ºê¹Û¾­¼Ãѧ£¨Ó¢ÎÄ£©£ºAnswers to Textbook Questions and Problems_Mankiw

Questions for Review 1. GDP measures both the total income of everyone in the economy and the total expendi- ture on the economy¡¯s output of goods and services. GDP can measure two things at once because both are really the same thing: for an economy as a whole, income must equal expenditure. As the circular flow diagram in the text illustrates, these are alter- native, equivalent ways of measuring the flow of dollars in the economy. 2. The consumer price index measures the overall level of prices in the economy. It tells us the price of a fixed basket of goods relative to the price of the same basket in the base year. 3. The Bureau of Labor Statistics classifies each person into one of the following three cat- egories: employed, unemployed, or not in the labor force. The unemployment rate, which is the percentage of the labor force that is unemployed, is computed as follows: Unemployment Rate = . Note that the labor force is the number of people employed plus the number of people unemployed. 4. Okun¡¯s law refers to the negative relationship that exists between unemployment and real GDP. Employed workers help produce goods and services whereas unemployed workers do not. Increases in the unemployment rate are therefore associated with decreases in real GDP. Okun¡¯s law can be summarized by the equation: %?Real GDP = 3% ¨C 2 × (?Unemployment Rate). That is, if unemployment does not change, the growth rate of real GDP is 3 percent. For every percentage-point change in unemployment (for example, a fall from 6 percent to 5 percent, or an increase from 6 percent to 7 percent), output changes by 2 percent in the opposite direction. Problems and Applications 1. A large number of economic statistics are released regularly. These include the follow- ing: Gross Domestic Product¡ªthe market value of all final goods and services produced in a year. The Unemployment Rate¡ªthe percentage of the civilian labor force who do not have a job. Corporate Profits¡ªthe accounting profits remaining after taxes of all manufacturing corporations. It gives an indication of the general financial health of the corporate sec- tor. The Consumer Price Index (CPI)¡ªa measure of the average price that consumers pay for the goods they buy; changes in the CPI are a measure of inflation. The Trade Balance¡ªthe difference between the value of goods exported abroad and the value of goods imported from abroad. 5 Number of Unemployed × 100 Labor Force CHAPTER 2 The Data of Macroeconomics 2. Value added by each person is the value of the good produced minus the amount the person paid for the materials necessary to make the good. Therefore, the value added by the farmer is $1.00 ($1 ¨C 0 = $1). The value added by the miller is $2: she sells the flour to the baker for $3 but paid $1 for the flour. The value added by the baker is $3: she sells the bread to the engineer for $6 but paid the miller $3 for the flour. GDP is the total value added, or $1 + $2 + $3 = $6. Note that GDP equals the value of the final good (the bread). 3. When a woman marries her butler, GDP falls by the amount of the butler¡¯s salary. This happens because measured total income, and therefore measured GDP, falls by the amount of the butler¡¯s loss in salary. If GDP truly measured the value of all goods and services, then the marriage would not affect GDP since the total amount of economic activity is unchanged. Actual GDP, however, is an imperfect measure of economic activ- ity because the value of some goods and services is left out. Once the butler¡¯s work becomes part of his household chores, his services are no longer counted in GDP. As this example illustrates, GDP does not include the value of any output produced in the home. Similarly, GDP does not include other goods and services, such as the imputed rent on durable goods (e.g., cars and refrigerators) and any illegal trade. 4. a. government purchases b. investment c. net exports d. consumption e. investment 5. Data on parts (a) to (g) can be downloaded from the Bureau of Economic Analysis (www.bea.doc.gov¡ªfollow the links to GDP and related data). Most of the data (not necessarily the earliest year) can also be found in the Economic Report of the President. By dividing each component (a) to (g) by nominal GDP and multiplying by 100, we obtain the following percentages: 1950 1975 2000 a. Personal consumption expenditures 65.5% 63.0% 68.2% b. Gross private domestic investment 18.4% 14.1% 17.9% c. Government consumption purchases 15.9% 22.1% 17.6% d. Net exports 0.2% 0.8% ¨C3.7% e. National defense purchases 6.7% 6.6% 3.8% f. State and local purchases 7.1% 12.8% 11.7% g. Imports 3.9% 7.5% 14.9% (Note: These data were downloaded February 5, 2002 from the BEA web site.) Among other things, we observe the following trends in the economy over the period 1950¨C2000: (a) Personal consumption expenditures have been around two-thirds of GDP, although the share increased about 5 percentage points between 1975 and 2000. (b) The share of GDP going to gross private domestic investment fell from 1950 to 1975 but then rebounded. (c) The share going to government consumption purchases rose more than 6 percentage points from 1950 to 1975 but has receded somewhat since then. (d) Net exports, which were positive in 1950 and 1975, were substantially negative in 2000. (e) The share going to national defense purchases fell from 1975 to 2000. (f) The share going to state and local purchases rose from 1950 to 1975. (g) Imports have grown rapidly relative to GDP. 6 Answers to Textbook Questions and Problems 6. a. i. Nominal GDP is the total value of goods and services measured at current prices. Therefore, Nominal GDP 2000 = (P × Q ) + (P × Q ) = ($50,000 × 100) + ($10 × 500,000) = $5,000,000 + $5,000,000 = $10,000,000. Nominal GDP 2010 = (P × Q ) + (P × Q ) = ($60,000 × 120) + ($20 × 400,000) = $7,200,000 + $8,000,000 = $15,200,000. ii. Real GDP is the total value of goods and services measured at constant prices. Therefore, to calculate real GDP in 2010 (with base year 2000), multi- ply the quantities purchased in the year 2010 by the 2000 prices: Real GDP 2010 = (P × Q ) + (P × Q ) = ($50,000 × 120) + ($10 × 400,000) = $6,000,000+ $4,000,000 = $10,000,000. Real GDP for 2000 is calculated by multiplying the quantities in 2000 by the prices in 2000. Since the base year is 2000, real GDP 2000 equals nominal GDP 2000 , which is $10,000,000. Hence, real GDP stayed the same between 2000 and 2010. iii. The implicit price deflator for GDP compares the current prices of all goods and services produced to the prices of the same goods and services in a base year. It is calculated as follows: Implicit Price Deflator 2010 = . Using the values for Nominal GDP 2010 and real GDP 2010 calculated above: Implicit Price Deflator 2010 = = 1.52. This calculation reveals that prices of the goods produced in the year 2010 increased by 52 percent compared to the prices that the goods in the economy sold for in 2000. (Because 2000 is the base year, the value for the implicit price deflator for the year 2000 is 1.0 because nominal and real GDP are the same for the base year.) iv. The consumer price index (CPI) measures the level of prices in the economy. The CPI is called a fixed-weight index because it uses a fixed basket of goods over time to weight prices. If the base year is 2000, the CPI in 2010 is an average of prices in 2010, but weighted by the composition of goods produced in 2000. The CPI 2010 is calculated as follows: CPI 2010 = = = = 1.6. Chapter 2 The Data of Macroeconomics 7 2000 cars 2000 cars 2000 bread 2000 bread 2010 cars 2010 cars 2010 bread 2010 bread 2000 cars 2010 cars 2000 bread 2010 bread Nominal GDP 2010 Real GDP 2010 $15,200,000 $10,000,000 $16,000,000 $10,000,000 ($60,000 × 100) + ($20 × 500,000) ($50,000 × 100) + ($10 × 500,000) ()( ) ( PQ PQ PQ cars cars bread bread cars c 2010 2000 2010 2000 2000 ×+× × aars bread bread 2000 2000 2000 )( )+×PQ This calculation shows that the price of goods purchased in 2010 increased by 60 percent compared to the prices these goods would have sold for in 2000. The CPI for 2000, the base year, equals 1.0. b. The implicit price deflator is a Paasche index because it is computed with a chang- ing basket of goods; the CPI is a Laspeyres index because it is computed with a fixed basket of goods. From (5.a.iii), the implicit price deflator for the year 2010 is 1.52, which indicates that prices rose by 52 percent from what they were in the year 2000. From (5.a.iv.), the CPI for the year 2010 is 1.6, which indicates that prices rose by 60 percent from what they were in the year 2000. If prices of all goods rose by, say, 50 percent, then one could say unam- biguously that the price level rose by 50 percent. Yet, in our example, relative prices have changed. The price of cars rose by 20 percent; the price of bread rose by 100 percent, making bread relatively more expensive. As the discrepancy between the CPI and the implicit price deflator illus- trates, the change in the price level depends on how the goods¡¯ prices are weight- ed. The CPI weights the price of goods by the quantities purchased in the year 2000. The implicit price deflator weights the price of goods by the quantities pur- chased in the year 2010. The quantity of bread consumed was higher in 2000 than in 2010, so the CPI places a higher weight on bread. Since the price of bread increased relatively more than the price of cars, the CPI shows a larger increase in the price level. c. There is no clear-cut answer to this question. Ideally, one wants a measure of the price level that accurately captures the cost of living. As a good becomes relatively more expensive, people buy less of it and more of other goods. In this example, consumers bought less bread and more cars. An index with fixed weights, such as the CPI, overestimates the change in the cost of living because it does not take into account that people can substitute less expensive goods for the ones that become more expensive. On the other hand, an index with changing weights, such as the GDP deflator, underestimates the change in the cost of living because it does not take into account that these induced substitutions make people less well off. 7. a. The consumer price index uses the consumption bundle in year 1 to figure out how much weight to put on the price of a given good: CPI 2 = = = 2. According to the CPI, prices have doubled. b. Nominal spending is the total value of output produced in each year. In year 1 and year 2, Abby buys 10 apples for $1 each, so her nominal spending remains con- stant at $10. For example, Nominal Spending 2 = (P × Q ) + (P × Q ) = ($2 × 0) + ($1 × 10) = $10. 8 Answers to Textbook Questions and Problems ($2 × 10) + ($1 × 0) ($1 × 10) + ($2 × 0) 2 red 2 red 2 green 2 green ()( ) ()( PQ P Q PQ P Q red red green green red red green gre 21 2 1 11 1 ×+ × ×+ × een 1 ) c. Real spending is the total value of output produced in each year valued at the prices prevailing in year 1. In year 1, the base year, her real spending equals her nominal spending of $10. In year 2, she consumes 10 green apples that are each valued at their year 1 price of $2, so her real spending is $20. That is, Real Spending 2 = (P × Q ) + (P × Q ) = ($1 × 0) + ($2 × 10) = $20. Hence, Abby¡¯s real spending rises from $10 to $20. d. The implicit price deflator is calculated by dividing Abby¡¯s nominal spending in year 2 by her real spending that year: Implicit Price Deflator 2 = = = 0.5. Thus, the implicit price deflator suggests that prices have fallen by half. The rea- son for this is that the deflator estimates how much Abby values her apples using prices prevailing in year 1. From this perspective green apples appear very valu- able. In year 2, when Abby consumes 10 green apples, it appears that her con- sumption has increased because the deflator values green apples more highly than red apples. The only way she could still be spending $10 on a higher consumption bundle is if the price of the good she was consuming feel. e. If Abby thinks of red apples and green apples as perfect substitutes, then the cost of living in this economy has not changed¡ªin either year it costs $10 to consume 10 apples. According to the CPI, however, the cost of living has doubled. This is because the CPI only takes into account the fact that the red apple price has dou- bled; the CPI ignores the fall in the price of green apples because they were not in the consumption bundle in year 1. In contrast to the CPI, the implicit price defla- tor estimates the cost of living has halved. Thus, the CPI, a Laspeyres index, over- states the increase in the cost of living and the deflator, a Paasche index, under- states it. This chapter of the text discusses the difference between Laspeyres and Paasche indices in more detail. 8. a. Real GDP falls because Disney does not produce any services while it is closed. This corresponds to a decrease in economic well-being because the income of work- ers and shareholders of Disney falls (the income side of the national accounts), and people¡¯s consumption of Disney falls (the expenditure side of the national accounts). b. Real GDP rises because the original capital and labor in farm production now pro- duce more wheat. This corresponds to an increase in the economic well-being of society, since people can now consume more wheat. (If people do not want to con- sume more wheat, then farmers and farmland can be shifted to producing other goods that society values.) c. Real GDP falls because with fewer workers on the job, firms produce less. This accurately reflects a fall in economic well-being. d. Real GDP falls because the firms that lay off workers produce less. This decreases economic well-being because workers¡¯ incomes fall (the income side), and there are fewer goods for people to buy (the expenditure side). e. Real GDP is likely to fall, as firms shift toward production methods that produce fewer goods but emit less pollution. Economic well-being, however, may rise. The economy now produces less measured output but more clean air; clean air is not Chapter 2 The Data of Macroeconomics 9 1 red 2 red 1 green 2 green Nominal Spending 2 Real Spending 2 $10 $20 traded in markets and, thus, does not show up in measured GDP, but is neverthe- less a good that people value. f. Real GDP rises because the high-school students go from an activity in which they are not producing market goods and services to one in which they are. Economic well-being, however, may decrease. In ideal national accounts, attending school would show up as investment because it presumably increases the future produc- tivity of the worker. Actual national accounts do not measure this type of invest- ment. Note also that future GDP may be lower than it would be if the students stayed in school, since the future work force will be less educated. g. Measured real GDP falls because fathers spend less time producing market goods and services. The actual production of goods and services need not have fallen, however. Measured production (what the fathers are paid to do) falls, but unmea- sured production of child-rearing services rises. 9. As Senator Robert Kennedy pointed out, GDP is an imperfect measure of economic per- formance or well-being. In addition to the left-out items that Kennedy cited, GDP also ignores the imputed rent on durable goods such as cars, refrigerators, and lawnmowers; many services and products produced as part of household activity, such as cooking and cleaning; and the value of goods produced and sold in illegal activities, such as the drug trade. These imperfections in the measurement of GDP do not necessarily reduce its usefulness. As long as these measurement problems stay constant over time, then GDP is useful in comparing economic activity from year to year. Moreover, a large GDP allows us to afford better medical care for our children, newer books for their education, and more toys for their play. 10 Answers to Textbook Questions and Problems Questions for Review 1. Factors of production and the production technology determine the amount of output an economy can produce. Factors of production are the inputs used to produce goods and services: the most important factors are capital and labor. The production technology determines how much output can be produced from any given amounts of these inputs. An increase in one of the factors of production or an improvement in technology leads to an increase in the economy¡¯s output. 2. When a firm decides how much of a factor of production to hire, it considers how this decision affects profits. For example, hiring an extra unit of labor increases output and therefore increases revenue; the firm compares this additional revenue to the addition- al cost from the higher wage bill. The additional revenue the firm receives depends on the marginal product of labor (MPL) and the price of the good produced (P). An addi- tional unit of labor produces MPL units of additional output, which sells for P dollars. Therefore, the additional revenue to the firm is P × MPL. The cost of hiring the addi- tional unit of labor is the wage W. Thus, this hiring decision has the following effect on profits: ?Profit = ?Revenue ¨C ?Cost = (P × MPL) ¨C W. If the additional revenue, P × MPL, exceeds the cost (W) of hiring the additional unit of labor, then profit increases. The firm will hire labor until it is no longer profitable to do so¡ªthat is, until the MPL falls to the point where the change in profit is zero. In the equation above, the firm hires labor until ?profit = 0, which is when (P × MPL) = W. This condition can be rewritten as: MPL = W/P. Therefore, a competitive profit-maximizing firm hires labor until the marginal product of labor equals the real wage. The same logic applies to the firm¡¯s decision to hire capi- tal: the firm will hire capital until the marginal product of capital equals the real rental price. 