Chaos in Non-Linear Business Cycle Models

 Chaotic Dynamics Within Kaldor's Non-Linear Business Cycle Model

This week’s post is shorter than average unfortunately, but I found it worth demonstrating one of the wide array of conditions in which economic models can generate unstable chaotic motions, e.g., Kaldor’s/Goodwin’s non-linear business cycle model. This brief argument laid out for the existence of unstable chaotic motions within the economy’s business cycles can further research on the relationships of chaos with business cycles, primarily attractor merging crises and the transition from weak to strong chaos, consequently. The Kaldorian system may demonstrate chaotic motions when the perturbation contains a strange attractor, leading to the perturbed system (Kaldor’s business cycle model in this case) displaying irregular, aperiodic, and highly sensitive behavior. Multiple arguments with Kaldor’s non-linear investment and savings functions have outlined the existence of complex phenomena that occasionally generate chaotic phenomena, too. 

A general defining characteristic of strange attractors is sensitive dependence on initial conditions, we can demonstrate this with the following system: 

Z = f (a, γ), a ∈ R, γ ∈ R

The system generates “sensitive dependence on initial conditions” if any variations in a(n) lead to major differences in the time series of the system in a short period of time. These systems, generated by a dynamic function, are deterministic in a crude sense. The evolution of the system is hypothetically predictable granted complete knowledge of initial conditions, i.e., the variances of a(n) generating new time series that are considered “chaotic”; along with future random perturbations that may influence the evolution of the deterministic non-linear system in a stochastic manner. A lesser known phenomena with chaotic systems is aperiodicity, i.e., the chaotic time series remain bounded within certain values but remain aperiodic, to be distinguished from functions such as f(x) = 2x, in which as x → ∞ the values never repeat themselves. Remaining confined in a certain space yet never repeating itself, aperiodic, is the reason for certain shapes created that resemble a “butterfly” when graphed in phase space. The phase space trajectory is incredibly complex for the original systems Lorenz studied which exhibited chaotic motion, and due to being bounded generated the “butterfly” sort shape for which the colloquial name of the “butterfly effect” follows. Now consider the model for an economy, Kaldor’s model: 

Y ′ = α[I(Y, K) − S(Y, K)]

K′ = I(Y, K) − δK

Which follows Y being income, I being gross investment, K is the capital stock, and S is savings.  δ being the depreciation rate and α being the adjustment coefficient.  The second equation is a general capital accumulation equation with not many distinct properties for the example being studied currently. Now following Akhmet et al.[1] we can specify the above system even more, given S(Y, K) = sY & I(Y, K) = Y −aY3+bK as they did: 

Y ′ = α[(1 − s)Y − aY3 + bK]

K′ = Y − aY3 + bK − δK

The specifications for the constant parameters remain the same as in the example provided by Akhmet et al., we can recognize a steady state solution if only δs < δ + bs. For the sake of the gross formatting mistakes that occur on Blogger occasionally, I will refrain from applying the transformations derived to the system above. Now we may also explore Kaldor’s model given foriegn capital investment and the consequent Hopf bifurcations that follow said system: 

S `= αY + pS(k − Y2 )

Y` = v(S + F)

F` = mS − rY

S being the savings of households, Y  being GDP, K being potential GDP, F being foreign capital, and t is time. If we set k to 1, potential output, the above identities are given as multiples of potential output, k. We also have α being marginal propensity to save, 1/V the K/Y ratio, and m capital inflow/savings ratio. Now that we have specified: 

S` = αY + 0.1S(1 − Y2) 

Y` = 0.5(S + F)

F` = 0.19S − 0.25Y

According to Pribylova (2009)[2] the system given above generates a Hopf bifurcation at α = α(0) ≡ 0.25. A Hopf bifurcation is generally described as a point in which the stability of a system switches through a change in the stability properties of the dynamical equation.[3] This is quite important for Kaldor’s business cycle, as the general business cycle model with the non-linear investment and savings function as the original model proposed by Kaldor was attempting to demonstrate the shifts from unstable equilibria. Demonstration of chaotic dynamics within the Kaldorian non-linear investment and savings functions results in a realization of the possibilities of changes in the local stability conditions, leading to Hopf bifurcations and other chaotic motions. Endogenous chaotic business cycles can either be generated by an attractor merging crises, which I will explain in the following paragraph, or by a change in the stability conditions followed by a Hopf bifurcations at certain values.  

