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Unformatted text preview: Econ 4721 Money and Banking Problem Set 2 Due Thursday, Feb. 24, before class Problem 1 (35 points)
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 Nt = nNt-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 (c1,t , c2,t+1 ) = log c1,t + log c2,t+1 with = 0.5. Members of the initial old generation only live for one period and have utility u (c0,1 ) = log c0,1 . The government expands the money supply by a factor of z each period, Mt = zMt-1 . Assume that z = 1.5. The money created each period is used to finance a lump-sum subsidy of at+1 goods to each old person. (a) Solve for the (stationary) Pareto efficient allocation. The answer should be two numbers cP O , cP O . 1 2 (c) Define a competitive equilibrium with money for this economy. (d) Solve for the rate of return of money (vt+1 /vt ) and the growth rate of the price level (pt+1 /pt ) in a stationary equilibrium. The answer should be two numbers. (e) Solve for the consumption allocation (c , c ) and a lump-sum subsidy a in a stationary equilibrium. The 1 2 answer should be three numbers. (f) Verify that agents prefer the Pareto efficient allocation to the competitive equilibrium allocation with inflation. (g) Illustrate the Pareto efficient allocation cP O , cP O and the competitive equilibrium allocation (c , c ) on the 1 2 1 2 (c1 , c2 ) plane. Your graph should also include the feasibility line, the lifetime budget constraint, and their indifference curves. (1) (b) Write the government's budget constraint in period t. Problem 2 (30 points)
This problem mimics Exercise 3.4 from the CF book. Consider the following modification of our overlapping generations model with consumers living for 2 periods. As usual, individuals are endowed with y units of consumption 1 good when young and with nothing when old, and the good is not storable. The government keeps the fiat money stock constant, i.e. z = 1. The population in the economy grows at rate n > 1. In every period, the government imposes a lump-sum tax each young person for units of consumption good. The total proceeds of the tax are then distributed equally among the old population in this period. We assume that the subsidy to the old is less than they would be happy to consume when old (so that there are still reasons for people to hold fiat money). Do the following: (a) Write down the first and second period budget constraints that a regular person born in period t faces. (Hint: remember that in every period there are more young people alive than old people.) Derive the lifetime budget constraint. (b) Derive the rate of return on fiat money in a stationary monetary equilibrium. (c) Is the stationary monetary equilibrium allocation Pareto efficient? Discuss. (d) Does this government policy have any effect on consumers' welfare? Explain. (e) Does your answer to part (d) of this question change if we assume that the subsidy to the old exceeds the quantity they would prefer to consume? (f) Assume now that tax collection is costly, so for every unit of consumption collected from the young, only 0.5 units end up being available for distribution to the old? Does your answer to part (d) change? Comment. Problem 3 (35 points)
There are two countries, USA (labeled a) and China (labeled b). Each country is described by our standard overlapping generations model with consumers that live for two periods. As usual, individuals in US are endowed with y a units of consumption good when young and with nothing when old, and the good is not storable. (For China, the corresponding quantities are y b and zero, respectively). Population in US grows according to the
a b following law of motion: Nta = na Nt-1 . Similarly, for China we have Ntb = nb Nt-1 . Each country has its own currency, dollars and renminbi. The supply of dollars follows the following equation: Mta
a b = z a Mt-1 . And the supply of renminbi follows: Mtb = z b Mt-1 . As usual, we are interested in stationary monetary equilibria. Do the following: (a) Suppose each country implements foreign currency controls. Derive the real rate of return on dollars and on renminbi. 2 a b (b) Explain why the exchange rate et has to be equal to the ratio of vt and vt . Suppose that population in China grows faster than in US and that both countries expand their money supply at the same rate. Will the US dollar appreciate or depreciate over time against the renminbi? Explain. (c) Suppose that all foreign currency controls were lifted. Demonstrate that we will not be able to determine the exchange rate et any more. Discuss. For the rest of the problem, we assume that na = nb = 1 and that N a = N b = 100. We will also assume that M a = M b = 600 and that the initial money stock is divided equally among the initial old generations in each country (so z a = z b = 1). In addition, we assume that every young person wants to hold real money balances that are worth 18 units of consumption (so y a - ca = y b - cb = 18). Finally, the exchange rate is fixed at e = 2, so 1 1 1 dollar can be exchanged for 2 renminbis. There are no foreign currency controls. (d) Find the value (measured in goods) of a dollar and of a renminbi. What is the consumption of an old person? (Hint: use the global money market clearing condition from part (c)). (e) Suppose that every initial old person in US and in China decides to reduce his holdings of China's money by 2 renminbi. Every person turns 2 renminbi to the Chinese monetary authorities wishing to exchange it for 1 dollar. Assume that the US monetary authorities are cooperating with their Chinese counterparts and are willing to print as many new dollars as necessary. What will happen to the total money stock of dollars and renminbi? How will the values of each currency change? (f) Now suppose that the same situation happens as in part (e), but the US monetary authorities refuse to cooperate. Instead, the Chinese government decides to honor its pledge for the fixed exchange rate by taxing every old Chinese citizen equally. What will be the values of each currency? How many goods must each old Chinese person be taxed? How much does an old US citizen consume, and how much does the old Chinese citizen consume? Who benefits from this policy? 3 ...
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This note was uploaded on 01/18/2012 for the course ECON 4721 taught by Professor Staff during the Spring '08 term at Minnesota.
- Spring '08