Luiggi Donayre Summer Washington University in St. Louis Section 2, Session 2. Problem Set # 3 Suggested Solutions Due: June 30 th, in class

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1 uiggi Donayre Summer 2009 Department of Economics Economics 104B Washington University in St. ouis Section 2, Session 2 Problem Set # 3 Suggested Solutions Due: June 30 th, in class Instructions: The problem set is due at the beginning of the class. Electronic copies will not be accepted. Feel free to discuss with classmates, but each person should hand in his/her own answers. 1. Choose two countries that interest you one rich and one poor. What is the income per person in each country? Find some data on country characteristics that might help explain the difference in income: investment rates, population growth rates, educational attainment, saving rates, etc. (Hint: from the blog, you can use the NBER or the OECD sites. Also, you can consider the site of the World Bank, to find such data). How might you figure out which of these factors I most responsible for the observed income differences? There is no unique way to answer this question. For the purpose of this question (and much to your surprise, I did not pick Peru :P), the U.S. and Pakistan were compared. From the Quick Reference Tables link in the Data and Statistics section of the World Bank s website, income per person was $46,060 in the U.S. in 2007 and $860 in Pakistan for the same year. As the text notes, the most important factors to explain this 46-fold difference in income per capita are differences in capital, labor and/or technology. The Solow growth model provides a framework for thinking about the importance of these factors. One clear difference across countries is in educational attainment, which could reflect differences in broad human capital or in the level of technology (e.g. if the work force is more educated, better technologies can be implemented). Here, education will be thought of as technology in that it allows more output per worker for any given level of physical capital per worker. Table 1 summarizes these data for both countries considered: Table 1 United States and Pakistan: Macroeconomic performance (selected variables) abor Force growth ( ) Investment/ GDP (1990) - in % Illiteracy (% of population 15+) United States Pakistan

2 It seems unlikely that the small difference in investment/gdp explains the large difference in per capita income, leaving labor-force growth and illiteracy (i.e., technology) as the likely culprits. To understand this in a more formal way, the Solow model can be used. As in the book, assume 1/ 2 1/ 2 that the two countries have the same production technology: Y K (this will determine whether differences in saving and population growth can explain differences in income per capita; if not, then differences in technology will remain as the likely explanation). The steady-state value of capital is derived from sf ( k) ( n ) k 0. Since then: s 2 k ( ) n y 1/ 2 f ( k) k, s so that y ( ). Assuming that the United States and Pakistan are in steady-state and have n the same rate of depreciation say, 5 percent- then the ratio of income per capita in the two countries is: y y US Pakis tan s s US Pakis tan n n Pakis tan US From this equation, it can be inferred that if the U.S. saving rate had been twice that of Pakistan, then U.S. income per worker would be twice Pakistan s level (ceteris paribus). Given that the U.S. has 46-times higher income per worker but very similar levels of investment relative to GDP, this variable is not a major factor in the comparison. Even population growth can only explain a factor of 2. The remaining culprit is technology, and the high level of illiteracy in Pakistan is consistent with this conclusion. 2. Consider an economy described by the following Cobb-Douglass production function: Y F 0.7 ( K, ) K a. What is the per-worker production function? The per-worker production function is given by: y Y F( K, ) K 0.7 K K k b. Assuming no population growth, find the steady-state capital stock per worker, output per worker, and consumption per worker as a function of the saving rate and depreciation rate.

3 In the steady-state, k 0 so that sf ( k) k. Plugging in f(k), it follows that: k f ( k ) k ( k ) 0. 3 s So that: 1/ 0.7 s ( ) 0.7 s k k Since y f ( k ) and / 0.7 s c ( 1 s). c 1 s ( ) y, it follows that 1/ 0.7 / 0.7 s s y and c. Assume that the depreciation rate is 10 percent ( 0. 1) per year. Make a table showing steady-state capital per worker, output per worker, and consumption per worker for saving rates of 0 percent, 10 percent, 20 percent, 30 percent and so on. What saving rate maximizes output per worker? What saving rate maximizes consumption per worker? Table 2 below shows k, y, and c for the saving rate in the left column, using the equations from part (b). The depreciation rate is assumed to be 0.1 (the last column shows the MPK, derived in part (d) below). Table 2 s k y c MPK Note that a saving rate of 100 percent maximizes output per worker. In that case, nothing is ever consumed, so c 0. Consumption per worker is maximized when s=, that is when s equals capital s share of output. This is the Golden rule level of s.

