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

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.

Appendices

Appendices

1.1 A Getting the reduced form of the system

We assume that the productive sector is endowed with CES technology so that

$$\begin{aligned} Y = A \left[ b K^{-\eta } + (1-b) (e^{a_l t} L)^{-\eta } \right] ^{- \frac{1}{\eta }}. \end{aligned}$$

(13)

Additionally, we make the assumption that wages are set at marginal rate of productivity, so that:

$$\begin{aligned} \frac{\partial Y}{\partial L} = \text {w} . \end{aligned}$$

For simplicity, we define $L^e := e^{a_l t} L$, so that the following relationship holds

$$\begin{aligned} \frac{\partial Y}{\partial L^e} = \frac{\partial Y}{\partial L} e^{ - a_l t} . \end{aligned}$$

(14)

By using Eqs. (13) and (14)

$$\begin{aligned} \frac{\partial Y}{\partial L^e} = \frac{(1-b)}{A^\eta } \left( \frac{Y}{L^e} \right) ^{1+\eta } . \end{aligned}$$

By taking the derivative of (13) and using (14)

$$\begin{aligned}&\left( \frac{\omega }{1-b} \right) ^{\frac{1}{\eta }} A = \frac{Y}{L^e} \nonumber \\\Leftrightarrow & {} \left( \frac{\omega }{1-b} \right) ^{\frac{1}{\eta }} A e^{a_l t} = \frac{Y}{L} \end{aligned}$$

(15)

with $\omega$, the share of total real wages ($W := \text {w}L$) in the production:

$$\begin{aligned} \omega := \frac{\text {w} L }{Y} . \end{aligned}$$

Let $a := Y/L$ be the labor productivity, one has $\omega = \text {w}/a$. The growth rate of the wage share is given by

$$\begin{aligned} \frac{{\dot{\omega }}}{\omega } = \frac{\dot{\text {w}}}{\text {w}}- \frac{{\dot{a}}}{a} . \end{aligned}$$

Using Eq. (15), one gets the following growth rate for labor productivity

$$\begin{aligned} \frac{{\dot{a}}}{a} = \frac{1}{\eta } \frac{{\dot{\omega }}}{\omega } + a_l . \end{aligned}$$

Suppose that the growth rate of wages is given by a short-term Phillips Curve

$$\begin{aligned} \frac{\dot{\text {w}}}{\text {w}} = \Phi (\lambda ) . \end{aligned}$$

The dynamic of the wage share is given by

$$\begin{aligned} \boxed {\frac{{\dot{\omega }}}{\omega } = \left( \frac{\eta }{1 + \eta } \right) \left[ \Phi (\lambda ) - a_l \right] }. \end{aligned}$$

The population grows according to

$$\begin{aligned} \frac{{\dot{N}}}{N} = \beta \ge 0 . \end{aligned}$$

The employment rate is defined by $\lambda := \frac{L}{N}$, while the capital-output ratio is given by $\nu := \frac{K}{Y}$. Hence, the employment rate dynamic

$$\begin{aligned} \frac{{\dot{\lambda }}}{\lambda }= & {} \frac{{\dot{L}}}{L} - \frac{{\dot{N}}}{N} \\= & {} \frac{{\dot{K}}}{K} - \frac{{\dot{a}}}{a} - \frac{{\dot{\nu }}}{\nu } - \frac{{\dot{N}}}{N} . \end{aligned}$$

The profit share in the production is given by

$$\begin{aligned} \pi := 1 - \omega - rd, \end{aligned}$$

where r is the short-term interest rate set by the central bank, and paid by producers, while d is the ratio of real debt-to-production (i.e $\frac{D}{Y}$). The capital accumulation is given by

$$\begin{aligned}&{\dot{K}} = \kappa (\pi ) Y - \delta K , \\&\frac{{\dot{K}}}{K} = \frac{\kappa (\pi )}{\nu } - \delta \end{aligned}$$

where $\delta$ is the depreciation rate of capital, $\kappa (\pi )$ is a function of the profit share, and $\nu$ is the time-varying capital-to-output ratio. From the expression of $\frac{\partial Y}{\partial K}$ and knowing that Y is homogeneous of degree one, we obtain

$$\begin{aligned} \nu = \left( \frac{1-\omega }{b} \right) ^{- \frac{1}{\eta }} \frac{1}{A} . \end{aligned}$$

