ECON 4721, Summer 2022 Problem Set 3
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Problem Set 3 - Answer Key
ECON 4721, SUmmer 2022
Question 1 - Money Aggregates (50 points). Consider an economy with a constant population in which people wish to hold bank checking deposits worth a total of 5,000 goods in every period. The economy has a total endowment of 10,000 goods in each period. There is a total stock of unintermediated capital of 1,000 goods in each period. Bank deposits are the only form of money in the economy. Deposits at banks are subject to a reserve requirement of 20 percent. The real rate of return on capital is 北 = 1.10 per period. After meeting the reserve requirement, banks invest the remainder of all deposits into capital. Individuals do not hold capital. The fiat money stock (monetary base) is $2,000 in every period. Calculate the following variables:
1. The price of a good (in dollars).
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Solution: The money market clearing condition is: Ut Mt = VNt ht |
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Ut 2000 = 0.2 × 5000 2Ut = 1 =⇒ pt = 2 |
2. The gross real rate of return on deposits that will be offered by banks in a competitive economy.
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r = V + (1 − V)北 = 0.2 × 1 + 0.8 × 1.1 = 1.08 X |
3. The total nominal money stock M1
M1 = = = $10000
4. The Money Multiplier
M1/M = 1/V = 1/0.2 = 5
5. The total capital stock.
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Solution: The total capital stock consists of capital intermediated by banks [(1 − V)Nt ht] = 0.8 × 5000 = 4000 plus unintermediated capital stock Nt kt = 1000 =⇒ Kt = 4000 + 1000 = 5000 |
Real GDP.
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Solution: Real GDP is this model is the total endowment plus the returns from unintermediated capital and intermediated capital: GDP = Nty + xNt −2kt −2 + Nt −2x[1 − V]ht −2 |
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Since capital pays a one-period gross real rate of return of 1.1 , its two-period gross real rate of return is x = ”2 = 1.21. GDP = 10000 + 1.21 × 1000 + 1.21 × (1 − 0.2)(5000) = 16050 |
From now on, assume that the central bank allows banks to borrow up to one-half of required reserves at the gross interest rate of 1.08. Calculate the following variables:
7. The price of a good (in dollars).
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Solution: The demand for fiat money comes from the reserve requirement, which with central bank lending can come from borrowed reserves (6VNt ht )and nonbor- rowed reserves Ut Mt . The money market-clearing condition: 6VNt ht + Ut Mt = VNt ht
=⇒ Ut = =⇒ pt = 4 |
8. The gross real rate of return on deposits that will be offered by banks in a competitive
economy.
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Solution:
X |
9. The total nominal money stock M1
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Solution:
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10. The Money Multiplier
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Solution:
N1/N = |
11. The total capital stock.
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Solution: The amount of capital intermediated by banks expands due to the cen- tral bank lending to [1 − 儿(1 − 9)]Nt Vt = [1 − 0.2(1 − 0.5)]5000 = 4500 Adding to this the stock of unintermediated capital (1000), yields the total capital stock of: 5500. |
12. Real GDP.
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Solution: We merely need to modify the expression for real GDP to incorporate the fact that the amount of intermediated capital is expanded by central bank lending: Oad = Ntn + XNt −2以t −2 + Nt −2X[1 − 儿(1 − 9)]Vt −2 Oad = 10000 + 1.21 × 1000 + 1.21 × (1 − 0.2(1 − 0.5))(5000) = 16655 |
Now consider the case in which the Central Bank is willing to loan any quantity of reserves to bank.
13. What amount would a a bank choose to borrow if the Central Bank interest rate is less than 北?
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Solution: Banks will borrow an infinite amount and the equilibrium price level will be infinity. |
14. Would your last answer change if capital exhibits diminishing marginal returns?
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Solution: For {\ (以) > 九 borrowing occurs. But with each additional unit of capital financed by the bank, capital also expands, lowering {\ (以) and therefore the market rate of return. Therefore, borrowing will continue up until {\ (以) = 九 |
Question 2 - Money Stock Fluctuations (30 points). Consider an economy in which there are 100 workers. One-half of the workers are endowed with 200 units of the consumption good when young and nothing when old. The remaining workers are endowed with 20 units
of the consumption good when young and nothing when old. Each worker saves 30 percent of their endowment when young. Let the gross real return on capital by 1.25. Money supply grows according to the following rule: Mt = 1.1Mt −1 . Assume that each worker uses 10 goods to identify themselves and make a withdrawal from a bank.
1. For all the agents, compute the return on money.
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Solution: Ut+1 Ut |
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2022-07-23