Dynamics

The thesis ended with two chapters on dynamics. Samuelson introduced this discussion by, once again, criticizing previous generations of econo­mists for failing to go beyond arguing that there were “laws” govern­ing economics to investigating the character of those laws.

This idea had been suggested to him by Wilson, in response to a paper Samuelson had written toward the end of 1938.²⁹ Wilson had complained that Samuelson’s analysis was not “so general in some respects as Willard

k. The contrast with Marshall, who had emphasized the continuity between his work and that of his predecessors, was stark.

Gibbs would have desired.” He told Samuelson that Gibbs used to lay great stress on the fact that it was important “to remain within the limits of sta­bility.” What Wilson appeared to be telling Samuelson was that he had not stated the conditions for optimization correctly and that these condi­tions were related to the conditions for stability.

Wilson also reminded Samuelson of the need to consider the more general case where functions were not continuously differentiable—where functions might have kinks or discontinuities.^m Wilson was describing the mathemat­ics of convex sets (a form of analysis more general than differential calculus), a type of mathematics that became important in economics in the 1950s, when economists sought more general proofs of results in general equilib­rium theory. His final advice was that Samuelson needed to explain himself better: “a little more text and not so many formulae in proportion to the text might make the whole decidedly easier reading.”

Samuelson tackled the problem of deriving comparative statics results in three stages. Given that the existing literature contained little explicit discussion of the topic, Samuelson had to begin by defining dynamics and related notions such as equilibrium (which had a different meaning in this context). He defined a dynamic theory as one that determined how, starting from arbitrary initial conditions, all variables would change over time. This took the mathematical analysis to an even higher level, for it might be mod­eled using “differential, difference, mixed differential-difference, integral, integro-differential and still more general” sets of equations.³⁰ Even if some of the ideas would have been familiar to economists, Samuelson was using mathematical language in a way that only a tiny minority of mathematical economists would have previously encountered. Given a definition of equi­librium influenced by Ragnar Frisch, Samuelson defined two concepts of stability: a variable might get ever closer to its equilibrium value (“perfect stability of the first kind”) or its motion might be bounded, meaning that it never remained on one side of the equilibrium for more than a finite time interval.

Though his discussion was to focus on stability of the first kind, Samuelson motivated stability of the second kind by noting that “no con­servative dynamical system of the type met in theoretical physics possesses stability of the first kind.” Given that he did not explain that “conser­vative” meant a system in which energy was conserved, few economist readers would have understood this. A footnote citing George Birkhoff explained that a system with friction (in which energy is dissipated as heat) might possess stability of the first kind. Though he did not develop it, Samuelson thus drew an analogy between stable economic systems and friction, implying that frictions were necessary to ensure stability of eco­nomic systems.

Comparative statics, the method he had advocated earlier in the thesis, was, Samuelson argued, a special case of a general dynamic analysis. While it might be possible to abstract from dynamic analysis, as Samuelson had done in his earlier chapters, it was important to take account of dynamics. He made this point by drawing out a series of examples from the literature, formulating each of them in the mathematical language of dynamic models. The first was supply and demand in a single market, where economists gen­erally assumed that if supply exceeded demand, the price would fall, and if demand exceeded supply, it would rise. This was formulated as a differential equation that was then solved to obtain price as a function of time. Stability could be proved to depend on the relative slopes of the supply and demand curves.

Samuelson's second example was an alternative to the first. Instead of assuming that price changed in response to the difference between supply and demand, he made the “Marshallian” assumption that the quantity of goods traded rose or fell according to whether the demand price (the price

n.

A third dynamic model, also found in the previous literature although Samuelson cited no sources, involved postulating that demand and supply responded to prices with a lag: that they depended on the price prevailing one period earlier. The fourth model would also have been very familiar to economists, because it corresponded to the diagram Marshall had used to analyze international trade, and it involved countries adjusting the quantities they traded in response to the difference between what they were trading and what they wanted to trade. His final example, attributed to Francis Dresch, whose mathematical economics thesis had been submitted at Berkeley in 1937, differed radically from the other examples, in that it assumed that prices changed in response to accumulated stocks of goods: if producers failed to sell all their output, their inventories would increase and companies would respond by reducing prices.

