Abstract
Frisch’s 1933 macroeconomic model for business cycles has been extensively studied. The present study is the first comprehensive mathematical analysis of Frisch’s model. It provides a detailed reconstruction of how the model was built. We demonstrate the workability of Frisch’s PPIP model without adding hypotheses or changing the value of Frisch’s parameters. We prove that (1) the propagation model oscillates; (2) the PPIP model is mathematically incomplete; (3) the latter could have been calibrated by Frisch; (4) Frisch’s analysis and demonstration are based on Poincaré’s methodology
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30 December 2022
A Correction to this paper has been published: https://doi.org/10.1007/s10699-022-09895-5
Notes
Frisch’s essay was published in the book Economic Essays in Honor of Gustav Cassel for which contributors were invited to write a chapter. In other words, Frisch did not have to follow the selection process as we have in a scientific journal, and he did not have to format his article according to the standards of a scientific article. Moreover, Frisch sent his chapter with a delay, and consequently it could not be reviewed by the editors as originally planned (Bjerkholt 2007, pp. 473–474).
It is worth reminding that Frisch met Divisia and other French mathematicians during his stay in Paris in 1921–1923. He also defended his doctorate on a subject in mathematical statistics rather than in economics (Bjerkholt 2007, p. 13).
Frisch had also his own time series, but he did not provide them (Bjerkholt 2007, p. 457).
In the particular case where the impressed frequency of the periodic force is equal to the natural (undamped) frequency of the system, amplitude of oscillations increase beyond all bounds as time passes. Such phenomenon is called resonance. As an example, in 1831, when an army was marching across a drawbridge in Britain, the drawbridge vibrated and the amplitude of the vibration grew rapidly. All soldiers were then thrown.
We can also include Tinbergen who explained that “The first publication of this sort was a paper in Polish by Kalecki in Proba Teorij Konjunktury, Warszawa, 1933. A few months later appeared Frisch’s “Propagation Problems and Impulse Problems”. Both theories were presented at the Leyden meeting of the Econometric Society in 1933” (Tinbergen 1935, p. 268).
The influence of the early work in econometrics on the theory of oscillations developed in mathematics and physics would deserve a separate investigation. However, it is worth mentioning that Rocard proposed in 1941 an economic model based on the analyses of Frisch and Kalecki and on relaxation oscillations. Rocard’s 1941 chaotic econometric model preceded Lorenz’ butterfly of 22 years. Ginoux (2017) established that this “new old” three-dimensional autonomous dynamical system is a new jerk system whose solution exhibits a chaotic attractor the topology of which varies, from a double scroll attractor to a Möbius-strip and then to a toroidal attractor, according to the values of a control parameter.
Frisch clarified that “For a more detailed mathematical analysis the reader is referred to a paper to appear in one of the early numbers of Econometrica” (Frisch 1933, p. 199). Unfortunately, such a paper has never been published. While Frisch did not provide an explicit mathematical formulation for his “propagation-impulse model”, his model is complete in the sense that Frisch found the values of the periods of the economic cycles by calibrating this model. Therefore, when Duque (2009, p. 48) argued that this model is incomplete, it must be understood that it is mathematically incomplete.
The annual investment is determined by the accelerator principle (Dupont-Kieffer 2012).
We can notice that \(\alpha\) is necessary non-null otherwise the system would not oscillate.
As Duque (2009, p. 23) explained “these values were the result of endless calculations and trial-and-error attempts as evidenced by his notes taken while preparing the manuscript”.
For an extended analysis of Zambelli’s error, see Jovanovic and Ginoux (2020).
Frisch (1933, 199) referred to a paper to appear in Econometrica, but such a paper has never been published.
Let’s notice that although the perturbations produced by such erratic shocks make vary the damped period and the amplitude, such variations, taking place “within such limits that it is reasonable to speak of an average period and an average amplitude”, do not correspond to the moving average as it can be easily evidenced.
Of course, the increase of the number n of erratic shocks would improve the accuracy of the modeling.
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Ginoux, JM., Jovanovic, F. Frisch’s Propagation-Impulse Model: A Comprehensive Mathematical Analysis. Found Sci 28, 57–84 (2023). https://doi.org/10.1007/s10699-021-09827-9
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DOI: https://doi.org/10.1007/s10699-021-09827-9