¹ Three works which survey the history of thought on the problem of the nature of time, which the writer has found useful, are: Cleugh, M. F., Time (London, 1937)Google Scholar; Gunn, J. Alexander, The Problem of Time (London, 1929)Google Scholar; Heath, L. R., The Concept of Time (Chicago, 1936).Google Scholar

² It is true that physics has no need of the concept of Absolute Time, and that in this field too, time is inextricably bound to events. Newton's concept of Absolute Time, as a reality known only to a deity, emerged as a result of his attempt to link physics with theology.

³ Furthermore, the ontological question, whether time is real and a part of existence, need not concern the economist. Time may be real as held by Galileo, Newton, Berkeley, Alexander, and others, or there may be doubts as to the reality of time as in the views of such philosophers as Plato, Leibnitz, Hegel, Bradley, and McTaggart; but the ultimate nature of reality is the metaphysician's problem and not the economist's.

⁴ The fact that Leibnitz, an idealist, denied reality, either subjective or objective, to time is of no concern here.

⁵ Kant too, among others, viewed time as a principle by which events are given as connected in experience, though his discussion, fraught with inconsistencies, delves into realms which need not concern the economist. It is a moot question for epistemology and metaphysics whether the notion of time is arrived at through intuition prior to an empirical examination of the universe (as with Kant) or is elaborated after such empirical examination in order to formulate thoughts and expressions about the universe (as with Leibnitz).

⁶ Some writers have been disposed to limit their concept of time to the succession of ideas in the mind; this was true of Locke and, to a greater extent, of Berkeley, to mention but two.

⁷ Bridgman, P. W., The Intelligent Individual and Society (New York, 1938), p. 91.Google Scholar

⁸ For an amusing and instructive example of the use of the t of physics see the story of the schoolboy required to solve the equation 368 = 30t + 2t ², in Cleugh, , Time, p. 48.Google Scholar An admirable statement of the nature of the t of physics is given in Merrill, A. A., “The ‘t’ of Physics” (Journal of Philosophy, vol. XIX, 1922, pp. 238–41).CrossRefGoogle Scholar

⁹ This must surely be true of all concepts of time, except certain narrow metaphysical ones.

¹⁰ Gunn, , The Problem of Time, p. 397 (my italics).Google Scholar

¹¹ See his The Logic of Modern Physics (New York, 1927)Google Scholar; The Nature of Physical Theory (Princeton, 1936)Google Scholar; and The Intelligent Individual and Society.

¹² For a rather popular account of a physicist's views on the irreversibility of time see SirEddington, Arthur, The Nature of the Physical World (1st ed., Cambridge, Eng., 1928) especially chap. IV.Google Scholar

¹³ For present purposes it will be sufficient to regard endogenous variables as all those which the economist as such, in the most general of his theories, seeks to explain, and exogenous variables as those which the economist always takes as given.

¹⁴ A process analysis may be defined as one in which values of the variables in disequilibrium situations are determinate. As will be seen this is, in the present terminology, synonymous with dynamic analysis. These parameters are of course the volume of “fixed” factors employed by the firms and the number of firms.

¹⁵ Oxford, 1939.

¹⁶ Hicks has chosen to call that part of his analysis which considers “dated” variables dynamic, even though it may not treat processes. The present writer has reserved the name statics for this case, though “quasi-dynamic analysis” might be a more useful term.

¹⁷ Hicks, , Value and Capital, p. 122.Google Scholar

¹⁸ Ibid., p. 246 (my italics). It should be noted that in his last two chapters Hicks engaged in what is here called sequence analysis.

¹⁹ Opie, Redvers, “Marshall's Time Analysis” (Economic Journal, vol. XLI, 1931, pp. 198–9).Google Scholar

²⁰ Samuelson, Compare P. A.: “The problem of stability of equilibrium cannot be discussed except with reference to dynamical considerations, however implicit and rudimentary.” Foundations of Economic Analysis (Cambridge, Mass., 1947), p. 262.Google Scholar

²¹ Chapter IX, “The Stability of Equilibrium: Comparative Statics and Dynamics,” in Foundations of Economic Analysis.

²² The term “sequence analysis” is still variously used in the literature. The present writer prefers to define sequence analysis as that branch of dynamic analysis in which change, within a given system, is regarded as discontinuous.

²³ Lundberg, Erik, Studies in the Theory of Economic Expansion (London, 1937), p. 46.Google Scholar

²⁴ Of course arbitrary parameter changes can be introduced into any dynamic models, whether sequential or not, and the effects on the time course of variables may be studied by the method of comparative dynamics. This happy term is due to Samuelson, , Foundation of Economic Analysis, p. 351.Google Scholar

²⁵ It may also be the period within which some variable completes a cycle, but this amounts to much the same thing.

²⁶ The entire ex ante, ex post analysis of the Swedish school stems from this concept of the unit period; Hicks's periods are of this type, as are Lundberg's and Metzler's in their theories of inventories, in Lundberg, Studies in the Theory of Economic Expansion, and Metzler, L. A., “The Nature and Stability of Inventory Cycles” (Review of Economic Statistics, vol. XXIX, 1947, pp. 1–15)Google Scholar, Robertson's “day” may be given this interpretation, although he himself interpreted it slightly differently.

²⁷ Frisch, Kalecki, Tinbergen, and others, in many of their works, have used this lag.

²⁸ While these are the lags which different writers on the employment problem have used, still others may be significant, such as others in the income circuit, or other production lags.