3. A production function has constant returns to scale if an equal percentage increase in all factors of production causes an increase in output of the same percentage. For exam- ple, if a firm increases its use of capital and labor by 50 percent, and output increases by 50 percent, then the production function has constant returns to scale. If the production function has constant returns to scale, then total income (or equivalently, total output) in an economy of competitive profit-maximizing firms is divided between the return to labor, MPL × L, and the return to capital, MPK × K. That is, under constant returns to scale, economic profit is zero. 4. Consumption depends positively on disposable income¡ªthe amount of income after all taxes have been paid. The higher disposable income is, the greater consumption is. The quantity of investment goods demanded depends negatively on the real inter- est rate. For an investment to be profitable, its return must be greater than its cost. Because the real interest rate measures the cost of funds, a higher real interest rate makes it more costly to invest, so the demand for investment goods falls. 5. Government purchases are those goods and services purchased directly by the govern- ment. For example, the government buys missiles and tanks, builds roads, and provides 11 CHAPTER 3 National Income: Where It Comes From and Where It Goes services such as air traffic control. All of these activities are part of GDP. Transfer pay- ments are government payments to individuals that are not in exchange for goods or services. They are the opposite of taxes: taxes reduce household disposable income, whereas transfer payments increase it. Examples of transfer payments include Social Security payments to the elderly, unemployment insurance, and veterans¡¯ benefits. 6. Consumption, investment, and government purchases determine demand for the econo- my¡¯s output, whereas the factors of production and the production function determine the supply of output. The real interest rate adjusts to ensure that the demand for the economy¡¯s goods equals the supply. At the equilibrium interest rate, the demand for goods and services equals the supply. 7. When the government increases taxes, disposable income falls, and therefore consump- tion falls as well. The decrease in consumption equals the amount that taxes increase multiplied by the marginal propensity to consume (MPC). The higher the MPC is, the greater is the negative effect of the tax increase on consumption. Because output is fixed by the factors of production and the production technology, and government pur- chases have not changed, the decrease in consumption must be offset by an increase in investment. For investment to rise, the real interest rate must fall. Therefore, a tax increase leads to a decrease in consumption, an increase in investment, and a fall in the real interest rate. Problems and Applications 1. a. According to the neoclassical theory of distribution, the real wage equals the mar- ginal product of labor. Because of diminishing returns to labor, an increase in the labor force causes the marginal product of labor to fall. Hence, the real wage falls. b. The real rental price equals the marginal product of capital. If an earthquake destroys some of the capital stock (yet miraculously does not kill anyone and lower the labor force), the marginal product of capital rises and, hence, the real rental price rises. c. If a technological advance improves the production function, this is likely to increase the marginal products of both capital and labor. Hence, the real wage and the real rental price both increase. 2. A production function has decreasing returns to scale if an equal percentage increase in all factors of production leads to a smaller percentage increase in output. For example, if we double the amounts of capital and labor, and output less than doubles, then the production function has decreasing returns to capital and labor. This may happen if there is a fixed factor such as land in the production function, and this fixed factor becomes scarce as the economy grows larger. A production function has increasing returns to scale if an equal percentage increase in all factors of production leads to a larger percentage increase in output. For example, if doubling inputs of capital and labor more than doubles output, then the pro- duction function has increasing returns to scale. This may happen if specialization of labor becomes greater as population grows. For example, if one worker builds a car, then it takes him a long time because he has to learn many different skills, and he must constantly change tasks and tools; all of this is fairly slow. But if many workers build a car, then each one can specialize in a particular task and become very fast at it. 3. a. According to the neoclassical theory, technical progress that increases the margin- al product of farmers causes their real wage to rise. b. The real wage in (a) is measured in terms of farm goods. That is, if the nominal wage is in dollars, then the real wage is W/PF, where PF is the dollar price of farm goods. 12 Answers to Textbook Questions and Problems c. If the marginal productivity of barbers is unchanged, then their real wage is unchanged. d. The real wage in (c) is measured in terms of haircuts. That is, if the nominal wage is in dollars, then the real wage is W/PH, where PH is the dollar price of a hair- cut. e. If workers can move freely between being farmers and being barbers, then they must be paid the same wage W in each sector. f. If the nominal wage W is the same in both sectors, but the real wage in terms of farm goods is greater than the real wage in terms of haircuts, then the price of haircuts must have risen relative to the price of farm goods. g. Both groups benefit from technological progress in farming. 4. The effect of a government tax increase of $100 billion on (a) public saving, (b) private saving, and (c) national saving can be analyzed by using the following relationships: National Saving = [Private Saving] + [Public Saving] = [Y ¨C T ¨C C(Y ¨C T)] + [T ¨C G] = Y ¨C C(Y ¨C T) ¨C G. a. Public Saving¡ªThe tax increase causes a 1-for-1 increase in public saving. T increases by $100 billion and, therefore, public saving increases by $100 billion. b. Private Saving¡ªThe increase in taxes decreases disposable income, Y ¨C T, by $100 billion. Since the marginal propensity to consume (MPC) is 0.6, consumption falls by 0.6 × $100 billion, or $60 billion. Hence, ?Private Saving = ¨C $100b ¨C 0.6 ( ¨C $100b) = ¨C $40b. Private saving falls $40 billion. c. National Saving¡ªBecause national saving is the sum of private and public sav- ing, we can conclude that the $100 billion tax increase leads to a $60 billion increase in national saving. Another way to see this is by using the third equation for national saving expressed above, that national saving equals Y ¨C C(Y ¨C T) ¨C G. The $100 billion tax increase reduces disposable income and causes consumption to fall by $60 bil- lion. Since neither G nor Y changes, national saving thus rises by $60 billion. d. Investment¡ªTo determine the effect of the tax increase on investment, recall the national accounts identity: Y = C(Y ¨C T) + I(r) + G. Rearranging, we find Y ¨C C(Y ¨C T) ¨C G = I(r). The left-hand side of this equation is national saving, so the equation just says the national saving equals investment. Since national saving increases by $60 billion, investment must also increase by $60 billion. Chapter 3 National Income: Where It Comes From and Where It Goes 13 How does this increase in investment take place? We know that investment depends on the real interest rate. For investment to rise, the real interest rate must fall. Figure 3¨C1 graphs saving and investment as a function of the real inter- est rate. The tax increase causes national saving to rise, so the supply curve for loan- able funds shifts to the right. The equilibrium real interest rate falls, and invest- ment rises. 5. If consumers increase the amount that they consume today, then private saving and, therefore, national saving will fall. We know this from the definition of national saving: National Saving = [Private Saving] + [Public Saving] = [Y ¨C T ¨C C(Y ¨C T)] + [T ¨C G]. An increase in consumption decreases private saving, so national saving falls. Figure 3¨C2 graphs saving and investment as a function of the real interest rate. If national saving decreases, the supply curve for loanable funds shifts to the left, thereby raising the real interest rate and reducing investment. 14 Answers to Textbook Questions and Problems S 1 S 2 I (r) I, S Investment, Saving r 1 r 2 r Real interest rate Figure 3¨C1 Figure 3¨C1 S 2 S 1 I (r) I, S Investment, Saving r 1 r 2 r Real interest rate Figure 3¨C2 Figure 3¨C2 6. a. Private saving is the amount of disposable income, Y ¨C T, that is not consumed: S private = Y ¨C T ¨C C = 5,000 ¨C 1,000 ¨C (250 + 0.75(5,000 ¨C 1,000)) = 750. Public saving is the amount of taxes the government has left over after it makes its purchases: S public = T ¨C G = 1,000 ¨C 1,000 = 0. Total saving is the sum of private saving and public saving: S = S private + S public = 750 + 0 = 750. b. The equilibrium interest rate is the value of r that clears the market for loanable funds. We already know that national saving is 750, so we just need to set it equal to investment: S = I 750 = 1,000 ¨C 50r Solving this equation for r, we find: r = 5%. c. When the government increases its spending, private saving remains the same as before (notice that G does not appear in the S private above) while government saving decreases. Putting the new G into the equations above: S private = 750 S public = T ¨C G = 1,000 ¨C 1,250 = ¨C250. Thus, S = S private + S public = 750 + (¨C250) = 500. d. Once again the equilibrium interest rate clears the market for loanable funds: S = I 500 = 1,000 ¨C 50r Solving this equation for r, we find: r = 10%. 7. To determine the effect on investment of an equal increase in both taxes and govern- ment spending, consider the national income accounts identity for national saving: National Saving = [Private Saving] + [Public Saving] = [Y ¨C T ¨C C(Y ¨C T)] + [T ¨C G]. We know that Y is fixed by the factors of production. We also know that the change in consumption equals the marginal propensity to consume (MPC) times the change in disposable income. This tells us that ?National Saving = [¨C ?T ¨C (MPC × ( ¨C ?T))] + [?T ¨C ?G] = [¨C ?T + (MPC ×?T)] + 0 = (MPC ¨C 1) ?T. Chapter 3 National Income: Where It Comes From and Where It Goes 15 The above expression tells us that the impact on saving of an equal increase in T and G depends on the size of the marginal propensity to consume. The closer the MPC is to 1, the smaller is the fall in saving. For example, if the MPC equals 1, then the fall in consumption equals the rise in government purchases, so national saving [Y ¨C C(Y ¨C T) ¨C G] is unchanged. The closer the MPC is to 0 (and therefore the larger is the amount saved rather than spent for a one-dollar change in disposable income), the greater is the impact on saving. Because we assume that the MPC is less than 1, we expect that national saving falls in response to an equal increase in taxes and govern- ment spending. The reduction in saving means that the supply of loanable funds curve shifts to the left in Figure 3¨C3. The real interest rate rises, and investment falls. 8. a. The demand curve for business investment shifts out because the subsidy increas- es the number of profitable investment opportunities for any given interest rate. The demand curve for residential investment remains unchanged. b. The total demand curve for investment in the economy shifts out since it repre- sents the sum of business investment, which shifts out, and residential invest- ment, which is unchanged. As a result the real interest rate rises as in Figure 3¨C4. 16 Answers to Textbook Questions and Problems S 2 S 1 I (r) I, S Investment, Saving r 1 r 2 r Real interest rate Figure 3¨C2 Figure 3¨C3 c. The total quantity of investment does not change because it is constrained by the inelastic supply of savings. The investment tax credit leads to a rise in business investment, but an offsetting fall in residential investment. That is, the higher interest rate means that residential investment falls (a shift along the curve), whereas the outward shift of the business investment curve leads business invest- ment to rise by an equal amount. Figure 3¨C5 shows this change. Note that 9. In this chapter, we concluded that an increase in government expenditures reduces national saving and raises the interest rate; it therefore crowds out investment by the full amount of the increase in government expenditure. Similarly, a tax cut increases disposable income and hence consumption; this increase in consumption translates into a fall in national saving¡ªagain, it crowds out investment. Chapter 3 National Income: Where It Comes From and Where It Goes 17 S I 2 I 1 I, S A B 1. An increase in desired investment . . . 2. . . . raises the interest rate. Investment, Saving Real interest rate r Figure 3¨C4 Business Investment I 2 B I 1 B Residential Investment I 2 R I 1 R r r 1 r 2 r r 1 r 2 Figure 3¨C5 I IIIS 11 22 BR BR +=+= Business investment Residential investment Figure 3¨C4 If consumption depends on the interest rate, then these conclusions about fiscal policy are modified somewhat. If consumption depends on the interest rate, then so does saving. The higher the interest rate, the greater the return to saving. Hence, it seems reasonable to think that an increase in the interest rate might increase saving and reduce consumption. Figure 3¨C6 shows saving as an increasing function of the interest rate. Consider what happens when government purchases increase. At any given level of the interest rate, national saving falls by the change in government purchases, as shown in Figure 3¨C7. The figure shows that if the saving function slopes upward, investment falls by less than the amount that government purchases rise; this happens because consumption falls and saving increases in response to the higher interest rate. Hence, the more responsive consumption is to the interest rate, the less government purchases crowd out investment. 18 Answers to Textbook Questions and Problems Figure 3¨C7 S 2 (r) S 1 (r) I(r) I, SI 1 I Investment, Saving ?G r 1 r r Real interest rate S(r) S Saving Real interest rate Figure 3¨C6 r Figure 3¨C6 Figure 3¨C7 More Problems and Applications to Chapter 3 1. a. A Cobb¨CDouglas production function has the form Y = AK ¦Á L 1 ¨C ¦Á . In the appendix we showed that the marginal products for the Cobb¨CDouglas production function are: MPL = (1 ¨C ¦Á)Y/L. MPK = ¦ÁY/K. Competitive profit-maximizing firms hire labor until its marginal product equals the real wage, and hire capital until its marginal product equals the real rental rate. Using these facts and the above marginal products for the Cobb¨CDouglas production function, we find: W/P = MPL = (1 ¨C ¦Á)Y/L. R/P = MPK = ¦ÁY/K. Rewriting this: (W/P)L = MPL × L = (1 ¨C ¦Á)Y. (R/P)K = MPK × K = ¦ÁY. Note that the terms (W/P)L and (R/P)K are the wage bill and total return to capi- tal, respectively. Given that the value of ¦Á = 0.3, then the above formulas indicate that labor receives 70 percent of total output, which is (1 ¨C 0.3), and capital receives 30 percent of total output. b. To determine what happens to total output when the labor force increases by 10 percent, consider the formula for the Cobb¨CDouglas production function: Y = AK ¦Á L 1 ¨C ¦Á . Let Y 1 equal the initial value of output and Y 2 equal final output. We know that ¦Á = 0.3. We also know that labor L increases by 10 percent: Y 1 = AK 0.3 L 0.7 . Y 2 = AK 0.3 (1.1L) 0.7 . Note that we multiplied L by 1.1 to reflect the 10-percent increase in the labor force. To calculate the percentage change in output, divide Y 2 by Y 1 : = = (1.1) 0.7 = 1.069. That is, output increases by 6.9 percent. To determine how the increase in the labor force affects the rental price of capital, consider the formula for the real rental price of capital R/P: R/P = MPK = ¦ÁAK ¦Á ¨C 1 L 1 ¨C ¦Á . We know that ¦Á = 0.3. We also know that labor (L) increases by 10 percent. Let (R/P) 1 equal the initial value of the rental price of capital, and (R/P) 2 equal the final rental price of capital after the labor force increases by 10 percent. To find (R/P) 2 , multiply L by 1.1 to reflect the 10-percent increase in the labor force: (R/P) 1 = 0.3AK ¨C 0.7 L 0.7 . (R/P) 2 = 0.3AK ¨C 0.7 (1.1L) 0.7 . Chapter 3 National Income: Where It Comes From and Where It Goes 19 AK 0.3 (1.1L) 0.7 AK 0.3 L 0.7 Y 2 Y 1 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 ¨C ¦Á)AK ¦Á L ¨C ¦Á . We know that ¦Á = 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 ¨C 0.3)AK 0.3 L ¨C 0.3 . (W/P) 2 = (1 ¨C 0.3)AK 0.3 (1.1L) ¨C 0.3 . To calculate the percentage change in the real wage, divide (W/P) 2 by (W/P) 1 : = (1.1) ¨C 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 Y 1 = AK 0.3 L 0.7 . Y 2 = A(1.1K) 0.3 L 0.7 . Therefore, we have: = = (1.1) 0.3 = 1.029. This equation shows that output increases by 2 percent. Notice that ¦Á < 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) ¨C0.7 = 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 ¨C0.7 (1.1L) 0.7 0.3AK ¨C0.7 L 0.7 (R/P) 2 (R/P) 1 A(1.1K) 0.3 L 1.7 AK 0.3 L 0.7 Y 2 Y 1 0.3A(1.1K) ¨C0.7 L 0.7 0.3AK ¨C0.7 L 0.7 (R/P) 2 (R/P) 1 ( () (.) (.) (.) .. .. W/P) W/P AK L AK L 2 1 03 03 03 03 103 11 103 = ? ? ? ? 