《宏观经济学》课程教学资源（英文版）Lecture 2 Answers to Textbook Questions and Problems_Mankiw

Answers to Textbook questions and Problems The rental price increases by the ratio BP203AK-7(1L07 (R/P10.3AK-L07 =(1.1) =1.069 So the rental price increases by 6.9 percent. To determine how the increase in the labor force affects the real wage, con- sider the formula for the real wage W/P W/P=MPL=(1-OAK L We know that a=0.3. We also know that labor(L)increases by 10 percent. Let (W/P)I equal the initial value of the real wage and (w/P)2 equal the final value of the real wage. To find (W/P)2, multiply L by 1.1 to reflect the 10-percent increase in the labor force (W/P1=(1-0.3AK"L (W/P2=(1-03AK0(11LD)03 To calculate the percentage change in the real wage, divide(w/p)2 by (W/P)1 P1(1-0.3)AK03 0.972. That is, the real wage falls by 2.8 percent c. We can use the same logic as(b) to set Y,=AKO.LO Y? Therefore we have Y2=A1.x°L (1.1 =1.029. This equation shows that output increases by 2 percent. Notice that o <0.5 means that proportional increases to capital will increase output by less than the same proportional increase to labor. logic as(b)for the change in the real rental price of (RP2034(11K)L0 (RP10.3AK07 0.935 The real rental price of capital falls by 6.5 percent because there are diminishing returns to capital; that is, when capital increases, its marginal product falls

The rental price increases by the ratio = = (1.1)0.7 = 1.069. So the rental price increases by 6.9 percent. To determine how the increase in the labor force affects the real wage, con￾sider the formula for the real wage W/P: W/P = MPL = (1 – α)AKα L – α . We know that α = 0.3. We also know that labor (L) increases by 10 percent. Let (W/P)1 equal the initial value of the real wage and (W/P)2 equal the final value of the real wage. To find (W/P)2, multiply L by 1.1 to reflect the 10-percent increase in the labor force: (W/P)1 = (1 – 0.3)AK0.3L– 0.3. (W/P)2 = (1 – 0.3)AK0.3(1.1L) – 0.3. To calculate the percentage change in the real wage, divide (W/P)2 by (W/P)1: = (1.1) – 0.3 = 0.972. That is, the real wage falls by 2.8 percent. c. We can use the same logic as (b) to set Y1 = AK0.3L0.7. Y2 = A(1.1K) 0.3L0.7. Therefore, we have: = = (1.1)0.3 = 1.029. This equation shows that output increases by 2 percent. Notice that α < 0.5 means that proportional increases to capital will increase output by less than the same proportional increase to labor. Again using the same logic as (b) for the change in the real rental price of capital: = = (1.1)–0.7 = 0.935. The real rental price of capital falls by 6.5 percent because there are diminishing returns to capital; that is, when capital increases, its marginal product falls. 20 Answers to Textbook Questions and Problems 0.3AK –0.7(1.1L) 0.7 0.3AK –0.7L0.7 (R/P)2 (R/P)1 A(1.1K) 0.3L1.7 AK0.3L0.7 Y2 Y1 0.3A(1.1K) –0.7L0.7 0.3AK–0.7L0.7 (R/P)2 (R/P)1 ( ( ) ( .) (. ) ( .) . . . . W/P) W/P AK L AK L 2 1 03 03 03 03 1 03 11 1 03 = − − − −

Chapter 3 National Income: Where It Comes From and Where It Goes Finally, the change in the real wage is..L7 (W/P)2 (WIP) (11)93 1.029 Hence, real wages increase by 2.9 percent because the added capital increases the marginal productivity of the existing workers. (Notice that the wage and output have both increased by the same amount, leaving the labor share unchanged-a feature of Cobb-Douglas technologies. d. Using the same formula, we find that the change in output is (1.1AKL =1.1. This equation shows that output increases by 10 percent. Similarly, the rental price of capital and the real wage also increase by 10 percent (RP20.3(1.1AK0L07 (RP10.3AK0 1.1 (W/P20.7(11AK3L03 (WP10.7AK3L03 2. a. The marginal product of labor MPL is found by differentiating the production function with respect to labor MPL- dy =oKHI2L-23 This equation is increasing in human capital because more human capital makes b. The marginal product of human capital MPH is found by differentiating the pro- duction function with respect to human capital dY MPH= This equation is decreasing in human capital because there are diminishing returns c. The labor share of output is the proportion of output that goes to labor. The total amount of output that goes to labor is the real wage(which, under perfect compe- tition, equals the marginal product of labor)times the quantity of labor This quantity is divided by the total amount of output to compute the labor share K8H43L23 Labor share

