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Unformatted text preview: Final Exam, Macro II, Spring 2006 Prof. Dean Corbae. Question 1. Based on Lucas and Stokey (1987) ”Money and Interest in a cash in advance econ omy,” Econometrica, p. 491513. The economy is populated by a continuum of identical, in f nitely lived agents. Time is discrete. There are two consumption goods available each period: “cash goods” (denoted c 1 ,t ) which are subject to a cashin advance constraint, and “credit goods” (denoted c 2 ,t ) which are not. Preferences are: E ( ∞ X t =0 β t [ α ln( c 1 ,t ) + (1 − α ) ln( c 2 ,t )] ) (1) with β ∈ (0 , 1). Goods are not storable, and the technology in each period is simply c 1 ,t + c 2 ,t ≤ y ( ω t ), where y ( ω t ) ∈ [ y , ¯ y ], the endowment, is a function of the current shock. Shocks to the system in any period, denoted by ω t ∈ Ω f nite, form a f rstorder Markov process with stationary transition function π ( ω , ω ). There is also a government in this economy. Its only activity is to supply money, injected as lump sum transfers, and the net money growth rate in any period t is f xed at g (i.e. M s t +1 = (1 + g ) M s t ). Therefore, if M s t is the aggregate money supply prior to period t , nominal transfers in period t are given by Υ t = M s t +1 − M s t = gM s t . The timing within a period is as follows: Υ is distributed, the market for c 1 opens, y ( ω ) is realized, and f nally the market for c 2 opens. a. (10 points) Why is the relative price of cash goods in terms of credit goods equal to 1? Write down the household’s constraints. State the government budget constraint. Hint: to keep things stationary, normalize all nominal variables by M s t (that is, if P t is the nominal price of cash goods ($/good), let p t ( S t ) ≡ P t /M s t where S t = ( ω t , M s t )). Denote the normalized money holdings of a household by m t and the normalized transfers by τ t . Answer. The technology in this economy allows the agents to transform cash goods into credit goods one for one. Hence in each period cash and credit goods will sell at the same nominal price. Otherwise, if the prices di f ered, agents would transform more of their endowment into the “expen sive” good, sell it, obtain more income, and buy more of both. But the absence of arbitrage rules out this price disparity. The household purchases goods ( c 1 , c 2 ) at price p t (expressed as a ratio to the current period’s money supply). P t M s t c 1 ,t ≤ M t + Υ t M s t ⇐⇒ p t c 1 ,t ≤ m t + τ t (2) 1 M t +1 M s t + P t M s t [ c 1 ,t + c 2 ,t ] ≤ P t M s t y ( ω t ) + M t + Υ t M s t ⇐⇒ m t +1 (1 + g ) + p t [ c 1 ,t + c 2 ,t ] ≤ p t y ( ω t ) + m t + τ t (3) m t +1 = p t y ( ω t ) + m t + τ t − p t [ c 1 ,t + c 2 ,t ] (1 + g ) Government budget constraint: Υ t M s t = M s t +1 − M s t M s t ⇐⇒ τ t = g (4) b. (5 points) Taking prices p ( S ) as given, write down the household problem in recursive form....
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 Spring '07
 CORBAE
 Economics, Equilibrium, Inflation, Mts

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