4721_HW4_AK - Econ 4721 Money and Banking Fall 2008...

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Econ 4721: Money and Banking, Fall 2008 Homework 4 Answer Key Problem 1. Bank Runs Consider an economy in which consumers live for three periods (called period 1, 2, and 3). Each consumer has an endowment of 20 goods when young, and nothing when middle-aged and when old. Utility from consumption depends on what ”type” a consumer is, as follows: u ( c 1 ,c 2 ,c 3 ) = log [ c 2 ] for type 1 — impatient log [ c 2 + c 3 ] for type 2 — patient There are 200 consumers, and half will become each type. However, consumers only learn their type in period 2. Assume no one else can observe the types (an agent knows her type only). There are two assets available in the economy, money and capital, offering the following rates of return: One-Period Return Two-Periods Return Fiat Money 1 1 Capital 0 . 9 1 . 5 (a) Suppose a consumer, on her own, saves half her goods in each asset in period 1. After finding out her type, she consumes all the savings in the appropriate period. Calculate the consumption the consumer would receive in either period. (b) Suppose a competitive bank offers demand deposit contracts to all consumers, specifying interest rates r 1 and r 2 for deposits withdrawn in periods 2 and 3, respectively. Which assets, and in what amounts, would the bank need to hold in order to offer r 1 = 1 and r 2 = 1 . 5? (c) Calculate the consumption a consumer would receive in each period from depositing with a bank. (d) Suppose a bank run occurs in this economy (all type 2 people pretend to be type 1 people and withdraw early). How many people, in total, would the bank be able to pay in period 2 at the promised rate of return before the bank runs out of assets? How many additional goods would the bank need in order to pay back all its depositors? (e) Suppose that in the period after you made your deposit at the bank you turn out to be a type 2 person and you learn that all the other type 2 people are about to pretend to be type 1 people so that they can withdraw early. Is it in your self-interest to also try to withdraw early? 1
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(f) Suppose that there are two cities, A and B , each with consumers and a bank as described above. There are no bank runs, but in each city the true distribution of consumer types turns out to be different from what people initially expected. Specifically, in city A only 25% of the people learn they are type 2 and in city B 75% of the people learn they are type 2 (instead of 50% in each city). The banks in each city, however, expected that half of their customers would be of each type, so
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4721_HW4_AK - Econ 4721 Money and Banking Fall 2008...

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