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¡ªa feature of Cobb¨CDouglas 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 = = K 1/3 H 1/3 L ¨C2/3 . 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= = K 1/3 L 1/3 H ¨C2/3 . 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) ¨C0.7 L 0.7 0.7AK ¨C0.7 L 0.7 (W/P) 2 (W/P) 1 0.7(1.1A)K 0.3 L ¨C0.3 0.7AK 0.3 L ¨C0.3 (1.1A)K 0.3 L 0.7 AK 0.3 L 0.7 Y 2 Y 1 0.3(1.1A)K ¨C0.7 L 0.7 0.3AK ¨C0.7 L 0.7 (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 - 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 ( K 1/3 L 1/3 H ¨C2/3 ) H K 1/3 H 1/3 L 1/3 1 3 1 3 K 1/3 L ¨C2/3 H 1/3 + K 1/3 L 1/3 H ¨C2/3 K 1/3 L ¨C2/3 H 1/3 L H 1 3 1 3 1 3 MPL + MPH MPL W skilled W unskilled 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¡ªoften 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¡ªthe 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¡ªnominal GDP. V represents the income velocity of money¡ªthe 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 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¡ªthat 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 ¦Ð: i = r + ¦Ð. 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¡ªby reducing government spending and collecting more taxes¡ªcan 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¡ªfor exam- ple, they might be measured in ¡°constant dollars.¡± That is, the units are adjusted for 24 Answers to Textbook Questions and Problems inflation. A nominal variable is one that is measured in current dollars; the value of the variable is not adjusted for inflation. For example, a real variable could be a Hershey¡¯s candy bar; the nominal variable is the current-price value of the Hershey¡¯s candy bar¡ª 5 cents in 1960, say, and 75 cents in 1999. The interest rate you are quoted by your bank¡ª8 percent, say¡ªis a nominal rate, since it is not adjusted for inflation. If infla- tion is, say, 3 percent, then the real interest rate, which measures your purchasing power, is 5 percent. Problems and Applications 1. Money functions as a store of value, a medium of exchange, and a unit of account. a. A credit card can serve as a medium of exchange because it is accepted in exchange for goods and services. A credit card is, arguably, a (negative) store of value because you can accumulate debt with it. A credit card is not a unit of account¡ªa car, for example, does not cost 5 VISA cards. b. A Rembrandt painting is a store of value only. c. A subway token, within the subway system, satisfies all three functions of money. Yet outside the subway system, it is not widely used as a unit of account or a medium of exchange, so it is not a form of money. 2. The real interest rate is the difference between the nominal interest rate and the infla- tion rate. The nominal interest rate is 11 percent, but we need to solve for the inflation rate. We do this with the quantity identity expressed in percentage-change form: % Change in M + % Change in V = % Change in P + % Change in Y. Rearranging this equation tells us that the inflation rate is given by: % Change in P + % Change in M + % Change in V ¨C % Change in Y. Substituting the numbers given in the problem, we thus find: % Change in P = 14% + 0% ¨C 5% = 9%. Thus, the real interest rate is 2 percent: the nominal interest rate of 11 percent minus the inflation rate of 9 percent. 3. a. Legislators wish to ensure that the real value of Social Security and other benefits stays constant over time. This is achieved by indexing benefits to the cost of living as measured by the consumer price index. With indexing, nominal benefits change at the same rate as prices. b. Assuming the inflation is measured correctly (see Chapter 2 for more on this issue), senior citizens are unaffected by the lower rate of inflation. Although they get less money from the government, the goods they purchase are cheaper; their purchasing power is exactly the same as it was with the higher inflation rate. 4. The major benefit of having a national money is seigniorage¡ªthe ability of the govern- ment to raise revenue by printing money. The major cost is the possibility of inflation, or even hyperinflation, if the government relies too heavily on seigniorage. The benefits and costs of using a foreign money are exactly the reverse: the benefit of foreign money is that inflation is no longer under domestic political control, but the cost is that the domestic government loses its ability to raise revenue through seigniorage. (There is also a subjective cost to having pictures of foreign leaders on your currency.) The foreign country¡¯s political stability is a key factor. The primary reason for using another nation¡¯s money is to gain stability. If the foreign country is unstable, then the home country is definitely better off using its own currency¡ªthe home econo- my remains more stable, and it keeps the seigniorage. 5. A paper weapon might have been effective for all the reasons that a hyperinflation is bad. For example, it increases shoeleather and menu costs; it makes relative prices more variable; it alters tax liabilities in arbitrary ways; it increases variability in rela- Chapter 4 Money and Inflation 25 tive prices; it makes the unit of account less useful; and finally, it increases uncertainty and causes arbitrary redistributions of wealth. If the hyperinflation is sufficiently extreme, it can undermine the public¡¯s confidence in the economy and economic policy. Note that if foreign airplanes dropped the money, then the government would not receive seigniorage revenue from the resulting inflation, so this benefit usually associ- ated with inflation is lost. 6. One way to understand Coolidge¡¯s statement is to think of a government that is a net debtor in nominal terms to the private sector. Let B denote the government¡¯s outstand- ing debt measured in U.S. dollars. The debt in real terms equals B/P, where P is the price level. By increasing inflation, the government raises the price level and reduces in real terms the value of its outstanding debt. In this sense we can say that the govern- ment repudiates the debt. This only matters, however, when inflation is unexpected. If inflation is expected, people demand a higher nominal interest rate. Repudiation still occurs (i.e., the real value of the debt still falls when the price level rises), but it is not at the expense of the holders of the debt, since they are compensated with a higher nominal interest rate. 7. A deflation means a fall in the general price level, which is the same as a rise in the value of money. Under a gold standard, a rise in the value of money is a rise in the value of gold because money and gold are in a fixed ratio. Therefore, after a deflation, an ounce of gold buys more goods and services. This creates an incentive to look for new gold deposits and, thus, more gold is found after a deflation. 8. An increase in the rate of money growth leads to an increase in the rate of inflation. Inflation, in turn, causes the nominal interest rate to rise, which means that the oppor- tunity cost of holding money increases. As a result, real money balances fall. Since money is part of wealth, real wealth also falls. A fall in wealth reduces consumption, and, therefore, increases saving. The increase in saving leads to an outward shift of the saving schedule, as in Figure 4¨C1. This leads to a lower real interest rate. The classical dichotomy states that a change in a nominal variable such as infla- tion does not affect real variables. In this case, the classical dichotomy does not hold; the increase in the rate of inflation leads to a decrease in the real interest rate. The Fisher effect states that i = r + ¦Ð. In this case, since the real interest rate r falls, a 1- percent increase in inflation increases the nominal interest rate i by less than 1 per- cent. 26 Answers to Textbook Questions and Problems Real interest rate r A B I, S Investment, Saving I (r) S 1 S 2 r 2 r 1 Figure 4¨C1 Most economists believe that this Mundell¨CTobin effect is not important because real money balances are a small fraction of wealth. Hence, the impact on saving as illustrated in Figure 4¨C1 is small. 9. The Economist (www.economist.com) is a useful site for recent data. [Note: unfortu- nately, as of this writing (February 2002), you need a paid subscription to get access to many of the tables online.] Alternatively, the International Monetary Fund has links to country data sources (www.imf.org; follow the links to standards and codes and then to data dissemination). For example, in the twelve months ending in November 2001, consumer prices in Turkey rose 69 percent from a year earlier, M1 rose 55 percent while M2 rose 52 per- cent, and short-term interest rates were 54 percent. By contrast, in the United States in the twelve months ending in December 2001, consumer prices rose about 2 percent, M1 rose 8 percent, M2 rose 14 percent; and short-term interest rates were a little under 2 percent. These data are consistent with the theories in the chapter, in that high-infla- tion countries have higher rates of money growth and also higher nominal interest rates. More Problems and Applications to Chapter 4 1. With constant money growth at rate ¦Ì, the question tells us that the Cagan model implies that p t = m t + ¦Ã¦Ì. This question draws out the implications of this equation. a. One way to interpret this result is to rearrange to find: m t ¨C p t = ¨C¦Ã¦Ì. That is, real balances depend on the money growth rate. As the growth rate of money rises, real balances fall. This makes sense in terms of the model in this chapter, since faster money growth implies faster inflation, which makes it less desirable to hold money balances. b. With unchanged growth in the money supply, the increase in the level of the money supply m t increases the price level p t one-for-one. c. With unchanged current money supply m t , a change in the growth rate of money ¦Ì changes the price level in the same direction. d. When the central bank reduces the rate of money growth ¦Ì, the price level will immediately fall. To offset this decline in the price level, the central bank can increase the current level of the money supply m t , as we found in part (b). These answers assume that at each point in time, private agents expect the growth rate of money to remain unchanged, so that the change in policy takes them by sur- prise¡ªbut once it happens, it is completely credible. A practical problem is that the private sector might not find it credible that an increase in the current money supply signals a decrease in future money growth rates. e. If money demand does not depend on the expected rate of inflation, then the price level changes only when the money supply itself changes. That is, changes in the growth rate of money ¦Ì do not affect the price level. In part (d), the central bank can keep the current price level p t constant simply by keeping the current money supply m t constant. Chapter 4 Money and Inflation 27 Questions for Review 1. By rewriting the national income accounts identity, we show in the text that S ¨C I = NX. This form of the national income accounts identity shows the relationship between the international flow of funds for capital accumulation, S ¨C I, and the international flow of goods and services, NX. Net foreign investment refers to the (S ¨C I) part of this identity: it is the excess of domestic saving over domestic investment. In an open economy, domestic saving need not equal domestic investment, because investors can borrow and lend in world finan- cial markets. The trade balance refers to the (NX) part of the identity: it is the differ- ence between what we export and what we import. Thus, the national accounts identity shows that the international flow of funds to finance capital accumulation and the international flow of goods and services are two sides of the same coin. 2. The nominal exchange rate is the relative price of the currency of two countries. The real exchange rate, sometimes called the terms of trade, is the relative price of the goods of two countries. It tells us the rate at which we can trade the goods of one country for the goods of another. 3. A cut in defense spending increases government saving and, hence, increases national saving. Investment depends on the world rate and is unaffected. Hence, the increase in saving causes the (S ¨C I) schedule to shift to the right, as in Figure 5¨C1. The trade bal- ance rises, and the real exchange rate falls. 28 ? ? ? Real exchange rate A B NX NX Net exports ( ?) NX 2 NX 1 S 1 ¨C IS 2 ¨C I 1 2 Figure 5¨C1 CHAPTER 5 The Open Economy 4. If a small open economy bans the import of Japanese VCRs, then for any given real exchange rate, imports are lower, so that net exports are higher. Hence, the net export schedule shifts out, as in Figure 5¨C2. The protectionist policy of banning VCRs does not affect saving, investment, or the world interest rate, so the S ¨C I schedule does not change. Because protectionist policies do not alter either saving or investment in the model of this chapter, they cannot alter the trade balance. Instead, a protectionist policy drives the real exchange rate higher. 5. We can relate the real and nominal exchange rates by the expression Nominal Real Ratio of Exchange = Exchange × Price Rate Rate Levels e = × (P * /P). Let P * be the Italian price level and P be the German price level. The nominal exchange rate e is the number of Italian lira per German mark (this is as if we take Germany to be the ¡°domestic¡± country). We can express this in terms of percentage changes over time as % Change in e = % Change in + (¦Ð * ¨C ¦Ð), where ¦Ð * is the Italian inflation rate and ¦Ð is the German inflation rate. If Italian infla- tion is higher than German inflation, then this equation tells us that a mark buys an increasing amount of lira over time: the mark rises relative to the lira. Alternatively, viewed from the Italian perspective, the exchange rate in terms of marks per lira falls. Chapter 5 The Open Economy 29 ? ? ? ? Real exchange rate B A NX NX 2 ( ) ?NX 1 ( ) Net exports 2 1 S - I ? ? Figure 5¨C2 Problems and Applications 1. a. An increase in saving shifts the (S ¨C I) schedule to the right, increasing the supply of dollars available to be invested abroad, as in Figure 5¨C3. The increased supply of dollars causes the equilibrium real exchange rate to fall from 1 to 2 . Because the dollar becomes less valuable, domestic goods become less expensive relative to foreign goods, so exports rise and imports fall. This means that the trade balance increases. The nominal exchange rate falls following the movement of the real exchange rate, because prices do not change in response to this shock. b. The introduction of a stylish line of Toyotas that makes some consumers prefer foreign cars over domestic cars has no effect on saving or investment, but it shifts the NX( ) schedule inward, as in Figure 5¨C4. The trade balance does not change, but the real exchange rate falls from 1 to 2 . Because prices are not affected, the nominal exchange rate follows the real exchange rate. 30 Answers to Textbook Questions and Problems Real exchange rate NX 2 NX 1 (S 1 ¨C I)(S 2 ¨C I) NX ( ) A B NX Net exports ? ? ? 