Samuelson finished his chapter by applying dynamic analysis to the sys­tems found in two very recent books. The first was the attempt by John Hicks, in Value and Capital (1939b), to generalize the stability conditions for a single market to multiple markets. Though Hicks had derived sta­bility conditions, he had not derived them from explicit dynamic systems. Drawing on mathematical techniques he had used earlier in the thesis, as well as those relevant only to dynamic models, Samuelson analyzed stabil­ity more rigorously, showing why the absence of explicit dynamics was a problem for Hicks. The second was the Keynesian system, as formulated by interpreters such as Meade, Hicks, and Lange, comprising three equations: a consumption function, a marginal efficiency of capital, and a liquidity prefer­ence schedule.

Clearly this related to the system he and Hansen had previously analyzed, but it was different in that he did not incorporate the accelerator, thereby keeping closer to the issues then being debated concerning the coherence and meaning of the General Theory. Perhaps more important, though it was not the same type of supply and demand system, and though it originated in business cycles and not in “economic theory,” Samuelson was treating the Keynesian system as something analogous to the other market systems he had been discussing in the thesis: he derived explicit comparative statics results (of precisely the sort that Hicks and others were trying to obtain).

o. Samuelson pointed out that although these two types of processes had become associated with Marshall and Walras, this involved a historical error.

This was the example in which he showed most clearly that dynamics—the assumption of a stable system—was closely related to the comparative statics results that were the goal of economic theory.

Samuelson achieved a number of things with these examples. The first was to show that dynamic processes were implicit in familiar economic examples, implying that economists could not argue that dynamics were unnecessary. They might not talk of explicit dynamic systems, but they were nonetheless using them implicitly. The second was to illustrate some of the different types of mathematics that could be used: differential, dif­ference, and integral equations. One of his examples also illustrated sta­bility of the second kind and the notion that systems might be subject to random shocks—that they might “take a random walk.” In a random walk, a variable does not converge on any equilibrium; it simply moves up or down with given probabilities. For example, it was later argued that stock prices followed a random walk: in each day they might rise or fall, that day's price being the starting point for the next day's movement. Thus if, though chance, a stock price rose for several successive days, it could depart a long way from its initial value.

Samuelson's second chapter on dynamics, chapter 9, “Foundations of Dynamical Theory,” moved further from economics and into mathematics, citing mathematicians more frequently than economists. He distinguished between “causal” systems, in which the system was completely determined by the initial conditions, and “historical” systems, in which one also needed to know the historical date at which the system started. The latter were, he explained, incomplete causal systems.

The notion of a causal system led directly to the analysis of certain prop­erties: whether the system could ever return to its initial point and whether any patterns might emerge in terms of relationships between variables. Though this argument was at a highly abstract level, it permitted discussion of problems with which economists were familiar, including the choice of variables to model and the fact that some variables changed more slowly than others. Dynamic analysis also made it possible for Samuelson to introduce randomness (also in one of his earlier examples) and to relate economic theory to econometrics (as the term eventually came to be understood). He provided a justification for representing economic equilibrium “as simply a statisti­cally fitted trend,” implying that the approach to estimating demand func­tions represented by Henry Ludwell Moore and Henry Schultz might have rigorous theoretical justification.^p

Economists understood the notion of a stationary equilibrium, but Samuelson argued that there was a case for thinking in terms of moving equilibria—of equilibrium as something that changed over time. Citing Lotka's Elements of Physical Biology (1925), he noted that a dynamic equi­librium of supply and demand was “essentially identical with the moving equilibrium of a biological or chemical system undergoing slow change.”³² Though some economists, such as Frisch, would have had no problem with such ideas, it was far from the thinking of most economists. As has been explained already, he was moving away from economics into the realm of mathematics and physical systems in general, sketching directions in which economic analysis might develop.^q