# Coupled Dynamics, Optimal Planning, Stabilization Policy, and Iteration

‘If you can get [the World Bank] to use *Eigenvectors* and *Eigenvalues,* I will privately award you a Nobel!! No one ever seems to see the great advantages of *separating dynamics from interdependence in large systems* (Goodwin to Velupillai, 5 July 1993; italics added).

Every contribution by Goodwin, to the problem of‘interdependence in large systems’, from the problem of dynamical coupling in 1947 (Goodwin 1947) to the problem of value and distribution theory in Sraffa-like systems (Goodwin 1974, 1977), and their dynamics, was by means of one or another form of decoupling with the use of coordinate transformations. He used engineering terminology here—normal coordinates or normalized general coordinates,

Figure 36.1 The 'one-bend' characteristic

Figure 36.2 A limit cycle generated for a 'one-bend' characteristic

unilateral coupling,^{[1]} and so on—to harness deep theorems in matrix theory (or the theory of linear transformations) to solve, iteratively, for possible stable, equilibrium values of such large interdependent systems.

Thus, his pioneering invoking of the Perron—Frobenius theorems; the use of the Duhamel, Faltung, or convolution sum; and the mastery of similarity transformations (from Matrix Theory) were all with the aim of making explicit an iterative mechanism that could be given some kind of algorithmic form. Whether it was for the purpose of computing the dynamic multiplier in a Leontief system (Goodwin 1949a), developing an explicit iterative procedure for a meaningful Walrasian *tatonnement *(Goodwin 1953), or determining the standard system and the standard ratio in a Sraffa-like system (Goodwin 1974, 1977), or, above all, calculating, transparently and pedagogically, stabilization regimes in Keynesian macrodynamic models, using the Phillips Machine (the MONIAC; see Goodwin 2000), it was always for enlightened policy purposes on the basis of rigorously approximate theory.

It was, in fact, in this remarkable paper (Goodwin 1953) linking Walrasian *tatonnement* with an iteration that would converge, that the Frobenius generalization of the Perron result was first reported, and used (contrary to the statement in Desai and Ormerod 1998: 1,433). It was ‘discovered’ by Goran Ohlin, a student in Goodwin’s graduate classes at Harvard (see Goodwin 1953: 83, fn. 7). The footnote, in Goodwin (1953), referring to Ohlin has the following interesting background. In a letter to Bjorn Thalberg dated 5 October 1976:^{[2]}

I had no idea that you [Thalberg] met Richard Goodwin in Cambridge. He was really a wonderful teacher. My acquaintance with him was unfortunately relatively short since he disappeared to Cambridge [UK] quite soon after I came to Harvard. I did not have any special status as his student, except in one sense — he formulated a problem (I seem to recall that it was about showing that there was only one real root to the characteristic equation for a certain type of matrix) and promised, somewhat jocularly, and in a lighthearted fashion, an *A* to anyone who could solve it, and I did it, which resulted in the footnote about me.

Unlike the dynamic way Goodwin used, for example, the Peron- Frobenius theorems on non-negative square matrices, in conjunction with similarity transformations into normalized general coordinates, economic theorists with a mathematical bent have appealed to these theorems for proving pure existence theorems. Similarly, the geometric basis of Sraffa- like systems was highlighted by the same kind of coordinate transformations to determine conceptually slippery terms like the standard ratio or the composition of the standard commodity.

Two neglected points in relation to Walrasian *tatonnement* should be observed, and they are best highlighted by Goodwin’s stubborn and persistent emphasis on the difference between proving pure existence theorems and coupling them to methods for finding solutions, on the one hand, and whether or not the hypothetical method for finding solutions—the algorithm—was meant to be the actual process by which the heterogeneous agents or institutions ‘groped’ towards the solution that was proved to exist, on the other. Goodwin never separated these two issues, and in this he was unique even in a Cambridge dominated, in his time there, by the first-generation pupils of Maynard Keynes. Perhaps this was because he remained a Schumpeterian, even while becoming an adherent of Cambridge Keynesian and Sraffian economics and methodology.

In any case, as he noted (Goodwin 1953: 59-60; bold text and italics in original):

[Walras] thoroughly confused two related but distinct questions: the question of the *existence of a solution*^{20} and *how to find it,* with the question of the reality of *a dynamic process* and its stability ... I incline to the opinion that he *wished merely* to show the existence of a solution and to assert that this is the actual one realized, but *not* the hypothetical process by which we might discover this solution is the same as the motion by which an actual economy reaches equilibrium.

I find support for this in Schumpeter’s statement that ‘I remember a conversation with Walras in which *I tried but completely failed to elicit the slightest symptom of interest* both *in dynamical approaches* and in a theory of economic evolution’.^{21}

Walras, actually, ‘thoroughly confused’, not two, but *four* ‘related but distinct questions’: *existence* of a solution, a *method of finding it* (if ‘proved’ to exist), the ‘reality’ of the *method* considered as *a dynamic process* and its *stability; ^{[3]} ^{[4]}*

the latter could be rephrased in terms of a question about the convergence of a dynamic process to the equilibrium which had, apparently, been shown to exist. The separation between a proof of existence and a method for finding the solution (or equilibrium) shown quite separately to ‘exist’ is characteristic of the kind of mathematics that dominates orthodox mathematical economics. Goodwin never indulged in this kind of separation. In this sense, he was an intrinsic computable and constructive economist.

