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Erschienen in: Mathematics and Financial Economics 3/2019

24.11.2018

Minskyan classical growth cycles: stability analysis of a stock-flow consistent macrodynamic model

verfasst von: Daniel Bastidas, Adrien Fabre, Florent Mc Isaac

Erschienen in: Mathematics and Financial Economics | Ausgabe 3/2019

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Abstract

This paper follows van der Ploeg (Metroeconomica 37(2):221–230, 1985)’s research program in testing both its extension of Goodwin (in: Feinstein (ed) Socialism, capitalism and economic growth, Cambridge University Press, Cambridge, 4, 54–58, 1967) predator–prey model and the Minsky Financial Instability Hypothesis (FIH) proposed by Keen (J Post Keynes Econ 17(4):607–635, 1995). By endowing the production sector with CES technology rather than Leontief, van der Ploeg showed that the possible substitution between capital and labor transforms the close orbit into a stable focus. Furthermore, Keen (1995)’s model relaxed the assumption that profit is equal to investment by introducing a nonlinear investment function. His aim was to incorporate Minsky’s insights concerning the role of debt finance. The primary goal of this paper is to incorporate additional properties, inspired by van der Ploeg’s framework, into Keen’s model. Additionally, we outline possibilities for production technology that could be considered within this research program. Using numerical techniques, we show that our new model keeps the desirable properties of Keen’s model. However, we also demonstrate that when the economy is endowed with a class of CES production function that includes the Cobb–Douglas and the linear technology as limit cases, the unique stable equilibrium is an economically desirable one. Finally, we propose a modified extension that includes speculative component in the economy as in Grasselli and Costa-Lima (Math Financ Econ 6(3):191–210, 2012) and investigate its effect on the dynamics. We conclude that CES production function is a more suitable assumption for empirical purposes than the Leontief counterpart. Finally, we show, using numerical simulations, that under plausible calibration, the model endowed with CES production function eventually lose the cyclical property of Goodwin’s model with and without the speculative component.

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Fußnoten
1
In the sense of the GDP at factor cost, where the income approach of the GDP is summarized as the distribution of wages and profits.
 
2
For a complete overview of Goodwin’s modern dynamics and its economic interpretation, we refer the reader to the paper of Grasselli and Costa-Lima [6].
 
3
The superscript k stands for Keen.
 
4
We confine ourselves to \(\eta \in ]0,+\infty [\) (that is an elasticity of substitution that lies in the set ]0, 1[). The reason is twofold: (i) such short term elasticity would imply an above unity substitution between capital and labor in a very short term that is very unlikely (see Klump et al. [10]); and (ii) such values values would break the predator–prey logic of the clockwise behavior suggested by Goodwin and shown by the data (Solow [17], Harvie [7], Mohun et al. [12], or Mc Isaac [11]).
 
5
Throughout the article, the consumption price is normalized to 1.
 
6
This minimal rationality argument is analogous to the assumption in Goodwin’s model that the allocation of capital and labor is always at the diagonal of the \((K^k,L^k)\)-plan, so that we have not only \(Y^k = \min \left( \frac{K^k}{\nu }, a L^k \right) \) but also \(Y^k = \frac{K^k}{\nu } = a L^k\).
 
7
See “Appendix A” for the computation.
 
8
Studying the consequences of dropping Say’s law will be the task of a forthcoming paper.
 
9
We note that according to Nguyen-Huu and Pottier [14], the channel of debt financing is not fully determined by the model. Indeed, it does not distinguish between loanable funds and endogenous money creation since both rationales induce the same set of equations.
 
10
Figure 5 includes the evolution of the capital-to-output ratio for benchmark parameters and \(\eta =100\).
 
11
In addition, it can be noticed that the equilibrium points differ slightly depending on the value of \(\eta \).
 
12
It can be shown that, with the simplest possible case of an affine function, a closed-form expression for the equilibrium is not available.
 
13
Thus, if the elasticity of substitution is too high, i.e. above that of Cobb–Douglas (as in the linear case e.g.), the Bad equilibrium is unstable.
 
14
See “Appendix B” for the details.
 
15
It can be noticed that the variation of \(\lambda \) is unbounded as \(\omega \rightarrow 1^-\). Therefore, it is very likely that the Lyapunov function shows unbounded variation making the variational domain be in \({\mathcal {D}}_{(\omega ,\lambda ,d)} = [0,1]\times [0,1]\times {\mathbb {R}}\). We leave the proof for further research. For more insights see Grasselli and Costa-Lima [6] or Costa-Lima and Ngyen-Huu [13].
 
16
The case \(\eta = 0.5\) is identified as being the closest to the Cobb–Douglas. However, as shown in “Appendix F”, when we derive the model with Cobb–Douglas production technology, we found that the wage share is no longer time-varying and equals, at all times, the output elasticity \(1-b\). Therefore, the original Goodwin prey–predator (between the employment rate and the wage share) logic does not hold anymore, as previously eluded.
 
18
Klump et al. [10] surveyed a number of studies intended for developed countries in various timeframes (ca. 1800–2000). Almost 75% of the estimated elasticities showed a value between 0.5 and 1.
 
19
Note that in Grasselli et al. [6], the growth rate of output equals that of capital, as \(\nu \) is constant. We chose to align the growth in speculation with the growth rate of capital, as its objects is precisely existing assets. However, aligning the speculation with output would have been equally compatible with Grasselli et al. [6].
 
20
For \(\eta <0\), \(\zeta _1\left( 0\right) \) takes high values (often above 1), so that the previous inequality holds.
 
21
This equivalence would be the opposite for a negative value of \(\eta \).
 
22
In the simulation, the value 0 would lead to numerical errors, therefore we choose the value 0.1 as the lowest value that does not show numerical error. Similarly, \(+\infty \) has been approached by \(10^{15}\) to show the behavior of the the model near the Leontief case.
 
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Metadaten
Titel
Minskyan classical growth cycles: stability analysis of a stock-flow consistent macrodynamic model
verfasst von
Daniel Bastidas
Adrien Fabre
Florent Mc Isaac
Publikationsdatum
24.11.2018
Verlag
Springer Berlin Heidelberg
Erschienen in
Mathematics and Financial Economics / Ausgabe 3/2019
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI
https://doi.org/10.1007/s11579-018-0231-6

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