A familiar condition of chaotic dynamics is the inability to forecast them outside of the very immediate future, e.g., within economic systems, 1-2 days. If a system is demonstrated to have a chaotic attractor, the behavior of the economic system is near impossible to forecast outside of the immediate short term. Another property of chaotic systems worth discussing quite briefly for such a short post is the existence of attractor merging crises highlighted by Pribylova (2009) and Chian (2007)[4]. Following Chian (2007), we can reduce relaxation oscillations that exist within Goodwin’s non-linear accelerator/multiplier model and Kaldor’s non-linear business cycle model:

x+μ(x2 −1)x`+ x=0

Which is the forced Van der Pol model used originally to describe oscillations within an electrical circuit model of a heartbeat. We may also attempt to understand the use of the forced Van der Pol using Goodwin’s non-linear accelerator-multiplier model, in contrast to the Samuelsonian vulgar linear accelerator-multiplier model: 

x`+ A(x)X̂  + B(x) = I(t) 

A(x) being an even function with A(0) > 0, and B(x) an odd function with B(0) = 0, following Chian (2007). Continuing:

I(t) = a sin(ωt) 

a being the amplitude of exogenous force, and ω, its frequency: 

A(x) = μ(x2 − 1), B(x) = x

We again obtain a forced Van der Pol model of non-linear business cycles: 

x`+ μ(x2 − 1)X̂  + x = a sin(ωt)

Using the forced Van der Pol model we are enabled with the tools that allow for us to comprehend the phenomena of an attractor merging crises, where an attractor finds an unstable fixed point that is inside a basin of attraction. In this example, when this occurs, two (or more) chaotic attractors merge together to form a stronger chaotic attractor, this can be used to explain the transition from booms to recessions within Kaldorian unstable equilibria transitions and Goodwin’s non-linear accelerator-multiplier model of the business cycle. The ability to forecast, and consequently uncertainty, tends to increase along with volatility during an attractor merging crises which may be the upper turning point within these models of the business cycle, leading to a consequent recession or a flight of capital from an economy. 

In this (very) brief blog post I have demonstrated the possibility of endogenous business cycles following from Goodwin’s and Kaldor’s non-linear models of the business cycle and introducing chaotic dynamics following from the work of other more knowledgeable mathematicians. Growth models and chaotic dynamics will most likely remain the focus of this blog for the next few weeks as there is a lot left unanswered with such a short blogpost. Chaotic dynamics offers a more physical argument of fundamental uncertainty in the market economy, contrasted to the a priori “ontological uncertainty” of present day fundamentalist Post-Keynesians, considering the great deal of theoretical conditions of which chaotic dynamics can occur and its implications for market stability that I have very briefly considered above. I hope that in the future I can clarify some of the open ended questions that were briefly addressed such as the implications for financial instability or the use of the forced Van der Pol model. 

References:

1.  Akhmet, M., Akhmetova, Z., & Fen, M. O. (n.d.). Exogenous versus endogenous for chaotic business cycles. https://arxiv.org/pdf/1509.01034. 

2.  Pribylova, L. (2009), Bifurcation routes to chaos in an extended Van der Pol’s equation applied to economic models, Electronic Journal of Differential Equations

3.  Mohd, Mohd Hafiz, et al. Dynamical Systems, Bifurcation Analysis and Applications. Springer Nature, 2019

4.  A. Chian, 2007. "Complex Systems Approach to Economic Dynamics," Lecture Notes in Economics and Mathematical Systems, Springer, number 978-3-540-39753-3, June.