2 1 ? ? ? ? ? Figure 5¨C3 Real exchange rate A B NX Net exports ? ? ? ? NX 2 ( ) ?NX 1 ( ) 2 1 (S ¨C I) Figure 5¨C4 c. In the model we considered in this chapter, the introduction of ATMs has no effect on any real variables. The amounts of capital and labor determine output Y. The world interest rate r * determines investment I(r * ). The difference between domes- tic saving and domestic investment (S ¨C I) determines net exports. Finally, the intersection of the NX( ) schedule and the (S ¨C I) schedule determines the real exchange rate, as in Figure 5¨C5. The introduction of ATMs, by reducing money demand, does affect the nomi- nal exchange rate through its effect on the domestic price level. The price level adjusts to equilibrate the demand and supply of real balances, so that M/P = (M/P) d . If M is fixed, then a fall in (M/P) d causes an increase in the price level: this reduces the supply of real balances M/P and restores equilibrium in the money market. Now recall the formula for the nominal exchange rate: e = × (P * /P). We know that the real exchange rate remains constant, and we assume that the foreign price level P * is fixed. When the domestic price level P increases, the nomi- nal exchange rate e depreciates. Chapter 5 The Open Economy 31 Real exchange rate NX Net exports ? ? ? NX ( ) S ¨C I ? ? ? Figure 5¨C5 2. a. National saving is the amount of output that is not purchased for current con- sumption by households or the government. We know output and government spending, and the consumption function allows us to solve for consumption. Hence, national saving is given by: S= Y ¨C C ¨C G = 5,000 ¨C (250 + 0.75(5,000 ¨C 1,000)) ¨C 1,000 = 750. Investment depends negatively on the interest rate, which equals the world rate r * of 5. Thus, I = 1,000 ¨C 50 × 5 = 750. Net exports equals the difference between saving and investment. Thus, NX = S ¨C I = 750 ¨C 750 = 0. Having solved for net exports, we can now find the exchange rate that clears the foreign-exchange market: NX = 500 ¨C 500 × ¦Å 0 = 500 ¨C 500 × ¦Å ¦Å = 1. b. Doing the same analysis with the new value of government spending we find: S= Y ¨C C ¨C G = 5,000 ¨C (250 + 0.75(5,000 ¨C 1,000)) ¨C 1,250 = 500 I = 1,000 ¨C 50 × 5 = 750 NX = S ¨C I = 500 ¨C 750 = ¨C250 NX = 500 ¨C 500 × ¦Å ¨C250 = 500 ¨C 500 × ¦Å ¦Å = 1.5. The increase in government spending reduces national saving, but with an unchanged world real interest rate, investment remains the same. Therefore, domestic investment now exceeds domestic saving, so some of this investment must be financed by borrowing from abroad. This capital inflow is accomplished by reducing net exports, which requires that the currency appreciate. 32 Answers to Textbook Questions and Problems c. Repeating the same steps with the new interest rate, S= Y ¨C C ¨C G = 5,000 ¨C (250 + 0.75(5,000 ¨C 1,000)) ¨C 1,000 = 750 I = 1,000 ¨C 50 × 10 = 500 NX = S ¨C I = 750 ¨C 500 = 250 NX = 500 ¨C 500 × ¦Å 250 = 500 ¨C 500 × ¦Å ¦Å = 0.5. Saving is unchanged from part (a), but the higher world interest rate lowers investment. This capital outflow is accomplished by running a trade surplus, which requires that the currency depreciate. 3. a. When Leverett¡¯s exports become less popular, its domestic saving Y ¨C C ¨C G does not change. This is because we assume that Y is determined by the amount of cap- ital and labor, consumption depends only on disposable income, and government spending is a fixed exogenous variable. Investment also does not change, since investment depends on the interest rate, and Leverett is a small open economy that takes the world interest rate as given. Because neither saving nor investment changes, net exports, which equal S ¨C I, do not change either. This is shown in Figure 5¨C6 as the unmoving S ¨C I curve. The decreased popularity of Leverett¡¯s exports leads to a shift inward of the net exports curve, as shown in Figure 5¨C6. At the new equilibrium, net exports are unchanged but the currency has depreciated. Even though Leverett¡¯s exports are less popular, its trade balance has remained the same. The reason for this is that the depreciated currency provides a stimulus to net exports, which overcomes the unpopularity of its exports by mak- ing them cheaper. Chapter 5 The Open Economy 33 S ¨C I NX(¦Å) 1 NX(¦Å) 2 NX 1 = NX 2 { NX Net exports ¦Å 1 ¦Å 2 ¦Å Real e xchange rate Figure 5¨C6 b. Leverett¡¯s currency now buys less foreign currency, so traveling abroad is more expensive. This is an example of the fact that imports (including foreign travel) have become more expensive¡ªas required to keep net exports unchanged in the face of decreased demand for exports. c. If the government reduces taxes, then disposable income and consumption rise. Hence, saving falls so that net exports also fall. This fall in net exports puts upward pressure on the exchange rate that offsets the decreased world demand. Investment and the interest rate would be unaffected by this policy since Leverett takes the world interest rate as given. 4. The increase in government spending decreases government saving and, thus, decreas- es national saving; this shifts the saving schedule to the left, as in Figure 5¨C7. Given the world interest rate r * , the decrease in domestic saving causes the trade balance to fall. 34 Answers to Textbook Questions and Problems Real interest rate r * Current-account deficit = Capital-account surplus I, S Investment, Saving I (r) S 2 S 1 r Trade deficit Figure 5¨C7 Figure 5¨C8 shows the impact of this increase in government purchases on the real exchange rate. The decrease in national saving causes the (S ¨C I) schedule to shift to the left, lowering the supply of dollars to be invested abroad. The lower supply of dol- lars causes the equilibrium real exchange rate to rise. As a result, domestic goods become more expensive relative to foreign goods, which causes exports to fall and imports to rise. In other words, as we determined in Figure 5¨C7, the trade balance falls. Chapter 5 The Open Economy 35 NX B A Net exports Change in current account Real exchange rate ? ? 1 ? 2 ?NX ( ) (S 2 ¨C I )(S 1 ¨C I ) NX Net exports NX 2 NX 1 the trade balance Figure 5¨C8 The answer to this question does depend on whether this is a local war or a world war. A world war causes many governments to increase expenditures; this increases the world interest rate r * . The effect on a country¡¯s external accounts depends on the size of the change in the world interest rate relative to the size of the decrease in sav- ing. For example, an increase in the world interest rate could cause a country to have a trade deficit, as in Figure 5¨C9, or a trade surplus, as in Figure 5¨C10. 36 Answers to Textbook Questions and Problems r Real interest rate Investment, Saving I, S Current-account deficit I(r) r 2 * r 1 * S 1 S 2 Trade deficit Figure 5¨C9 Current- account surplus r Real interest rate I, S Investment, Saving I (r) S 2 S 1 r 2 * r 1 * Trade surplus Figure 5¨C10 5. Clinton¡¯s policy would not affect net exports because it does not affect national saving (because it would not affect Y, C, or G) or investment. It would, however, shift the NX curve by decreasing U.S. demand for Japanese auto imports. This shift of the curve, shown in Figure 5¨C11, would raise the exchange rate. Although net exports would not change, the volume of both imports and exports would fall by the same amount. There are also important compositional effects of this policy. On the production side, the higher exchange rate increases imports and puts pressure on the sales of American companies with the exception of American luxury car production, which is shielded by the tariff. Also American exporters will be hurt by the higher exchange rate, which makes their goods more expensive to foreign countries. Consumers of Japanese luxury cars will be hurt by the tariffs while all other consumers will benefit from the appreci- ated dollar, which allows them to purchase goods more cheaply. In sum, the policy would shift demand to American luxury car producers at the expense of the rest of American production and also shift consumption from Japanese luxury cars to all other imports. Chapter 5 The Open Economy 37 S ¨C I NX(¦Å) 2 NX(¦Å) 1 NX 1 = NX 2 { NX Net exports ¦Å 1 ¦Å 2 ¦Å Real e xchange rate Figure 5¨C11 6. a. If the countries that institute an investment tax credit are large enough to shift the world investment demand schedule, then the tax credits shift the world investment demand schedule upward, as in Figure 5¨C12. b. The world interest rate increases from r to r because of the increase in world investment demand; this is shown in Figure 5¨C12. (Remember that the world is a closed economy.) c. The increase in the world interest rate increases the required rate of return on investments in our country. Because the investment schedule slopes downward, we know that a higher world interest rate means lower investment, as in Figure 5¨C13. 38 Answers to Textbook Questions and Problems * 1 * 2 S I, S World investment, World saving Real interest rate B A I 1 (r) I 2 (r) r 2 * r 1 * r Figure 5¨C12 r Real interest rate Investment II 1 I 2 I(r) r 2 * r 1 * Figure 5¨C13 d. Given that our saving has not changed, the higher world interest rate means that our trade balance increases, as in Figure 5¨C14. e. To bring about the required increase in the trade balance, the real exchange rate must fall. Our goods become less expensive relative to foreign goods, so that exports increase and imports decrease, as in Figure 5¨C15. 7. The easiest way to tell if your friend is right or wrong is to consider an example. Suppose that ten years ago, a cup of American coffee cost $1, while a cup of Italian espresso cost 1,000 lira. Since $1 bought 1,000 lira ten years ago, it cost the same amount of money to buy a cup of coffee in both countries. Since total U.S. inflation has been 25 percent, the American cup of coffee now costs $1.25. Total Italian inflation has been 100 percent, so the Italian cup of espresso now costs 2,000 lira. This year, $1 buys 1,500 lira, so that the cup of espresso costs 2,000 lira/[1,500 lira/dollar] = $1.33. This means that it is now more expensive to purchase espresso in Italy than coffee in the United States. Chapter 5 The Open Economy 39 Current-account surplus = Capital-account deficit S r Real interest rate I, S Investment, Saving r 2 * I (r) r 1 * Trade surplus Figure 5¨C14 ? ? ? Real exchange rate A B NX NX ( ) Net exports 1 ? 2 NX 2 (S ¨C I) 1 (S ¨C I) 2 NX 1 Figure 5¨C15 Thus, your friend is simply wrong to conclude that it is cheaper to travel in Italy. Even though the dollar buys more lira than it used to, the relatively rapid inflation in Italy means that lira buy fewer goods than they used to¡ªit is more expensive now for an American to travel there. 8. a. The Fisher equation says that i = r + ¦Ð e where i = the nominal interest rate r = the real interest rate (same in both countries) ¦Ð e = the expected inflation rate. Plugging in the values given in the question for the nominal interest rates for each country, we find: 12 = r + ¦Ð e Can* 8 = r + ¦Ð e US* This implies that €€€€€€€€€€€€€¦Ð e Can ¨C ¦Ð e US = 4. Because we know that the real interest rate r is the same in both countries, we conclude that expected inflation in Canada is four percentage points higher than in the United States. b. As in the text, we can express the nominal exchange rate as e = ¦Å× (P Can /P US ), where ¦Å = the real exchange rate P Can = the price level in Canada P US = the price level in the United States. The change in the nominal exchange rate can be written as: % change in e = % change in ¦Å + (¦Ð Can ¨C ¦Ð US ). We know that if purchasing-power parity holds, than a dollar must have the same purchasing power in every country. This implies that the percent change in the real exchange rate ¦Å is zero because purchasing-power parity implies that the real exchange rate is fixed. Thus, changes in the nominal exchange rate result from differences in the inflaction rates in the United States and Canada. In equation form this says % change in e = (¦Ð Can ¨C ¦Ð US ). Because economic agents know that purchasing-power parity holds, they expect this relationship to hold. In other words, the expected change in the nominal exchange rate equals the expected inflation rate in Canada minus the expected inflation rate in the United States. That is, Expected % change in e = ¦Ð e Can ¨C ¦Ð e US* In part (a), we found that the difference in expected inflation rates is 4 percent. Therefore, the expected change in the nominal exchange rate e is 4 percent. c. The problem with your friend¡¯s scheme is that it does not take into account the change in the nominal exchange rate e between the U.S. and Canadian dollars. Given that the real interest rate is fixed and identical in the United States and Canada, and given purchasing-power parity, we know that the difference in nomi- 40 Answers to Textbook Questions and Problems nal interest rates accounts for the expected change in the nominal exchange rate between U.S. and Canadian dollars. In this example, the Canadian nominal interest rate is 12 percent, while the U.S. nominal interest rate is 8 percent. We conclude from this that the expected change in the nominal exchange rate is 4 percent. Therefore, e this year = 1 C$/US$. e next year = 1.04 C$/US$. Assume that your friend borrows 1 U.S. dollar from an American bank at 8 percent, exchanges it for 1 Canadian dollar, and puts it in a Canadian Bank. At the end of the year your friend will have $1.12 in Canadian dollars. But to repay the American bank, the Canadian dollars must be converted back into U.S. dollars. The $1.12 (Canadian) becomes $1.08 (American), which is the amount owed to the U.S. bank. So in the end, your friend breaks even. In fact, after paying for transaction costs, your friend loses money. More Problems and Applications to Chapter 5 1. a. As shown in Figure 5¨C16, an increase in government purchases reduces national saving. This reduces the supply of loans and raises the equilibrium interest rate. This causes both domestic investment and net foreign investment to fall. The fall in net foreign investment reduces the supply of dollars to be exchanged into for- eign currency, so the exchange rate appreciates and the trade balance falls. Chapter 5 The Open Economy 41 A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Net foreign investment Figure 5¨C16 b. As shown in Figure 5¨C17, the increase in demand for exports shifts the net exports schedule outward. Since nothing has changed in the market for loanable funds, the interest rate remains the same, which in turn implies that net foreign invest- ment remains the same. The shift in the net exports schedule causes the exchange rate to appreciate. The rise in the exchange rate makes U.S. goods more expensive relative to foreign goods, which depresses exports and stimulates imports. In the end, the increase in demand for American goods does not affect the trade balance. 42 Answers to Textbook Questions and Problems A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI Net foreign investment NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Figure 5¨C17 c. As shown in Figure 5¨C18, the U.S. investment demand schedule shifts inward. The demand for loans falls, so the equilibrium interest rate falls. The lower inter- est rate increases net foreign investment. Despite the fall in the interest rate, domestic investment falls; we know this because I + NFI does not change, and NFI rises. The rise in net foreign investment increases the supply of dollars in the market for foreign exchange. The exchange rate depreciates, and net exports rise. Chapter 5 The Open Economy 43 A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI Net foreign investment NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Figure 5¨C18 d. As shown in Figure 5¨C19, the increase in saving increases the supply of loans and lowers the equilibrium interest rate. This causes both domestic investment and net foreign investment to rise. The increase in net foreign investment increases the supply of dollars to be exchanged into foreign currency, so the exchange rate depreciates and the trade balance rises. 44 Answers to Textbook Questions and Problems Figure 7-17 A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI Net foreign investment NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Figure 5¨C19 e. The reduction in the willingness of Americans to travel abroad reduces imports, since foreign travel counts as an import. As shown in Figure 5¨C20, this shifts the net exports schedule outward. Since nothing has changed in the market for loan- able funds, the interest rate remains the same, which in turn implies that net for- eign investment remains the same. The shift in the net exports schedule causes the exchange rate to appreciate. The rise in the exchange rate makes U.S. goods more expensive relative to foreign goods, which depresses exports and stimulates imports. In the end, the fall in Americans¡¯ desire to travel abroad does not affect the trade balance. Chapter 5 The Open Economy 45 A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI Net foreign investment NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Figure 5¨C20 f. As shown in Figure 5¨C21, the net foreign investment schedule shifts in. This reduces demand for loans, so the equilibrium interest rate falls and investment rises. Net foreign investment falls, despite the fall in the interest rate; we know this because I + NFI is unchanged and investment rises. The fall in net foreign investment reduces the supply of dollars to be exchanged into foreign currency, so the exchange rate appreciates and the trade balance falls. 46 Answers to Textbook Questions and Problems Figure 7-19 A. The Market for Loanable Funds B. Net Foreign Investment I + NFI NFI(r) Loans S, I + NFI Net foreign investment NFI Real interest rate Real exchange rate S rr C. The Market for Foreign Exchange Net exports NX NX NFI ? ?