|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.
|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 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|
- 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.