Dynamics and the Business Cycle

Though Samuelson started Foundations with static analysis, dynamics were just as important. The obvious reason was that some economic problems, such as the business cycle, were inherently dynamic.

r. It should be noted that Robbins held views that were less austere in relation to welfare judgments than the position with which he came to be associated in the light of his book.

statical system [could] be utilized to derive information concerning the dynamic properties of a system.”⁶⁴ This was what justified elevating an idea to which he had not attached a name when writing his thesis to the status of a “principle.”^s Most of the material on dynamics was first presented in a series of articles in Econometrica and the Review of Economics and Statistics, only the first of which had been a chapter in his thesis.⁶⁵ It was in the second of these articles that he introduced the Correspondence Principle, making the claim that, with it, a radical change was taking place in economics.

An understanding of this principle [the Correspondence Principle] is all the more important at a time when pure economic theory has undergone a revolution of thought—from statical to dynamical modes. While many earlier foreshadowings can be found in the literature, we may date this upheaval from the publication of Ragnar Frisch's Cassel Volume essay of only a decade ago. The resulting change in outlook can be compared to that of the transition from classical to quantum mechanics.

Samuelson would have had in mind Keynes's comparison of the revolution to be wrought by his General Theory with that brought about by Einstein in physics. Samuelson was not claiming to have initiated this revolution, poten­tially even more fundamental than the Keynesian, but he staked a claim to be playing a vital role in it.^t

The third of Samuelson's three articles on dynamics, “Dynamics, Statics and the Stationary State,” did not become a chapter of Foundations, but it was the source for a substantial section titled “Statics and Dynamics.” In this sec­tion, Samuelson provided less technical definitions and explanations of basic terminology, arguing that given recent progress in the field, it had become possible to provide a rigorous differentiation between “statics and station- ariness, between dynamics and history.”⁶⁷ However, the details of these dis­tinctions were perhaps less important than his criticism of economists who

s. This was a point he had learned from Wilson in 1938. See chapter 14 this volume.

t. It is surprising that he cited “an unpublished manuscript” dealing with the distinction between “complete causal systems” and “historical or incomplete causal ones” (Samuelson 1942f, p. 2), but did not cite his thesis. The unpublished manuscript could have been either the thesis or the article published the following year. In Foundations, the latter was reproduced unchanged, aside from a three-sentence summary of his results and an indication of his future work.

treated the word dynamic as nothing more than a synonym for good, complex, or realistic. The problem was that though economists might make analo­gies with theoretical physics, they were generally hampered by their lack of technical knowledge, causing them to get “bogged down in the search for economic concepts corresponding to mass, energy, inertia, momentum, force and space.” Here, he was conveying something he had learned from Wilson: the reason why methods taken from physics could be useful was that the economic and physical problems could exhibit common mathematical structures, and it was a mistake to look for detailed parallels between eco­nomics and physics.

Similarly, just as it was wrong to look for exact analogies with physical systems, it was also wrong to look for exact analogies in biology. Here, his main target was Marshall, whose use of biological analogies he found very vague. There were no differences in principle between physical and biologi­cal sciences:

if one examines the more exact biological sciences, one looks in vain for any new weapon, secret or otherwise, for discovering scientific truths. If the bloodstream is capable of a simple, abstract, rigorous description in terms of the usual laws of physical thermodynamics, so much the better; if not, one must be content with more complicated unwieldy explanations.⁶⁸

Indeed, Lawrence Henderson had pointed out that the idea of a stable equi­librium, central to physics, had first been formulated in relation to the body's resistance to disease. This undermined Marshall's contention that mechanical analogies needed to be replaced with biological, for there was no difference in principle.

Samuelson proposed a fourfold classification of economic systems, using this to criticize Hicks for being too vague in saying that dynamic analysis was where variables must be dated.^u He was offering something more rigor­ous. In the article, he went on to use this terminology to tackle the concept, discussed by many economists, of a stationary state and the problem, impor­tant to both Knight and Schumpeter, of whether the rate of interest would be

u. Samuelson argued that systems could be grouped under four headings: (ι) static and stationary; (2) static and historical; (3) dynamic and causal (nonhistorical); (4) dynamic and causal.

zero in a stationary state. However, in Foundations, he omitted all discussion of stationary states and the rate of interest, and simply included his nontech­nical classification of different types of system, using it as the prelude to the more technical account of the same concepts, using mathematics, drawn from his thesis.