( ) Figure 5¨C21 2. Gingrich¡¯s statement has no immediate effect on any of the ¡°fundamentals¡± in the econ- omy: consumption, government purchases, taxes, and output are all unchanged. International investors, however, will be more reluctant to invest in the American econ- omy, particularly to purchase U.S. government debt, because of the default risk. As both Americans and foreigners move their money out of the United States, the NFI curve shifts outward (there is more foreign investment), as shown in Figure 5¨C22(B). This raises the interest rate in order to keep I + NFI equal to the unchanged S, shown in Figure 5¨C22(A). The increase in NFI raises the supply in the market for foreign exchange, which lowers the equilibrium exchange rate as shown in Figure 5¨C22(C). Chapter 5 The Open Economy 47 S NX(¦Å) NX Loans ¦Å Figure 8¨C22 Real interest rate NFI S, I + NFI NFI(r) I + NFI Net exports Net foreign investment r r A. The Market for Loanable Funds B. Net Foreign Investment C. The Market for Foreign Exchange Real e xchange rate NFI Figure 5¨C22 Questions for Review 1. The rates of job separation and job finding determine the natural rate of unemploy- ment. The rate of job separation is the fraction of people who lose their job each month. The higher the rate of job separation, the higher the natural rate of unemployment. The rate of job finding is the fraction of unemployed people who find a job each month. The higher the rate of job finding, the lower the natural rate of unemployment. 2. Frictional unemployment is the unemployment caused by the time it takes to match workers and jobs. Finding an appropriate job takes time because the flow of informa- tion about job candidates and job vacancies is not instantaneous. Because different jobs require different skills and pay different wages, unemployed workers may not accept the first job offer they receive. In contrast, wait unemployment is the unemployment resulting from wage rigidity and job rationing. These workers are unemployed not because they are actively search- ing for a job that best suits their skills (as in the case of frictional unemployment), but because at the prevailing real wage the supply of labor exceeds the demand. If the wage does not adjust to clear the labor market, then these workers must ¡°wait¡± for jobs to become available. Wait unemployment thus arises because firms fail to reduce wages despite an excess supply of labor. 3. The real wage may remain above the level that equilibrates labor supply and labor demand because of minimum wage laws, the monopoly power of unions, and efficiency wages. Minimum-wage laws cause wage rigidity when they prevent wages from falling to equilibrium levels. Although most workers are paid a wage above the minimum level, for some workers, especially the unskilled and inexperienced, the minimum wage raises their wage above the equilibrium level. It therefore reduces the quantity of their labor that firms demand, and an excess supply of workers¡ªthat is, unemployment¡ªresults. The monopoly power of unions causes wage rigidity because the wages of union- ized workers are determined not by the equilibrium of supply and demand but by col- lective bargaining between union leaders and firm management. The wage agreement often raises the wage above the equilibrium level and allows the firm to decide how many workers to employ. These high wages cause firms to hire fewer workers than at the market-clearing wage, so wait unemployment increases. Efficiency-wage theories suggest that high wages make workers more productive. The influence of wages on worker efficiency may explain why firms do not cut wages despite an excess supply of labor. Even though a wage reduction decreases the firm¡¯s wage bill, it may also lower worker productivity and therefore the firm¡¯s profits. 4. Depending on how one looks at the data, most unemployment can appear to be either short term or long term. Most spells of unemployment are short; that is, most of those who became unemployed find jobs quickly. On the other hand, most weeks of unemploy- ment are attributable to the small number of long-term unemployed. By definition, the long-term unemployed do not find jobs quickly, so they appear on unemployment rolls for many weeks or months. 5. Economists have proposed at least two major hypotheses to explain the increase in the natural rate of unemployment in the 1970s and 1980s, and the decrease in the natural rate in the 1990s. The first is the changing demographic composition of the labor force. Because of the post¨CWorld-War-II baby boom, the number of young workers rose in the 48 CHAPTER 6 Unemployment 1970s. Young workers have higher rates of unemployment, so this demographic shift should tend to increase unemployment. In the 1990s, the baby-boom workers aged and the average age of the labor force increased, thus lowering the average unemployment rate. The second hypothesis is based on changes in the prevalence of sectoral shifts. The greater the amount of sectoral reallocation of workers, the greater the rate of job separation and the higher the level of frictional unemployment. The volatility of oil prices in the 1970s and 1980s is a possible source of increased sectoral shifts; in the 1990s, oil prices have been more stable. The proposed explanations are plausible, but neither seems conclusive on its own. Problems and Applications 1. a. In the example that follows, we assume that during the school year you look for a part-time job, and that on average it takes 2 weeks to find one. We also assume that the typical job lasts 1 semester, or 12 weeks. If it takes 2 weeks to find a job, then the rate of job finding in weeks is: f = (1 job/2 weeks) = 0.5 jobs/weeks. b. If the job lasts for 12 weeks, then the rate of job separation in weeks is: s = (1 job/12 weeks) = 0.083 jobs/week. c. From the text, we know that the formula for the natural rate of unemployment is (U/L) = (s/(s+ f)), where U is the number of people unemployed and L is the number of people in the labor force. Plugging in the values for f and s that were calculated in part (b), we find: (U/L) = (0.083/(0.083 + 0.5)) = 0.14. Thus, if on average it takes 2 weeks to find a job that lasts 12 weeks, the natural rate of unemployment for this population of college students seeking part-time employment is 14 percent. 2. To show that the unemployment rate evolves over time to the steady-state rate, let¡¯s begin by defining how the number of people unemployed changes over time. The change in the number of unemployed equals the number of people losing jobs (sE) minus the number finding jobs (fU). In equation form, we can express this as: U t + 1 ¨C U t = ?U t+ 1 = sE t ¨C fU t . Recall from the text that L = E t + U t , or E t = L ¨C U t , where L is the total labor force (we will assume that L is constant). Substituting for E t in the above equation, we find: ?U t + 1 = s(L ¨C U t ) ¨C fU t . Dividing by L, we get an expression for the change in the unemployment rate from t to t + 1: ?U t + 1 /L = (U t + 1 /L) ¨C (U t /L) = ?[U/L] t + 1 = s(1 ¨C U t /L) ¨C fU t /L. Rearranging terms on the right-hand side, we find: ?[U/L] t + 1 = s ¨C (s + f)U t /L = (s + f)[s/(s + f) ¨C U t /L]. Chapter 6 Unemployment 49 The first point to note about this equation is that in steady state, when the unemploy- ment rate equals its natural rate, the left-hand side of this expression equals zero. This tells us that, as we found in the text, the natural rate of unemployment (U/L) n equals s/(s + f). We can now rewrite the above expression, substituting (U/L) n for s/(s + f), to get an equation that is easier to interpret: ?[U/L] t + 1 = (s + f)[(U/L) n ¨C U t /L]. This expression shows the following: If U t /L > (U/L) n (that is, the unemployment rate is above its natural rate), then ?[U/L] t + 1 is negative: the unemployment rate falls. If U t /L < (U/L) n (that is, the unemployment rate is below its natural rate), then ?[U/L] t + 1 is positive: the unemployment rate rises. This process continues until the unemployment rate U/L reaches the steady-state rate (U/L) n . 3. Call the number of residents of the dorm who are involved I, the number who are unin- volved U, and the total number of students T = I + U. In steady state the total number of involved students is constant. For this to happen we need the number of newly unin- volved students, (0.10)I, to be equal to the number of students who just became involved, (0.05)U. Following a few substitutions: (0.05)U = (0.10)I = (0.10)(T ¨C U), so = = . We find that two-thirds of the students are uninvolved. 4. Consider the formula for the natural rate of unemployment, = . If the new law lowers the chance of separation s, but has no effect on the rate of job finding f, then the natural rate of unemployment falls. For several reasons, however, the new law might tend to reduce f. First, raising the cost of firing might make firms more careful about hiring workers, since firms have a harder time firing workers who turn out to be a poor match. Second, if searchers think that the new legislation will lead them to spend a longer period of time on a par- ticular job, then they might weigh more carefully whether or not to take that job. If the reduction in f is large enough, then the new policy may even increase the natural rate of unemployment. 5. a. The demand for labor is determined by the amount of labor that a profit-maximiz- ing firm wants to hire at a given real wage. The profit-maximizing condition is that the firm hire labor until the marginal product of labor equals the real wage, MPL =. 50 Answers to Textbook Questions and Problems U T U L W P s s + f 0.10 0.10 + 0.05 2 3 The marginal product of labor is found by differentiating the production function with respect to labor (see the appendix to Chapter 3 for more discussion), MPL =. = = K 1/3 L ¨C1/3 . In order to solve for labor demand, we set the MPL equal to the real wage and solve for L: K 1/3 L ¨C1/3 = L = K ( ) ¨C3 . Notice that this expression has the intuitively desirable feature that increases in the real wage reduce the demand for labor. b. We asume that the 1,000 units of capital and the 1,000 units of labor are supplied inelastically (i.e., they will work at any price). In this case we know that all 1,000 units of each will be used in equilibrium, so we can substitute them into the above labor demand function and solve for . 1,000 = 1,000 ( ) ¨C3 = . In equilibrium, employment will be 1,000, and multiplying this by 2/3 we find that the workers earn 667 units of output. The total output is given by the production function: Y= K 1/3 L 2/3 = 1,000 1/3 1,000 2/3 = 1,000. Notice that workers get two-thirds of output, which is consistent with what we know about the Cobb¨CDouglas production function from the appendix to Chapter 3. c. The congressionally mandated wage of 1 unit of output is above the equilibrium wage of 2/3 units of output. d. Firms will use their labor demand function to decide how many workers to hire at the given real wage of 1 and capital stock of 1,000: L = 1,000(1) ¨C3 = 296, so 296 workers will be hired for a total compensation of 296 units of output. e. The policy redistributes output from the 704 workers who become involuntarily unemployed to the 296 workers who get paid more than before. The lucky workers benefit less than the losers lose as the total compensation to the working class falls from 667 to 296 units of output. f. This problem does focus the analysis of minimum-wage laws on the two effects of these laws: they raise the wage for some workers while downward-sloping labor demand reduces the total number of jobs. Note, however, that if labor demand is Chapter 6 Unemployment 51 dY dL d(K 1/3 L 2/3 ) dL 2 3 2 3 8 27 8 27 8 27 W P W P W P W P W P 2 3 less elastic than in this example, then the loss of employment may be smaller, and the change in worker income might be positive. 6. a. The labor demand curve is given by the marginal product of labor schedule faced by firms. If a country experiences a reduction in productivity, then the labor demand curve shifts downward as in Figure 6¨C1. If labor becomes less productive, then at any given real wage, firms demand less labor. b. If the labor market is always in equilibrium, then, assuming a fixed labor supply, an adverse productivity shock causes a decrease in the real wage but has no effect on employment or unemployment, as in Figure 6¨C2. 52 Answers to Textbook Questions and Problems Real wage L Labor w/p L 2 D L 1 D Figure 6¨C1 L Labor w/p Real wage L L S L 2 D L 1 D w/p 2 w/p 1 Figure 6¨C2 c. If unions constrain real wages to remain unaltered, then as illustrated in Figure 6¨C3, employment falls to L 1 and unemployment equals L ¨C L 1 . This example shows that the effect of a productivity shock on an economy depends on the role of unions and the response of collective bargaining to such a change. 7. The vacant office space problem is similar to the unemployment problem; we can apply the same concepts we used in analyzing unemployed labor to analyze why vacant office space exists. There is a rate of office separation: firms that occupy offices leave, either to move to different offices or because they go out of business. There is a rate of office finding: firms that need office space (either to start up or expand) find empty offices. It takes time to match firms with available space. Different types of firms require spaces with different attributes depending on what their specific needs are. Also, because demand for different goods fluctuates, there are ¡°sectoral shifts¡±¡ªchanges in the com- position of demand among industries and regions¡ªthat affect the profitability and office needs of different firms. Chapter 6 Unemployment 53 LL Unemployment Labor Real wage w/p w/p 1 L 1 L S L 2 D L 1 D AB Figure 6¨C3 Questions for Review 1. In the Solow growth model, a high saving rate leads to a large steady-state capital stock and a high level of steady-state output. A low saving rate leads to a small steady- state capital stock and a low level of steady-state output. Higher saving leads to faster economic growth only in the short run. An increase in the saving rate raises growth until the economy reaches the new steady state. That is, if the economy maintains a high saving rate, it will also maintain a large capital stock and a high level of output, but it will not maintain a high rate of growth forever. 2. It is reasonable to assume that the objective of an economic policymaker is to maximize the economic well-being of the individual members of society. Since economic well-being depends on the amount of consumption, the policymaker should choose the steady state with the highest level of consumption. The Golden Rule level of capital represents the level that maximizes consumption in the steady state. Suppose, for example, that there is no population growth or technological change. If the steady-state capital stock increases by one unit, then output increases by the marginal product of capital MPK; depreciation, however, increases by an amount ¦Ä, so that the net amount of extra output available for consumption is MPK ¨C ¦Ä. The Golden Rule capital stock is the level at which MPK = ¦Ä, so that the marginal product of capital equals the depreciation rate. 3. When the economy begins above the Golden Rule level of capital, reaching the Golden Rule level leads to higher consumption at all points in time. Therefore, the policymaker would always want to choose the Golden Rule level, because consumption is increased for all periods of time. On the other hand, when the economy begins below the Golden Rule level of capital, reaching the Golden Rule level means reducing consumption today to increase consumption in the future. In this case, the policymaker¡¯s decision is not as clear. If the policymaker cares more about current generations than about future gener- ations, he or she may decide not to pursue policies to reach the Golden Rule steady state. If the policymaker cares equally about all generations, then he or she chooses to reach the Golden Rule. Even though the current generation will have to consume less, an infinite number of future generations will benefit from increased consumption by moving to the Golden Rule. 54 CHAPTER 7 Economic Growth I 4. The higher the population growth rate is, the lower the steady-state level of capital per worker is, and therefore there is a lower level of steady-state income. For example, Figure 7¨C1 shows the steady state for two levels of population growth, a low level n 1 and a higher level n 2 . The higher population growth n 2 means that the line represent- ing population growth and depreciation is higher, so the steady-state level of capital per worker is lower. The steady-state growth rate of total income is n + g: the higher the population growth rate n is, the higher the growth rate of total income is. Income per worker, however, grows at rate g in steady state and, thus, is not affected by population growth. Problems and Applications 1. a. A production function has constant returns to scale if increasing all factors of pro- duction by an equal percentage causes output to increase by the same percentage. Mathematically, a production function has constant returns to scale if zY = F(zK, zL) for any positive number z. That is, if we multiply both the amount of capital and the amount of labor by some amount z, then the amount of output is multi- plied by z. For example, if we double the amounts of capital and labor we use (set- ting z = 2), then output also doubles. To see if the production function Y = F(K, L) = K 1/2 L 1/2 has constant returns to scale, we write: F(zK, zL) = (zK) 1/2 (zL) 1/2 = zK 1/2 L 1/2 = zY. Therefore, the production function Y = K 1/2 L 1/2 has constant returns to scale. b. To find the per-worker production function, divide the production function Y = K 1/2 L 1/2 by L: If we define y = Y/L, we can rewrite the above expression as: y = K 1/2 /L 1/2 . Defining k = K/L, we