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4721_HW2_AK

# 4721_HW2_AK - Econ 4721 Money and Banking Fall 2008...

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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 + β log c 2 ,t +1 (1) with β = 0 . 5. Members of the initial old generation only live for one period and have utility u ( c 0 , 1 ) = log c 0 , 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 lump-sum 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 P O 1 , c P O 2 ) . (b) Write the government’s 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 lump-sum 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 P O 1 , c P O 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 life-time 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 + β log c 2 (2) 1

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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 I’ve discussed in class at some point: c P O 1 = 1 1 + β y = 40 (5) c P O 2 = β 1 + β y 1 n = 24 (6) So your answer for part (a) is ( c P O 1 , c P O 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 lump-sum 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 * 0 , 1 for the initial old, prices of money v * t for all t 1 and government transfers a * t for all t 1 such that: 1. [1.] 2. For every t 1 , every person born at t , taking v * t , v * t +1 and a * t +1 as given, chooses ( c * 1 ,t , c * 2 ,t +1 ) to solve: max c 1 ,t ,c 2 ,t +1 ,m t log c 1 ,t + β log c 2 ,t +1 (8) s.t.
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4721_HW2_AK - Econ 4721 Money and Banking Fall 2008...

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