Knowledge that Transforms

ALMOST ALL of the phenomena of economic life, like many other processes, social, meteorological, and others, occur in sequences of rising and falling movements, like waves. Just as waves following each other on the sea do not repeat each other perfectly, so economic cycles never repeat earlier ones exactly either in duration or in amplitude. Nevertheless, in both cases, it is almost always possible to detect, even in the multitude of individual peculiarities of the phenomena, marks of certain approximate uniformities and regularities. The eye of the observer instinctively discovers on waves of a certain order other smaller waves, so that the idea of harmonic analysis, viz., that of the possibility of expressing the irregularities of the form and the spacing of the waves by means of the summation of regular sinusoidal fluctuations, presents itself to the mind almost spontaneously. If the results of the analysis happen sometimes not to be completely satisfactory, the discrepancies usually will be interpreted as casual deviations superposed on the regular waves. If the analyses of the first and of the second halves of a series give considerably divergent results (such as, for example, were found by Schuster while analyzing sunspot periodicity),' it is, even then, possible to find the solution without giving up the basic concept. Such a discrepancy may be the result of the interference of certain factors checking the continuous movement of the process and substituting for the former regularity a new one which sometimes may

THE purpose of the present note is to state the mathematical and physical definition of invariant in such a form as will make this notion generally applicable to observations ordered in time, and in particular applicable to economic variables. In mathematics we are taught to call an invariant a quantity which retains its numerical values in spite of transformations of coordinates. Take, for example, the length of a bar in a Euclidean space. If the coordinate system chosen is x, y, z, a Cartesian one, then the length of the bar in question will be

E. H. Phelps Brown, The Profit-Experience of Producers and Their Response to Price: Third Report of the Econometrica Committee on Source Materials for Quantitative Production Studies, Econometrica, Vol. 4, No. 4 (Oct., 1936), pp. 310-323

STATEMENTS of the proportions in which total costs at a given rate of output are divided between different categories are available in greater quantity than those which show how total costs vary with It is, therefore, of interest to point out that the former kind of statement does not merely set out the state of affairs at one and only one rate of output, but diffuses a halo of light upon the behavior of costs within a range extending some little way above and below the given rate. The light is seen only by reflection from assumptions that we choose to introduce, and it becomes dimmer the further we depart from the starting-point; yet the following considerations may show that it does exist. Thus, (a), costs may be divided simply into fixed and proportional, where is used in its strict sense as meaning bearing a constant proportion to output. Such a classification cannot be suitable over the whole range of output, for if it were, average cost would continue to fall until the output became infinite; but within a certain range it may give a fair approximation. Suppose, then, that we are told of one factory that, at a given rate of output, the fixed element of cost makes up 40 per cent and the proportional 60 per cent of the whole: we can infer that at a rate of output 10 per cent less, average cost will be increased in the ratio of 94 to 90. More generally, we assume as an approximation within a certain range the function,