Mathematical Economics and the Dynamics of Capitalism

Goodwin's Legacy Continued

Peter Flaschel, Michael Landesmann, Peter Flaschel, Michael Landesmann

• Publisher: Routledge
• Year: 2016
• Print ISBN:
  9780415762014
• eBook ISBN:
  9781134057573
• Topic: Business
• Subtopic: Business General
• Index: Business

Richard Goodwin was a pioneer in the use of mathematical tools to understand the dynamics of capitalist economies. This book contains contributions which focus on the rigorous extension of Goodwin's modelling of macro-dynamics and the micro-structures underlying them, and also research with a wider perspective related to Goodwin's vision of an integrated Marx-Keynes-Schumpeter (M-K-S) system of the dynamics of capitalist economies.

The variety of approaches in this book range from detailed business cycle analyses to Schumpeterian processes of creative destruction. They include

• thorough theoretical analysis of delayed dynamical systems.
• empirical studies of Goodwin's classical growth cycle model and the integration of Keynesian aspects of effective demand and of financial mechanisms that impact the real macro-economy.
• micro-economic structural analysis.
• expectations driven aspects of micro-founded business cycle modelling

Part I
Nonlinear macrodynamics: theory

1 Coexistence of multiple business cycles in Goodwin’s 1951 model

Akio Matsumoto and Mami Suzuki

Introduction

The contribution of Goodwin (1951) is reconsidered in this study. Goodwin developed a nonlinear accelerator business cycle model and showed that it could generate a stable limit cycle when a stationary point was locally unstable. Considerable effort has been devoted to investigate the dynamic structure of Goodwin’s model since then. However, in the existing literature, not much has yet been revealed with respect to the circumstances under which the stationary point is locally stable. In particular, it is not yet known whether cyclical global dynamics may appear in the stable case. We draw attention to this unexplored case and exhibit the coexistence of multiple limit states, namely a stable stationary point, an unstable limit cycle and a stable limit cycle. Since very few explorations have been made in the global dynamics of the stable case, this study is intended as an investigation of an unexplored aspect of Goodwin’s nonlinear business cycle model.

Goodwin proposed five different versions of his business cycle model. The first version assumes a piecewise linear function with three levels of investment, which can be thought as the crudest or simplest of the nonlinear accelerator. This is a textbook model that can give a simple exhibition of how nonlinearities give rise to endogenous cycles without relying on structurally unstable parameters, exogenous shocks, etc. The second version replaces the piecewise linear investment function with a smooth nonlinear investment function. Although persistent cyclical oscillations of output are shown to exist, the second version includes a unfavorable phenomenon, namely, discontinuous investment jumps, which is not observed in the real economic world. “In order to come close to reality” (Goodwin 1951: 11), a production lag is introduced in the third version. However, no dynamic considerations are given to this third version by Goodwin. The existence of a stable limit cycle is examined in the fourth version, which is a linear approximation of the third version with respect to the production lag. Goodwin’s final modification makes the amount of autonomous expenditure alter over time. This fifth version is recently reconsidered by Lorenz (1987) as a forced oscillator system in which the emergence of chaotic motion is demonstrated. More recently Sasakura (1996) gave an elegant proof of the stability and uniqueness of Goodwin’s cycle for the fourth version. Thus it has been confirmed that Goodwin’s nonlinear accelerator model possesses a unique stable limit cycle. Since all these results are obtained when the stationary point is locally unstable, we can ask a basic question: Is cyclical behavior robust under locally stable circumstances?

The main result of this study is to provide a positive answer to this question. For this purpose, we augment Goodwin’s fourth version by introducing a nonlinear investment function of arctangent type and demonstrate the coexistence of multiple cycles. We will combine the Poincaré-Bendixson theorem with the Hopf bifurcation theorem and characterize the global dynamics when the stationary state is locally stable. The coexistence of multiple cycles is also shown for Kaldor’s business cycle model in Grasman and Wentzel (1994) and for a Metzlerian inventory cycle model in Matsumoto (1996) using an approach which we continue here.

We also examine the dynamics of the third version with an unstable stationary point. Only limited efforts has been devoted to this version in the past. Since it is a nonlinear differential equation, attempts at analytical solution seem fruitless. Hence, we perform numerical simulations to find what effects the production lag produces on the characteristics (i.e. length and amplitude) of cyclical oscillations of the output.

The following section “Goodwin business cycle model” overviews three versions (i.e. the second, third and fourth versions) of Goodwin’s business cycle model and reveals the characteristics of Goodwin’s cycle. The section “Coexistence of multiple cycles” presents the new results which shows that the linear approximated version exhibits corridor stability in which the solution is stable for small shocks but is unstable and generates multiple limit cycles for large shocks. Concluding remarks are made in the final section.

Goodwin business cycle model

This section is divided into three parts. Each of three versions of Goodwin’s model are reviewed in each subsection. In particular, we recapitulate the basic elements of the second version and numerically simulate the model to see what dynamics it can generate in “Basic model”. We then introduce a production lag into the second version to get the third version and perform, again, numerical simulations to find out how the lag affects the characteristics (i.e. the length of a period and the amplitude) of endogenous cycles in “Delayed model”. Finally, we derive the most popular version, the fourth version, by expanding the third version with respect to the lag and reveal its stability condition in “Approximated model”.

Basic model

The second model, which we call the basic model , is summarized as follow...

Table of contents

• Cover
• Half Title
• Routledge Frontiers of Political Economy
• Title Page
• Copyright Page
• Table of Contents
• List of Figures
• List of Tables
• Contributors
• Dedication
• Introduction: mathematical economics and the dynamics of capitalism
• Part I Nonlinear macrodynamics: theory
• Part II Nonlinear macrodynamics: empirical analysis
• Part III Multisectoral and microfounded approaches
• Index