can rewrite the above expression as: y = k 1/2 . Chapter 7 Economic Growth I 55 (¦Ä + n 2 )k (¦Ä + n 1 )k sf (k) Investment, break-even investment Capital per worker k Figure 4-1 k 2 * k 1 * Figure 7¨C1 Y L KL L = 12 12 . c. We know the following facts about countries A and B: ¦Ä = depreciation rate = 0.05, s a = saving rate of country A = 0.1, s b = saving rate of country B = 0.2, and y = k 1/2 is the per-worker production function derived in part (b) for countries A and B. The growth of the capital stock ?k equals the amount of investment sf(k), less the amount of depreciation ¦Äk. That is, ?k = sf(k) ¨C ¦Äk. In steady state, the capital stock does not grow, so we can write this as sf(k) = ¦Äk. To find the steady-state level of capital per worker, plug the per-worker pro- duction function into the steady-state investment condition, and solve for k * : sk 1/2 = ¦Äk. Rewriting this: k 1/2 = s/¦Ä k = (s/¦Ä) 2 . To find the steady-state level of capital per worker k * , plug the saving rate for each country into the above formula: Country A: k = (s a /¦Ä) 2 = (0.1/0.05) 2 = 4. Country B: k = (s b /¦Ä) 2 = (0.2/0.05) 2 = 16. Now that we have found k * for each country, we can calculate the steady-state lev- els of income per worker for countries A and B because we know that y = k 1/2 : y = (4) 1/2 = 2. y = (16) 1/2 = 4. We know that out of each dollar of income, workers save a fraction s and con- sume a fraction (1 ¨C s). That is, the consumption function is c = (1 ¨C s)y. Since we know the steady-state levels of income in the two countries, we find Country A: c = (1 ¨C s a )y = (1 ¨C 0.1)(2) = 1.8. Country B: c = (1 ¨C s b )y = (1 ¨C 0.2)(4) = 3.2. d. Using the following facts and equations, we calculate income per worker y, con- sumption per worker c, and capital per worker k: s a = 0.1. s b = 0.2. ¦Ä = 0.05. k o = 2 for both countries. y = k 1/2 . c = (1 ¨C s)y. 56 Answers to Textbook Questions and Problems * a * b * a * b * a * a * b * b Chapter 7 Economic Growth I 57 Country A Year ky= k 1/2 c = (1 ¨C s a )yi= s a y ¦Äk ?k = i ¨C ¦Äk 1 2 1.414 1.273 0.141 0.100 0.041 2 2.041 1.429 1.286 0.143 0.102 0.041 3 2.082 1.443 1.299 0.144 0.104 0.040 4 2.122 1.457 1.311 0.146 0.106 0.040 5 2.102 1.470 1.323 0.147 0.108 0.039 Country B Year ky= k 1/2 c = (1 ¨C s a )yi= s a y ¦Äk ?k = i ¨C ¦Äk 1 2 1.414 1.131 0.283 0.100 0.183 2 2.183 1.477 1.182 0.295 0.109 0.186 3 2.369 1.539 1.231 0.308 0.118 0.190 4 2.559 1.600 1.280 0.320 0.128 0.192 5 2.751 1.659 1.327 0.332 0.138 0.194 Note that it will take five years before consumption in country B is higher than consumption in country A. 2. a. The production function in the Solow growth model is Y = F(K, L), or expressed terms of output per worker, y = f(k). If a war reduces the labor force through casu- alties, then L falls but k = K/L rises. The production function tells us that total output falls because there are fewer workers. Output per worker increases, howev- er, since each worker has more capital. b. The reduction in the labor force means that the capital stock per worker is higher after the war. Therefore, if the economy were in a steady state prior to the war, then after the war the economy has a capital stock that is higher than the steady- state level. This is shown in Figure 7¨C2 as an increase in capital per worker from k * to k 1 . As the economy returns to the steady state, the capital stock per worker falls from k 1 back to k * , so output per worker also falls. Investment, break-even investment Capital per worker k Figure 4-2 k * k 1 sf (k) (¦Ä + n) k Figure 7¨C2 Hence, in the transition to the new steady state, output growth is slower. In the steady state, we know that technological progress determines the growth rate of output per worker. Once the economy returns to the steady state, output per worker equals the rate of technological progress¡ªas it was before the war. 3. a. We follow Section 7-1, ¡°Approaching the Steady State: A Numerical Example.¡± The production function is Y = K 0.3 L 0.7 . To derive the per-worker production func- tion f(k), divide both sides of the production function by the labor force L: Rearrange to obtain: Because y = Y/L and k = K/L, this becomes: y = k 0.3 . b. Recall that ?k = sf(k) ¨C ¦Äk. The steady-state value of capital k * is defined as the value of k at which capital stock is constant, so ?k = 0. It follows that in steady state 0 = sf(k) ¨C ¦Äk, or, equivalently, For the production function in this problem, it follows that: Rearranging: or Substituting this equation for steady-state capital per worker into the per-worker production function from part (a) gives: Consumption is the amount of output that is not invested. Since investment in the steady state equals ¦Äk * , it follows that 58 Answers to Textbook Questions and Problems Y L KL L = 0.3 0 7. . Y L K L = ? ? ? ? ? ? 03. . k fk * (*) = s ¦Ä . k k s* (*) . 0.3 = ¦Ä (*) , . k s 07 = ¦Ä k s * /. = ? ? ? ? ? ? ¦Ä 107 y s * ./. = ? ? ? ? ? ? ¦Ä 03 07 cfk k ss *(*) * ./. /. =?= ? ? ? ? ? ? ? ? ? ? ? ? ? ¦Ä ¦Ä ¦Ä ¦Ä 03 07 1 07 (Note: An alternative approach to the problem is to note that consumption also equals the amount of output that is not saved: Some algebraic manipulation shows that this equation is equal to the equation above.) c. The table below shows k * , y * , and c * for the saving rate in the left column, using the equations from part (b). We assume a depreciation rate of 10 percent (i.e., 0.1). (The last column shows the marginal product of capital, derived in part (d) below). k * y * c * MPK 0 0.00 0.00 0.00 0.1 1.00 1.00 0.90 0.30 0.2 2.69 1.35 1.08 0.15 0.3 4.80 1.60 1.12 0.10 0.4 7.25 1.81 1.09 0.08 0.5 9.97 1.99 1.00 0.06 0.6 12.93 2.16 0.86 0.05 0.7 16.12 2.30 0.69 0.04 0.8 19.50 2.44 0.49 0.04 0.9 23.08 2.56 0.26 0.03 1 26.83 2.68 0.00 0.03 Note that a saving rate of 100 percent (s = 1.0) maximizes output per worker. In that case, of course, nothing is ever consumed, so c * =0. Consumption per work- er is maximized at a rate of saving of 0.3 percent¡ªthat is, where s equals capital¡¯s share in output. This is the Golden Rule level of s. d. We can differentiate the production function Y = K 0.3 L 0.7 with respect to K to find the marginal product of capital. This gives: The table in part (c) shows the marginal product of capital for each value of the saving rate. (Note that the appendix to Chapter 3 derived the MPK for the general Cobb¨CDouglas production function. The equation above corresponds to the special case where ¦Á equals 0.3.) 4. Suppose the economy begins with an initial steady-state capital stock below the Golden Rule level. The immediate effect of devoting a larger share of national output to invest- ment is that the economy devotes a smaller share to consumption; that is, ¡°living stan- dards¡± as measured by consumption fall. The higher investment rate means that the capital stock increases more quickly, so the growth rates of output and output per worker rise. The productivity of workers is the average amount produced by each work- er¡ªthat is, output per worker. So productivity growth rises. Hence, the immediate effect is that living standards fall but productivity growth rises. In the new steady state, output grows at rate n + g, while output per worker grows at rate g. This means that in the steady state, productivity growth is independent of the rate of investment. Since we begin with an initial steady-state capital stock below the Golden Rule level, the higher investment rate means that the new steady state has a higher level of consumption, so living standards are higher. Thus, an increase in the investment rate increases the productivity growth rate in the short run but has no effect in the long run. Living standards, on the other hand, fall immediately and only rise over time. That is, the quotation emphasizes growth, but not the sacrifice required to achieve it. Chapter 7 Economic Growth I 59 MPK KL K Y K y k ===03 03 03 03 07 ... .. csfk sk s s * ( ) ( *) ( )( *) ( ) . ./. =? =? =? ? ? ? ? ? ? 11 1 03 03 07 ¦Ä 5. As in the text, let k = K/L stand for capital per unit of labor. The equation for the evolu- tion of k is ?k = Saving ¨C (¦Ä + n)k. If all capital income is saved and if capital earns its marginal product, then saving equals MPK × k. We can substitute this into the above equation to find ?k = MPK × k ¨C (¦Ä + n)k. In the steady state, capital per efficiency unit of capital does not change, so ?k = 0. From the above equation, this tells us that MPK × k = (¦Ä + n)k, or MPK = (¦Ä + n). Equivalently, MPK ¨C ¦Ä = n. In this economy¡¯s steady state, the net marginal product of capital, MPK ¨C ¦Ä, equals the rate of growth of output, n. But this condition describes the Golden Rule steady state. Hence, we conclude that this economy reaches the Golden Rule level of capital accumu- lation. 6. First, consider steady states. In Figure 7¨C3, the slower population growth rate shifts the line representing population growth and depreciation downward. The new steady state has a higher level of capital per worker, k , and hence a higher level of output per worker. What about steady-state growth rates? In steady state, total output grows at rate n + g, whereas output per-person grows at rate g. Hence, slower population growth will lower total output growth, but per-person output growth will be the same. Now consider the transition. We know that the steady-state level of output per person is higher with low population growth. Hence, during the transition to the new steady state, output per person must grow at a rate faster than g for a while. In the decades after the fall in population growth, growth in total output will fall while growth in output per person will rise. 60 Answers to Textbook Questions and Problems Capital per worker Investment, break-even investment Figure 4-3 k (¦Ä + n 1 ) k (¦Ä + n 2 ) k sf (k) k 2 * k 1 * * 2 Figure 7¨C3 7. If there are decreasing returns to labor and capital, then increasing both capital and labor by the same proportion increases output by less than this proportion. For exam- ple, if we double the amounts of capital and labor, then output less than doubles. This may happen if there is a fixed factor such as land in the production function, and it becomes scarce as the economy grows larger. Then population growth will increase total output but decrease output per worker, since each worker has less of the fixed fac- tor to work with. If there are increasing returns to scale, then doubling inputs of capital and labor more than doubles output. This may happen if specialization of labor becomes greater as population grows. Then population growth increases total output and also increases output per worker, since the economy is able to take advantage of the scale economy more quickly. 8. a. To find output per capita y we divide total output by the number of workers: where the final step uses the definition k = . Notice that unemployment reduces the amount of per capita output for any given capital¨Cperson ratio because some of the people are not producing anything. The steady state is the level of capital per person at which the increase in capital per capita from investment equals its decrease from depreciation and pop- ulation growth (see Chapter 4 for more details). sk *¦Á (1 ¨C u) 1¨C¦Á = (¦Ä + n)k* Unemployment lowers the marginal product of capital and, hence, acts like a neg- ative technological shock that reduces the amount of capital the economy can reproduce in steady state. Figure 7¨C4 shows this graphically: an increase in unem- ployment lowers the sf(k) line and the steady-state level of capital per person. Chapter 7 Economic Growth I 61 y kuL L = ?[] ? ¦Á ¦Á (*)1 = ? ? ? ? ? ? ? ? K L u ¦Á ¦Á (*)1 1 =? ? ku ¦Á¦Á (*),1 1 K L sy n k* =+()*¦Ä ku s n *( *)=? ? ? ? ? ? ? ? 1 1 1 ¦Ä+ ¦Á Finally, to get steady-state output plug the steady-state level of capital into the production function: Unemployment lowers output for two reasons: for a given k, unemployment lowers y, and unemployment also lowers the steady-state value k * . b. Figure 7¨C5 below shows the pattern of output over time. As soon as unemploy- ment falls from u 1 to u 2 , output jumps up from its initial steady-state value of y*(u 1 ). The economy has the same amount of capital (since it takes time to adjust the capital stock), but this capital is combined with more workers. At that moment the economy is out of steady state: it has less capital than it wants to match the increased number of workers in the economy. The economy begins its transition by accumulating more capital, raising output even further than the original jump. Eventually the capital stock and output converge to their new, higher steady-state levels. 62 Answers to Textbook Questions and Problems In v estment, Break-e v en in v estment Capital per person (¦Ä + n)k sf(k, u 2 ) * sf(k, u 1 ) * k 2 * k 1 * Figure 7¨C4 yu s n u*( *) ( *)=? + ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 11 1 1 1 ¦Ä ¦Á ¦Á ¦Á =? + ? ? ? ? ? ? ? (*)1 1 u s n¦Ä ¦Á ¦Á 9. There is no unique way to find the data to answer this question. For example, from the World Bank web site, I followed links to "Data and Statistics." I then followed a link to "Quick Reference Tables" (http://www.worldbank.org/data/databytopic/GNPPC.pdf) to find a summary table of income per capita across countries. (Note that there are some subtle issues in converting currency values across countries that are beyond the scope of this book. The data in Table 7¨C1 use what are called ¡°purchasing power parity.¡±) As an example, I chose to compare the United States (income per person of $31,900 in 1999) and Pakistan ($1,860), with a 17-fold difference in income per person. How can we decide what factors are most important? As the text notes, differences in income must come from differences in capital, labor, and/or technology. The Solow growth model gives us a framework for thinking about the importance of these factors. One clear difference across countries is in educational attainment. One can think about differences in educational attainment as reflecting differences in broad ¡°human capital¡± (analogous to physical capital) or as differences in the level of technology (e.g., if your work force is more educated, then you can implement better technologies). For our purposes, we will think of education as reflecting ¡°technology,¡± in that it allows more output per worker for any given level of physical capital per worker. From the World Bank web site (country tables) I found the following data (down- loaded February 2002): Labor Force Investment/GDP Illiteracy Growth (1990) (percent of (1994¨C2000) (percent) population 15+) United States 1.5 18 0 Pakistan 3.0 19 54 How can we decide which factor explains the most? It seems unlikely that the small difference in investment/GDP explains the large difference in per capital income, leaving labor-force growth and illiteracy (or, more generally, technology) as the likely culprits. But we can be more formal about this using the Solow model. We follow Section 7-1, ¡°Approaching the Steady State: A Numerical Example.¡± For the moment, we assume the two countries have the same production technology: Y=K 0.5 L 0.5 . (This will allow us to decide whether differences in saving and population growth can explain the differences in income per capita; if not, then differences in tech- nology will remain as the likely explanation.) As in the text, we can express this equa- tion in terms of the per-worker production function f(k): y = k 0.5 . Chapter 7 Economic Growth I 63 tu * falls y * (u ) ? y * (u ) * y 2 1 Figure 7¨C5 In steady-state, we know that The steady-state value of capital k* is defined as the value of k at which capital stock is constant, so ?k = 0. It follows that in steady state or, equivalently, For the production function in this problem, it follows that: Rearranging: or Substituting this equation for steady-state capital per worker into the per-worker production function gives: If we assume that the United States and Pakistan are in steady state and have the same rates of depreciation¡ªsay, 5 percent¡ªthen the ratio of income per capita in the two countries is: This equation tells us that if, say, the U.S. saving rate had been twice Pakistan's saving rate, then U.S. income per worker would be twice Pakistan's level (other things equal). Clearly, given that the U.S. has 17-times higher income per worker but very similar levels of investment relative to GDP, this variable is not a major factor in the compari- son. Even population growth can only explain a factor of 1.2 (0.08/0.065) difference in levels of output per worker. The remaining culprit is technology, and the high level of illiteracy in Pakistan is consistent with this conclusion. 64 Answers to Textbook Questions and Problems ?¦Äksfk n k=?+() ( ). 0 =?+sf k n() ( ¦Ä)k, k fk s n * (*) .= +¦Ä k k s n * * , . () = + 05 ¦Ä k s n *, . ()= + 05 ¦Ä k s n *.= + ? ? ? ? ? ? ¦Ä 2 y s n *.= + ? ? ? ? ? ? ¦Ä y y s s n n US Parkistan US Pakistan Pakistan US = ? ? ? ? ? ? + + ? ? ? ?005 005 . . ? ? Questions for Review 1. In the Solow model, we find that only technological progress can affect the steady-state rate of growth in income per worker. Growth in the capital stock (through high saving) has no effect on the steady-state growth rate of income per worker; neither does popula- tion growth. But technological progress can lead to sustained growth. 