final387s03ans - Econ 387L Macro II Spring 2003 University...

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Econ 387L: Macro II Spring 2003, University of Texas Instructor: Dean Corbae Final Exam Answers Consider the following cash-in-advance economy where fi rms are monopolistically competitive and must set prices for their goods one period in advance. The cash-in-advance constraints mean that expected in fl ation cause agents to inef fi ciently economize on money holdings. The monopolistic competition means that equilibrium output falls short of the ef fi cient level. Sticky prices mean that unanticipated money changes have real effects. Thus the government faces a tradeoff between the costs of expected in fl ation and the bene fi ts of unexpected in fl ation. Speci fi cally, the economy consists of a representative family (composed of a worker/shopper household and a continuum of fi rms indexed by i [0 , 1] ) and a government. Each fi rm produces a distinct, nonstorable good i with the technology y t ( i ) = n t ( i ) where n t ( i ) is the labor input at fi rm i. The representative household (HH) consumes a basket of goods (call it c t ) and supplies labor to each of the fi rms (call the total labor supply n t ). HH preferences are given by X t =0 β t ½ c α t α n t ¾ (1) where 0 < β < 1 , 0 < α < 1 and the composite goods are de fi ned by c t = ·Z 1 0 c t ( i ) ( θ 1) di ¸ θ/ ( θ 1) (2) where 1 < θ and n t = Z 1 0 n t ( i ) di. The HH trades bonds as well as money. Bonds costing B t +1 /R t dollars at time t return B t +1 dollars at time t + 1 where R t is the gross nominal interest rate between t and t + 1 (i.e. the nominal bond price Q t = 1 /R t ). Bonds are available in zero net supply so B t +1 = 0 must hold in equilibrium. Let M t +1 be a HH’s choice of money balances in period t and M s t +1 is the government supply of money. The money supply evolves according to M s t +1 = x t M s t via government lump sum taxes/transfers τ t = ( x t 1) M s t (note that this is also the government budget constraint). We assume the growth rate of the money supply lies in the set x t [ β, x ] where x > 1 so that the government might actually shrink the money supply. Timing in any period is given by: 1. The HH enters t with M t and B t chosen at t 1 . Each fi rm enters t taking as given nominal price for its goods P t ( i ) chosen at t 1 . 2. The HH receives/pays nominal transfer/tax τ t and its bonds mature (generating nominal balances of M t + τ t + B t ). 3. The HH purchases B t +1 bonds and pays B t +1 /R t out of its nominal balances. 1
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4. The shopper must use its nominal balances to purchase consumption goods from each of the fi rms Z P t ( i ) c t ( i ) di M t + τ t + B t B t +1 /R t . (3) 5. The fi rm hires n t ( i ) worker hours, pays nominal wages W t n t ( i ) to the worker, and chooses next period’s price P t +1 ( i ) to maximize future pro fi ts. 6. The fi rm brings home any current nominal pro fi ts D t ( i ) = P t ( i ) y t ( i ) W t ( i ) n t ( i ) .
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