Optimal Growth: Introduction

Back

At least as far back as Eugen von Böhm-Bawerk (1889), economists had entertained the idea that people are "myopic" in the sense that they tend to underestimate their future needs and desires and therefore "discount" their future utilities. This was seen by Böhm-Bawerk and many of his contemporaries as an irrationality, a result of a deficient cognitive process.

From this proposition, the Cambridge economist Arthur C. Pigou (1920) posed an interesting conundrum: if, indeed, agents tend to underestimate their future utility, they will probably not make proper provision for their future wants and thus personally save less than they would have wished had they made the calculation correctly. In other words, Pigou proposed, the very fact that people possess defective "telescopic faculties" probably means that savings, as a whole, are less than what is "optimal". This, Pigou conjectured, implies that there is a "market failure" of sorts in the market for savings.

"Generally speaking, everybody prefers present pleasures or satisfactions of given magnitude to future pleasures or satisfactions of equal magnitude, even when the latter are perfectly certain to occur. But this preference for present pleasures does not -- the idea is self-contradictory -- imply that a present pleasure of given magnitude is any greater than a future pleasure of the same magnitude. It implies only that our telescopic faculty is defective, and that we, therefore, see future pleasures, as it were, on a diminished scale....This reveals a far-reaching economic disharmony. For it implies that people distribute their resources between the present, the near future, and the remote future on the basis of a wholly irrational preference."

(A.C. Pigou, Economics of Welfare, 1920: p.24-5)

Yet in order to confirm that the rate of savings thrown up by a market system with myopic agents was indeed suboptimal, one must first determine what the optimal savings rate might be. It is at this point that the Cambridge philosopher Frank P. Ramsey (1928) picked up on Pigou's pregnant suggestion. Ramsey applied standard Benthamite utilitarian calculus to derive the "optimal rate of savings" for a society. He proposed an intertemporal social welfare function and then tried to obtain the "optimal" rate of savings as the rate which maximized "social utility" subject to some underlying economic constraints. Of course, Ramsey deliberately excluded discounting of future utility form this social welfare function: just because people are individually short-sighted, does not mean that society should be similarly "short-sighted". This is a normative, not a positive exercise.

Ramsey's conclusion was to confirm Pigou's suggestion: the optimal rate of savings is higher than the rate that myopic agents in a market economy would choose. Yet Ramsey's arguments fell largely on deaf ears for three reasons. Firstly, Ramsey's use of the calculus of variations in his argument was quite beyond the mathematical understanding of most contemporary economists. Secondly, even the economic parts of the argument were subtle and unfamiliar. Recall that Irving Fisher's Theory of Interest was only written in 1930, so most economists understanding of the concept of "intertemporality" was still rudimentary.  Thirdly, and perhaps more importantly, Ramsey's exercise was an unfashionable one. The 1930s were the years of the Paretian revival of ordinalism and the "unholy alliance" of economics and Benthamite utilitarianism was gradually unraveling. The "New Welfare Economics" stayed clear of anything which implied any sort of interpersonal comparisons of cardinal utility. Ramsey's social utility function was certainly regarded as a damnable construction.

During the 1950s and 1960s, capital and growth theory was emerging into its own and questions about "efficient" programs of accumulation were being asked (e.g. Malinvaud, 1953). Some people seemed to recollect that Ramsey had a thing or two to say about this. In a rather anachronistic, but highly commendable effort, Paul Samuelson and Robert Solow (1956) brought forth an extension of Ramsey's original model to a multi-commodity scenario.

Overall, it was practical considerations that resurrected the question of the "optimal" savings rate. For development economists -- who were in those days obsessed with government planning and accumulation -- the question was both natural and urgent. Jan Tinbergen (1956) was perhaps the first to try his hand. 

Yet, almost from the outset, the exercise was attacked. As Peter T. Bauer (1957) and Jan de Van Graaff (1957) argued, the determination of the "optimal" savings rate is irrelevant for policy on at least two grounds. Firstly, it is not implementable -- people save what they will save, period. Secondly, even if could be implemented (e.g. via Social Security schemes and what not), it is not up for a "social planner" (i.e. government) to dictate it according to some simple ethical criterion set up by economists. If anything, it is a political decision, and will be the outcome of the political culture of a nation. And the "social planner", envisaged in optimal growth theory, is frighteningly authoritarian. 