Samuelson completed this chapter with a discussion of business cycle theories. After explaining that part of his purpose had been to show that the problem of dynamics was not synonymous with that of the cycle, he explained that he was not going to provide a survey based on economic characteristics but, rather, was going to focus on the “analytical differences involved.”⁶⁹ In other words, the “nature” of the business cycle was defined by the mathemat­ics used to model it. The result was a section that was less a survey of business cycle theories than a survey of different mathematical methods that could be used to model the cycle. The basic distinction was between endogenous models (which explained the cycle as self-generating, determined by factors within the system being analyzed) and exogenous theories (which explained fluctuations in terms of factors outside the model).

The problem of endogenous theories was that they required that there be no damping—that the parameters of the economic system were such as to generate a system where fluctuations neither faded away nor exploded. In physics there were constants that might generate such systems, but there was no reason to assume such constants in economics. Thus, he was critical of the Polish economist Michal Kalecki for imposing the condition that cycles not be damped.⁷⁰ This was a milder version of an even more critical appraisal of Kalecki that he had made privately in a letter to Hurwicz:

By the way, have you read Kalecki's most recent “Studies in Economic Dynamics?” He has a chapter on “pure” business cycles which, in my humble opinion, hits the low as far as method is concerned. In order to obtain his favorite mixed difference-differential equations he approxi­mates differences by derivatives, but not all the way thru—then he would have had a simple differential equation.

Kalecki, Samuelson believed, did not understand the mathematics he was using.

Samuelson then went on to say that the problem with linear endogenous models was that they could not explain the amplitude of the cycle: as with a pendulum, the amplitude could be of any magnitude, depending on where the system started. One way out was to drop the assumption of a purely endogenous cycle and to assume that external factors kept the system going (though he did not cite it at this point, this was Ragnar Frisch's “rocking horse” model of the cycle, according to which a rocking horse hit periodically by outside shocks would exhibit a continuing cycle). The other was to go for a nonlinear model, exemplified by “billiard table” theories in which out­put bounced up and down between a full-employment ceiling and a floor.⁷² A problem here was that, as Hansen had shown, there was “no (relevant) natural bottom to the economic system.” Not surprisingly, he favored mixed endogenous-exogenous systems, this being the section in which he cited his own work. His multiplier—accelerator model was an endogenous model in that it could generate cycles, but it could be augmented with external shocks.

Finally, he turned to “Mixed Systems of a Linear Stochastic Type”—linear models subject to random shocks. Such models had been analyzed by the Russian economist Eugen Slutsky and by Ragnar Frisch, the latter whose paper he described as brilliant, but Samuelson linked the method to his MIT colleague Norbert Wiener, the key figure in the development of cyber­netics and whose informal seminar he had been attending.^w He also linked Frisch's approach to the cycle with the problem of estimation, citing Trygve Haavelmo's “The Probability Approach in Econometrics.”^{[57] [58] Though such models were much more difficult to handle, he also outlined the problem of modeling the cycle as a nonlinear stochastic system.}

Leaving aside the Correspondence Principle, the chapters on dynamics were perhaps the most original and also the most unfinished; and for many economists, the most difficult parts of the book. In the chapters on statics, Samuelson was refining a body of ideas that had been around for decades— treating it rigorously, correcting errors, and codifying it. He could do this only to a very limited extent with dynamic theory, and as a result he drew on a much more disparate mathematical literature. To a much greater extent than in the chapters on optimization, the chapters on dynamic analysis were about the mathematics, explaining its relevance for economics. The reason for this was that there was much less dynamic economic theory to codify, and much of what did exist was based on different conceptual foundations. For example, Keynesian models were conceptually distinct from dynamic Walrasian general equilibrium theory, and the many business cycle models in circulation were conceptually heterogeneous. Under these circumstances, it was inevitable that Samuelson would focus on the mathematics and that his discussion of dynamics and the business cycle would have a more unfin­ished appearance. It was probably also inevitable that this section of the book would contain much material that many economists would not understand.

Samuelson also discussed the demand for money, an important component of many business cycle theories, at the end of his chapter on the theory of consumer behavior.⁷³ This is notable for his decision to start not with recent literature, such as Keynes, but with Walras, who had analyzed the demand for cash balances in the context of his theory of general equilibrium. In part, this discussion has the feel of an exercise designed to show that his theory of demand could be applied to money, as well as to ordinary goods and services.

He did not engage with the literature on money. Given his growing knowledge of probability theory and statistics, it is perhaps remarkable that this section contains no suggestion that mathematical analysis is needed to resolve the issues at stake, such as whether liquidity preference (the term used by Keynes to refer to the holding of money because of uncertainty about what might happen to interest rates and bond prices) was necessary to ensure a positive rate of interest.