Markets, Productivity, and Happiness in a Historical Perspective

The quest for relaxation oscillations in macrodynamics from 1928 to 1941

Jovanovic Franck, TELUQ University
Ginoux Jean-Marc, Centre de Physique Théorique, Université de Toulon, France

This article examines one aspect of the history of nonlinear macrodynamic models from 1928 to 1941: the attempts to model macrodynamic fluctuations with relaxation oscillations and the reason for their subsequent abandon. Thanks to Ludwig Hamburger’s and Philippe Le Corbeiller’s research, members of the Econometric Society became aware of the application of Van der Pol’s relaxation oscillations to business cycles in the 1930s. Hamburger and Le Corbeiller shared their viewpoints particularly with Ragnar Frisch who, in his 1933 book chapter “Propagation problems and impulse problems in dynamic economics” (PPIP), attempted to integrate self-maintained oscillations in a dynamic macroeconomic model. In 1974, Paul Samuelson, while reviewing aspects of Frisch’s PPIP, explained that, by accepting Frisch’s criticism of the Kalecki’s article, “it slowed down our recognition of the importance of nonlinear auto-relaxation models of the Van der Pol-Rayleigh type.” Samuelson’s mention indirectly referred to Richard Goodwin’s work. In 1951, the latter proposed a nonlinear macrodynamic model with self-maintained oscillations for studying business cycles. In his work, Goodwin thanked Le Corbeiller for his help. Knowing that Le Corbeiller was aware of relaxation oscillations as early as the 1920s, we can suppose that such an approach could have been developed earlier. This seems to be one of the reasons for Samuelson’s comment. But were economists really facing a dogma, as Samuelson claimed? Is this a sufficient reason to explain the slow recognition of the importance of “nonlinear auto-relaxation models of the Van der Pol-Rayleigh typ”? In this study, we challenge Samuelson’s perspective. To date, and despite recent investigations, no satisfactory reason has been given to explain why no author succeeded in proposing a business cycle model with nonlinear oscillations before Goodwin. The present work sheds new light on the early history of macrodynamics by investigating some challenges economists faced for modeling business cycles. It starts by clarifying a frequent confusion in the literature between relaxation oscillations and self-maintained oscillations. By using a precise definition of these two kinds of oscillations, our work contributes to the history of macrodynamics on three counts. Firstly, we show that Van der Pol’s 1926 articles on relaxation oscillations opened three research programs in economics for exploring the extension of such oscillations to macrodynamic and business cycle theories: relaxation oscillations, self-maintained oscillations, and forced oscillations, i.e., damped oscillations maintained by exogenous and erratic impulses. Secondly, we establish that in 1941 the French physicist Yves Rocard demonstrated that the application of relaxation oscillations for modeling economic fluctuations was a dead end. Thirdly, we show that ten years before Goodwin, Rocard provided an econometric model with nonlinear self-maintained oscillations.


Keywords: Ragnar Frisch; Ludwig Hamburger; Philippe Le Corbeiller; Michal Kalecki; Richard Goodwin; Yves Rocard; theory of oscillations; relaxation oscillations; history of econometrics; nonlinear macrodynamics

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