Paul Samuelson’s ways to macroeconomic dynamics

Abstract

Samuelson kept optimisation-based problems separated from macroeconomic dynamics in his Foundations, where dynamics was defined in terms of difference and differential equations. Despite some criticism of his “correspondence principle” of stability analysis by D.F. Gordon, D. Patinkin and others, it was only in the 1970s that Samuelson’s separation was effectively challenged, particularly by R. E. Lucas. After the Foundations, Samuelson developed dynamic optimisation models, sometimes featuring representative agents, but he did not extend that to the study of macroeconomic fluctuations. Neither did he accept market clearing inter-temporal maximisation as a solution to the micro-foundations problem that beset his models of macroeconomic dynamics. His 1988 nonlinear non-optimising business cycle model was his last contribution to dynamics. Eventually, Samuelson disentangled his 1965 “efficient market hypothesis” of financial economics from rational expectations and claimed that the former should form one of the pillars of macroeconomic dynamics, together with imperfectly competitive markets for goods and labour.

Keywords:

• Samuelson
• optimization
• macroeconomic dynamics
• stability
• multiplier-accelerator

JEL Codes:

• B22
• B31
• B41

Acknowledgements

I would like to thank Edward Nelson, Bruna Ingrao, David Laidler, Wade Hands, Rogerio Arthmar, Hans-Michael Trautwein, Rodolphe Ferreira, Michaël Assous, Sylvie Rivot, Pedro Duarte, Gilberto Lima and (other) participants at the conference “Macroeconomics: dynamic stories – when statics is no longer enough” (Université Halte-Alsace, Colmar, May 16–18, 2019) and at a seminar at the University of São Paulo (August 15, 2019) for helpful comments on an early draft. I have also benefitted from useful suggestions by two anonymous referees. Research funding from CNPq and permission to quote from the Patinkin Papers (Rubenstein Library, Duke University) are gratefully acknowledged.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 I owe the last remark in this paragraph to correspondence with Bruna Ingrao.

2 On the Scandinavian pedigree of Frisch’s macro-dynamics see Boianovsky and Trautwein (2007). Samuelson (1939a, p. 78, n.) benefitted also from another Scandinavian contribution: Lundberg’s (1937) numerical analysis of the interaction between the multiplier and the accelerator through a second-order difference equation, which, unlike Samuelson, Lundberg did not solve analytically – see Siven (2003), and Lindbeck and Persson’s (1990, p. 276) remark that “Alvin Hansen gave a young talented student with mathematical skills the assignment to formalize Lundberg’s argument. The student was Paul Samuelson” (quoted and translated by Trautwein 2014, p. 866). On the other hand, Lindahl’s Swedish inter-temporal equilibrium approach to dynamics and expectations influenced Hicks deeply, but not Samuelson.

3 Baumol’s book opens with a lengthy discussion of what he famously called the “magnificent dynamics” of classical economists, Marx, Schumpeter and Harrod, which does not easily fit into Samuelsonian dynamics.

4 See articles collected in Lucas 1981.

5 See also Michael Brady (2018).

6 Samuelson did not use that model to perform comparative static exercises of the effect of money-wage changes on employment equilibrium. This might be ascribed, as suggested by Assous and Carret (2020, section 2), to Samuelson’s implicit assumption that full employment equilibrium may be unstable (because of expectation effects etc.) and therefore unfit for comparative statics, but there is no clear textual evidence to support that interpretation.

7 As put by Patinkin (1982, 10), the equilibrating effect of the contraction in aggregate income is the nucleus of Keynes’s theory of effective demand, expressed by the stability of the equation dY/dt = f [F(Y) – Y], where F(Y) is aggregate demand and f’ $>$ 0.

8 Albert Hart (1951, viii) would comment on the absence of anticipations in Samuelson’s (1948a) survey of economic dynamics (see Kregel 1980, 27–29).

9 Samuelson’s multiplier-accelerator model, therefore, is not a fully endogenous model, since the outcome depends crucially on the values of the parameters α and β because of the linear relations involved. This is well illustrated by an exercise in his Economics (Samuelson 1976, 268, question 9).

10 See also Brown and Rogers (1978) for a survey of critical assessments of the correspondence principle from the 1950s to early 1970s. Negishi (1982) compares Samuelson’s stability analysis with non-Walrasian disequilibrium macroeconomics, without mentioning Patinkin though.