Finally, the change in the real wage is: = (1.1)0.3 = 1.029. Hence, real wages increase by 2.9 percent because the added capital increases the marginal productivity of the existing workers. (Notice that the wage and output have both increased by the same amount, leaving the labor share unchanged—a feature of Cobb–Douglas technologies.) d. Using the same formula, we find that the change in output is: = = 1.1. This equation shows that output increases by 10 percent. Similarly, the rental price of capital and the real wage also increase by 10 percent: = = 1.1. = = 1.1. 2. a. The marginal product of labor MPL is found by differentiating the production function with respect to labor: MPL = = K1/3H1/3L–2/3. This equation is increasing in human capital because more human capital makes all the existing labor more productive. b. The marginal product of human capital MPH is found by differentiating the pro￾duction function with respect to human capital: MPH= = K1/3L1/3H–2/3. This equation is decreasing in human capital because there are diminishing returns. c. The labor share of output is the proportion of output that goes to labor. The total amount of output that goes to labor is the real wage (which, under perfect compe￾tition, equals the marginal product of labor) times the quantity of labor. This quantity is divided by the total amount of output to compute the labor share: Labor Share = = . Chapter 3 National Income: Where It Comes From and Where It Goes 21 0.7A(1.1K) –0.7L0.7 0.7AK–0.7L0.7 (W/P)2 (W/P)1 0.7(1.1A)K0.3L–0.3 0.7AK0.3L–0.3 (1.1A)K0.3L0.7 AK0.3L0.7 Y2 Y1 0.3(1.1A)K–0.7L0.7 0.3AK–0.7L0.7 (R/P)2 (R/P)1 dY dL 1 3 dY dH 1 3 1 3 (W/P)2 (W/P)1 ( ) 1 3 13 13 23 13 13 13 KHL L KHL -

Answers to Textbook questions and Problems We can use the same logic to find the human capital share ()H Human Capital share KIshI3L13 1 so labor gets one-third of the output, and human capital gets one-third of the out put. Since workers own their human capital(we hope!), it will appear that labor gets two-thirds of output. d. The ratio of the skilled wage to the unskilled wage is MPL MPh KLH18+1K1③L1aH2 K L-23HI3 Notice that the ratio is always greater than 1 because skilled workers get paid more than unskilled workers. Also, when h increases this ratio falls because the diminishing returns to human capital lower its return, while at the same time increasing the marginal product of unskilled workers. e. If more college scholarships increase H, then it does lead to a more egalitarian society. The policy lowers the returns to education, decreasing the gap between che wages of more and less educated workers. More importantly, the policy even raises the absolute wage of unskilled workers because their marginal product rises when the number of skilled workers rises

We can use the same logic to find the human capital share: Human Capital Share = = , so labor gets one-third of the output, and human capital gets one-third of the out￾put. Since workers own their human capital (we hope!), it will appear that labor gets two-thirds of output. d. The ratio of the skilled wage to the unskilled wage is: = = = 1 + . Notice that the ratio is always greater than 1 because skilled workers get paid more than unskilled workers. Also, when H increases this ratio falls because the diminishing returns to human capital lower its return, while at the same time increasing the marginal product of unskilled workers. e. If more college scholarships increase H, then it does lead to a more egalitarian society. The policy lowers the returns to education, decreasing the gap between the wages of more and less educated workers. More importantly, the policy even raises the absolute wage of unskilled workers because their marginal product rises when the number of skilled workers rises. 22 Answers to Textbook Questions and Problems ( K1/3L1/3H–2/3) H K1/3H1/3L1/3 1 3 1 3 K1/3L–2/3H1/3 + K1/3L1/3H–2/3 K1/3L–2/3H1/3 L H 1 3 1 3 1 3 MPL + MPH MPL Wskilled Wunskilled

