The search model of Diamond (1982)[9] and its elaboration by Diamond and Fudenberg (1989) have been influential, as evidenced by frequent citations in the search literature.

Diamond begins his analysis by assuming infinitely many agents in the model. Consequently, his dynamical analysis is entirely deterministic, and cycles of the model are generated by a set of deterministic differential equations.

This is not to fault the model for lack of realism. Nevertheless, one would like to know how his model behaves in a finite-agent version: Does it produce substantial or negligible fluctuations about a locally stable state? Which of the multiple equilibria is chosen?

In this chapter, we recast their model in a framework of the modeling strategy advocated in this book.^[10] We have two objectives in recasting their model this way: One is to obtain information on fluctuations about the equilibria, and to provide a simpler explanation than Diamond and Fudenberg did for cyclical behavior. The other is to provide a new and more natural basis for equilib­rium selection for models with multiple equilibria than those available in the economic literature on equilibrium selection.

By our reformulation a different view of cycles emerges. We will show that the model has multiple equilibria, and that stochastic fluctuations cause the fraction of the employed to move from one basin of attraction to another with positive probabilities. These stochastic asymmetrical cycles are quite differ­ent from the deterministic cycles generated by a set of Diamond-Fudenberg nonlinear differential equations. Even when we take the number of agents to infinity, we show below that we gain new information about the limit­ing probability distribution over the steady states, and that this provides a natural basis for considering equilibrium selection in models with multiple equilibria.

The dynamical behavior of the model is now described by the master equa­tion.

When we let the number of agents go to infinity, the equation for the mean re­produces the equation for the fraction derived by Diamond. The Fokker-Planck equation is new. The critical points of the ordinary differential equation and the endogenously determined reservation-cost expression jointly yield information on the equilibria. These are the same as in Diamond.

Additionally, we derive information on the asymmetrical cyclical behavior, which is substantially different from his. Fluctuations about aggregate dynamics occur in our analysis because microshocks intrinsic to our models do not vanish when the number of agents in the model is finite.^[11] With positive probabilities, net effects of arrivals of production and trading opportunities do not vanish, but accumulate to change the fraction of employed from one basin of attraction to the other.

In Section 9.1, we describe the model as a jump Markov process with the transition rates specified in Section 9.2. The transition rates are endogenously determined, because they depend on the reservation cost for accepting produc­tion opportunities, which is determined by comparing value functions for the alternative choices of becoming employed by accepting the production oppor­tunity and of remaining unemployed by rejecting the opportunity as being too costly. Recall our introductory discussion on value functions in Section 6.2. After we discuss the dynamics for the mean of the fraction, and a Fokker- Planck equation for the fluctuations about the mean, in Sections 9.3 and 9.4, the value functions are evaluated in Section 9.5. Then, in Section 9.6, we discuss the expected first-passage times between two locally stable equilibria when the model dynamics have multiple locally stable equilibria. We also discuss our equilibrium selection criterion, which is based on the relative sizes of basins of attractions and the probabilities that the model state is in the basins. The larger the basin of attraction for a stationary state, or the smaller the fluctuation around the stationary state, the more likely it is that such a stationary state will be selected as the equilibrium.

9.1