| Abstract: | This paper proposes a general modelling framework for macroeconomic dynamic behaviour of a large collection of interacting agents, and illustrates the procedure for a simple case of agents having the same binary decision set. Evolution over time of the fractions of agents using the same technology is modelled as a birth-and-death stochastic process with endogenized transition rates, and its master equation (Chapman-Kolmogorov backward equation) is expanded into power series in the total number of agents in the model. The first term of the expansion is the aggregate (macroeconomic) dynamic model. Stochastic dynamics of the proportions of agents using the same technology can exhibit multiple equilibria; this paper discusses an example with one locally unstable and two locally stable equilibria, with some simulation computer runs. There are two results of broad implication. One is the demonstration that stochastic dynamics allocates positive probabilities to all locally stable equilibria. Stochastic dynamics thus stands in sharp contrast to deterministic dynamics. In the latter, history (initial conditions) and expectations (often exogenously introduced discontinuous changes in trajectories) uniquely select the basin of attraction in which trajectories are eventually located. The second is the illustration of the usefulness of jump Markov processes in general, and birth-and-death processes in particular in economic modelling. Among several results specific to these processes, we show that the mean first passage time from one locally stable equilibrium to another is proportional to the exponential function of the height of the potential barrier separating the two basins of attraction of these two locally stable equilibria. |
