ps12macro2sp08ans

# ps12macro2sp08ans - Econ 387L MacroII Spring 2008...

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Econ 387L: MacroII Spring 2008, University of Texas at Austin Instructor : Dean Corbae Solutions to Problem Set #13 , April 29, 2008 1. Consider the following search model of money. Time is discrete and there is a continuum of agents with population normalized to 1. Any particular agent specializes in the production of one service (a nonstorable good) but likes other services in an interval of size x (0 , 1) . She derives utility u ( q )= q 1 / 2 from consuming q R + units of the service provided it falls in her desired interval. An agent discounts the future at rate 1 (1+ r ) . There is a constant disutility q to producing q units of a service. Production and consumption occur at the end of the period (and hence should be appropriately discounted). At the beginning of time, a fraction of agents M (0 , 1) are randomly given one unit of currency. Currency is indivisible and can be stored only one unit at a time. Agents are exogenously matched in the following way. Agents with money (we will term them buyers) are randomly matched in pairs with agents without money. Thus, the probability that a buyer is matched with a seller whose good she desires is X (1 M ) .A l so , the probability that a seller is matched with a buyer who wants her good is XM . Every agent’s trading history is private information. Finally, assume that buyers submit take-it-or-leave-it o f ers (which amount to a trade of 1 unit of money for Q units of the seller’s service). (a) Taking the quantity of services bargained for Q as given, write down the value functions for a buyer V b ( Q ) and a seller V S ( Q ) respectively . Answer: V b ( Q 1 1+ r n X (1 M ) ³ Q 1 / 2 + V s ( Q ) ´ +(1 X (1 M )) V b ( Q ) o (1) rV b ( Q X (1 M ) ³ Q 1 / 2 + V s V b ´ (2) V s ( Q 1 r { ( Q + V b ( Q )) + (1 ) V s ( Q ) } (3) s ( Q ( Q + V b V s ) (4) (b) Taking the value functions V b and V S

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ps12macro2sp08ans - Econ 387L MacroII Spring 2008...

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