Abstract
This seven-part Introduction to the contributions of Stefano Zambelli to modern macroeconomic theory and simulations is nothing more than a scratching of the surface of his rich input. In particular, the first and last parts are personal introductions; the middle four parts are reasonably ‘professional’ summaries of his work in the following areas: Frisch’s Rocking Horse Does Not Rock, Busy Beavers and the Phillips Machine, solving and simulations of coupled continuous dynamical systems and Sraffian economics. The underpinning theoretical technology is classical computability theory. The penultimate section is a tentative summarizing of the contents of this chapter with the final section, largely, notes on the contributions to this volume.
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Notes
- 1.
In July 1985 with cum laude.
- 2.
Simultaneously, he was a member of the Macroeconomics and Monetary Policy Research Group of the BCI, cooperating with Bocconi University.
- 3.
On an American Field Service (AFS) scholarship, in 1976.
- 4.
The story of Lambretta can easily be accessed via the Internet. I should add that in distant Colombo, where I grew up, the more ‘popular’ Italian post-war scooter was the Vespa.
- 5.
I am sure he was not an admirer of Marinetti’s Italian futurism , and the machine age which it extolled (and supported Mussolini’s fascism , of which Stefano was an implacable opponent—in spite of the fact that Il Duce was born and buried in Predappio , a small town in the once ‘red’ region of Emilia-Romagna ).
- 6.
Not that there is always a pedagogical element in all of Zambelli’s writings!
- 7.
However, it is not Wicksell’s Rocking Horse, many times Frisch’s incorrect allusion is invoked by latter-day business cycle theorists (Zambelli 2007, fn. 4, p. 147).
- 8.
This is an absolute exception to the rule that I will not refer to any joint work by Zambelli!
- 9.
In Italian, as Teorie Matematiche del Ciclo Economico.
- 10.
I was immensely pleased to read Zambelli writing (on 24 February 2020—Richard Goodwin’s birthday ):
As you surely remember, the first time …. I met you was when you delivered a seminar where you discussed … Frisch’s PPIP (Propagation Problems and Impulse Problems) paper—in relation with Lucas.
- 11.
Sensitiveness to initial values and therefore the need to investigate, for example, the initial value problem (ivp) of ordinary differential equations (ode) required a kind of perturbation analysis, but strictly within the framework of solutions of deterministic dynamical systems.
- 12.
Richard Day’s felicitous description of Lucasian—or newclassical—Real Business Cycle (RBC) methodology (Day 1992, p. 180), which has its origins, unfortunately, in Frisch’s Rocking Horse metaphor in PPIP, which Zambelli, with precise and incisive analysis, debunked most elegantly.
- 13.
I hasten to add that the exception to this ‘rule’ was his adherence to de Finetti’s view of subjective probability with countable additivity .
- 14.
PPIP: Propagation Problems and Impulse Problems.
- 15.
I differentiate—as does Zambelli—experimental from empirical; the latter approximates ‘reality’ (whatever that may mean) and the former is an approximation involved in, for example, numerical integration. A fairly similar distinction is made in the case of digital and analogue computation.
- 16.
Linear, first-order, difference equations ‘can potentially account for free oscillations’, as, for example, the simplest cobweb model in economics shows.
- 17.
At this point the text refers to ‘the appendix’ in Zambelli, ibid. ‘Toward the equilibrium’ is a statement that is common in business cycle theory (of any variety)—but there are equilibrium business cycle theories , usually of newclassical persuasions, but not exclusively so.
- 18.
Clearly, both of Zambelli’s published papers of 2011 (Zambelli 2011a, b) belong to the next section on solving and simulating coupled dynamic models of national—that is, aggregative—economies. Whereas the paper on the Phillips Machine is more explicit on the machine’s analogue computing nature, the remarks on this kind of computing are not very prevalent in the Zambelli 2011b paper (but see note 10, p. 630).
- 19.
Not too much subtlety is required to understand the parallels with Production of Commodities by Means of Commodities (Sraffa 1960)! But there is no explicit reference to Sraffa’s book in this article by Zambelli.
