Cobweb cycle

In the cobweb cycle, as drawn in Fig. X.2, the equilibrium is unstable. Any small perturbation will set up an ever-widening cycle. By the same token, if the farmers begin out of equilibrium, they will never come near it. If we draw the diagram differently, with the supply curve steeper than the demand curve, the opposite is true. After a while, the farmers converge to the equilibrium and return to it after any accidental perturbation. A preliminary conclusion might be that the realization of an equilibrium depends on details of the interaction. Some deviations from equilibrium correct themselves, while others get out of hand. 

There are many arguments for the rational-expectations hypothesis. In the simple cobweb cycle, and even with adaptive expectations, we must assume that each agent believes himself to be the only one to adjust rationally to the circumstances and that others act in a more or less mechanical way But this is an irrational belief, which we should not impute to people without evidence. It is surely more plausible to assume that people believe others to be as rational as themselves. Also, in a rapidly changing world people would be silly to pay much attention to the past. When oil prices quadrupled in 1973, the prices of oil before 1973 lost all relevance as guides to future prices. And if ordinary people understood much less of the economy than economists do, we would expect the latter to make much more money than they in fact do. The reason economists don t make a killing by outguessing the market is that the market has access to whatever information they have and can use it just as efficiently. ... 

This applies even when decisions are made simultaneously, if simultaneous choices have to be made on many successive occasions. The decision whether to vote in a given election might depend on the turnout in the last election. In this way, political cobweb cycles could be generated. 

Turning now to unintended consequences that arise because of interaction among several persons, let me begin with a famous example from economic theory, the "cobweb," also called the "hog cycle" because it was first put forward as an explanation of cyclical fluctuations in hog production. It has a much wider application, however. Fluctuations in the shipbuilding industry in recent decades had very much the same pattern, with a seller s market followed by overinvestment and glut. 

Another line of argument is to stipulate rational expectations. In essence, this means that persons living in a society use the same models and the same information as the social scientist studying them. If he can anticipate what will happen, so can they. To achieve equilibrium, farmers do not have to go through a long sequence of cycles and learning. Using the cobweb model, they... 

The cobweb shows that the fixed point r = 1 is stable and unique. No surprise, since we knew from Example 7.1.1 that this system has a stable limit cycle at r = 1. ... 

If the graph of f is concave down near x, the cobweb tends to produce a small, stable 2-cycle close to the fixed point. B ut like pitchfork bifurcations, flip bifurcations can also be subcritical, in which case the 2-cycle exists below the bifurcation and is unstable—see Exercise 10.3.11. 

A superstable cycle) Consider the logistic map with r= 3.7389149. Plot the cobweb diagram, starting from x = y (the maximum of the map). You should find a superstable cycle. What is its period ... 

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