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Unformatted text preview: Econ 4721: Money and Banking, Fall 2008 Homework 2 Answer Key 1 Problem 1. Inflation Consider an overlapping generations model in which consumers live for two periods. The number of people born in each generation grows in each period, according to N t = nN t 1 , where n = 1 . 2. In each period, young consumers are endowed with y = 60 and old consumers are endowed with 0 units of the single consumption good. Each member of the generations born in period 1 and later have the following utility function: u ( c 1 ,t ,c 2 ,t +1 ) = log c 1 ,t + log c 2 ,t +1 (1) with = 0 . 5. Members of the initial old generation only live for one period and have utility u ( c , 1 ) = log c , 1 . The government expands the money supply by a factor of z each period, M t = zM t 1 . Assume that z = 1 . 5. The money created each period is used to finance a lumpsum subsidy of a t +1 goods to each old person. (a) Solve for the (stationary) Pareto efficient allocation. The answer should be two numbers ( c PO 1 ,c PO 2 ) . (b) Write the governments budget constraint in period t + 1. (c) Define a competitive equilibrium with money for this economy. (d) Solve for the rate of return of money ( v t +1 /v t ) and the growth rate of the price level ( p t +1 /p t ) in a stationary equilibrium. The answer should be two numbers. (e) Solve for the consumption allocation ( c * 1 ,c * 2 ) and a lumpsum subsidy a * in a stationary equilibrium. The answer should be three numbers. (f) Verify that agents prefer the Pareto Optimal allocation to the competitive equilibrium allocation with inflation. (g) Illustrate the Pareto Optimal allocation ( c PO 1 ,c PO 2 ) and the competitive equilibrium allocation ( c * 1 ,c * 2 ) on the ( c 1 ,c 2 ) plane. Your graph should also include the feasibility line, the lifetime budget con straint, and their indifference curves. 1.1 Solution To find the Pareto efficient allocation, we set up the following problem, explicitly incorporating the fact its solution will be stationary: max c 1 ,c 2 log c 1 + log c 2 (2) 1 s.t. N t c 1 + N t 1 c 2 = N t y (3) Using the fact that N t 1 = 1 n N t , and getting rid of N t terms, we rewrite constraint as: c 1 + 1 n c 2 = y (4) You can solve this problem in multiple ways. I use the quick solution formula Ive discussed in class at some point: c PO 1 = 1 1 + y = 40 (5) c PO 2 = 1 + y 1 n = 24 (6) So your answer for part (a) is ( c PO 1 ,c PO 2 ) = (40 , 24). The government budget constraint is very simple: v t +1 ( M t +1 M t ) = a t +1 N t (7) where the left hand side (LHS) is the seignorage revenue, and the RHS is the total value of lumpsum rebates that are given to the people. We define the monetary equilibrium for this economy as follows: Definition 1 A competitive equilibrium with money for this economy would be an allocation ( c * 1 ,t ,c * 2 ,t +1 ) for every person born at time t 1 , c * , 1 for the initial old, prices of money v * t for all t 1 and government transfers a * t...
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This note was uploaded on 02/20/2011 for the course ECON 4721 taught by Professor Staff during the Fall '08 term at Minnesota.
 Fall '08
 Staff

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