The master equation
Once states have been assigned to a collection of economic agents, their behavior over time is specified by the dynamics for the joint probabilities of the states. Dynamics are set up by taking account of the probability fluxes into and out of a specified state over a small interval of time.
We use the backward Chapman-Kolmogorov equation to do this accounting. We adopt the shorter name that is used in the physics and ecology literature and call the dynamic equation the master equation. This is an appropriate name because everything of importance we need to know about the dynamic behavior can be deduced from this equation. In particular we derive the dynamics for aggregate variables, which we call the aggregate dynamics (roughly corresponding to macroeconomic dynamics) and the dynamics for the fluctuations of state variables about the mean, or aggregate, values. The latter is called the Fokker-Planck equation.It should be emphasized that the master equation describes the time evolution of the probability distribution of states, not the time evolution of the states themselves. This distinction may seem unimportant to the reader, but it is a crucial one and helps to avoid some technical difficulties. For example, in a model with two types of agents of a fixed total number, the fraction of one type of agents is often used as the state. The master equation describes how the probability for the fraction of one type evolves with time, not the time evolution of the fraction itself. The latter may exhibit some abnormal behavior at the extreme values of zero and one, but the probability distribution cannot.
When the master equations admit stationary solutions, as some models in this book do, we can deduce much from those distributions. Some nonstationary distributions may be obtained by the method of probability generating functions, or information on moments derived from cumulant generating functions. These are discussed in Chapters 3 and 4.
In general, we use Taylor series expansion in inverse powers of some measure of the model size, such as the number of agents in the model. We can show that in the limit of an infinite number of agents we recover traditional macroeconomic results. This is illustrated by reworking the well-known Diamond (1982) model in our framework in Chapter 9.
1.5