Economics 102 Introductory Macroeconomics - Spring 2006 ...



Economics 102 Introductory Macroeconomics - Spring 2006, Professor J. Wissink

Problem Set 5 – ANSWERS

1. Consider the super simple linear frugal governed open economy with the marginal propensity to consume being the only non-zero marginal propensity. Ralph understands why the multiplier for G’ is opposite in sign of the multiplier for T’, but he can’t seem to intuitively figure out why its value would be smaller. That is, why would a change in G’ work “better” or be “stronger” than an equivalent and opposite change in T’? Help Ralph out.

Changes in government spending (∆G) directly boost the economy. By the full amount in the first period, by

c ∆G in the second period (assuming the super simple model), etc. (To get the total change in output at the end of the day, just sum the infinite series.)

On the other hand, changes in taxation (∆T) boost economy indirectly and less. The reason is that households save part of their tax rebate. Thus, in the first period, (assuming the super simple model) output changes by c∆T, in the second period by c2 ∆T, etc. (To get the total change in output at the end of the day, just sum the inifinite series.)

2. The Merriam-Webster Online Dictionary defines a spendthrift as, “a person who spends improvidently or wastefully.” Suppose this actually means a person who never saves. Suppose we had a linear spendthrift UNgoverned closed economy with no investment. What is the value of the aggregate desired expenditure multiplier in this world?

The multiplier would be infinite.

Initial change in autonomous consumption will increase output by the whole amount in each period. There will be no “equilibrium” level of output, it will grow till infinity.

3. In a governed model, if taxes are not lump-sum, but depend on income, what happens to the size of the government spending multiplier, and why?

The easiest answer is to derive the multiplier.

Denoting by C’ an autonomous consumption and by t a marginal tax rate, let us write:

[pic]

[pic]

If taxes are not lump-sum, the multiplier is smaller. The reason is that the effect of a change in government expenditures on equilibrium output is dampened by taxation. (So, the initial ∆G will change output by ∆G in the first period, but only by c(1-t)∆G in the second period, etc. Sum the series to get the total change in output at the end of the day on your own.)

4. Explain what will happen to the size of both M1 and M2 in each of the following situations. Assume that all transactions take place in the U.S. and in commercial banks in the U.S. Federal Reserve system.

a. Jane, a millionaire, withdraws $500,000 from her money market account and gets cash to buy a famous painting from Vincent. Vincent takes the cash and puts it under his bed.

b. Paul transfers $10,000 from his NOW account to his savings account.

c. Sarah takes $5,000 out of her checking account to buy IBM stock. IBM deposits the money they got from Sarah into a short term Certificate of Deposit.

A: If Jane withdraws funds from her money market account for cash M2 will not change at all and M1 will increase by $500,000.

B: M1 will decrease by $10,000. M2 will not change at all.

C: If Sarah withdraws from her checking account and IBM puts the money into a CD, then M1 will decrease by $5000 and M2 will stay the same.

5. Consider the following system of equations that fully describe the economy of Mesopotamia and use this information to answer the following questions:

AEd = C + I + G + EX - IM

C = 1000 + 0.8Yd

I = 1600 – 3000r where r is the interest rate

G = T = 200

EX = 400 - .05Yd

IM = 0

Msupply = 650 and Mdemand = 800 – 1000r where r is the interest rate

rrr = 0.2

Total Reserves = $130 – and there are no excess reserves or loose cash

a. What is the equilibrium interest rate, r*?

MS = (1/rrr)Res = 650

650 = 800-1000r

r* = 15%

b. What is the value of Y*?

Y* = 1000+.8Y*-0.8(200)+1600-3000(0.15)+200+400-0.05(Y* -200)

0.25Y = 2600

Y* = 10400

c. Suppose that the Mesopotamian government believes that full employment level of output is YFE = $10,550. Can the government achieve this level of Y by a balanced budget fiscal policy (ie where G=T)? Explain and show the numbers.

Yes, government can achieve this goal.

First we need to derive the balanced budget multiplier.

Let X stand for exports and M for imports. In our case:

[pic]

[pic]

Clearly the balanced budget multiplier is 1 in our case as: [pic]

To increase Y by 150 we must increase G&T by $ 150.

d. Suppose that the fiscal policy is abandoned entirely and instead the government FED guys decide to reach a Y* = $10,000 by monetary policy. By how much must interest rates change?

Change in Y is $ -400. What is the investment multiplier? We know it from previous derivation: [pic]

Plug numbers to get: 1/(1-0.8+0.05)=4. Change in investment is therefore$ -100.

So I=1150-100=1050.

1050 =1600-3000r

r*= 18.33 % and change in r* is 3.33 %.

e. In order to force interest rates to the level you found in the part above, by how much must the money supply change?

Md = 800-1000r = 616.7.

Money supply must decrease by (650-616.7) = $ 33.3.

f. How many bonds must the Fed sell to reach the desired money supply?

(Change in money supply) = (1/rrr)(Change in reserves)

-33.3 *rrr = (change in reserves)

(change in reserves) = -6.66.

The FED has to sell bonds for $ 6.66.