2. To decide whether an economy has more or less capital than the Golden Rule, we need to compare the marginal product of capital net of depreciation (MPK ¨C ¦Ä) with the growth rate of total output (n + g). The growth rate of GDP is readily available. Estimating the net marginal product of capital requires a little more work but, as shown in the text, can be backed out of available data on the capital stock relative to GDP, the total amount of depreciation relative to GDP, and capital¡¯s share in GDP. 3. Economic policy can influence the saving rate by either increasing public saving or pro- viding incentives to stimulate private saving. Public saving is the difference between government revenue and government spending. If spending exceeds revenue, the gov- ernment runs a budget deficit, which is negative saving. Policies that decrease the deficit (such as reductions in government purchases or increases in taxes) increase pub- lic saving, whereas policies that increase the deficit decrease saving. A variety of gov- ernment policies affect private saving. The decision by a household to save may depend on the rate of return; the greater the return to saving, the more attractive saving becomes. Tax incentives such as tax-exempt retirement accounts for individuals and investment tax credits for corporations increase the rate of return and encourage pri- vate saving. 4. The rate of growth of output per person slowed worldwide after 1972. This slowdown appears to reflect a slowdown in productivity growth¡ªthe rate at which the production function is improving over time. Various explanations have been proposed, but the slowdown remains a mystery. In the second half of the 1990s, productivity grew more quickly again in the United States and, it appears, a few other countries. Many com- mentators attribute the productivity revival to the effects of information technology. 5. Endogenous growth theories attempt to explain the rate of technological progress by explaining the decisions that determine the creation of knowledge through research and development. By contrast, the Solow model simply took this rate as exogenous. In the Solow model, the saving rate affects growth temporarily, but diminishing returns to capital eventually force the economy to approach a steady state in which growth depends only on exogenous technological progress. By contrast, many endogenous growth models in essence assume that there are constant (rather than diminishing) returns to capital, interpreted to include knowledge. Hence, changes in the saving rate can lead to persistent growth. 65 CHAPTER 8 Economic Growth II Problems and Applications 1. a. To solve for the steady-state value of y as a function of s, n, g, and ¦Ä, we begin with the equation for the change in the capital stock in the steady state: ?k = sf(k) ¨C (¦Ä + n + g)k = 0. The production function y = k can also be rewritten as y 2 = k. Plugging this pro- duction function into the equation for the change in the capital stock, we find that in the steady state: sy ¨C (¦Ä + n + g)y 2 = 0. Solving this, we find the steady-state value of y: y * = s/(¦Ä + n + g). b. The question provides us with the following information about each country: Developed country: s = 0.28 Less-developed country: s = 0.10 n = 0.01 n = 0.04 g = 0.02 g = 0.02 ¦Ä = 0.04 ¦Ä = 0.04 Using the equation for y * that we derived in part (a), we can calculate the steady- state values of y for each country. Developed country: y * = 0.28/(0.04 + 0.01 + 0.02) = 4. Less-developed country: y * = 0.10/(0.04 + 0.04 + 0.02) = 1. c. The equation for y * that we derived in part (a) shows that the less-developed coun- try could raise its level of income by reducing its population growth rate n or by increasing its saving rate s. Policies that reduce population growth include intro- ducing methods of birth control and implementing disincentives for having chil- dren. Policies that increase the saving rate include increasing public saving by reducing the budget deficit and introducing private saving incentives such as I.R.A.¡¯s and other tax concessions that increase the return to saving. 2. To solve this problem, it is useful to establish what we know about the U.S. economy: A Cobb¨CDouglas production function has the form y = k ¦Á , where ¦Á is capital¡¯s share of income. The question tells us that ¦Á = 0.3, so we know that the production function is y = k 0.3 . In the steady state, we know that the growth rate of output equals 3 percent, so we know that (n + g) = 0.03. The depreciation rate ¦Ä = 0.04. The capital¨Coutput ratio K/Y = 2.5. Because k/y = [K/(L × E)]/[Y/(L × E)] = K/Y, we also know that k/y = 2.5. (That is, the capital¨Coutput ratio is the same in terms of effec- tive workers as it is in levels.) a. Begin with the steady-state condition, sy = (¦Ä + n + g)k. Rewriting this equation leads to a formula for saving in the steady state: s = (¦Ä + n + g)(k/y). Plugging in the values established above: s = (0.04 + 0.03)(2.5) = 0.175. The initial saving rate is 17.5 percent. b. We know from Chapter 3 that with a Cobb¨CDouglas production function, capital¡¯s share of income ¦Á = MPK(K/Y). Rewriting, we have: MPK = ¦Á/(K/Y). 66 Answers to Textbook Questions and Problems ¡Ì Plugging in the values established above, we find: MPK = 0.3/2.5 = 0.12. c. We know that at the Golden Rule steady state: MPK = (n + g + ¦Ä). Plugging in the values established above: MPK = (0.03 + 0.04) = 0.07. At the Golden Rule steady state, the marginal product of capital is 7 percent, whereas it is 12 percent in the initial steady state. Hence, from the initial steady state we need to increase k to achieve the Golden Rule steady state. d. We know from Chapter 3 that for a Cobb¨CDouglas production function, MPK = ¦Á (Y/K). Solving this for the capital¨Coutput ratio, we find: K/Y = ¦Á/MPK. We can solve for the Golden Rule capital¨Coutput ratio using this equation. If we plug in the value 0.07 for the Golden Rule steady-state marginal product of capi- tal, and the value 0.3 for ¦Á, we find: K/Y = 0.3/0.07 = 4.29. In the Golden Rule steady state, the capital¨Coutput ratio equals 4.29, compared to the current capital¨Coutput ratio of 2.5. e. We know from part (a) that in the steady state s = (¦Ä + n + g)(k/y), where k/y is the steady-state capital¨Coutput ratio. In the introduction to this answer, we showed that k/y = K/Y, and in part (d) we found that the Golden Rule K/Y = 4.29. Plugging in this value and those established above: s = (0.04 + 0.03)(4.29) = 0.30. To reach the Golden Rule steady state, the saving rate must rise from 17.5 to 30 percent. 3. a. In the steady state, we know that sy = (¦Ä + n + g)k. This implies that k/y = s/(¦Ä + n + g). Since s, ¦Ä, n, and g are constant, this means that the ratio k/y is also constant. Since k/y = [K/(L × E)]/[Y/(L × E)] = K/Y, we can conclude that in the steady state, the capital¨Coutput ratio is constant. b. We know that capital¡¯s share of income = MPK × (K/Y). In the steady state, we know from part (a) that the capital¨Coutput ratio K/Y is constant. We also know from the hint that the MPK is a function of k, which is constant in the steady state; therefore the MPK itself must be constant. Thus, capital¡¯s share of income is constant. Labor¡¯s share of income is 1 ¨C [capital¡¯s share]. Hence, if capital¡¯s share is constant, we see that labor¡¯s share of income is also constant. c. We know that in the steady state, total income grows at n + g¡ªthe rate of popula- tion growth plus the rate of technological change. In part (b) we showed that labor¡¯s and capital¡¯s share of income is constant. If the shares are constant, and total income grows at the rate n + g, then labor income and capital income must also grow at the rate n + g. d. Define the real rental price of capital R as: R = Total Capital Income/Capital Stock = (MPK × K)/K = MPK. We know that in the steady state, the MPK is constant because capital per effec- tive worker k is constant. Therefore, we can conclude that the real rental price of capital is constant in the steady state. Chapter 8 Economic Growth II 67 To show that the real wage w grows at the rate of technological progress g, define: TLI = Total Labor Income. L = Labor Force. Using the hint that the real wage equals total labor income divided by the labor force: w = TLI/L. Equivalently, wL = TLI. In terms of percentage changes, we can write this as ?w/w + ?L/L = ?TLI/TLI. This equation says that the growth rate of the real wage plus the growth rate of the labor force equals the growth rate of total labor income. We know that the labor force grows at rate n, and from part (c) we know that total labor income grows at rate n + g. We therefore conclude that the real wage grows at rate g. 4. How do differences in education across countries affect the Solow model? Education is one factor affecting the efficiency of labor, which we denoted by E. (Other factors affect- ing the efficiency of labor include levels of health, skill, and knowledge.) Since country 1 has a more highly educated labor force than country 2, each worker in country 1 is more efficient. That is, E 1 > E 2 . We will assume that both countries are in steady state. a. In the Solow growth model, the rate of growth of total income is equal to n + g, which is independent of the work force¡¯s level of education. The two countries will, thus, have the same rate of growth of total income because they have the same rate of population growth and the same rate of technological progress. b. Because both countries have the same saving rate, the same population growth rate, and the same rate of technological progress, we know that the two countries will converge to the same steady-state level of capital per efficiency unit of labor k * . This is shown in Figure 8¨C1. Hence, output per efficiency unit of labor in the steady state, which is y * = f(k * ), is the same in both countries. But y * = Y/(L × E) or Y/L = y * E. We know that y * will be the same in both countries, but that E 1 > E 2 . Therefore, y * E 1 > y * E 2 . This 68 Answers to Textbook Questions and Problems Investment, break-even investment k Capital per efficiency unit k * (¦Ä + n + g) k sf (k) Figure 8¨C1 implies that (Y/L) 1 > (Y/L) 2 . Thus, the level of income per worker will be higher in the country with the more educated labor force. c. We know that the real rental price of capital R equals the marginal product of cap- ital (MPK). But the MPK depends on the capital stock per efficiency unit of labor. In the steady state, both countries have k = k = k * because both countries have the same saving rate, the same population growth rate, and the same rate of tech- nological progress. Therefore, it must be true that R 1 = R 2 = MPK. Thus, the real rental price of capital is identical in both countries. d. Output is divided between capital income and labor income. Therefore, the wage per efficiency unit of labor can be expressed as: w = f(k) ¨C MPK k. As discussed in parts (b) and (c), both countries have the same steady-state capital stock k and the same MPK. Therefore, the wage per efficiency unit in the two countries is equal. Workers, however, care about the wage per unit of labor, not the wage per efficiency unit. Also, we can observe the wage per unit of labor but not the wage per efficiency unit. The wage per unit of labor is related to the wage per efficiency unit of labor by the equation Wage per Unit of L = wE. Thus, the wage per unit of labor is higher in the country with the more educated labor force. 5. a. In the two-sector endogenous growth model in the text, the production function for manufactured goods is Y = F(K,(1 ¨C u) EL). We assumed in this model that this function has constant returns to scale. As in Section 3-1, constant returns means that for any positive number z, zY = F(zK, z(1 ¨C u) EL). Setting z = 1/EL, we obtain: Using our standard definitions of y as output per effective worker and k as capital per effective worker, we can write this as y = F(k,(1 ¨C u)). b. To begin, note that from the production function in research universities, the growth rate of labor efficiency, ?E / E, equals g(u). We can now follow the logic of Section 8-1, substituting the function g(u) for the constant growth rate g. In order to keep capital per effective worker (K/EL) constant, break-even investment includes three terms: ¦Äk is needed to replace depreciating capital, nk is needed to provide capital for new workers, and g(u) is needed to provide capital for the greater stock of knowledge E created by research universities. That is, break-even investment is (¦Ä + n + g(u))k. c. Again following the logic of Section 8-1, the growth of capital per effective worker is the difference between saving per effective worker and break-even investment per effective worker. We now substitute the per-effective-worker production func- tion from part (a), and the function g(u) for the constant growth rate g, to obtain: ?k = sF(k,(1 ¨C u)) ¨C (¦Ä + n + g(u))k. In the steady state, ?k = 0, so we can rewrite the equation above as: sF(k,(1 ¨C u)) = (¦Ä + n + g(u))k Chapter 8 Economic Growth II 69 Y EL F K EL u=? ? ? ? ? ? ? ,( ) .1 * 2 * 1 As in our analysis of the Solow model, for a given value of u we can plot the left- and right-hand sides of this equation: The steady state is given by the intersection of the two curves. d. The steady state has constant capital per effective worker k as given by Figure 8¨C2 above. We also assume that in the steady state, there is a constant share of time spent in research universities, so u is constant. (After all, if u were not con- stant, it wouldn¡¯t be a ¡°steady¡± state!). Hence, output per effective worker y is also constant. Output per worker equals yE, and E grows at rate g(u). Therefore, out- put per worker grows at rate g(u). The saving rate does not affect this growth rate. However, the amount of time spent in research universities does affect this rate: as more time is spent in research universities, the steady-state growth rate rises. e. An increase in u shifts both lines in our figure. Output per effective worker falls for any given level of capital per effective worker, since less of each worker¡¯s time is spent producing manufactured goods. This is the immediate effect of the change, since at the time u rises, the capital stock K and the efficiency of each worker E are constant. Since output per effective worker falls, the curve showing saving per effective worker shifts down. 70 Answers to Textbook Questions and Problems In v estment, break-e v en in v estment Capital per effective worker [¦Ä + n + g(u)]k sF (k, 1 ¨C u) Figure 8¨C2 At the same time, the increase in time spent in research universities increas- es the growth rate of labor efficiency g(u). Hence, break-even investment [which we found above in part (b)] rises at any given level of k, so the line showing break- even investment also shifts up. Figure 8¨C3 below shows these shifts: In the new steady state, capital per effective worker falls from k 1 to k 2 . Output per effective worker also falls. f. In the short run, the increase in u unambiguously decreases consumption. After all, we argued in part (e) that the immediate effect is to decrease output, since workers spend less time producing manufacturing goods and more time in research universities expanding the stock of knowledge. For a given saving rate, the decrease in output implies a decrease in consumption. The long-run steady-state effect is more subtle. We found in part (e) that out- put per effective worker falls in the steady state. But welfare depends on output (and consumption) per worker, not per effective worker. The increase in time spent in research universities implies that E grows faster. That is, output per worker equals yE. Although steady-state y falls, in the long run the faster growth rate of E necessarily dominates. That is, in the long run, consumption unambiguously rises. Nevertheless, because of the initial decline in consumption, the increase in u is not unambiguously a good thing. That is, a policymaker who cares more about current generations than about future generations may decide not to pursue a pol- icy of increasing u. (This is analogous to the question considered in Chapter 7 of whether a policymaker should try to reach the Golden Rule level of capital per effective worker if k is currently below the Golden Rule level.) Chapter 8 Economic Growth II 71 In v estment, break-e v en in v estment Capital per effective worker [¦Ä + n + g(u 2 )]k sF (k, 1 ¨C u 1 ) B A [¦Ä + n + g(u 1 )]k sF (k, 1 ¨C u 2 ) k 2 k 1 Figure 8¨C3 More Problems and Applications to Chapter 8 1. a. The growth in total output (Y) depends on the growth rates of labor (L), capital (K), and total factor productivity (A), as summarized by the equation: ?Y/Y = ¦Á?K/K + (1 ¨C ¦Á)?L/L + ?A/A, where ¦Á is capital¡¯s share of output. We can look at the effect on output of a 5-per- cent increase in labor by setting ?K/K = ?A/A = 0. Since ¦Á = 2/3, this gives us ?Y/Y = (1/3) (5%) = 1.67%. A 5-percent increase in labor input increases output by 1.67 percent. Labor productivity is Y/L. We can write the growth rate in labor productivity as = ¨C . Substituting for the growth in output and the growth in labor, we find ?