4 3. Do problems 7 and 9 (only parts a, b and e) from the problems and applications part of Chapter 13, on pp (Note: problem 7 is new to the 5 th edition and problem 9 in the 5 th edition is labeled as problem 8 in the 4 th edition). Problem 7 Private saving is equal to (Y C T) = 10,000 6,000 1,500 = 2,500. Public saving is equal to (T G) = 1,500 1,700 = National saving is equal to (Y C G) = 10,000 6,000 1,700 = 2,300. Investment is equal to saving = 2,300. The equilibrium interest rate is found by setting investment equal to 2,300 and solving for r: 3, r = 2, r = 1,000. r = 10 percent. Problem 9 a. Figure 1 illustrates the effect of the $20 billion increase in government borrowing. Initially, the supply of loanable funds is curve S 1, the equilibrium real interest rate is i 1, and the quantity of loanable funds is 1. The increase in government borrowing by $20 billion reduces the supply of loanable funds at each interest rate by $20 billion, so the new supply curve, S 2, is shown by a shift to the left of S 1 by exactly $20 billion. As a result of the shift, the new equilibrium real interest rate is i 2. The interest rate has increased as a result of the increase in government borrowing. Figure 1

5 b. Because the interest rate has increased, investment and national saving decline and private saving increases. The increase in government borrowing reduces public saving. From the figure you can see that total loanable funds (and thus both investment and national saving) decline by less than $20 billion, while public saving declines by $20 billion and private saving rises by less than $20 billion. e. If households believe that greater government borrowing today implies higher taxes to pay off the government debt in the future, then people will save more so they can pay the higher future taxes. Thus, private saving will increase, as will the supply of loanable funds. This will offset the reduction in public saving, thus reducing the amount by which the equilibrium quantity of investment and national saving decline, and reducing the amount that the interest rate rises. If the rise in private saving was exactly equal to the increase in government borrowing, there would be no shift in the national saving curve, so investment, national saving, and the interest rate would all be unchanged. This is the case of Ricardian equivalence. 4. Do problems 5, 10 and 13 from the problems and applications part of Chapter 16, on pp (Note: problems 10 and 13 in the 5 th edition are labeled as problems 9 and 11 in the 4 th edition. Problem 5 is the same in both editions). Problem 5 a. Here is BSB's T-account: Beleaguered State Bank Assets iabilities Reserves $25 million Deposits $250 million oans $225 million b. When BSB's largest depositor withdraws $10 million in cash and BSB reduces its loans outstanding to maintain the same reserve ratio, its T-account is now: Beleaguered State Bank Assets iabilities Reserves $24 million Deposits $240 million oans $216 million c. Because BSB is cutting back on its loans, other banks will find themselves short of reserves and they may also cut back on their loans as well. d. BSB may find it difficult to cut back on its loans immediately, because it cannot force people to pay off loans. Instead, it can stop making new loans. But for a time it might find itself with more loans than it wants. It could try to attract additional deposits to get additional reserves, or borrow from another bank or from the Fed.