Its growth rate is given by

$$\begin{aligned} \frac{{\dot{\nu }}}{\nu } = \frac{{\dot{\omega }}}{(1-\omega )\eta } . \end{aligned}$$

Therefore, the growth rate of employment is

$$\begin{aligned} \boxed {\frac{{\dot{\lambda }}}{\lambda } = \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } - \delta - a_l - \beta - \frac{1}{\eta (1-\omega )} \frac{{\dot{\omega }}}{\omega }} . \end{aligned}$$

The debt dynamic is the difference between investment and the profits made by the corporate sector, in other words

$$\begin{aligned} {\dot{D}} = \kappa (\pi ) Y - (\pi ) Y . \end{aligned}$$

The growth rate of production is given by

$$\begin{aligned} g := \frac{{\dot{Y}}}{Y} = \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } - \delta - \frac{{\dot{\omega }}}{(1-\omega )\eta }. \end{aligned}$$

Thus, the ratio of real debt on production is

$$\begin{aligned} \frac{{\dot{d}}}{d} = \frac{{\dot{D}}}{D} - \frac{{\dot{Y}}}{Y} = \frac{\kappa (\pi ) - \pi }{d} - \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } + \delta + \frac{{\dot{\omega }}}{(1-\omega )\eta }. \end{aligned}$$

Hence, its dynamic is

$$\begin{aligned} \boxed {{\dot{d}} = d \left\{ r - \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } + \delta + \frac{{\dot{\omega }}}{(1-\omega )\eta } \right\} + \kappa (\pi ) - (1-\omega )}. \end{aligned}$$

To wrap up, and for the sake of clarity, the tree-dimensional system is

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\omega }} &{} = \omega \left[ \left( \frac{\eta }{1 + \eta } \right) \left[ \Phi (\lambda ) - a_l \right] \right] \\ \\ {\dot{\lambda }} &{} = \lambda \left[ \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } - \delta - a_l - \beta - \frac{1}{\eta (1-\omega )} \frac{{\dot{\omega }}}{\omega } \right] \\ \\ {\dot{d}} &{} = d \left[ r - \kappa (\pi ) A \left( \frac{1-\omega }{b} \right) ^{1/\eta } + \delta + \frac{{\dot{\omega }}}{(1-\omega )\eta } \right] + \kappa (\pi ) - (1-\omega ) \end{array}\right. } \end{aligned}$$

1.2 B Parameter values

See Fig. 14.

The calibration is almost entirely borrowed from Keen [9] to the exception of the speculation function that is borrowed from Grasselli and Costa-Lima [6]. The time frequency between t and $t+1$ is considered to be one year.

Fig. 14

Phenomenological functions behaviors according to the parameters in Table 5

Fig. 15

Phenomenological functions behaviors according to the parameters in Table 5

The same generalized exponential function is used for both the relationship between investment as a share of output, and the short term Phillips Curve

$$\begin{aligned} \Phi (\lambda , \phi _0, \phi _1, \phi _2) = \Phi (\lambda )=\phi _0 + \phi _1*e^{\phi _2*\lambda }). \end{aligned}$$

Figure 15 displays the behaviors of the phenomenological functions using the calibration given by Table 5.

Table 5 Calibration for the numerical estimations

1.3 C Proofs for Propositions 1 and 2

Proposition 1

The Good equilibrium exists if $\kappa \left( 1-\frac{r}{r-a_l-\beta }\left( 1-\zeta _1\left( 0\right) \right) \right) < \zeta _1\left( 0\right)$.

