Knowledge that Transforms

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-

I am glad to be able to state, right from the beginning, that I fully agree with Dr. Preinreich as concerns the relativity of depreciation technique. I never had the intention of stating that there is a true method of depreciation, so much the less, as I do not accept any true capital value. This, I hope, will be clear from the following addition to my paper. In my study of depreciation schemes, I have tried to introduce great simplifications. But I do not think this has veiled fundamental difficulties of the problem, as Dr. Preinreich seems to suspect. I should also like to emphasize that my introduction of the distribution, which I may denote by (,u, s) and which is characterized by a slight constant mortality during s years followed by a catastrophical mortality after this time of all buildings left, is something more than the well-known annuity method of depreciation, although it may be interpreted formally in such a way. To replace the vague risk margin in the valuation rate of interest by ,i seems to be a theoretical improvement of the scheme. For with it, depreciation insurance is introduced in practice. But the advantages of the scheme are perhaps more easily grasped when the method is applied to numerical work, and I shall permit myself to come back to this question at a later occasion with the support of practical examples. The simplifications I have introduced (or the restrictions as to generalizations that I have found it advisable to impose on my theory) are evidently, as I have already stated in my paper, a matter of taste, and de gustibus non est disputandum. But in choosing these first approximations, I tried of course to profit by a long experience of practical mathematics. Personally, I am convinced that the choice will prove itself to have been happy, but such statements can be demonstrated only by practical work. I also have the impression that the double line of thought which is illustrated by my two approximations one ad-

STATISTICIANS are familiar with the moving average. For example, when monthly prices are given, it may be desirable to eliminate the seasonal variation. The average of the monthly prices for a calendar year may be found, then the average of the prices from the February of this year to the January of the next, then from March to February, and so on. Each average thus formed involves just once each of the twelve months; and such averaging would seem a good method for eliminating the effects of the seasonal cycle. A question, however, arises: When we take out one cycle, such as the twelve-month cycle, are we likely to put in another cycle, with substantial waves? Under certain conditions, the answer is: Yes. But to understand why such an unwelcome cycle intrudes itself, some explanation is required. As a basis for studying cycles, it is often assumed that data contain an additive chance constituent. That is, it is assumed that the rth measurement Ur may be analyzed thus: