How Much Should a Nation Save? A New Answer

We introduce a formula for the optimal savings rate in an economy driven by an investment policy reflecting competitive equilibrium. The reasonable numbers generated by the formula should be of help not only to assess our present situation, but also to prepare our future. Moreover, this paper provides two theorems correcting a widely spread error in economic growth theory, namely that a steady state can be asymptotically reached only if technical progress is laboraugmenting. We finally show that the magnitudes of the optimal savings rates are highly robust to very different, S-shaped evolutions of population and technology. The paper closes with a daring conjecture. ∗Olivier de La Grandville, visiting professor, Management Science and Engineering, Stanford University, Stanford, CA 94305-4026; email: lagrandvil@aol.com. I am deeply grateful to Kenneth Arrow, Giuseppe De Archangelis, Robert Feicht, Erich Gundlach, Bjarne Jensen, Rainer Klump, Peter McAdam, Bernardo Maggi, Martin Paldam, Enrico Saltari, Robert Solow, Wolfgang Stummer, Juerg Weber and Alpo Willman, as well as to participants in seminars at Frankfurt, Nürnberg, Rome (La Sapienza) and Stanford for their highly helpful remarks. My heartfelt thanks also go to my colleague Ernst Hairer who checked the accuracy of the numerical calculation of the definite integral (27), using an advanced, Gaussian-type method he had developed earlier, with 15-decimal precision. Thy letters have transported me beyond This ignorant present, and I feel now The future in the instant. Lady MacBeth Enrich the time to come with smooth-faced peace, With smiling plenty and fair prosperous days! Richmond (King Richard III) Throughout civilisations, our future has been an object of passionate inquiry. In the last centuries, societies became increasingly aware that they had not only the opportunity but also the responsibility to shape it. This is perhaps why early classical treaties gave central importance to the subject of economic growth. We may then wonder why such a fundamental issue for our future as the determination of the optimal savings rate has not yet received a convincing, if not precise, answer. Textbooks are highly discreet about the issue, as if they wanted to avoid some embarassment about the strange and typically exceedingly high savings pro les generated by traditional approaches. No debate about methodology or numbers takes place. Needless to say, no attempt has ever been made at evaluating the bene ts generated for society by an optimal savings policy. This dire situation, in our opinion, seems to owe as much to the way the problem has been tackled as to the results attained. Tradition would command that an optimality criterion for society be de ned as follows: rst, convert future consumption ows into utility units through a strictly concave function; choose a discounting factor for the future utility ows, and then sum up those discounted ows. Finally, maximize this sum by nding an optimal time path of capital, under the constraint that consumption plus investment equals the production potential of the economy. Two families of utility functions are declared t for service not necessarily because they showed any special merit, nor because they have been approved in some national referendum, but most probably because they had the good idea of being strictly concave: they are the a¢ ne transform of a power function (C 1)= ; and the negative of a decreasing exponential, e = : This is where serious di¢ culties loom up. A detailed analysis of the time paths resulting from the rst of those families reveals that the initial 1 de La Grandville: How Much Should a Nation Save? A New Answer Published by De Gruyter, 2012 savings rate necessary to set the economy on the lone, highly unstable path to equilibrium is inordinately high  sometimes in the order of 50%. To obtain acceptable initial savings rates, one must resort to assigning negative values to ; entailing a concavity of the utility function no one would ever suggest nor accept: after a sharp bend already at a very low consumption level, it becomes hardly distinguishable from a horizontal line. As to the negative exponential function, the situation is even worse since in that case an equilibrium point does not exist any more1. The model becomes completely unstable. A rst warning light could have been ashed already nearly half a century ago, thanks to Richard Goodwins paper "The Optimal Growth Path for an Underdeveloped Economy"(1961), where at some point of time the optimal savings reached around 60% in both models he considered. And even Frank Ramsey, in his brilliant, path-breaking essay "A Mathematical Theory of Saving" (1928), trying to put some numbers on his own model, recognised that "the rate of saving which the rule requires is greatly in excess of that which anyone would suggest [. ] the amount that should be saved out of a family income of £ 500 would be about £ 300". One can almost feel that Ramsey was not overwhelmingly happy with his result since he added that the concave utility function he used was "put forward merely as an illustration"(op. cit., p. 548). We may venture to explain why, when optimal growth theory made a strong revival at the end of the fties and in the sixties, no one analysed in a systematic way the actual time paths generated by optimal growth theory. It may be due to the following, technical reason. The central model, as described above, incorporating strictly concave utility and production functions, almost invariably yields a Euler equation in the form of a second order, non linear di¤erential equation which does not possess an analytical solution. And in those days it would have taken not only a specialist in numerical analysis but also a good friend to do a complete, qualitative analysis of the results. Today we have very e¢ cient programs as well as excellent friends that can help us in this task, so there are no more reasons not to recognize and deal with the problem at stake. In this paper we propose a competitive equilibrium model of optimal 1See La Grandville, Economic Growth: A Uni ed Approach, with two special contributions by Robert M. Solow, Cambridge University Press, Cambridge, 2009 (in particular pp. 224-229 and 240-256). growth with a precise aim: to provide a convincing answer to the problem of 2 Studies in Nonlinear Dynamics & Econometrics Vol. 16 [2012], No. 2, Article 1