Knowledge that Transforms

PROFESSOR HOTELLING'S PAPER, Relations between Two Sets of Variates,'l should be widely known and his method used by practical statisticians. Yet, few practical statisticians seem to know of the paper, and perhaps those few are inclined to regard it as a mathematical curiosity rather than an important and useful method of analyzing concrete problems. This may be due to two reasons: first, that Hotelling's paper makes use of rather complicated mathematics and does not spell out in detail the methods of numerical computation; and, second, that although the paper applied the methods to two sets of statistical data, the major emphasis is on mathematical theory, and only rather incidental consideration is given to the meaning of the results obtained in actual statistical work. This paper will try to do two things: first, and most important, it will apply these methods to two different kinds of problems in the hope that this will suggest other practical applications; second, it will develop the methods of analysis in somewhat simpler terms, and discuss numerical computation in greater detail than Hotelling's paper.

Also published in: A.H. Hansen and R.V. Clemence (Eds), Readings in Business Cycles and National Income, Allen & Unwin, London, 1953, pp. 357-375

IN AN EARLIER PAPER1 I derived conditions for the stability of equilibrium in monopolistic competition for two competitors. The extension of that analysis to cover more than two competitors is by no means obvious, and it is a matter of importance to know how the number of competitors affects the question of stability. It is therefore to the solution of the problem for n competitors that the present paper will be devoted. Although the economic problem will be limited to the question of monopolistic price competition, the methods employed can be used to test the stability of any equilibrium determined by the-solution of a system of linear equations. In Section I we shall formulate a demand function for n competitors in an imperfect market, and also their cost functions. In Section II we shall derive the conditions for the existence of equilibrium and in Section III we shall determine the conditions for its stability in the cases both of noncontinuous and continuous adjustment on the basis of a given set of expectations. In Section IV we shall consider the implications of our results for the general cases of two, three, and n competitors, while in Section V we solve the problem completely for n identical competitors. Finally in Section VI, we shall adumbrate the problems involved when the stability of the expectations themselves is brought into question.

IN THE JANUARY issue of this journal Mr. Elliott B. Woolley presented a method of determining a straight-line regression by the summed absolute values of the areas of rectangles formed by the projections of each observation upon the regression line.' The resulting line possesses the usual property of passing through the point of means, and its slope is a simple average of the elementary regression slopes derived by in each direction; it is the geometric mean of the elementary regression coefficients, each referred to the same axis, and has their algebraic sign. It should be pointed out that this is nothing other than Frisch's regression (cf. Statistical Confluence Analysis . . .), and a statistical parameter which has long appeared in the literature. In terms of a correlation surface it represents the major axis of the concentric ellipses of equal frequency. While Mr. Woolley has made an interesting contribution in proving this minimizing property of the diagonal regression,2 his further argument that it is to be preferred in any sense as a method of determining regression lines seems to require brief comment. (a) The lack of consistency between the elementary regressions is a necessary property of a linear multivariate frequency surface. It is expressed in the purely formal statistical law of regression towards the average. The elementary regressions are not thereby illogical. (b) If the aim of the investigation is not simply a characterization of the properties of the multivariate distribution, but rather the search for a hypothetical true (in some sense) linear relationship, upon which has been superimposed a distribution of errors, then no definite method of determining the regression equation can be specified until some assumptions have been made concerning the nature of the disturbing causes. These assumptions must be in the nature of postulates; by no possible method can they be determined inductively from an examination of the data, even in an infinitely large sample. This last statement must be emphasized since some of the recent literature seems at first sight to suggest otherwise. This is because seemingly innocent, but in fact highly restrictive and often arbitrary, assumptions of noncorrela-