²⁹ Cf. Metzler's comment: “… As a practical matter, the solutions of such hybrid systems are quite complex, and useful results from them are difficult to derive. Moreover, if the length of one of the lag periods is quite short compared with the other, either the Robertson or the Lundberg sequence will give a close approximation to the true situation, the selection depending upon which lag is the shorter.” The longer lag, Metzler argues should be chosen for the analysis, “The Nature and Stability of Inventory Cycles,” p. 114.

³⁰ Hicks, , Value and Capital, p. 122.Google Scholar

³¹ Cf. Lundberg, , Studies in the Theory of Economic Expansion, p. 187.Google Scholar

³² Machlup, F., in International Trade and the National Income Multiplier (Philadelphia, 1943)Google Scholar, specifically assumes that the lag is of the same duration in both countries, p. 22. Metzler, L. A., in his analyses in “The Transfer Problem Reconsidered” (Journal of Political Economy, vol. L, 1942, pp. 397–414)Google Scholar, and “Underemployment Equilibrium in International Trade” (Econometrica, vol. X, 1942, pp. 97–112)Google Scholar, is not specific on this point. Stopler, W. F. assumes a different time lag in the two countries but does not justify the assumption in “The Volume of Foreign Trade and the Level of Income” (Quarterly Journal of Economics, vol. LXI, 1946–1947, p. 289).Google Scholar

³³ For small changes in x, the rate of change of y with respect to x may be approximated by a constant. Unit changes in x may be taken as small as the analysis requires (or, if it be preferred, the calculus of finite differences rather than the infinitesimal calculus may be used). If dy/dx is not a constant for all relevant ranges of the function then of course y does not always move one unit in a half of the unit time period, but given the function, the acceleration as well as the velocity of y with respect to x is given. Of course one could choose to measure time in terms of changes in y rather than x, but given the functional relation between y and x it is analytically a matter of indifference which time scale is used since the relation between the two time scales is given by the function.

³⁴ Cf. Samuelson, , Foundations of Economic Analysis, p. 263 Google Scholar; Lange, Oscar, Price Flexibility and Employment (Bloomington, 1944)Google Scholar, appendix, section 2; Reder, M. W., Studies in the Theory of Welfare Economics (New York, 1947), pp. 106 ff.Google Scholar

³⁵ So far as this writer knows, this has not been done in the literature. Criticism of the literature in this respect is not intended. Equation (4) is a necessary first step in the development of the theory. The concern here is to stress, what is no doubt well realized by the writers who have used it, that an endogenous time measure is implied.

³⁶ This particular example is one of a complete causally-determinate system in Samuelson's terminology (Foundations of Economic Analysis, pp. 317–18) since its behaviour depends only on the initial conditions and the time which elapsed after that at which the initial conditions were specified. Samuelson refers to an historical system as one which also depends on the time at which the initial conditions are specified. It should perhaps be pointed out that the word “historical” does not necessarily imply the calendar, but in the first instance, a date given by the values of economic variables.

³⁷ Kalecki, M., “A Macrodynamic Theory of Business Cycles” (Econometrica, vol. IV, 1936–1937, pp. 77–97)Google Scholar; “Essai d'une Theorie du Mouvement Cyclique des Affaires” ( Revue d'Économie Politique, vol. XLIX, 1935, pp. 287–305).Google Scholar

³⁸ The necessity of assuming uniformity of gestation periods for all capital goods, or a significant average, may be noted.

³⁹ The problem has received scant explicit attention in the literature. In a rather different connexion, Samuelson briefly discusses the matter in a footnote, Foundations of Economic Analysis, p. 318, n. 16. It will be seen below in Section III that some attempts at relating models to reality can be viewed as attempts to relate the endogenous time scale to the calendar.

⁴⁰ It is a matter of indifference which variable is chosen as independent.

⁴¹ It ought to be stated explicitly, that a widening of the scope of static analysis does not involve, in any significant sense, a more extensive accounting for time. This is true because static analysis abstracts from the succession of events (except as noted above) and hence from time. But a widening of the scope of dynamic analysis in which the succession of events is treated explicitly does entail a more extensive treatment of time.

⁴² Cf. for example, Machlup, F., “Period Analysis and Multiplier Theory” reprinted in Readings in Business Cycle Theory (Philadelphia, 1944), pp. 203–34.Google Scholar

⁴³ See for example Haavelmo, Trygve, “The Probability Approach in Econometrics” (Econometrica, vol. XII, 1944, supplement)Google Scholar; also “Statistical Testing of Business Cycle Theories” (Review of Economic Statistics, vol. XXV, 1943, pp. 13–18)Google Scholar; Cooper, Gershon, “The Role of Econometric Models in Economic Theory” (Journal of Farm Economics, vol. XXX, 1948, pp. 101–16)CrossRefGoogle Scholar; Koopmans, T., “Statistical Estimation of Simultaneous Economic Relations” (Journal of the American Statistical Association, vol. XL, 1945, pp. 448–66)CrossRefGoogle Scholar; the forthcoming Cowles Commission monograph, Statistical Inference in Dynamic Economic Models.

⁴⁴ Samuelson, , Foundations of Economic Analysis, pp. 340–1.Google Scholar

⁴⁵ For discussions of the problem of time in sociology, see Sorokin, P. A. and Merton, R. K., “Social Time: A Methodological and Functional Analysis” (American Journal of Sociology, vol. XLII, 1937, pp. 615–29)CrossRefGoogle Scholar; Sorokin, P. A., Social and Cultural Dynamics, Basic Problems, Principles and Methods (New York, 1941), vol. IV Google Scholar; Durkheim, Emile, The Elementary Forms of Religious Life, translated from the French by Swan, Joseph Ward (New York, 1926).Google Scholar

⁴⁶ Innis, Harold A., Political Economy in the Modern State (Toronto, 1946), p. 34.Google Scholar