A Chaotic Business Cycle

Figure 1: A Chaotic Attractor in Kaldor's Model of the Business Cycle

I might as well post another interim result from my analyses of formalizations of Kaldor's business cycle model. (Today, Noah Smith also posts about chaotic dynamics.) Figure 1 is based on Figure 3 in a 2006 paper from Orlando Gomes. Table 1 shows the parameter values for the model used to generate this figure. For these parameters, Kaldor's model has one attractor, and that attractor is chaotic. The figure shows 1,000,000 (presumably non-transient) points on a single orbit. Although maybe not apparent from the figure, the orbit rotates around the origin in a clockwise direction.

Table 1: Values of Model Parameters

Parameter	Value
Speed of adjustment (α)	12
Depreciation rate (δ)	0.2
Propensity to Save (σ)	0.13
Expected level of output (μ)	200
Cost to adjust capital stock (γ)	0.6

In this model, for these parameters, every business cycle looks somewhat different from the previous one. Yet the model is deterministic. Variations among business cycles, in this model, are not coming from a source of random shocks. Furthermore, the figure shows a fractal structure across business cycles that may not be apparent to agents in the model living through five or ten cycles.

In Figure 1, I've also shown the model's fixed points and indicated their stability. The stability of fixed points in a dynamical system can be analyzed by looking at the eigenvalues of a linear approximation to the system at each fixed point (Figure 2). Methods exist to determine the stability of a fixed point without actually calculating eigenvalues. But the calculation of eigenvalues and eigenvectors is needed to numerically determine the location of the stable and unstable sets at interesting fixed points (albeit I do not show such sets in Figure 1).

Figure 2: Eigenvalues and Stability

References

• Andronov, A. A., E. A. Leontovich, I. I. Gordon, and A. G. Maier (1971). Theory of Bifurcations of Dynamic Systems On a Plane (Translated from Russian), National Aeronautics and Space Administration.
• Gomes, Orlando (2006). "Routes to Chaos in Macroeconomic Theory", Journal of Economic Studies, V. 33, N. 6: 437-468.
• Kuznetsov, Y. A. (1998). Elements of Applied Bifurcation Theory, Second edition. Springer-Verlag.