However, these objections were largely overruled by the dirigiste spirit of the times.  The optimal savings question was first applied on Keynesian (Harrod-Domar) growth models by Jan Tinbergen (1956, 1960) and Richard Goodwin (1961). But the Neoclassical (Solow-Swan) growth model had recently become available too. This model was particularly interesting because its steady-state path was generally "consumption-inefficient". Since the rate of savings is one of the critical parameters in determining the Solow-Swan steady-state, the question of "what is the optimal savings rate?" emerged quite naturally.

In the early 1960s, numerous researchers independently examined the question of optimal savings for the Neoclassical model. The answer seemed simple: the optimal rate of savings will be that which makes the rate of return on capital equal to the natural rate of population growth. This "Golden Rule" for efficient growth, as it has been called, was set forth simultaneously by Edmund S. Phelps (1961), Jacques Desrousseaux (1961), Maurice Allais (1962), Joan Robinson (1962), Christian von Weizsäcker (1962) and Trevor Swan (1963).

The derivation of the Golden Rule did not employ Ramsey's old Benthamite trappings of "social utility" and all that. These were, however, brought back into prominence after the subtle but influential work of Tjalling Koopmans (1960). Taking Ramsey's construction seriously (and piling on generous coats of "ordinalist" polish), Koopmans made it hip to consider intertemporal social welfare functions once again. Across the hallway in the two-sector growth model world, Hirofumi Uzawa (1964) and T.N. Srinivasan (1964) demonstrated how intertemporal optimality and growth theory could be combined fruitfully.

After some false starts and a flurry of activity, David Cass (1965), Tjalling Koopmans (1965), Edmond  Malinvaud (1965), James A. Mirrlees (1967), Karl Shell (1967) and others finally pieced together the canonical one-sector optimal growth model. Although this is sometimes (and erroneously) called the "Ramsey" model, we prefer to refer to it by its other name, the "Cass-Koopmans" optimal growth model.

Optimal growth theory began to recede in the 1970s for a variety of reasons. Firstly, the inconsistencies in capital theory unearthed during the Cambridge Controversy were a source of despair for growth theorists across the board and the optimal growth theorists were not immune to it. Furthermore, economists realized that the "social planner" did not really exist and developments in microeconomic theory indicated that any appeal to "representative agents" should be greeted with suspicion. But, above everything, it was the "saddlepoint dynamics" of optimal growth models made them seem inherently inapplicable. There was no good economic reason to suppose that an economy would "stumble" upon the optimal growth path. Consequently, as the 1970s progressed, optimal growth models were discarded as ultimately inapplicable constructions, however beautiful and utopian they may seem.

The tune changed in the 1980s with the rise of the rational expectations revolution. Saddlepoint dynamics began being regarded as an asset rather than a liability of a model. Specifically, rational expectations were precisely the mechanism by which an economy would jump onto the stable arm of a saddlepoint. Indeed, saddlepoints were necessary if one were to obtain a precise solution to a model with rational expectations!

It was also during the 1980s that the "decentralization" argument began being put forth more forcefully. As a result, optimal growth models stopped being conceived of as "normative" exercises about the way the economy should work, and started being regarded as a "positive" exercise about the way the economy does work. Real business cycle theory, one of the principal macroeconomic enterprises of the late 1980s and 1990s, built itself up precisely on that premise.

Our brief survey of optimal growth theory concentrates almost exclusively on its connection with one-sector, Neoclassical growth theory. We begin with Ramsey's 1928 exercise and then jump a quarter-century to the 1960s  the "Golden Rule" of growth. We then  take a rather leisurely digression on intertemporal social welfare functions and the ethical implications of time preference. All this leads us to the Cass-Koopmans optimal growth model, the version of optimal growth theory that is closest to Solow-Swan. Finally, we turn to a brief discussion of turnpikes and the infamous "decentralization" argument.

Back

Top

Selected References

Next