Harold Hotelling

Stability in Competition

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.

Relations Between Two Sets of Variates

Concepts of correlation and regression may be applied not only to ordinary one-dimensional variates but also to variates of two or more dimensions. Marksmen side by side firing simultaneous shots at targets, so that the deviations are in part due to independent individual errors and in part to common causes such as wind, provide a familiar introduction to the theory of correlation; but only the correlation of the horizontal components is ordinarily discussed, whereas the complex consisting of horizontal and vertical deviations may be even more interesting. The wind at two places may be compared, using both components of the velocity in each place. A fluctuating vector is thus matched at each moment with another fluctuating vector. The study of individual differences in mental and physical traits calls for a detailed study of the relations between sets of correlated variates. For example the scores on a number of mental tests may be compared with physical measurements on the same persons. The questions then arise of determining the number and nature of the independent relations of mind and body shown by these data to exist, and of extracting from the multiplicity of correlations in the system suitable characterizations of these independent relations. As another example, the inheritance of intelligence in rats might be studied by applying not one but s different mental tests to N mothers and to a daughter of each

The Generalization of Student’s Ratio

The accuracy of an estimate of a normally distributed quantity is judged by reference to its variance, or rather, to an estimate of the variance based on the available sample. In 1908 “Student” examined the ratio of the mean to the standard deviation of a sample.1 The distribution at which he arrived was obtained in a more rigorous manner in 1925 by R.A. Fisher,2 who at the same time showed how to extend the application of the distribution beyond the problem of the significance of means, which had been its original object, and applied it to examine regression coefficients and other quantities obtained by least squares, testing not only the deviation of a statistic from a hypothetical value but also the difference between two statistics.

The General Welfare in Relation to Problems of Taxation and of Railway and Utility Rates

This classic paper defends the use of marginal cost pricing for railways. The author views his contributions as an extension of Dupuits fundamental insights concerning the relation between the general welfare and railway and utility rates. The basic purpose of the paper is to justify the public subsidization of declining cost industries. In discussing the theoretical results, the author strongly supports federal investments in the Tennessee Valley Authority. Although various externalities would accrue to those living outside the region, the author acknowledges that taxpayers not living in the region would be expected to lose on account of the project, but a rough randomness in distribution should be ample to ensure such a distribution of benefits that most persons in every part of the country would be better off by reason of the program as a whole. Two groups are singled out that might expect not to benefit from the consistent application of total benefit maximization: the very wealthy, who would likely be asked to pay considerable taxes, and the land speculators, who would likely face increases in property taxes.

A General Mathematical Theory of Depreciation

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.

Edgeworth’s Taxation Paradox and the Nature of Demand and Supply Functions

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!

The Relation of Prices to Marginal Costs in an Optimum System

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.

Effects of Pregnancy, Mouth Acidity, and Age on Dental Caries

The widespread idea that pregnancy is a cause of dental decay has been questioned. In the present study an attempt was made to throw further light on the problem. An index of caries based on the number of decayed tooth surfaces (Bodeckers Caries Index) was calculated for each of 324 pregnant women referred routinely from the Sloan Hospital for Women (fig. 1). Each patient received full mouth radiographs, allowances were made for missing teeth, and occlusal grooves large enough to admit the sharp point of an explorer were set down as cavities. The caries noted is of course an accumulation (see Bodeckers Caries Index) so that allowance must be made for age in comparing the observations on teeth with the number of past pregnancies. A further supposedly relevant variable is the hydrogen ion concentration of the saliva, which was recorded in each case because of the oft presumed relationship of caries to mouth acidity. In recording the salivary pH, unstimulated saliva was analyzed by the colorimetric method. In addition to these pregnant women, 31 women who had never been pregnant were examined in the same way for comparison. The nonpregnant group consisted of a miscellany of hospital employees, patients coming in for reasons other than caries, et al. It is not possible to obtain a strictly comparable control of this nature, and the statistical results may perhaps be interpreted to mean that these 31 are not a random sample with respect to caries distribution, despite all precautions. Their mean caries index is substantially higher than that of the pregnant group, which may be interpreted to mean either that pregnancy prevents caries, or that the selection of cases

A Final Note

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.

The collected economics articles of Harold Hotelling

The Life and Economic Thought of Harold Hotelling.- Bibliography of Harold Hotelling.- Reprints of the Published Economics Papers by Harold Hotelling.- 1925 A general mathematical theory of depreciation. Journal of the American Statistical Association, 20, 340-53..- 1929 Stability in competition. Economic Journal, 39, 41-57..- 1931 The economics of exhaustible resources. Journal of Political Economy, 39, 137-75..- 1932 Edgeworths taxation paradox and the nature of demand and supply functions. Journal of Political Economy, 40, 577-616..- 1933 Note on Edgeworths taxation phenomenon and Professor Garvers additional condition on demand functions. Econometrica, 1, 408-9..- 1935 Demand functions with limited budgets. Econometrica, 3, 66-78..- 1936 Curtailing production is anti-social. Columbia Alumni News, 28(4), 3 and 16..- 1938 The general welfare in relation to problems of taxation and of railway and utility rates. Econometrica, 6, 242-69..- 1939 The relation of prices to marginal costs in an optimum system. Econometrica, 7, 151-6..- 1939 A final note. Econometrica, 7, 158-60..- 1943 Income-tax revision as proposed by Irving Fisher. Econometrica, 11, 83-7..