CHAPTER Money and Inflation questions for Review 1. Money has three functions: it is a store of value, a unit of account, and a medium of exchange. As a store of value, money provides a way to transfer purchasing power from the present to the future. As a unit of account, money provides the terms in which prices are quoted and debts are recorded. As a medium of exchange, money is what we use to buy goods and services 2. Fiat money is established as money by the government but has no intrinsic value. For example, a U.S. dollar bill is fiat money. Commodity money is money that is based on a commodity with some intrinsic value. Gold, when used as money, is an example of com- 3. In many countries, a central bank controls the money supply. In the United States, the central bank is the Federal Reserve--often called the Fed. The control of the money supply is called monetary policy The primary way that the Fed controls the money supply is through open-market operations, which involve the purchase or sale of government bonds. To increase the money supply, the Fed uses dollars to buy government bonds from the public, putting more dollars into the hands of the public. To decrease the money supply, the Fed sells some of its government bonds, taking dollars out of the hands of the public 4. The quantity equation is an identity that expresses the link between the number of transactions that people make and how much money they hold. We write it as Money x velocity Price x Transactions MxV=P The right-hand side of the quantity equation tells us about the total number of transac tions that occur during a given period of time, say, a year T represents the total num- ber of times that any two individuals exchange goods or services for money. P repre- sents the price of a typical transaction. Hence, the product P x T represents the number of dollars exchanged in a year. The left-hand side of the quantity equation tells us about the money used to make these transactions. M represents the quantity of money in the economy. V represents the transactions velocity of money-the rate at which money circulates in the economy Because the number of transactions is difficult to measure, economists usually use a slightly different version of the quantity equation, in which the total output of the economy Y replaces the number of transactions T Money x velocity Price x Output M×V=P×Y. P now represents the price of one unit of output, so that px y is the dollar value of out- put--nominal GDP. V represents the income velocity of money-the number of times dollar bill becomes a part of someone s income. 5. If we assume that velocity in the quantity equation is constant, then we can view the quantity equation as a theory of nominal gDP. The quantity equation with fixed veloci- MV=PY If velocity V is constant, then a change in the quantity of money(M)causes a propor tionate change in nominal GDP (Py). If we assume further that output is fixed by the factors of production and the production technology, then we can conclude that the

Questions for Review 1. Money has three functions: it is a store of value, a unit of account, and a medium of exchange. As a store of value, money provides a way to transfer purchasing power from the present to the future. As a unit of account, money provides the terms in which prices are quoted and debts are recorded. As a medium of exchange, money is what we use to buy goods and services. 2. Fiat money is established as money by the government but has no intrinsic value. For example, a U.S. dollar bill is fiat money. Commodity money is money that is based on a commodity with some intrinsic value. Gold, when used as money, is an example of com￾modity money. 3. In many countries, a central bank controls the money supply. In the United States, the central bank is the Federal Reserve—often called the Fed. The control of the money supply is called monetary policy. The primary way that the Fed controls the money supply is through open-market operations, which involve the purchase or sale of government bonds. To increase the money supply, the Fed uses dollars to buy government bonds from the public, putting more dollars into the hands of the public. To decrease the money supply, the Fed sells some of its government bonds, taking dollars out of the hands of the public. 4. The quantity equation is an identity that expresses the link between the number of transactions that people make and how much money they hold. We write it as Money × Velocity = Price × Transactions M × V = P × T. The right-hand side of the quantity equation tells us about the total number of transac￾tions that occur during a given period of time, say, a year. T represents the total num￾ber of times that any two individuals exchange goods or services for money. P repre￾sents the price of a typical transaction. Hence, the product P × T represents the number of dollars exchanged in a year. The left-hand side of the quantity equation tells us about the money used to make these transactions. M represents the quantity of money in the economy. V represents the transactions velocity of money—the rate at which money circulates in the economy. Because the number of transactions is difficult to measure, economists usually use a slightly different version of the quantity equation, in which the total output of the economy Y replaces the number of transactions T: Money × Velocity = Price × Output M × V = P × Y. P now represents the price of one unit of output, so that P × Y is the dollar value of out￾put—nominal GDP. V represents the income velocity of money—the number of times a dollar bill becomes a part of someone’s income. 5. If we assume that velocity in the quantity equation is constant, then we can view the quantity equation as a theory of nominal GDP. The quantity equation with fixed veloci￾ty states that MV = PY. If velocity V is constant, then a change in the quantity of money (M) causes a propor￾tionate change in nominal GDP (PY). If we assume further that output is fixed by the factors of production and the production technology, then we can conclude that the 23 CHAPTER 4 Money and Inflation