6. Ben Bernanke, Chairman of the Board of Governors of the Federal Reserve System gives you the assignment of expanding the money supply by $2 million using open market operations as your policy tool. The legal reserve requirement for banks is 10%. Suppose that the balance sheet of the Fed prior to any policy action is as follows:

(in millions of dollars)

FED's Assets FED's Liabilities and Net Worth

Gold and Foreign Exchange 10,000 Federal Reserve Notes outstanding 15,000

U.S. Govt. Bonds 10,000 Banks∗ Deposits 5,000

Net worth 0

a. Assume that banks hold no excess reserves and that neither households nor businesses hold any currency. What dollar value of government bonds must be bought or sold to generate a $2 million increase in the money supply?

Money multiplier in our case is 1/0.1=10. To increase money supply by $2 million, the FED has to buy bonds from households or businesses for $ 200,000, i.e. $ 0.2 million.

b. Adjust the Fed∗’s balance sheet to accurately represent their levels of assets and liabilities after the open market operations are executed. (Assume that if bonds are purchased they are purchased with new currency and if bonds are sold they are sold for currency that is then destroyed.)

The new balance sheet will look like as follows:

(in millions of dollars)

FED's Assets FED's Liabilities and Net Worth

Gold and Foreign Exchange 10,000 Federal Reserve Notes outstanding 15000.2

U.S. Govt. Bonds 10,000.2 Banks∗ Deposits 5,000

Net worth 0

c. Assuming a vertical money supply curve and a downward sloping money demand curve, graphically show the effect of these open market operations on the equilibrium interest rate. (Hold income constant and assume zero inflation.)

As money supply increases, equilibrium interest rate declines.

[pic]

7. This exercise is on the multiple contraction of the money supply resulting from open market sales of securities by the Federal Reserve. Assume: (1) that the rrr = 20%; and (2) all commercial banks hold no excess reserves and (3) net worth in all banks is always zero so that assets = liabilities at all times. Suppose the Federal Reserve sells two $30,000 bonds, one to Maria and one to Peter. Maria pays for her bond with money she withdraws from her account at Nations Bank. Peter pays for his bond with money he stored under his mattress.

a. The following is Nations Bank's balance sheet before Maria withdraws $30,000 in cash. Complete the balance sheet.

Nations Bank: Balance Sheet

|Asset |Liabilities |

|Reserves | | 80,000 | |Demand Deposits: | | | |

|Loans | |280,000 | | of Maria | | 60,000 | |

|Securities | | 40,000 | | of others | |340,000 | |

|Total | |400,000 | |Total | |400,000 | |

b. Show the impact of Maria’s withdrawal on Nations Bank's balance sheet.

Nations Bank: Balance Sheet

|Asset |Liabilities |

|Reserves | | 50,000 | |Demand Deposits: | | | |

|Loans | |280,000 | | of Maria | | 30,000 | |

|Securities | | 40,000 | | of others | |340,000 | |

|Total | |370,000 | |Total | |370,000 | |

| | | | | | | | |

c. At this point, by how much do Nations Bank's reserves fall short of the legal requirement?

20% of 370,000=74,000.

They fall short of 74,000-50,000= $ 24,000.

d. Suppose Nations Bank replenishes its reserves by selling some of its securities to Lars (i.e. the bank sells Lars securities equal to the amount its reserves fall short). Lars has a checking account at The Bank of America. Complete The Bank of America's balance sheet before Lars purchases the securities.

The Bank of America: Balance Sheet

|Asset |Liabilities |

|Reserves | |130,000 | |Demand Deposits: | | | |

|Loans | |190,000 | | of Lars | |50,000 | |

|Securities | |330,000 | | of others | |600,000 | |

|Total | |650,000 | |Total | |650,000 | |

e. Show what happens on Nations Bank's and The Bank of America's balance sheets when Lars pays for those securities by giving Nations Bank a personal check drawn on his account at The Bank of America, which Nations Bank turns over to The Bank of America in exchange for cash.

Nations Bank: Balance Sheet

|Asset |Liabilities |

|Reserves | | 74,000 | |Demand Deposits: | | | |

|Loans | |280,000 | | of Maria | | 30,000 | |

|Securities | | 16,000 | | of others | |340,000 | |

|Total | |370,000 | |Total | |370,000 | |

The Bank of America: Balance Sheet

|Asset |Liabilities |

|Reserves | |106,000 | |Demand Deposits: | | | |

|Loans | |190,000 | | of Lars | | 26,000 | |

|Securities | |330,000 | | of others | |600,000 | |

|Total | |626,000 | |Total | |626,000 | |

f. At this point, what is the decrease in the money supply, due only to the original sale of two bonds to Maria and to Peter as well as the subsequent sale of Nations Bank's securities to Lars?

30,000+30,000+24,000=84,000.

g. If the Bank of America replenishes its shortage of reserves (caused by transactions described above) by selling some of its securities, how much must it sell?

20% of 626,000=125,000.

125,200-106,000=19,200.

h. What will be the ultimate decrease in the money supply resulting from this initial sale of two bonds by the Federal Reserve?

30,000+30,000/0.2=30,000+150,000=180,000.

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M1S

MD

M2S

r

money

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