- 20.
A reflection of his critique of Frisch’s methodological precepts as in PPIP!
- 21.
I remember vividly how, in Los Angeles in the late 1980s and early 1990s, our conversations were often in terms of Gödel numbers of sentences!
- 22.
I am reminded of the sentence in the Swedish translation (by Vibeke Emond) of Haruki Murakami’s Japanese original, Kishidanchōgoroshi (italics added):
Det är bara att låta metaforerna förbli metaforer, låta det krypterade förblir krypterat och låta sållen vara såll.
My ‘free’ translation from the Swedish of the original Japanese is:
It is best to leave metaphors be metaphorical, encryptions be encrypted and let sieves to sieve.
- 23.
As far as I know, Potts (1982) and Stuart and Humphries (1986) are ‘mainstays’ in Zambelli’s repertoire of approximation literature. The former is on the dangers of mindless approximations of differential to difference equations; the latter is about the more general problem of approximation, for computing with numerical methods, of dynamical systems as ODEs.
- 24.
I remember very well Goodwin placing buckets to collect the leaking water, when he lectured on Keynesian economic policy, using the original Cambridge Phillips Machine!
- 25.
See footnote 13, p. 1620, in Zambelli (2015).
- 26.
Of course, Zambelli knew of the parallels with Turing (ibid.), see footnote 9, p. 1615, in Zambelli (2015).
- 27.
In the case of Conway’s LIFE, for example, chapter 25 of Berlekamp et al. (1982).
- 28.
Post (1936, p. 105) called a thesis a natural law (I think in the sense of the 2nd Law of Phenomenological Thermodynamics)—‘a working hypothesis’ (ibid.); he could have referred to axioms in the same vein! I subscribe, wholly, to Post’s view—which is also Turing’s and, I dare say, Zambelli’s, too!
- 29.
I remember vividly, telling Zambelli, that Prescott (2005) had written (italics added):
In the 1960s there was the famous Cambridge capital controversy. This controversy bears on the issue “What is money?” The Cambridge capital controversy was a silly one, as pointed out so clearly by Arrow (1989).
Stefano was visibly upset by this ignorant remark by one of the Godfathers of RBC; he subsequently, but not only as the result of this stupid remark by Prescott , tore into RBC with a vengeance that was truly remarkable! By the way, Arrow never, particularly in Arrow (op.cit.), to my knowledge said anything Prescott attributes to him in the above quote.
- 30.
Shaikh, too!
- 31.
Zambelli (2018b), to which I shall devote the rest of this section, refers to Ganguli (1997). I would like to state categorically that Ganguli (1997) is not constructive in any sense, although he claims so (ibid., p. 534 & 541). He is also comprehensively incorrect in his statements about Steedman (1984), so-called equations, (10)–(12), ibid., p. 535. Steedman is very clear that (10) is a definitional identity—in fact he uses the symbol ≡ connecting the l.h.s to the r.h.s, apart from stating explicitly (ibid., p. 135), that ‘Relation (10) is just a definitional identity’. This is not a paper pointing out the infelicities in Ganguli (op.cit); there will be time and place for such an exercise!
- 32.
The whole of PCMC is in terms of equations—not inequalities, concomitant slack variables, assumptions (or axioms) of satiability, and the like.
- 33.
In the case of the uniformity assumption, it may well be justified to refer to rate of profits; but with non-uniformity, perhaps Zambelli’s modification to rates of profit may be more suitable!
- 34.
I have used precisely and exactly interchangeably; I mean it in the sense in which Aberth (2007) uses precise; Zambelli uses precise, coupled (sic!) to numerical methods. In this sense it is more in line with the title of Aberth’s book, which I have seen, well-thumbed, on the shelf in his study (see Fig. 1.8).
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Velupillai, K. (2021). Introduction to the Zambelli Festschrift . In: Velupillai, K. (eds) Keynesian, Sraffian, Computable and Dynamic Economics . Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-58131-2_1
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