William A. Brock

The concept of adaptively rational equilibrium (A.R.E.) is introduced. Agents adapt their beliefs over time by choosing from a finite set of different predictor or expectations functions. Each predictor is a function of past observations and has a performance or fitness measure which is publicly available. Agents make a rational choice concerning the predictors based upon their past performance. This results in a dynamics across predictor choice which is coupled to the equilibrium dynamics of the endogenous variables. As a simple, but typical, example we consider a cobweb type demand-supply model where agents can choose between rational and naive expectations. In an unstable market with (small) positive information costs for rational expectations, a high intensity of choice to switch predictors leads to highly irregular equilibrium prices converging to a strange attractor. The irregularity of the equilibrium time paths is explained by the existence of a so-called homoclinic orbit and its associated complicated dynamical phenomena. Thus local instability and global complicated dynamics may be a feature of a fully rational notion of equilibrium.

This paper provides an analysis of aggregate behavioural outcomes when individual utility exhibits social interaction effects. We study generalized logistic models of individual choice which incorporate terms reflecting the desire of individuals to conform to the behaviour of others in an environment of noncooperative decisionmaking. Laws of large numbers are generated in such environments. Multiplicity of equilibria in these models, which are equivalent to the existence of multiple self-consistent means for average choice behaviour, will exist when the social interactions exceed a particular threshold. Local stability of these multiple equilibria is also studied. The properties of the noncooperative economy are contrasted with the properties of an economy in which a social planner determines the set of individual choices. Finally, a likelihood function based on the theoretical model is given and conditions for the econometric identifiability of the model are established.

Many dynamical systems, including lakes, organisms, ocean circulation patterns, or financial markets, are now thought to have tipping points where critical transitions to a contrasting state can happen. Because critical transitions can occur unexpectedly and are difficult to manage, there is a need for methods that can be used to identify when a critical transition is approaching. Recent theory shows that we can identify the proximity of a system to a critical transition using a variety of so-called 'early warning signals', and successful empirical examples suggest a potential for practical applicability. However, while the range of proposed methods for predicting critical transitions is rapidly expanding, opinions on their practical use differ widely, and there is no comparative study that tests the limitations of the different methods to identify approaching critical transitions using time-series data. Here, we summarize a range of currently available early warning methods and apply them to two simulated time series that are typical of systems undergoing a critical transition. In addition to a methodological guide, our work offers a practical toolbox that may be used in a wide range of fields to help detect early warning signals of critical transitions in time series data.