The Phillips curve in a matching model

Following ideas in Hume, monetary shocks are embedded in the Lagos-Wright model in a new way: there are only nominal shocks that are accomplished by way of individual transfers and there is sufficient noise in individual transfers so that realizations of those transfers do not permit the agents to deduce much about the aggregate realization. The last condition is achieved by assuming that the distribution of aggregate shocks is almost degenerate. For such rare shocks, aggregate output increases with the growth rate of the stock of money—our definition of the Phillips curve. This almost-degeneracy assumption is far from being necessary; under some mild conditions, the Phillips curve result holds for
a large class of distributions.