24 Answers to Textbook questions and Problems quantity of money determines the price level. This is called the quantity theory of money. 6. The holders of money pay the inflation tax As prices rise, the real value of the money that people hold falls-that is, a given amount of money buys fewer goods and services since prices are higher 7. The Fisher equation expresses the relationship between nominal and real interest rates. It says that the nominal interest rate i equals the real interest rate r plus the inflation rateπ: This tells us that the nominal interest rate can change either because the real interest rate changes or the inflation rate changes. The real interest rate is assumed to be unaf- fected by inflation; as discussed in Chapter 3, it adjusts to equilibrate saving and investment. There is thus a one-to-one relationship between the inflation rate and the nominal interest rate: if inflation increases by 1 percent, then the nominal interest rate also increases by 1 percent. This one-to-one relationship is called the Fisher effect If inflation increases from 6 to 8 percent, then the Fisher effect implies that nominal interest rate increases by 2 percentage points, while the real interest rate remains constant 8. The costs of expected inflation include the following: a. Shoeleather costs. Higher inflation means higher nominal interest rates, which mean that people want to hold lower real money balances. If people hold lower money balances, they must make more frequent trips to the bank to withdraw money. This is inconvenient (and it causes shoes to wear out more quickly). b. Menu costs. Higher inflation induces firms to change their posted prices more often. This may be costly if they must reprint their menus and catalogs c. Greater variability in relative prices. If firms change their prices infrequently, then inflation causes greater variability in relative prices. Since free-market economies rely on relative prices to allocate resources efficiently, inflation leads to microeconomic inefficiencies d. Altered tax liabilities. Many provisions of the tax code do not take into account the effect of inflation. Hence. inflation can alter individuals and firms tax liabilities often in ways that lawmakers did not intend e. The inconvenience of a changing price level. It is inconvenient to live in a world with a changing price level. Money is the yardstick with which we measure eco- nomic transactions. Money is a less useful measure when its value is always There is an additional cost to unexpected inflation: f. Arbitrary redistributions of wealth. Unexpected inflation arbitrarily redistributes wealth among individuals. For example, if inflation is higher than expected, debtors gain and creditors lose. Also, people with fixed pensions are hurt because their dollars buy fewer goods 9. A hyperinflation always reflects monetary policy. That is, the price level cannot grow rapidly unless the supply of money also grows rapidly; and hyperinflations do not end unless the government drastically reduces money growth. This explanation, however, begs a central question: Why does the government start and then stop printing lots of money? The answer almost always lies in fiscal policy: When the government has a large budget deficit (possibly due to a recent war or some other major event)that it ca not fund by borrowing, it resorts to printing money to pay its bills. And only when this fiscal problem is alleviated--by reducing government spending and collecting more taxes-can the government hope to slow its rate of money growtl 10. A real uariable is one that is measured in units that are constant over time-for exam- ple, they might be measured in"constant dollars. "That is, the units are adjusted for

quantity of money determines the price level. This is called the quantity theory of money. 6. The holders of money pay the inflation tax. As prices rise, the real value of the money that people hold falls—that is, a given amount of money buys fewer goods and services since prices are higher. 7. The Fisher equation expresses the relationship between nominal and real interest rates. It says that the nominal interest rate i equals the real interest rate r plus the inflation rate π: i = r + π. This tells us that the nominal interest rate can change either because the real interest rate changes or the inflation rate changes. The real interest rate is assumed to be unaf￾fected by inflation; as discussed in Chapter 3, it adjusts to equilibrate saving and investment. There is thus a one-to-one relationship between the inflation rate and the nominal interest rate: if inflation increases by 1 percent, then the nominal interest rate also increases by 1 percent. This one-to-one relationship is called the Fisher effect. If inflation increases from 6 to 8 percent, then the Fisher effect implies that the nominal interest rate increases by 2 percentage points, while the real interest rate remains constant. 8. The costs of expected inflation include the following: a. Shoeleather costs. Higher inflation means higher nominal interest rates, which mean that people want to hold lower real money balances. If people hold lower money balances, they must make more frequent trips to the bank to withdraw money. This is inconvenient (and it causes shoes to wear out more quickly). b. Menu costs. Higher inflation induces firms to change their posted prices more often. This may be costly if they must reprint their menus and catalogs. c. Greater variability in relative prices. If firms change their prices infrequently, then inflation causes greater variability in relative prices. Since free-market economies rely on relative prices to allocate resources efficiently, inflation leads to microeconomic inefficiencies. d. Altered tax liabilities. Many provisions of the tax code do not take into account the effect of inflation. Hence, inflation can alter individuals’ and firms’ tax liabilities, often in ways that lawmakers did not intend. e. The inconvenience of a changing price level. It is inconvenient to live in a world with a changing price level. Money is the yardstick with which we measure eco￾nomic transactions. Money is a less useful measure when its value is always changing. There is an additional cost to unexpected inflation: f. Arbitrary redistributions of wealth. Unexpected inflation arbitrarily redistributes wealth among individuals. For example, if inflation is higher than expected, debtors gain and creditors lose. Also, people with fixed pensions are hurt because their dollars buy fewer goods. 9. A hyperinflation always reflects monetary policy. That is, the price level cannot grow rapidly unless the supply of money also grows rapidly; and hyperinflations do not end unless the government drastically reduces money growth. This explanation, however, begs a central question: Why does the government start and then stop printing lots of money? The answer almost always lies in fiscal policy: When the government has a large budget deficit (possibly due to a recent war or some other major event) that it can￾not fund by borrowing, it resorts to printing money to pay its bills. And only when this fiscal problem is alleviated—by reducing government spending and collecting more taxes—can the government hope to slow its rate of money growth. 10. A real variable is one that is measured in units that are constant over time—for exam￾ple, they might be measured in “constant dollars.” That is, the units are adjusted for 24 Answers to Textbook Questions and Problems