11 Donald F. Gordon (1923–2001) did his PhD at Cornell University and spent a substantial, part of his career at the University of Washington.

12 Under Gordon’s influence (his colleague at the University of Washington), Silberberg (1978, 527–528) criticized Samuelson’s stability hypothesis for replacing the explicit assertions of maximizing behavior for a weaker account of how the markets operate. This appeal to some “mystical stability properties” represented a “departure from the explicitly choice-theoretic microeconomic paradigm”. Moreover, according to Silberberg, Samuelson’s dynamics conflicted with the notion that there can be no disequilibrium in a world of utility or wealth maximizers.

13 Attempts to rehabilitate the correspondence principle under dynamic stochastic rational expectations models have been made by Brock (1987) and Evans and Honkapohja (2007). Apart from the dynamic issue, the content of the principle is dubious because any continuous function can be an excess demand function (the Sonnenschein-Mantel-Debreu Theorem; see Ingrao and Israel 1990, chapter XI).

14 Endogenous business cycle models, of the kind inaugurated by Samuelson’s (1939a, 1939b), were developed along different lines in the 1980s (see Grandmont 1985), since the 1939 model was perceived as lacking “rigorous micro-foundations” (Bénassy 2011, 206–207). Ironically enough, Grandmont based his model on the overlapping generations framework devised by Samuelson (1958).

15 As Samuelson (1971, 691) observed, were the Foundations “written today with knowledge of the revival of interest in Ramsey growth models, I would certainly have added a chapter on optimal-control theory and similar dynamic maximization matters. And then instead of being preoccupied with the problem of damped stability of dynamic motions, I would have been interested as well in stationary points which are saddle-points surrounded by dynamic motions of the catenary type that we associate with modern turnpike theory”.

16 Samuelson’s original multiplier-accelerator model did include, however, a particular case (region D of his well-known diagram) displaying unstable sustained economic growth. Anyway, it was only later that Samuelson realized that the multiplier-accelerator interaction could be expressed in terms of the instability results produced by Harrod’s natural and warranted growth rates (see e.g. Samuelson 1964, 263, 743–746).

17 Samuelson’s production turnpike theorem was a dynamic generalization of von Neumann’s closed system, in the sense that whatever composition of consumption and capital goods the planner would like to achieve, one obtains the most of all goods if the (efficient) growth path is close to the von Neumann path for most of the time. Samuelson (1965b) later established as well consumption turnpikes with the help of Ramsey’s (1928) model of optimal saving (for a comprehensive survey of the literature see Turnovsky 1970).

18 Samuelson (1956, 5, n. 2) referred to Wicksell’s ([1893] 1954, 2–74) early work on homothetic utility and preference aggregation. Wicksell used the notion of a representative agent in his discussion of optimal capital accumulation, but not in his macroeconomics (Boianovsky 2016). That suggests an interesting parallel with Samuelson.

19 However, as Edward Nelson pointed out to me in correspondence, under certain circumstances representative agent models may be used to illustrate fallacies of composition and the need to guard against such fallacies.

20 This confirms Hahn’s (1983, 51) feeling that Samuelson in the 1930s and 1940s had refrained from discussing the problem – that there is no meaning to ‘lack of effective demand’ in a world where agents can buy and sell as much as they wish at current prices – because he, “like everyone today, did not know how to proceed to a resolution”. In any event, pointed out Hahn, business cycle theory without prices (as in the multiplier-accelerator model) “now seems inconclusive”. Samuelson (1988, 12, n. 1) acknowledged as much.

21 That was close to Patinkin (1965), but Samuelson did not elaborate. Significantly enough, in his Economics the chapters on income determination and economic fluctuations come before the microeconomic chapters on the firm and the consumer.

22 Samuelson (1988) was published in the inaugural issue of Japan and the World Economy, edited by Ryuzo Sato, Samuelson’s co-author in papers about demand for money in the 1980s.

23 As Samuelson (2009, 26; italics in the original) put it, “Markets do tend to be micro efficient. Only when you know new correct news that others don’t yet know you can capture easy returns in micro-efficient markets. Does that mean that every rise or fall in the indexes of most stock prices are rational reactions to knowable correct news? Not at all. The big cumulative swings in mean prices that historians document – as in 1929–34 or 2007–08 – are well known features of historic business cycles … Again, what makes macro efficiency impossible is the hard fact that economic history is at best quasi-stationary time series. That quasi kills all uncertainties.”