Suppose a search economy with the possibility of double coincidences; that is, assume that when an agent produces a good, that she cannot consume it herself. In a random match where two agents meet and each has the good that they produced, the first agent has what the second consumes with probability x, the second has what the first consumes with probability x, and each has what the other consumes with probability x².
(a)  In this economy, show that there are three equilibria: a barter equilibrium where money is not accepted, an equilibrium where an agent with a good is indifferent between accepting and not accepting money, and an equilib-rium where agents with goods always accept money.
(b)  Show that x needs to be sufficiently small before having money in this economy actually increases welfare over having barter, and explain this result.