Cobwebs, chaos and bifurcations

The first equation states that the demand for goods qt a at time t is a function of the current price PI. Equation (2) states that the supply for goods q: at time t is a function of the expected price /3t for period t. The market clearing eq. (3) states that supply equals demand at each time t. Equation (4) describes the expectation mechanism. Producers have adaptive expectations about price: If the actual price Ptt is larger (smaller) than the previous expected price/3~_ 1 then the new expected price /3t is revised upwards (downwards). Note that the new expected price is just a weighted average of the old expected price and the old price. The parameter w is called the expectations weight factor. It will be useful to consider the case w = 1 first. We then have that ,fit = Pl 1, i.e. the expected price is just the previous price. This type of expectations is called naive expectations and was used in the original version of the cobweb model (which we will call the traditional cobweb model), see e.g. Ezekiel [3]. It is well known that, if the demand curve is decreasing and the supply curve is increasing, then only three types of dynamics are possible: convergence to a unique stable equilibrium price, convergence to stable period two oscillations or unbounded oscillations. Recently, it has been shown by Artstein [1] and Jensen and Urban [5] that in the traditional cobweb model chaotic price behaviour can occur if at least one of the supply and demand curves is non-monotonic, In order to understand the occurrence of chaos from a geometric point of view, recall that the price-quanti ty time paths in the traditional cobweb model can graphically be represented as a backand-forth movement, along horizontal and vertical line segments, between the supply