A legion of pioneering contributors to economic theory, all the way from Bortkiewicz and Edgeworth, via Pareto and Pantaleoni, to Lange, Patinkin, Morishima, and those two modern scholars par excellence of Walras, William Jaffe and Don Walker, have contributed to furthering the Walrasian confusions. Between Jaffe (1967, 1980, 1981) and Walker (1987, 1988), all the relevant historical references, related to the ‘thoroughly confused’ issues which Walras raised, are meticulously listed, discussed and interpreted, even if not always with the admirable clarity one expects from these two outstanding doctrinal scholars.

For almost the first decade and a half of his academic life at Cambridge, Goodwin was the custodian of the Phillips Machine, which he maintained and used for his lectures, as Michael Kuczynski (2011: 97, 99-100), a distinguished auditor of his lectures recalled, with considerable affection:

Goodwin’s lectures were on the ‘national income machine’ [the Phillips Machine] and delivered in [what is now called the Meade Room] ... Phillips’ bulky machine always stood there, despite its rollers, its brown wooden frame locked in a heavy glass cabinet whose awkward doors Goodwin would prise open at the start of each lecture.

At the start of each lecture Goodwin would set one or other of the levers controlling the functional relationships (tax to national income, imports to reserves, etc.), and then = as the flow got going—he would scamper around stylishly calibrating the others till the flushing and cascading of the liquid flow found its equanimity and the gurgling settled down to a quiet stream.

Above all, the imaginative way Goodwin coupled two Phillips Machines, with the help of Phillips himself, to generate what is now referred to as the quasi-stable paradox, gave transparent and pedagogical content to stabilization regimes and policies. As he wrote in the appendix to a letter addressed to Professor—now Lord—Nicholas Stern (Goodwin 1991; italics added):^{[5]}

[O]n coming to Cambridge, England from Cambridge, Mass., [imagine] my thrill to find Phillips’s extraordinary machine installed; I spent years using it for teaching both linear and nonlinear dynamics—and some time keeping it in reasonable working order (occasionally with help from Phillips).

I had long known about computers and had even written an article (Goodwin 1951b), which no one seems to have read, comparing the price-market economic [sic] to a gigantic computing machine. But I had never had access to a computer, so one can imagine my pleasure and thrill at having under my control that astonishingly effective electronic, mechanical, hydraulic Phillips’s contraption. One could draw an arbitrary nonlinear curve and make it *solve it;* also, I found that accidentally the machine embodied what I had formulated as a ‘flexible’ accelerator in place of the rather unsatisfactory simple accelerator.

Furthermore, I was very excited to find that *Phillips had two of his magical machines* in London, so I could reproduce what I had analyzed back in 1947 in my dynamical coupling paper (Goodwin 1947). If I remember correctly, Phillips did not believe we could produce *erratic behaviour* by coupling his machine — but we did.

Zambelli (2011) has comprehensively analysed the coupled dynamics of Phillips Machines, using macrodynamic models of generalized Goodwin- type dynamic multiplier, flexible accelerator, non-linear interdependent economies linked non-linearly. Goodwin’s intuition of being able to ‘produce erratic behavior by coupling machines’ were fully confirmed by Zambelli’s simulation results.

In the same period, mid-1950s to mid-1960s, Goodwin also lectured on Indian Planning, reflecting on, and based upon, his work at the Indian Statistical Institute, constructing India’s first input-output tables for use in that country’s second Five-Year Plan (cf. Velupillai 2015). He was, as a result of his deep interest in the development of the Indian economy, also the pioneer in the resurrection of Ramsey economics, that is, the application of calculus of variations, and its modern extensions. This came about as a result of his interest in determining the optimal path of development (cf. Goodwin 1961,^{[6]} and for the full details, also Velupillai, op. cit).

I give these three examples—the preoccupation with the practical meaning of a *tatonnement,* but underpinned by computational, simulational, and rigorously approximate theory; stabilization policy, investigated experimentally and with simulations on an analogue computer (which the Phillips Machine was); and the problems of underdevelopment, particularly of an important emerging nation such as India, based on actual experience of constructing tables of inter-industrial transaction flows—just to highlight that Goodwin never worked or speculated on pure theory in a vacuum. Theory was always the servant, especially mathematical theory; applied theory was the master, particularly with practical, macroeconomic, policy in mind.

- [1] Simon’s felicitous use of this concept (e.g. in Simon 1952), to develop a rich repertoire of concepts oncausality, identifiability, semi-decomposability, evolutionary dynamics—and much else—is a testimonyto the fertility and originality of Goodwin’s innovative theoretical contributions, even within the framework of linearity.
- [2] A ‘free’ translation, by me, from the original Swedish version of the letter, which was made available tome by the late Professor Thalberg, in 1976.
- [3] Goodwin was also always careful to choose the phrase ‘existence of a solution rather than the moreorthodox ‘existence of an equilibrium . Had this distinction been as carefully observed by computablegeneral equilibrium theorists, the nonsensical constructive and computable claims they—and theirdynamic stochastic general equilibrium (DSGE) followers—have been making would have shown to bethe true non sequiturs they actually are.
- [4] In a footnote to this Schumpeterian point, Goodwin adds (ibid.: 60; italics in original): ‘From theunpublished manuscript of his History of Economic Analysis . My reading of this monumental text bySchumpeter (1954) does not show any evidence of this important paragraph on his meeting with Walrasat least in the published version of the History of Economic Analysis. Modern readers of this classic should,perhaps, be reminded that Goodwin was the one who ‘put together.. .the material in Part IV, Chapter 7’,that is, the chapter on Equilibrium Analysis (see the concluding paragraph in the ‘Editor’s Introduction’by Elizabeth Boody Schumpeter).
- [5] To the best of my knowledge this appendix has never been published in its entirety.
- [6] In addition, see some of the essays republished in Goodwin (1982).