## Monday, February 12, 2007

### Bifurcations and "Perverse" Switch Points

1.0 Introduction
I seem to be slow in writing up these results into a working paper. So I thought I would just present them here.

The model in this post is an example of reswitching. The model is closed in a neoclassical framework, that is with an overlapping generations, representative agent approach. Nevertheless, two closures are presented. The details of the utility functions differ between the closures.

The variations of stationary states with utility function parameters are explored. This is an analysis of structural stability, in some sense. Multiple equilibria arise for some ranges of parameter values. But whether multiple equilibria are associated with the "normal" or the "perverse" switch varies between the two examples.

I think these results may be a challenge to Rosser's (1983) identification of reswitching with a cusp catastrophe. (I don't fully understand catastrophe theory. My favorite bifurcations are Hopf bifurcations and period-doubling, although even the chaotic mathematics of, say, the Lorenz equations is a stretch for me.) I think these results could be used to more strongly contrast with Rosser's if I considered two parameters in the first closure. The second parameter could be, for example, related to a coefficient of production.

This sensitivity to modeling details, of which switch point is associated with multiple equilibria, may also have implications for some sort of dynamic stability analysis. These results suggest that perhaps "perverse" switches are not necessarily associated with dynamic instability. A fuller analysis might lead me to come down disappointingly on Mandler's (2005) side in his debate with Garegnani and Schefold. (I'm uncomfortable with the emphasis in that debate being on tâtonnement stability, to the exclusion of the analysis of paths in models of temporary equilibria.)

2.0 First Closure
In this model, a single agent is born at the start of each each. The agents in each generation have identical utility functions, and they live for two years. In this closure, the agent is a worker for the first year of his life and retired in the second (Figure 1). He is paid wages at the end of his first year for the year of labor services he sells during that year. Out of those wages, he purchases some corn to consume immediately. The remainder he saves at the prevailing interest rate in the second year, for consumption of corn at the end of the last year of his life. The intertemporal consumption decision is modeled as a constrained maximization of a Cobb-Douglas utility function.
 Figure 1: Overlapping Generations
The utility function in this closure contains a single parameter. A higher value of this parameter is associated with agents less likely to defer consumption. Outdated neoclassical intuition would lead one to expect a higher value of this parameter to be associated with a smaller supply of "capital" and a corresponding higher stationary state interest rate. But the relation in this model between the stationary state interest rate and this parameter (Figure 2) is not non-decreasing.
 Figure 2: Equilibrium Interest Rates in First Closure

3.0 Second Closure
This closure differs from the first in that the agent chooses how much labor to supply each of the two years of his life. That is, the agent's utility-maximization problem embodies a trade-off between leisure and goods in each year, as well as intertemporal trade-offs. The utility function is of a different form than in the first closure. Two parameters of the utility function control the agent's decisions.

Accordingly stationary state values of endogenous values (for example, the interest rate) form a two-dimensional manifold in a three-dimensional space. Figure 3 shows a slice through such a manifold in which one parameter of the utility function is kept constant. A higher value of the varying parameter is associated with a lesser willingness to defer consumption, that is, a smaller supply of "capital", in some sense. One trained with outdated neoclassical intuition would expect the relation graphed in Figure 3 to be non-decreasing. That is, one so mistrained would expect a smaller supply of capital to be associated with a larger stationary-state interest rate. But here too the relation shown is sometimes decreasing.
 Figure 3: Equilibrium Interest Rates in Second Closure
Figure 4 shows another slice through a manifold. In this case, the parameter controlling the relative desirability of consumption goods and leisure in each year varies, while the other parameter of the utility function is kept constant. A higher value of the varying utility function parameter is associated with a smaller willingness to supply labor. Given outdated neoclassical intuition, one would expect the relation graph to be non-decreasing. Here too the relation shown demonstrates such intuition to be mistaken.
 Figure 4: Equilibrium Wages in Second Closure

References
• Garegnani, P. (2005a). "Capital and Intertemporal Equilibria: A Reply to Mandler", Metroeconomica, V. 56, Iss. 4 (Nov): 411-437.
• Garegnani, P. (2005b). "Further on Capital and Intertemporal Equilibria: A Rejoinder to Mandler", Metroeconomica, V. 56, Iss. 4 (Nov): 495-502.
• Mandler, Michael (2005). "Well-Behaved Production Economies", Metroeconomica, V. 56, Iss. 4 (Nov): 477-494.
• Parrinello, Sergio (2005). "Intertemporal Competitive Equilibrium, Capital and the Stability of Tatonnement Pricing Revisited", Metroeconomica, V. 56, Iss. 4 (Nov): 514-531.
• Rosser, J. B., Jr. (1983). "Reswitching as a Cusp Catastrophe", Journal of Economic Theory, V. 31, N. 1: 182-193.
• Schefold, B. (2005a). "Reswitching as a Cause of Instability of Intertemporal Equilibrium", Metroeconomica, V. 56, Iss. 4 (Nov): 438-476.
• Schefold, B. (2005b). "Zero Wages - No Problem? A Reply to Mandler", Metroeconomica, V. 56, Iss. 4 (Nov): 503-513.