7ROM among various recent contributions to the problem of risky investment decisions and portfolio diversification, the work of Harry Markowitz and that of James Tobin deserve particular attention.2 Both these authors interpret the as an individual who is attracted by certain characteristics of assets, and is repelled by other characteristics. A high mean of the expected frequency distribution of yields is viewed as an attractive feature; high dispersion (say, high variance) about the mean is considered a repelling feature for the investor; 3 and the question of how much of the attractive feature the individual is willing to trade for how much avoidance of the repelling feature is then said to depend on his preference system. At any rate, the with these tastes should abstain from acquiring portfolios (security-mixes) which have smaller yield and greater expected variance than other portfolios. Analytical frameworks of this kind suggest the use of indifference-functions. In particular the individual's gains from deciding how much of various assets to hold express themselves readily in a rise to a higher preference level, since he is essentially comparing objective marginal rates of transformation, which are provided by market opportunities, with subjective marginal rates of substitution between expected yield and avoidance of variance. For large numbers of securities the analysis becomes involved, but the basic principles remain the same and I will not go into further detail here. However, I would like to take my departure from the comments which Markowitz and Tobin have made on the probabilistic background of their analysis. Both authors feel that a set of axioms expressing the principles of operational utility and numerical subjective probability underlies their approach. My own position in this regard may be summarized in the following three points. (i) If we postulate strictly probabilistic behavior that is, if we interpret the decisionmaker as being guided by subjective degrees of belief that are consistent with one another by the specific standards of probability theory and as maximizing his utility-expectations then it is clearly desirable to develop the analysis in terms of alternative surpluses expressed in cardinal utility. In the present article I shall make such an attempt. A high degree of generality may be claimed for the validity of the results of such analysis, as long as one accepts strictly probabilistic basic assumptions. (2) I must, however, say that I do not regard it as generally fruitful to interpret the decisionmaking process in strictly probabilistic terms, say, in terms of L. J. Savage's axioms alone. One reason for this was expressed in articles by Daniel Ellsberg and by myself on an earlier occasion.4 In these two articles the reader will 'I am grateful to my colleague, John W. Hooper, for having read the manuscript of this article and for having made valuable suggestions. 2 Harry M. Markowitz, Journal of Finance, vii (March I952), and Portfolio Selection, Efficient Diversification of Investments, Cowles Foundation Monograph No. i6, New York: John Wiley & Sons, I959; James Tobin, Liquidity Preference as Behavior towards Risk, Review of Economic Studies, xxv (2) (February I958). 'The typical investor in this sense is a person with decreasing marginal utility of wealth. Such an will find a given mathematical expectation of money gains more attractive if it is associated with little dispersion than if the dispersion is great. However, acceptance of the variance as the uniquely relevant measure of dispersion implies specific (additional) constraints. It implies a quadratic aggregate utility function and/or a frequency distribution of expected returns which can be fully described by the mean and the second moment about it. See particularly Tobin, op. cit., and Marcel K. Richter, Cardinal Utility, Portfolio Selection and Taxation, Review of Economic Studies, XXVII (3) (June I960). See also p. I82 ff. These specific constraints on the utility functions and/or on the frequency distributions play a role in theories relating to the desirable degree of whenever the among which the diversifies have different probabilistic properties, but not so if these bets have the identical properties. In the latter case diversification will always diminish dispersion in the relevant sense, and in this latter case diversification will always be desirable to an with monotonically decreasing marginal utility. See section on limits of diversification, i8i if. 4See the symposium on Decisions under Uncertainty in

fore think that if the nation really wants to take the necessary steps, the present over-all inflationary pressure can be reduced to a relatively minor problem. 4. growth. The measures proposed above will also remove a number of the present barriers to achievement of an optimum growth rate. Elimination of government actions that favor special groups without commensurately increasing the national welfare, however, as in the case of most protective tariffs and of many other operations that are in fact subsidies, is also necessary. If, on the other hand, average growth is held back by persistent deficiencies in aggregate private demand, as indicated by rising average rates of unemployment, then government spending on socially desirable programs should obviously be accelerated, on the average, until the gap is filled. Optimum growth need not mean merely more automobiles, television sets, and advertising displays. for which productivity increases, as output grows, have typically been small or zero.

DR. JAMES TOBIN, in his interesting analLI ysis of the shape of the liquidity preference function and the effectiveness of alternative monetary-fiscal policies,' refers to my discussion of the same problems.2 I would like to develop more fully some statements contained in my earlier discussion and to comment on Dr. Tobin's views and findings. In a passage in which he criticizes my views on interest, Dr. Tobin argues in effect that, for logical reasons, it is necessary to assume that the liquidity preference function becomes infinitely elastic at sufficiently low rates of interest (or, in any event, before the rate becomes negative). For, at the zero rate there surely exists an unqualified preference for cash as against claims;3 and if we take into account the institutional costs of credit operations, then this floor is set not at the zero rate but at some very low positive rate. An economist who maintains that, in the range of changes in interest rates, the demand for idle balances may possess little elasticity to interest, should add that the liquidity function must become perfectly elastic at the institutional floor-level (i.e., above the zero rate, at a level determined by institutional costs) . I am in agreement with Dr. Tobin. In my book in which I suggested that liquidity functions may be rather inelastic in the range of changes in interest rates I made at least one explicit statement which is the precise equivalent of the foregoing (italicized) proposition (pp. I86-87). I should have made an explicit statement also in another passage, and I will do so in the forthcoming second edition of the book (on pp. I70-7I). However, throughout my analysis, I assumed the validity of this proposition. My statements concerning the small elasticity of liquidity functions are explicitly limited to the usual range of changes in interest rates. The diagrams in my book are drawn merely for positive interest rates, and I believe that it is made clear that the lower limit is not really meant to be the zero rate proper but some very low rate including the institutional costs in question. Negative rates are not included because it is maintained that negative rates are qualitatively different phenomena in that they express subsidies.4 No private individual or institution can be made to hold securities at negative rates, if money can be held free of cost.5 Acceptance of the italicized proposition in the preceding paragraph does not imply abandonment of views expressed in my book. But Dr. Tobin is right in insisting that the implications of this proposition should be made clear. The main point in this connection is that as long as the liquidity function is inelastic above the floor-level in question, the conclusion with respect to policy is that which was presented in my book.6 The conclusion with respect to pol-

We had divided the demand for money during a period of time according to the method used to satisfy that demand: sale of goods, sale of securities, and keeping money already held. Obviously the first two constituents involve flow-analysis in the usual sense money, goods, and securities must change hands during the period to satisfy the demand for money. Nor is it the intention of the demanders to

William J. Fellner, Harold M. Somers, Interest Theory--Supply and Demand for Loans or Supply and Demand for Cash: Comment on Dr. Lerner's Note, The Review of Economics and Statistics, Vol. 26, No. 2 (May, 1944), p. 92