fact that of all the purchasers of a commodity, some buy from one seller, some from another, in spite of moderate differences of price. If the purveyor of an article gradually increases his price while his rivals keep theirs fixed, the diminution in volume of his sales will in general take place continuously rather than in the abrupt way which has tacitly been assumed.

Contemplation of the world’s disappearing supplies of minerals, forests, and other exhaustible assets has led to demands for regulation of their exploitation. The feeling that these products are now too cheap for the good of future generations, that they are being selfishly exploited at too rapid a rate, and that in consequence of their excessive cheapness they are being produced and consumed wastefully has given rise to the conservation movement. The method ordinarily proposed to stop the wholesale devastation of irreplaceable natural resources, or of natural resources replaceable only with difficulty and long delay, is to forbid production at certain times and in certain regions or to hamper production by insisting that obsolete and inefficient methods be continued. The prohibitions against oil and mineral development and cutting timber on certain government lands have this justification, as have also closed seasons for fish and game and statutes forbidding certain highly efficient means of catching fish. Taxation would be a more economic method than publicly ordained inefficiency in the case of purely commercial activities such as mining and fishing for profit, if not also for sport fishing. However, the opposition of those who are making the profits, with the apathy of everyone else, is usually sufficient to prevent the diversion into the public treasury of any considerable part of the proceeds of the exploitation of natural recources.

That a tax imposed on the seller of a monopolized article may lead to an actual lowering of the price to the buyer has been shown by F. Y. Edgeworth.² His example was of a railway supplying two classes of passenger service at different prices and, unhindered by governmental interference, setting its rates so as to make its own profit a maximum. When the company is compelled to pay a tax on each first-class ticket, it finds it profitable, in Edgeworth’s example, to reduce rates on both classes of accommodations. Regarding this paradoxical conclusion, Professor Seligman writes:³

The mathematics which can show that the result of a tax is to cheapen the untaxed as well as the taxed commodities will surely be a grateful boon to the perplexed and weary secretaries of the treasury and ministers of finance throughout the world!

In a recent paper¹ I derived for demand functions for n commodities conditions which, I concluded, assure the satisfaction of all further conditions which may reasonably be applied to all sets of demand functions. Letting p₁, p₂, … p _n , and q₁, q₂, …, q _n be respectively the prices and quantities, the conditions are: These are equivalent to and were known, at least in part, by Edgeworth. Similar conditions in which the inequality signs are all > apply for supply functions.

The success of the government’s recovery program, or of any national economic program, must be judged, not in terms of price levels, but in terms of the quantities of physical goods and services which are put into the hands of consumers. With this is to be considered the effect of the program on the distribution of wealth among different classes. But the chief thing needed is to increase physical production. In this respect much that is being done at Washington is definitely in the wrong direction. The attempts to increase the prices and curtail the production of oil, agricultural products, and other commodities are anti-social.

In this paper we shall bring down to date in revised form an argument due essentially to the engineer Jules Dupuit, to the effect that the optimum of the general welfare corresponds to the sale of everything at marginal cost. This means that toll bridges, which have recently been reintroduced around New York, are inefficient reversions; that all taxes on commodities, including sales taxes, are more objectionable than taxes on incomes, inheritances, and the site value of land; and that the latter taxes might well be applied to cover the fixed costs of electric power plants, waterworks, railroads, and other industries in which the fixed costs are large, so as to reduce to the level of marginal cost the prices charged for the services and products of these industries. The common assumption, so often accepted uncritically as a basis of arguments on important public questions, that “every tub must stand on its own bottom,” and that therefore the products of every industry must be sold at prices so high as to cover not only marginal costs but also all the fixed costs, including interest on irrevocable and often hypothetical investments, will thus be seen to be inconsistent with the maximum of social efficiency. A method of measuring the loss of satisfactions resulting from the current scheme of pricing, a loss which appears to be extremely large, will emerge from the analysis. It will appear also that the inefficient plan of requiring that all costs, including fixed overhead, of an industry shall be paid out of the prices of its products is responsible for an important part of the instability which leads to cyclical fluctuations and unemployment of labor and other resources.

In the July issue of Econometrica¹ I gave a new proof, taking account of the interrelations of commodities by methods not available in the times of Dupuit and Marshall, that in a specified sense maximum welfare requires that the quantity of each good consumed or produced by an individual shall be that corresponding to all sales being at marginal cost. This proposition has revolutionary implications, for example in electric-power and railway economics, in showing that society would do well to cut rates drastically and replace the revenue thus lost by subsidies derived largely from income and inheritance taxes and the site value of land.

The proof of the fundamental theorems of my paper was on the basis of a set of assumptions including one to the effect that prices in the absence of the excise taxes were equal to marginal costs. It follows quite readily from these theorems that the attainment of the maximum of the general welfare, in the only sense that can apparently be given to this expression with the help only of rank ordering of a person’s satisfactions without interpersonal comparisons, requires that all sales be at such prices that the quantities bought will be exactly the same as if these sales were at marginal cost. In particular, in considering what prices should be charged by state-owned or regulated enterprises, we have marginal cost as the criterion, leaving no place for the criteria generally used, such as average cost.

What is ordinarily called income, and what the law calls income, may be divided into two parts, called savings and spendings. The exact distinction between these two is not so clear-cut as may at first appear, since spending is often for satisfactions spread over a considerable time in the future, and since savings yield a present satisfaction through the security, power, and independence they confer as well as the future satisfactions from the goods for which the savings may eventually be spent. Nevertheless the distinction can be made, somewhat roughly, when the satisfactions arising from the possession of savings are neglected, and when depreciation accounts are kept for such articles as family automobiles, furniture, and fur coats as well as for business property.