Chaos in the cobweb model with a new learning dynamic

Instability in the cobweb model under the BNN dynamic

“…For a proof of the above and a detailed analysis of all steady states, see Waters (2009). 5 Goeree and Hommes (2000) extend the work of Brock and Hommes (1997) to the case of non-linear supply and demand.…”

“…The origin is stable for b ≤ 1 but is not stable for b > 1 (Waters, 2009). The primary goal is to prove that the 2-cycle…”

“…Furthermore, the 2-cycle (±p * , q * ) under F from (ii) of Proposition 3 becomes a steady state (p * , q * ) under F, and the dynamics of the second iterates of both functions are identical. The phase diagram (Waters, 2009) shows counterclockwise rotation around (p * , q * ) under F.…”

“…For example, Brock and Hommes (1997) study the cobweb model where the multinomial logit dynamic describes switching between naive and rational forecasting strategies, while Branch and McGough (2008) use a version of the replicator dynamic in a similar framework. Waters (2009) shows that the cobweb model with the dynamic of Brown et al (1950) produces related dynamics, while avoiding problematic behavior when one strategy is eliminated from the population, as arises under multinomial logit as well as imitative dynamics such as the replicator. For all of these dynamics, an unstable 2-cycle associated with strange and periodic attractors arises when the ratio of the slopes of supply and demand is greater than one.…”

“…A consequence of both points is that there are intuitively appealing steady states where one strategy dominates under the BNN dynamic. Waters (2009) provides a more detailed discussion.…”