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,