Tomáš Holub Monetary Economics October 2012

1. Acceptability, Means of Payment and Media of ExchangeNobuhiro KiyotakiRandall WrightRandom Matching Model Tomáš Holub Monetary Economics October 2012

2. Random Matching Model • Kiyotaki, Wright (1991) • Simple theoretical model that determines acceptability of money endogenously. • Economy - assumptions: large number of  lived agents, large number of perfectly storable consumption goods. Each agent consumes a fraction x of goods, each good is consumed by fraction x of agents.

3. All goods produces by equal numbers of agents, but agents don’t produce goods that they themselves consume. Goods, money - indivisible (units of size 1). Good consumed  utility u, immediately a new good produced  cost c (disutility) (note - agents cannot produce without first consuming) Initial endowment of some agents (randomly chosen) with consumption goods and the rest with money. M … fraction of agents endowed with money (only 1 unit of money per agent, units indivisible) 1-M … fraction of agents endowed with commodity

4. Trade : Agents meet bilaterally (at random) once each period and trade iff it is mutually advantageous. ε … transaction cost (in terms of disutility) (carried by the receiver of the consumption good from the trade) no transaction cost for accepting money Nash equilibria in trading strategies of each agent (in order to maximize the expected discounted utility of consumption net of production and transaction costs, taking the strategies of other agents as given).

5. Detemination of Acceptability of money (I): •  ... probability of accepting money by a repre agent •  ... avg probability of accepting money by the others • V(c) ... payoff when the agent has a commodity at the end of each period • V(m) ... payoff when the agent has money at the end of each period •  ... discount factor between periods

6. Detemination of Acceptability of money (II): Determination of agent’s payoff : • agent with a commodity: V(c) = {(1-M)x2U + MxV(m) + (1- Mx)V(c)} • agent with money: V(m) = {(1-M)x(U+V(c)) + [1- (1-M)x ] V(m)} where U = u -  - c  ... transaction cost c ... production cost • condition of acceptability of money: V(m)  V(c)

7. Detemination of Acceptability of money (III): From V(c) and V(m) : • if  < x, then V(m) < V(c) (i.e. when m is less acceptable than com, the payoff from trading with money is less than from barter) result:  = 0 (never to trade com for money) • if  > x, then V(m) > V(c) (i.e. when m is more acceptable than com, the payoff from trading with money is more than from barter) result:  = 1 (always to trade com for money)

8. Detemination of Acceptability of money (IV): • if  = x, then V(m) = V(c) (i.e. when m is just as acceptable as com, the payoffs from trading with m and from bartering are equal) result: any  between 0 and 1  3 equilibria in the model:  = 0, when money not acceptable  = 1, money generally acceptable  = x, money partially acceptable

9. Graphical Illustration

10. Conclusions • Acceptability is key for circulation of money • Determined endogenously in the model • Money improves welfare (frequency of trade) • May circulate even with some opportunity costs (no rate-or-return dominance puzzle) • But may collapse with too high opportunity costs (e.g. hyperinflation) • Multiple equilibria - monetary system potentially fragile, based on trust

11. Random Matching Model vs. OLG