Proof

The right hand side of Eq. (3) is a function of $\omega _1$ that equals 0 at $\omega _1 = 1$ and $\zeta _1\left( 0\right) =\frac{a_l+\beta +\delta }{A}b^{\frac{1}{\eta }}$ at 0; while the left hand side equals $\kappa _0$ for $\omega _1=1$ and $\kappa \left( 1-\frac{r}{r-a_l-\beta }\left( 1-\zeta _1\left( 0\right) \right) \right)$ for $\omega _1=0$. Since both sides are continuous function of $\omega$, it suffices that $\kappa \left( 1-\frac{r}{r-a_l-\beta }\left( 1-\zeta _1\left( 0\right) \right) \right) < \zeta _1\left( 0\right)$ to ensure the existence of the Good equilibrium. As $\eta >0$, $1-\frac{r}{r-a_l-\beta }\left( 1-\zeta _1\left( 0\right) \right)$ is often negative $\big ($it is negative as long as $\frac{a_l+\beta +\delta }{A}>\frac{1}{1-\zeta _1\left( 0\right) }\big )$, so that $\kappa _0<\zeta _1\left( 0\right)$ ensures the existence of the equilibrium (because $\kappa$ is increasing).²⁰ As $\zeta _1\left( 0\right) \in \left[ 0;\frac{a_l+\beta +\delta }{A}\right]$ in this case, assuming e.g. $\kappa \left( 0\right) =0$ is a sufficient condition for the existence of the Good equilibrium.

Proposition 2

$\omega _1$ is positive if $\left( \frac{A\zeta _1}{a_l+\beta +\delta }\right) ^\eta >b$.

Proof

As Eq. (3) rewrites $\omega _1=1-b\left( \frac{A\zeta _1}{a_l+\beta +\delta }\right) ^{-\eta }$, one just needs that $\left( \frac{A\zeta _1}{a_l+\beta +\delta }\right) ^\eta >b$ to ensure a positive $\omega _1$. As $\eta > 0$, this is equivalent to $\zeta _1$ being above $\frac{a_l+\beta +\delta }{A}b^{\frac{1}{\eta }}$.²¹ Hence, as long as the image of $\kappa$ is entirely on the right side of this value, $\omega _1$ is positive. $\square$

1.4 D Numerical results for the stability of equilibrium

Table 6 The numerical eigenvalues of all the models at their Good equilibrium point $(\omega _1, \lambda _1, d_1)$

Table 6 displays the numerical eigenvalues of the Jacobian at the Good equilibrium point, where $(\omega _1, \lambda _1, d_1)$ are all finite. Remember, to be locally stable, the Jacobian matrix at this equilibrium point has to be negative definite. This would mean that the eigenvalues are all non-positive. In this exercise, they all show local stability at the Good equilibrium value.²²

Table 7 The numerical eigenvalues of all the models at their Trivial equilibrium point $({\bar{\omega }}_3, {\bar{\lambda }}_3, {\bar{d}}_3)$

Table 8 The numerical eigenvalues of all the models at the slavery equilibrium point $({\bar{\omega }}_2, {\bar{\lambda }}_2, {\bar{d}}_2)$

For the sake of completeness, Tables 7 and 8 show respectively the eigenvalues of the Trivial and the Slavery equilibria. We observe that, as expected, non of these trials display a local stability for that equilibrium point.

Table 9 The numerical eigenvalues of all the models at their Bad equilibrium point $(\omega _3, \lambda _3, d_3)$

The eigenvalues of the Bad equilibrium are shown in Table 9 and confirm that, whenever $\eta \in ]-1;0[$, the only stable equilibrium that may be asymptotically globally stable is the good.

1.5 E Existence of the slavery equilibrium

Let us take the last two equations of the main system when $\omega \rightarrow 0$.

$$\begin{aligned} {\dot{\lambda }}\simeq & {} \lambda \left[ \kappa \left( 1-rd\right) A b^{-\frac{1}{\eta }}-\frac{\Phi \left( \lambda \right) -a_{l}}{1+\eta }-\left( \delta +a_{l}+\beta \right) \right] \end{aligned}$$

(16)

$$\begin{aligned} {\dot{d}}\simeq & {} d\left( r-\kappa \left( 1-rd\right) A b^{-\frac{1}{\eta }}+\delta +\omega \frac{\Phi \left( \lambda \right) -a_{l}}{1+\eta }\right) +\kappa \left( 1-rd\right) -1 \end{aligned}$$

(17)

