² For details see chap. XIII of the present writer's Studies in the Theory of Welfare Economics (New York, 1947).Google Scholar

³ They also ignore those cases where the individual or firm is confronted with a demand or supply function which is less than infinitely elastic, although determinate; that is, cases of impure competition with zero “conjectural variations.” However, the extension of the theory to such cases is formally quite simple.

⁴ In a sense, von Neumann and Morgenstern's “Theory of Games” is such a generalized theory, except that its whole approach is so different from the usual economic theory that it would perhaps be better to think of it as simply another kind of theory and not force it into the mold of the other or vice versa.

⁵ Bronfenbrenner, M., “Applications of the Discontinuous Oligopoly Demand Curve” (Journal of Political Economy, 1940, pp. 420–7).Google Scholar

⁶ The fact that it could earn more (have a larger net worth) by borrowing less would not deter it, for, unless some limit were to be imposed, its owners would have the use of more funds in perpetuity, by this procedure than by behaving rationally in the usual sense of the word. A “rational” individual would, under these conditions, of course borrow an arbitrarily large amount and keep borrowing to repay interest and prinicpal etc. The key to this paradox is simply that the command of funds in perpetuity, and with certainty, is indistinguishable from ownership of them. It is not too much to say that without such an absolute limitation on borrowing, particularly on households, the “economic problem” cannot exist; for each household, in the absence of such limitation, will have at its disposal infinite resources and hence no economic problem. This possibility is sometimes mis-used to prove the logical absurdity of a zero rate of interest; that is, if the rate of interest were zero, people would borrow indefinitely, However, it should be apparent that without an absolute limit on the amount of borrowing, indefinite borrowing would occur at any finite rate of interest; but with a limit, it need not occur even though the rate of interest were zero or even negative.

⁷ For some alternatives see, for example, de Scitovsky, T., “A Note on Profit Maximization and Its Implications” (Review of Economic Studies, Winter, 1943, pp. 57–60)Google Scholar or Reder, M. W., “A Reconsideration of the Marginal Productivity Theory” (Journal of Political Economy, 10, 1947, pp. 450–8).Google Scholar

⁸ The method of valuation is not very important for this purpose; the reader may think of the valuation as determined by market valuations.

⁹ The reader may wonder a bit about the relation of long-run equilibrium to the dynamic production function of which we have been speaking. The relation is as follows: if the system were in a position of long-run equilibrium, then a new firm attempting to enter a given industry would have to undergo a preliminary period of losses (including entrepreneurial rents in cost) before it could begin to earn profits. In other words, factors would be so priced, in long-run equilibrium, that, at, first, a firm's operating costs would compel losses. These losses would have to be considered as investment and yield a return equal to the rate of interest in order to find entrepreneurs willing to bear them. After some finite time period, a new firm would have exactly the same cost function as any other; but the temporary losses it must bear in order to reach this position have to yield a return equal to the rate of interest, and hence explain “pure profit” in long-run equilibrium.

¹⁰ Cf. Reder, M. W., “Interest and Employment” (Journal of Political Economy, 06 1946, pp. 243–59).Google Scholar

¹¹ This problem has received considerable attention of late; cf. Klein, L. R., “Macroeconomics and the Theory of Rational Behavior” (Econometrica, vol. XIV, 1946, p. 93 et seq.)CrossRefGoogle Scholar and the discussion that followed among Klein, May, and Pu.

¹² To support this claim he cites an article of the reviewer's (“Welfare Economics and Rationing,” Quarterly Journal of Economics, vol. LVII) as an example. However, it would seem that he has misinterpreted the reference of the article which was to an individual and not, as he supposes; to an entire economy.

¹³ Hicks's mathematical definition would seem to be identical with that suggested by Samuelson (p. 184, no. 13), the verbal translation of which would be: Two commodities, i and j, are complementary if a decline in the price of i (j), income and all other prices constant, leads to an increase in the consumption of j (i) after the individual has been subjected to a tax on his income such that if he chose to buy the same amount of i (j) as before, he would exhaust his remaining income by purchasing the same amount of every other commodity as before. The converse holds in case of a price rise. This is not quite the same as Hicks's verbal definition on p. 44; but Hicks's intention, apparently, was to translate his mathematical definition, that is, to define complementarity in terms of the “substitution effect” of a price change. That he slipped provides a good argument for careful attention to the verbal meaning of symbolic operations.

¹⁴ Cf. Lange, O., “Say's Law: A Restatement” (in Essays in Mathematical Economics and Econometrics in Honor of Henry Schultz, edited by Lange, O., et al., Chicago, 1942, pp. 49–68).Google Scholar

¹⁵ Samuelson generously acknowledges his indebtedness to a paper by Bergson, A. “A Reformulation of Certain Aspects of Welfare Economics,” 1938, pp. 310–34Google Scholar, which he points out has not received due notice. As the author of a recent small volume on welfare economics, Studies in the Theory of Welfare Economics (New York, 1947)Google Scholar, in which Professor Bergson's contribution received its customary lack of mention, I should like to take this opportunity of making public apology. Since publication, I have re-read his article with profit and embarrassment.

¹⁶ de Scitovsky, T., “A Note on Welfare Propositions in Economics” (Review of Economic Studies, 1941–1942, pp. 77–88).Google Scholar

¹⁷ Reder, Studies in Welfare Economics, especially chaps. VIII and XIII.

¹⁸ While it is sufficient, in order for situation A to be superior to B, that only one person be better off in A than in B, with no one worse off, it is always possible to divide the money equivalent of the individual's gain (provided it is finite) among everyone, thus making everyone better off.

¹⁹ Chapters IX and X were originally published as articles in Econometrica (April 1, 1941 and January, 1942) and parts of Chapter XI were published in the Review of Economic Statistics, Feb., 1943.

²⁰ The distinction between static and dynamic theorems that we are making is essentially the same as Samuelson's (in chapters IX and XI), a dynamic system is one in which the value taken at any given moment of time by any variable of the system is determined partly or wholly by the values taken by that variable and/or other variables (given the parameters and functional form of the system) at some moment or moments of time in the past. A static system, on the other hand, is one where the value taken by one variable at any given moment is determined, given the parameters and functional form of the system, solely by the values taken by the other variables of the system at that moment.

²¹ Cf. also Reder, M. W., Studies in Welfare Economics, pp. 112–16, n. 13.Google Scholar