Wednesday, September 26, 2007

Chaotic Cobwebs (Part 1 1/2)

5.0 Return to a Special Case

This post continues the examination of a cobweb cycle with a non-linear demand curve. In this part, I talk again about the special case examined in Section 3 of that previous post. In that case, the parameters b and e of the model are zero. Furthermore, I continue to assume c is 3/5 and d is 21/20. Figure 6 shows attracting limiting behavior for a whole range of the parameter a. The abscissa in this graph is a, with a set to unity at the right edge. The ordinate is the limiting values of the normalized quantity, Q(t). As I point out towards the right, a two-period cycle shows up on the graph as a plot of two values of the ordinate for the value of a for which that cycle is generated. One can see the period-doubling scenario leading to chaos as one moves to the left on the graph. By the way, this figure is a fractal, repeating on an infinite number of scales. It has both qualitative and quantitative universal features for a certain family of one-dimensional maps (for example, characterized by the Feigenbaum constant).
Figure 6: Structural Dynamics of a Special Case
6.0 An Economically Relevant Special Case

The case above has both demand and supply functions through the origin in the quantity-price space. I want to consider a case in which the demand curves intersects the price axis at a strictly positive price and slopes downward in the first quadrant. Accordingly, consider the case where a is unity, b is 781/960, c is 3/5, d is 1/2, and e is zero. (I put aside questions of whether adaptive expectations make sense here or whether a partial equilibrium framework with monotonic supply and demand curves is justified - see implications of the Sonnenschein-Debreu-Mantel theorem.)

Figure 7 shows how the price and quantity evolve for selected initial values. The red line suggests the point equilibriating of supply and demand is unstable. It will never be observed in this model. Instead, the red line evolves to a two-period limit cycle. The blue line shows that points outside that cycle will evolve inward to that cycle. As a matter of fact, I chose parameter values such that Figures 2 and 3 in Section 3 of the previous post show the behavior of the normalized quantity for this case.
Figure 7: Temporal Dynamics with All Positive Prices and Quantities
I don't see that one can find positive parameters values such that period doubling can lead to chaos in this model with price and quantity remaining positive throughout. Although I am not sure of this, I am willing to accept that the maximum of the quadratic function in Equation 4 (see previous post) must be strictly positive for chaos to arise in the first quadrant. I don't understand what Richard Goodwin graphs in Figures 2.4 and 2.5 of his book. Perhaps he is considering a cubic demand curve or some other higher polynomials for supply and demand.

Update (28 Sep.): A Google search on "cobweb", "chaos", and "economics" shows lots of literature, mostly behind paywalls. I notice particularly work by Barkley Rosser, Jr. and in the Journal of Economic Behavior and Organization, which he edits. So I have reading I could do.


YouNotSneaky! said...

"Accordingly, consider the case where a is unity, b is 781/960, c is 3/5, d is 1/2, and e is zero... Instead, the red line evolves to a two-period limit cycle."

What initial value are you using and how many periods does it take you to get to the two period cycle?
q* = 11/24, right?

Robert Vienneau said...

Yes, the equilibrium quantity is 11/24.

The quantities in the 2-period cycle are, apparently, 1/24 and 19/24.

The red line starts at a quantity of 0.13377039. This more or less converges to the limit cycle in 93 iterations. Not shown are 49 iterations from a quantity of 1/2 to the quantity of 0.13377039.

YouNotSneaky! said...

I see. We have our terminology regarding periods off by a multiple of 2. Since odd-period cycles can't happen, what I meant was that you can't get something like figure 4 in the previous post.

And you're right - q_t does go to 1/24 and 19/24 rather than 0.