THE TIME is ripe for a fresh, modern look at the concept of complementarity. Whatever the intrinsic merits of the concept, forty years ago it helped motivate Hicks and Allen to perform their classical reconsideration of ordinal demand theory. And, as I hope to show, the last word has not yet been said on this ancient preoccupation of literary and mathematical economists. The simplest things are often the most complicated to understand fully. For this reason, I have redrafted the present paper along the following lines: The main discussion is primarily literary. Then comes a mathematical section. Finally, I give a brief survey of the history of the subject.

The fallacy that a many-period expected-utility maximizer should maximize (a) the expected logarithm of portfolio outcomes or (b) the expected average compound return of his portfolio is now understood to rest upon a fallacious use of the Law of Large Numbers. This paper exposes a more subtle fallacy based upon a fallacious use of the Central-Limit Theorem. While the properly normalized product of independent random variables does asymptotically approach a log-normal distribution under proper assumptions, it involves a fallacious manipulation of double limits to infer from this that a maximizer of expected utility after many periods will get a useful approximation to his optimal policy by calculating an efficiency frontier based upon (a) the expected log of wealth outcomes and its variance or (b) the expected average compound return and its variance. Expected utilities calculated from the surrogate log-normal function differ systematically from the correct expected utilities calculated from the true probability distribution. A new concept of ‘initial wealth equivalent’ provides a transitive ordering of portfolios that illuminates commonly held confusions. A non-fallacious application of the log-normal limit and its associated mean-variance efficiency frontier is established for a limit where any fixed horizon period is subdivided into ever more independent sub-intervals. Strong mutual-fund Separation Theorems are then shown to be asymptotically valid.

Paul A. Samuelson, Robert C. Merton, Generalized Mean-Variance Tradeoffs for Best Perturbation Corrections to Approximate Portfolio Decisions, The Journal of Finance, Vol. 29, No. 1 (Mar., 1974), pp. 27-40