handoutmoney08 - Spatial Models of Money The...

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Spatial Models of Money The cash-in-advance model took as an assumption that people must use money to buy goods. Why? 1 Lucas-Cass-Yaari There is an underlying environment which delivers such a constraint. Townsend (1980, Models of Monetary Economies, Section 4) calls it the Lucas-Cass-Yaari model. See Figure 1a. A household is a shopper-worker pair. HHs have preferences over good i +1 but produce good i. At the beginning of every period, workers travel to a market ( i 1 ,i ) while shoppers travel to market ( i, i +1) . In market ( i, i , shopper i buys goods from worker i . While shopper i likes i 0 s good, worker i doesn’t like shopper i 0 s good (his family likes good i . This is known as an absence of a double coincidence. It means that in the absence of money, there can be no barter (and since it is costly to produce, there can be no gifts). But with money earned by worker i in market ( i 1 ) in period t, the shopper can give worker i money in exchange for goods in t . In a symmetric equilibrium, i does not matter. Thus we have a spatial interpretation of a c-i-a constraint. 2 Townsend Turnpike Section 2 of Townsend (1980, Models of Monetary Economies). Sargent also puts Bewley model of money in this category. 2.1 Environment Pop. Unit measures of two types i { O,E } of agents at each of a count- able in f nity of locations. Prefs: P t =0 β t u ( c i t ) , with u 0 (0) = ,u concave and increasing, bounded. Nonstorable Endowments y O t = ½ 0 t even 1 t odd ,y E t = ½ 1 t even 0 t odd Thus we can interpret type with the date the agent gets paid. 1
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There is no commitment technology to force agents to turn over their good. Spatial Arrangement. Type E 0 s travel east and type O 0 s travel west and there is no communication across locations. Notice that type O and E in any given location, will never meet the other any time in the future or even a trading partner of the other. This model has the property that there is a temporal absence of a double coincidence of wants. At each point in time one person has what the other wants but the other doesn’t have (at that time) something the other wants (hew i l lhaveitnextper iod) . 2.2 First Best Planner’s problem (what is resource and individually rational if a planner could impose allocations across all locations): max { c A t ,c B t } t =0 X t =0 β t u ( c O t ) s.t.c O t + c E t =1 , t X t =0 β t u ( c E t ) U Letting ψ t be the multiplier on the resource constraint and κ the multiplier on type E 0 s utility, the foc are: β t u 0 ( c O t )= ψ t = β t κu 0 ( c E t ) , t But this implies u 0 ( c O t ) u 0 ( c E t ) = κ, t (1) so that between any two periods t and t +1 , say, a P.O. allocation must satisfy u 0 ( c O t ) u 0 ( c O t +1 ) = u 0 ( c E t ) u 0 ( c E t +1 ) so that hhs should have equal MRS.
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handoutmoney08 - Spatial Models of Money The...

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