At the equilibrium, whenever $\lambda >0$, one finds, by injecting 16 into 17:

$$\begin{aligned} 0\simeq & {} d\left( r-\kappa \left( 1-rd\right) A b^{-\frac{1}{\eta }}+\delta +\omega \left( \kappa \left( 1-rd\right) A b^{-\frac{1}{\eta }}-\delta -a_{l}-\beta \right) \right) +\kappa \left( 1-rd\right) -1 \end{aligned}$$

As $\kappa$ is bounded, one can neglect the term in $\omega$. Defining $s=A b^{-\frac{1}{\eta }}$, $e=r+\delta >0$ and $z\left( d\right) =-\kappa \left( 1-rd\right) \in \left[ -1;0\right]$ one thus obtains

$$\begin{aligned} d\left( sz\left( d\right) +e\right)= & {} 1+z\left( d\right) \in \left[ 0;1\right] \end{aligned}$$

If $s<e$, the left hand side is a continuous and non-negative function of d which passes through the origin and is equivalent to $d\left( e-s\kappa _{0}\right)$ at $+\infty$. Hence, the equation has a solution and $s<e$ is a sufficient condition for the existence of a Slavery equilibrium.

For $\eta <0$, s converges decreasingly towards 0 when $\eta$ tends to 0, so there exists an interval $]-\eta _{min};0[$ within which the existence of the equilibrium is insured.

Note that for $\eta >0$, as $b<1$, s decreases with $\eta$ and converges towards A, so that if there is a substitutability $\eta _{0}$ such that $s_{\eta _{0}}<e=r+\delta$, the existence of the Slavery equilibrium is insured below this substitutability (for $\eta >\eta _{0}$). That being said, for the benchmark specification, this inequality does not hold. One can then derive a less restrictive sufficient condition: $e-s\kappa _{0}>0$. Finally, if $\kappa _{0}<\nicefrac {b^{\frac{1}{\eta }}\left( r+\delta \right) }{A}$, a Slavery equilibrium exists (with an equilibrium value for the debt potentially very high). For our benchmark specification with $\eta =1$ (resp. $\eta =\infty$, resp. $\eta =-0.5$), it suffices that $\kappa _{0}<0.02$ (resp. $\kappa _{0}<0.15$, resp. $\kappa _{0}<8.2$) for the equilibrium to exist.

1.6 F Getting the reduced form of the system with Cobb–Douglas production function

We assume that the productive sector is endowed with a constant return to scale Cobb–Douglas technology so that

$$\begin{aligned} Y = A \left[ K^{b} (e^{a_l t} L)^{(1-b)} \right] . \end{aligned}$$

(18)

Additionally, we make, as previously, the assumption that wages are set at marginal rate of productivity, so that:

$$\begin{aligned} \frac{\partial Y}{\partial L} = \text {w} . \end{aligned}$$

For simplicity, we define $L^e := e^{a_l t} L$, so that the following relationship holds

$$\begin{aligned} \frac{\partial Y}{\partial L^e} = \frac{\partial Y}{\partial L} e^{ - a_l t} . \end{aligned}$$

(19)

By using Eqs. (18) and (19)

$$\begin{aligned} \frac{\partial Y}{\partial L^e} = (1-b) \left( \frac{Y}{L^e} \right) . \end{aligned}$$

By taking the derivative of (13) and using (14)

$$\begin{aligned} \omega = (1-b) \end{aligned}$$

(20)

with $\omega$, the share of total real wages ($W := \text {w}L$) in the production:

$$\begin{aligned} \omega := \frac{\text {w} L }{Y} . \end{aligned}$$

Let $a := Y/L$ be the labor productivity, one has $\omega = \text {w}/a$. The growth rate of the wage share is given by

$$\begin{aligned} \frac{{\dot{\omega }}}{\omega } = \frac{\dot{\text {w}}}{\text {w}}- \frac{{\dot{a}}}{a} . \end{aligned}$$

Using Eq. (20), one gets the following growth rate for labor productivity

$$\begin{aligned} \frac{{\dot{a}}}{a} = \frac{\dot{\text {w}}}{\text {w}} . \end{aligned}$$

