The instability of some non-full-support steady states in a random matching model of money

Zhu (2003) shows existence of full-support monetary steady states with strictly concave value functions in a random matching model with individual money holdings in {0, 1, 2, . . .. , B} for a general B. He also shows that corresponding to each such steady state is an l-replica steady state for each l∈N: money is traded in bundles of l units, the support is {0, l, 2. l, . . .. , lB}, and the value function is a step-function with jumps at points of the support. We show that such l-replicas are unstable if the underlying full-support steady state is a pure strategy steady state and if the support of the initial distribution is not {0, l, 2. l, . . .. , lB}.