6 Problem 10 a. With a required reserve ratio of 10% and no excess reserves, the money multiplier is 1/.10 = 10. If the Fed sells $1 million of bonds, reserves will decline by $1 million and the money supply will contract by 10 x $1 million = $10 million. b. Banks might wish to hold excess reserves if they need to hold the reserves for their day-today operations, such as paying other banks for customers' transactions, making change, cashing paychecks, and so on. If banks increase excess reserves such that there is no overall change in the total reserve ratio, then the money multiplier does not change and there is no effect on the money supply. Problem 13 a. If people hold all money as currency, the quantity of money is $2,000. b. If people hold all money as demand deposits at banks with 100% reserves, the quantity of money is $2,000. c. If people have $1,000 in currency and $1,000 in demand deposits, the quantity of money is $2,000. d. If banks have a reserve ratio of 10%, the money multiplier is 1/.10 = 10. So if people hold all money as demand deposits, the quantity of money is 10 x $2,000 = $20,000. e. If people hold equal amounts of currency (C) and demand deposits (D) and the money multiplier for reserves is 10, then two equations must be satisfied: (1) C = D, so that people have equal amounts of currency and demand deposits; and (2) 10 x ($2,000 C) = D, so that the money multiplier (10) times the number of dollar bills that are not being held by people ($2,000 C) equals the amount of demand deposits (D). Using the first equation in the second gives 10 x ($2,000 D) = D, or $20,000 10D = D, or $20,000 = 11 D, so D = $1, Then C = $1, The quantity of money is C + D = $3, Explain why currency is considered to be a perfectly liquid asset. Rank the following assets according to their liquidity and explain your ranking: a share of IBM stock, currency, a checking deposit, a used BMW, a passbook savings account, a U.S. government bond, a General Motors bond. The liquidity of an asset has two dimensions. First, an asset is more liquid if it can be easily transformed into money or "generalized purchasing power." In economics, it is typical to recast the concept of "easy" into quantitative terms and say "low cost." Therefore, a more precise statement would be that very liquid assets can be transformed into generalized purchasing power at low cost. These costs, however, include opportunity costs such as the time needed to go to a bank and cash a check, for example. Second, assets are considered more liquid if their value in

7 money terms is more certain. That is, when the asset is sold for money, people know how much money they will get. The most liquid asset mentioned in the problem is currency. It is already in the form of generalized purchasing power and it has a definite value. The next most liquid asset is a checking account. Its value is known with certainty and it is quite easy to use checks to get money or to pay for goods directly. In some cases, however, checks may not be accepted as payment, which implies that they are somewhat less liquid than currency. A passbook savings account (an oldstyle account) also has a known and certain value, but you have to go to the bank and present your passbook to get your funds, which reduces its liquidity relative to a checking account. The U.S. government bond probably ranks next on the liquidity scale. It is more difficult to get money for the bond, as it must be redeemed at a bank or Federal Reserve branch. Also, its value will fluctuate as interest rates change, so you don't know for certain how much you will get for it. The General Motors bond and the share of IBM stock are less liquid yet. They are more costly to sell for money, and their money values fluctuate. The GM bond is less liquid than the government bond, in part because the risk that GM will default on the bond (fail to re-pay) is higher than for the government bond. Usually, stock is considered less liquid than bonds because its value is often more volatile. The used BMW is probably the least liquid asset. It is costly to sell a used car, and the money value of the car is far from certain. 6. Do problems 1 and 2 from the problems and applications part of Chapter 17, on pp (Note: both problems are the same in both editions). Problem 1 In this problem, all amounts are shown in billions. a. Nominal GDP = P x Y = $10,000 and Y = real GDP = $5,000, so P = (P x Y )/Y = $10,000/$5,000 = 2. Because M x V = P x Y, then V = (P x Y )/M = $10,000/$500 = 20. b. If M and V are unchanged and Y rises by 5%, then because M x V = P x Y, P must fall by 5%. As a result, nominal GDP is unchanged. c. To keep the price level stable, the Fed must increase the money supply by 5%, matching the increase in real GDP. Then, because velocity is unchanged, the price level will be stable. d. If the Fed wants inflation to be 10%, it will need to increase the money supply 15%. Thus M x V will rise 15%, causing P x Y to rise 15%, with a 10% increase in prices and a 5% rise in real GDP.

8 Problem 2 a. If people need to hold less cash, the demand for money shifts to the left, because there will be less money demanded at any price level. b. If the Fed does not respond to this event, the shift to the left of the demand for money combined with no change in the supply of money leads to a decline in the value of money (1/P), which means the price level rises, as shown in Figure 2. Figure 2 c. If the Fed wants to keep the price level stable, it should reduce the money supply from S1 to S2 in Figure 3. This would cause the supply of money to shift to the left by the same amount that the demand for money shifted, resulting in no change in the value of money and the price level. Figure 3