Suppose that the growth rate of wages is given by a short-term Phillips Curve

$$\begin{aligned} \frac{\dot{\text {w}}}{\text {w}} = \Phi (\lambda ) . \end{aligned}$$

The dynamic of the wage share is given by

$$\begin{aligned} \boxed {\frac{{\dot{\omega }}}{\omega } = 0 } . \end{aligned}$$

The population grows according to

$$\begin{aligned} \frac{{\dot{N}}}{N} = \beta \ge 0 . \end{aligned}$$

The employment rate is defined by $\lambda := \frac{L}{N}$, while the capital-output ratio is given by $\nu := \frac{K}{Y}$. Hence, the employment rate dynamic

$$\begin{aligned} \frac{{\dot{\lambda }}}{\lambda }= & {} \frac{{\dot{L}}}{L} - \frac{{\dot{N}}}{N} \\= & {} \frac{{\dot{K}}}{K} - \frac{{\dot{a}}}{a} - \frac{{\dot{\nu }}}{\nu } - \frac{{\dot{N}}}{N} . \end{aligned}$$

The profit share in the production is given by

$$\begin{aligned} \pi:= & {} 1 - \omega - rd,\\= & {} b-rd \end{aligned}$$

where r is the short-term interest rate set by the central bank, and paid by producers, while d is the ratio of real debt-to-production (i.e $\frac{D}{Y}$). The capital accumulation is given by

$$\begin{aligned}&{\dot{K}} = \kappa (\pi ) Y - \delta K , \\&\frac{{\dot{K}}}{K} = \frac{\kappa (\pi )}{\nu } - \delta \end{aligned}$$

where $\delta$ is the depreciation rate of capital, $\kappa (\pi )$ is a function of the profit share, and $\nu$ is the time-varying capital-to-output ratio. By dividing Eq. 18, by the output, Y, we are able to obtain

$$\begin{aligned} \nu = \left( \frac{1}{A} \right) ^{1/b} \left( \frac{a}{e^{a_l t}} \right) ^{\frac{1-b}{b}} . \end{aligned}$$

Its growth rate is given by

$$\begin{aligned} \frac{{\dot{\nu }}}{\nu } = \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) . \end{aligned}$$

Therefore, the growth rate of employment is

$$\begin{aligned} \boxed {\frac{{\dot{\lambda }}}{\lambda } = \frac{\kappa (\pi )}{\nu } - \delta - \Phi (\lambda ) - \beta - \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) }. \end{aligned}$$

The debt dynamic is the difference between investment and the profits made by the corporate sector, in other words

$$\begin{aligned} {\dot{D}} = \kappa (\pi ) Y - (\pi ) Y . \end{aligned}$$

The growth rate of production is given by

$$\begin{aligned} g := \frac{{\dot{Y}}}{Y} = \frac{\kappa (\pi )}{\nu } - \delta - \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) . \end{aligned}$$

Thus, the ratio of real debt on production is

$$\begin{aligned} \frac{{\dot{d}}}{d} = \frac{{\dot{D}}}{D} - \frac{{\dot{Y}}}{Y} = \frac{\kappa (\pi ) - \pi }{d} -\frac{\kappa (\pi )}{\nu } + \delta + \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) . \end{aligned}$$

Hence, its dynamic is

$$\begin{aligned} \boxed {{\dot{d}} = d \left\{ r - \frac{\kappa (\pi )}{\nu } + \delta + \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) \right\} + \kappa (\pi ) - b} . \end{aligned}$$

To wrap up, and for the sake of clarity, the three-dimensional system is

$$\begin{aligned} {\left\{ \begin{array}{ll} {\dot{\nu }} &{} = \nu \left( \frac{1-b}{b} \left[ \Phi (\lambda ) - a_l \right] \right) \\ {\dot{\lambda }} &{} = \lambda \left[ \frac{\kappa (\pi )}{\nu } - \delta - \Phi (\lambda ) - \beta - \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) \right] \\ {\dot{d}} &{} = d \left\{ r - \frac{\kappa (\pi )}{\nu } + \delta + \frac{1-b}{b} \left( \Phi (\lambda ) - a_l\right) \right\} + \kappa (\pi ) - b \end{array}\right. } \end{aligned}$$