Equilibrium selection
As we increase the number of agents in the economy to infinity, our model converges to that of Diamond. This can be seen by the fact that the variance of the employed fraction of agents converges to zero as N is taken to infinity, because the density function for e is Gaussianwithmean φ and variance σ2(φ)/N.
This suggests that the stationary (invariant) distribution over the fraction of employed agents in the economy becomes a pair of spikes with probability masses of π1 and π2 assigned for employed fractions e = φ1 and e = φ2, respectivelyThese probability masses for each locally stable critical point provide the criteria for equilibrium selection for the model of multiple equilibria with an infinite number of agents. One can easily check that our special case given in Section 9.5 yields exactly the same stationary fractions of employed agents φ1 and φ2 in Diamond's model if we use the same matching function b and cost distribution function G.
We next calculate π1 when N is taken to infinity. As suggested earlier, we have
and
The larier the distance between the critical point and the boundary of the basins of attraction, or the smaller the variance of fluctuation around the critical point, the more likely it is that this critical point will be selected as an equilibrium in a model with an infinite number of agents.
9.8 Possible extensions of the model
There are several ways of extending this model. The reader is invited to try some. As an example, we sketch one such extension. This is related to Kiyotaki and Wright (1993), who introduced money traders who each hold a unit of money, which they can exchange with positive probability for a unit of commodity produced by the unemployed.
They also work with an infinite number of agents. In their extension, the amount of money, m units, is exogenously fixed, and stays the same throughout the time, even though the identities of agents who hold money change over time. For this reason, the dynamics is basically that of the Diamond model, because m appears in the dynamics as a fixed parameter, not as an additional state variable.We suggest to the reader that their model be modified so that money traders may hold multiple units of money This may be accomplished by making the process of changing money for commodities more efficient for agents staying in the cluster of money traders than joining the unemployed each time they exchange a unit of money for commodity. Let ai be the number of money traders who have exactly i units of money Then a = (a1,..., am) is a partition vector and is the state vector for the cluster of money traders. We have 
as the total amount of money in the cluster of money traders, and 
is the number of money traders.
See Chapter 10 (Section 10.5 in particular) for the dynamics of such clusters. Specify appropriate transition rates in terms of the partition vector, and establish the corresponding stationary distributions of money stocks among money traders. See Aoki (2000a) for further information and suggestions along this line of model modification.