|Figure 1: An Empirically-Constructed Bifurcation Diagram|
I thought I would try to summarize what I have learned about the qualitative behavior of Kaldor's business cycle model. (Although I had some preparation, I learned much of the mathematics in this post during this analysis.) Figure 1, above, shows (mostly) bifurcations found empirically, while Figure 2 below shows bifurcations that can be found analytically. The normalized model is specified by four parameters, the speed of adjustment of national income to the difference between aggregate demand and supply, the depreciation rate of capital, the (average and marginal) propensity to save out of income, and the cost of adjusting the capital stock to the desired level. Figures 1 and 2 are drawn for a constant depreciation rate (δ = 1/5) and a constant cost of adjustment (γ = 3/5). The abscissa in the figures represents different levels of the speed of adjustment (α), while the ordinate represents different levels of the propensity to save (σ).
|Figure 2: An Analytically-Constructed Bifurcation Diagram|
The Kaldor model has two (endogenously determined) state variables, the normalized level of the capital stock (kt) and the normalized flow of national income (yt). A bifurcation analysis determines regions in the parameter space in which the flows for the state variables qualitatively differ. A bifurcation is a manifold in the parameter space in which some such qualitative difference arises. The figures above show selected bifurcations, where each bifurcation is represented by a line or curve in the figure. (I do not claim that the bifurcations shown are the complete set of bifurcations that arises, even in the part of the parameter space shown. For example, I ignore homoclinic bifurcations of points along a limit cycle with the (in)stability of a saddle point.)
The horizontal line near the top of Figure 1 is shown near the bottom of Figure 2. This line represents a pitchfork bifurcation. The model has one fixed point (stationary state), at the origin of the state space, above the line. The model has three fixed points below the line, one at the origin, and the other two symmetrically located around the origin in the first and third quadrants. As I understand it, the origin always has saddle-point (in)stability below this line.
The blue curve in Figure 2 arcing upward to the right from the horizontal line is a Neimark-Sacker bifurcation. (The Neimark-Sacker bifurcation is the discrete time analog to the Hopf bifurcation.) To the left of this curve, the fixed point at the origin is stable. The origin loses its stability at the Niemark-Sacker bifurcation, and it throws out a stable limit cycle to the right. A limit cycle corresponds to a business cycle in the modeled economy.
I think that the bifurcations in the region with three fixed points, below the horizontal line representing the pitchfork bifurcation, are more complicated and more difficult to understand. In the lower left of Figure 1, the two symmetrical fixed points not at the origin are stable. One can find each fixed point's basin of attraction in the state space, that is, those values of the state variables such that a trajectory in the state space started with those values converges to the given fixed point. As I understand it the two basins of attraction cover the entire state space in this region of the parameter space.
Figure 1 shows three (hard to distinguish) curves coming down from the horizontal line and curving to the right. All of these curves intersect the horizontal line at the same point. And that point is also the intersection with that horizontal line of the curve above the horizontal line representing the Neimark-Sacker bifurcation I have previously described. (I have no idea how you would formally prove this.)
The lowest of these curves sloping downward to the right represents a bifurcation in which a limit cycle with saddle point stability appears. Along this curve in the parameter space, a region exists in the state space in which trajectories approach the limit cycle, only to ultimately diverge to one of the fixed points away from the origin.
The next higher curve sloping downward to the right in Figure 1 is almost impossible to tell apart from the curve that I have just described. This curve represents a homoclinic bifurcation. The basins of attraction of the fixed points away from the origin no longer combine to cover the state space. At the homoclinic bifurcation, the stable and unstable sets of the origin merge. Between the lower curve and this curve, the limit cycle with saddle-point stability bifurcates to form (at least) two limit cycles, one stable and one unstable. The unstable limit cycle is the boundary of the union of the basins of attraction of the two stable fixed points. Just above this curve, the border of the basins of attraction bifurcates, to form two disjoint unstable limit cycles. The stable limit cycle remains in the state space, enclosing the fixed point at the origin and these unstable limit cycles.
The highest curve sloping downward to the right in Figure 1 is another Neimark-Sacker bifurcation. The fixed points away from the origin lose their stability at this bifurcation. Their basins of attraction disappear. The unstable limit cycles forming the border of each basin of attraction are absorbed into the corresponding limit point. A stable limit cycle remains. So on the right of Figure 1, both above and below the horizontal line, a stable limit cycle exists, even though the number of fixed points varies with the propensity to save.
I was surprised at the diverse and complex behavior that economists have found in the Kaldor model, a model of the business cycle that is nearly three-quarters of a century old.