Existence of a monetary steady state in a matching model: divisible money

Existence of a monetary steady state is established in a random matching model with divisible goods, divisible money, an arbitrary bound on individual money holdings, and take-it-or-leave-it offers by consumers. The monetary steady state shown to exist has nice properties: the value function, defined on money holdings, is strictly increasing and strictly concave, and the distribution over money holdings has full support. The approach is to show that the “limit” of the nice steady states for indivisible money, existence of which was established in an earlier paper, as the unit of money goes to zero is a monetary steady state for divisible money. For indivisible money, the marginal utility of consumption at zero was assumed to be large; for divisible money it is assumed to be large and finite.