(Y/L)/(Y/L) = 1.67% ¨C 5.0% = ¨C3.34%. Labor productivity falls by 3.34 percent. To find the change in total factor productivity, we use the equation ?A/A = ?Y/Y ¨C ¦Á?K/K ¨C (1 ¨C ¦Á)?L/L. For this problem, we find ?A/A = 1.67% ¨C 0 ¨C (1/3) (5%) = 0. Total factor productivity is the amount of output growth that remains after we have accounted for the determinants of growth that we can measure. In this case, there is no change in technology, so all of the output growth is attributable to measured input growth. That is, total factor productivity growth is zero, as expect- ed. b. Between years 1 and 2, the capital stock grows by 1/6, labor input grows by 1/3, and output grows by 1/6. We know that the growth in total factor productivity is given by ?A/A = ?Y/Y ¨C ¦Á?K/K ¨C (1 ¨C ¦Á)?L/L. Substituting the numbers above, and setting ¦Á = 2/3, we find ?A/A = (1/6) ¨C (2/3)(1/6) ¨C (1/3)(1/3) = 3/18 ¨C 2/18 ¨C 2/18 = ¨C 1/18 = ¨C .056. Total factor productivity falls by 1/18, or approximately 5.6 percent. 2. By definition, output Y equals labor productivity Y/L multiplied by the labor force L: Y = (Y/L)L. Using the mathematical trick in the hint, we can rewrite this as = + . We can rearrange this as = ¨C . 72 Answers to Textbook Questions and Problems ?(Y/L) Y/L ?L L ?Y Y ?(Y/L) Y/L ?L L ?Y Y ?Y Y ?L L ?(Y/L) Y/L Substituting for ?Y/Y from the text, we find Using the same trick we used above, we can express the term in brackets as ?K/K ¨C ?L/L = ?(K/L)/(K/L). Making this substitution in the equation for labor productivity growth, we conclude that 3. We know the following: ?Y/Y = n + g = 3% ?K/K = n + g = 3% ?L/L = n = 1% Capital¡¯s share = ¦Á = 0.3 Labor¡¯s share = 1 ¨C ¦Á = 0.7. Using these facts, we can easily find the contributions of each of the factors, and then find the contribution of total factor productivity growth, using the following equa- tions: Output = Capital¡¯s + Labor¡¯s + Total Factor Growth Contribution Contribution Productivity ?Y ¦Á?K (1 ¨C ¦Á)?L ?A YK 3.0% = (0.3)(3%) + (0.7)(1%) + ?A/A We can easily solve this for ?A/A, to find that 3.0% = 0.9% + 0.7% + 1.4%. We conclude that the contribution of capital is 0.9% per year, the contribution of labor is 0.7% per year, and the contribution of total factor productivity growth is 1.4% per year. These numbers are qualitatively similar to the ones in Table 8¨C3 in the text for the United States although in Table 8¨C3, capital and labor contribute more and TFP contributed less from 1950¨C1999. Chapter 8 Economic Growth II 73 ?(Y/L) = ?A + ¦Á?K + (1 ¨C ¦Á) ?L ¨C ?L Y/L A K L L ?(Y/L) = ?A + ¦Á?(K/L) . Y/L A K/L = ?A + ¦Á?K ¨C ¦Á?L A K L =+ ? ? ? ? ? ? ? ? ¦Á ??A A K K L L . Questions for Review 1. The Keynesian cross tells us that fiscal policy has a multiplied effect on income. The reason is that according to the consumption function, higher income causes higher con- sumption. For example, an increase in government purchases of ?G raises expenditure and, therefore, income by ?G. This increase in income causes consumption to rise by MPC ×?G, where MPC is the marginal propensity to consume. This increase in con- sumption raises expenditure and income even further. This feedback from consumption to income continues indefinitely. Therefore, in the Keynesian-cross model, increasing government spending by one dollar causes an increase in income that is greater than one dollar: it increases by ?G/(1 ¨C MPC). 2. The theory of liquidity preference explains how the supply and demand for real money balances determine the interest rate. A simple version of this theory assumes that there is a fixed supply of money, which the Fed chooses. The price level P is also fixed in this model, so that the supply of real balances is fixed. The demand for real money balances depends on the interest rate, which is the opportunity cost of holding money. At a high interest rate, people hold less money because the opportunity cost is high. By holding money, they forgo the interest on interest-bearing deposits. In contrast, at a low interest rate, people hold more money because the opportunity cost is low. Figure 10¨C1 graphs the supply and demand for real money balances. Based on this theory of liquidity preference, the interest rate adjusts to equilibrate the supply and demand for real money balances. 82 Supply of real money balances L (r) Demand for real money balances = M/PM/P Real money balances r r Interest rate Figure 10¨C1 CHAPTER 10 Aggregate Demand I Why does an increase in the money supply lower the interest rate? Consider what happens when the Fed increases the money supply from M 1 to M 2 . Because the price level P is fixed, this increase in the money supply shifts the supply of real money bal- ances M/P to the right, as in Figure 10¨C2. The interest rate must adjust to equilibrate supply and demand. At the old interest rate r 1 , supply exceeds demand. People holding the excess supply of money try to con- vert some of it into interest-bearing bank deposits or bonds. Banks and bond issuers, who prefer to pay lower interest rates, respond to this excess supply of money by lower- ing the interest rate. The interest rate falls until a new equilibrium is reached at r 2 . 3. The IS curve summarizes the relationship between the interest rate and the level of income that arises from equilibrium in the market for goods and services. Investment is negatively related to the interest rate. As illustrated in Figure 10¨C3, if the interest rate rises from r 1 to r 2 , the level of planned investment falls from I 1 to I 2 . Chapter 10 Aggregate Demand I 83 L (r) M/P Real money balances Interest rate r r 1 r 2 M 1 /P M 2 /P Figure 10¨C2 r Interest rate I Investment I (r) r 2 r 1 I(r) I 2 I 1 Figure 10¨C3 The Keynesian cross tells us that a reduction in planned investment shifts the expendi- ture function downward and reduces national income, as in Figure 10¨C4(A). Thus, as shown in Figure 10¨C4(B), a higher interest rate results in a lower level of national income: the IS curve slopes downward. 84 Answers to Textbook Questions and Problems E 45? r Planned expenditure Interest rate ¨C ¨C YY 2 Y 1 Y = E E 1 = C (Y T ) + I (r 1 ) + G E 2 = C (Y T ) + I (r 2 ) + G ?I Income, output (A) YY 2 Y 1 Income, output (B) IS r 2 r 1 Figure 10¨C4 4. The LM curve summarizes the relationship between the level of income and the inter- est rate that arises from equilibrium in the market for real money balances. It tells us the interest rate that equilibrates the money market for any given level of income. The theory of liquidity preference explains why the LM curve slopes upward. This theory assumes that the demand for real money balances L(r, Y) depends negatively on the interest rate (because the interest rate is the opportunity cost of holding money) and positively on the level of income. The price level is fixed in the short run, so the Fed determines the fixed supply of real money balances M/P. As illustrated in Figure 10¨C5(A), the interest rate equilibrates the supply and demand for real money balances for a given level of income. Now consider what happens to the interest rate when the level of income increas- es from Y 1 to Y 2 . The increase in income shifts the money demand curve upward. At the old interest rate r 1 , the demand for real money balances now exceeds the supply. The interest rate must rise to equilibrate supply and demand. Therefore, as shown in Figure 10¨C5(B), a higher level of income leads to a higher interest rate: The LM curve slopes upward. Chapter 10 Aggregate Demand I 85 (A) (B) r Interest rate r M P/ M P/ Real money balances Y Income, output LM Interest rate r 2 r 1 L (r,Y 2 ) L (r,Y 1 ) Y 1 Y 2 r 1 r 2 Figure 10¨C5 Problems and Applications 1. a. The Keynesian cross graphs an economy¡¯s planned expenditure function, E = C(Y ¨C T) + I + G, and the equilibrium condition that actual expenditure equals planned expenditure, Y = E, as shown in Figure 10¨C6. An increase in government purchases from G 1 to G 2 shifts the planned expendi- ture function upward. The new equilibrium is at point B. The change in Y equals the product of the government-purchases multiplier and the change in govern- ment spending: ?Y = [1/(1 ¨C MPC)]?G. Because we know that the marginal propensity to consume MPC is less than one, this expression tells us that a one- dollar increase in G leads to an increase in Y that is greater than one dollar. b. An increase in taxes ?T reduces disposable income Y ¨C T by ?T and, therefore, reduces consumption by MPC × ?T. For any given level of income Y, planned expenditure falls. In the Keynesian cross, the tax increase shifts the planned- expenditure function down by MPC ×?T, as in Figure 10¨C7. 86 Answers to Textbook Questions and Problems Y A B G? 45? Income, output E Planned expediture Y 1 Y 2 E 2 = C(Y ¨C T) + I + G 2 E 1 = C(Y ¨C T) + I + G 1 Y = E Planned expenditure Figure 10¨C6 Y = E MPC T×? Y ¨C MPC 1 ¨C MPC ?T Income, output 45? B A E Planned expenditure E = C(Y ¨C T ) + I + G Y 2 Y 1 Income, output Figure 10¨C7 The amount by which Y falls is given by the product of the tax multiplier and the increase in taxes: ?Y = [ ¨C MPC/(1 ¨C MPC)]?T. c. We can calculate the effect of an equal increase in government expenditure and taxes by adding the two multiplier effects that we used in parts (a) and (b): ?Y = [(1/(1 ¨C MPC))?G] ¨C [(MPC/(1 ¨C MPC))?T]. Government Tax Spending Multiplier Multiplier Because government purchases and taxes increase by the same amount, we know that ?G = ?T. Therefore, we can rewrite the above equation as: ?Y = [(1/(1 ¨C MPC)) ¨C (MPC/(1 ¨C MPC))]?G = ?G. This expression tells us that an equal increase in government purchases and taxes increases Y by the amount that G increases. That is, the balanced-budget multi- plier is exactly 1. 2. a. Total planned expenditure is E = C(Y ¨C T)+I + G. Plugging in the consumption function and the values for investment I, govern- ment purchases G, and taxes T given in the question, total planned expenditure E is E = 200 + 0.75(Y ¨C 100) + 100 + 100 = 0.75Y + 325. This equation is graphed in Figure 10¨C8. b. To find the equilibrium level of income, combine the planned-expenditure equa- tion derived in part (a) with the equilibrium condition Y = E: Y = 0.75Y + 325 Y = 1,300. The equilibrium level of income is 1,300, as indicated in Figure 10¨C8. c. If government purchases increase to 125, then planned expenditure changes to E = 0.75Y + 350. Equilibrium income increases to Y = 1,400. Therefore, an Chapter 10 Aggregate Demand I 87 Y = E E = 0.75 Y + 325 Y Y * = 1,300 Income, output 45 325 E Planned e xpenditure Figure 10¨C8 increase in government purchases of 25 (i.e., 125 ¨C 100 = 25) increases income by 100. This is what we expect to find, because the government-purchases multiplier is 1/(1 ¨C MPC): because the MPC is 0.75, the government-purchases multiplier is 4. d. A level of income of 1,600 represents an increase of 300 over the original level of income. The government-purchases multiplier is 1/(1 ¨C MPC): the MPC in this example equals 0.75, so the government-purchases multiplier is 4. This means that government purchases must increase by 75 (to a level of 175) for income to increase by 300. 3. a. When taxes do not depend on income, a one-dollar increase in income means that disposable income increases by one dollar. Consumption increases by the marginal propensity to consume MPC. When taxes do depend on income, a one-dollar increase in income means that disposable income increases by only (1 ¨C t) dollars. Consumption increases by the product of the MPC and the change in disposable income, or (1 ¨C t)MPC. This is less than the MPC. The key point is that disposable income changes by less than total income, so the effect on consumption is smaller. b. When taxes are fixed, we know that ?Y/?G = 1/(1 ¨C MPC). We found this by con- sidering an increase in government purchases of ?G; the initial effect of this change is to increase income by ?G. This in turn increases consumption by an amount equal to the marginal propensity to consume times the change in income, MPC ×?G. This increase in consumption raises expenditure and income even fur- ther. The process continues indefinitely, and we derive the multiplier above. When taxes depend on income, we know that the increase of ?G increases total income by ?G; disposable income, however, increases by only (1 ¨C t)?G¡ªless than dollar for dollar. Consumption then increases by an amount (1 ¨C t) MPC ×?G. Expenditure and income increase by this amount, which in turn caus- es consumption to increase even more. The process continues, and the total change in output is ?Y = ?G {1 + (1 ¨C t)MPC + [(1 ¨C t)MPC] 2 + [(1 ¨C t)MPC] 3 + ....} = ?G [1/(1 ¨C (1 ¨C t)MPC)]. Thus, the government-purchases multiplier becomes 1/(1 ¨C (1 ¨C t)MPC) rather than 1/(1 ¨C MPC). This means a much smaller multiplier. For example, if the mar- ginal propensity to consume MPC is 3/4 and the tax rate t is 1/3, then the multipli- er falls from 1/(1 ¨C 3/4), or 4, to 1/(1 ¨C (1 ¨C 1/3)(3/4)), or 2. c. In this chapter, we derived the IS curve algebraically and used it to gain insight into the relationship between the interest rate and output. To determine how this tax system alters the slope of the IS curve, we can derive the IS curve for the case in which taxes depend on income. Begin with the national income accounts identi- ty: Y = C + I + G. The consumption function is C = a + b(Y ¨C T ¨C tY). Note that in this consumption function taxes are a function of income. The invest- ment function is the same as in the chapter: I = c ¨C dr. Substitute the consumption and investment functions into the national income accounts identity to obtain: Y = [a + b(Y ¨C T ¨C tY)] + c ¨C dr + G. Solving for Y: Y = + G + T + r. 88 Answers to Textbook Questions and Problems a + c 1 ¨C b(1 ¨C t) 1 1 ¨C b(1 ¨C t) ¨C b 1 ¨C b(1 ¨C t) ¨C d 1 ¨C b(1 ¨C t) Chapter 10 Aggregate Demand I 89 This IS equation is analogous to the one derived in the text except that each term is divided by 1 ¨C b(1 ¨C t) rather than by (1 ¨C b). We know that t is a tax rate, which is less than 1. Therefore, we conclude that this IS curve is steeper than the one in which taxes are a fixed amount. 4. a. If society becomes more thrifty¡ªmeaning that for any given level of income people save more and consume less¡ªthen the planned-expenditure function shifts down- ward, as in Figure 10¨C9 (note that C 2 < C 1 ). Equilibrium income falls from Y 1 to Y 2 . b. Equilibrium saving remains unchanged. The national accounts identity tells us that saving equals investment, or S = I. In the Keynesian-cross model, we assumed that desired investment is fixed. This assumption implies that invest- ment is the same in the new equilibrium as it was in the old. We can conclude that saving is exactly the same in both equilibria. c. The paradox of thrift is that even though thriftiness increases, saving is unaffect- ed. Increased thriftiness leads only to a fall in income. For an individual, we usu- ally consider thriftiness a virtue. From the perspective of the Keynesian cross, however, thriftiness is a vice. d. In the classical model of Chapter 3, the paradox of thrift does not arise. In that model, output is fixed by the factors of production and the production technology, and the interest rate adjusts to equilibrate saving and investment, where invest- ment depends on the interest rate. An increase in thriftiness decreases consump- tion and increases saving for any level of output; since output is fixed, the saving schedule shifts to the right, as in Figure 10¨C10. At the new equilibrium, the inter- est rate is lower, and investment and saving are higher. Thus, in the classical model, the paradox of thrift does not exist. Y = E E Planned expenditure A B Y Income, output E 1 = C 1 + c (Y ¨C T ) + I + G E 2 = C 2 + c (Y ¨C T ) + I + G Y 1 Y 2 Figure 10¨C9 A B I, S Investment, Saving r Real interest rate I(r) r 1 r 2 S 2 S 1 Figure 10¨C10 5. a. The downward sloping line in Figure 10¨C11 represents the money demand func- tion (M/P) d = 1,000 ¨C 100r. With M = 1,000 and P = 2, the real money supply (M/P) s = 500. The real money supply is independent of the interest rate and is, therefore, represented by the vertical line in Figure 10¨C11. b. We can solve for the equilibrium interest rate by setting the supply and demand for real balances equal to each other: 500= 1,000 ¨C 100r r = 5. Therefore, the equilibrium real interest rate equals 5 percent. c. If the price level remains fixed at 2 and the supply of money is raised from 1,000 to 1,200, then the new supply of real balances (M/P) s equals 600. We can solve for the new equilibrium interest rate by setting the new (M/P) s equal to (M/P) d : 600 = 1,000 ¨C 100r 100r = 400 r = 4. Thus, increasing the money supply from 1,000 to 1,200 causes the equilibrium interest rate to fall from 5 percent to 4 percent. d. To determine at what level the Fed should set the money supply to raise the inter- est rate to 7 percent, set (M/P) s equal to (M/P) d : M/P = 1,000 ¨C 100r. Setting the price level at 2 and substituting r = 7, we find: M/2 = 1,000 ¨C 100 × 7 M = 600. For the Fed to raise the interest rate from 5 percent to 7 percent, it must reduce the nominal money supply from 1,000 to 600. 90 Answers to Textbook Questions and Problems M/P1,000500 Real money balances r 10 5 Interest rate (M/P) s (M/P) d Figure 10¨C11 Questions for Review 1. The aggregate demand curve represents the negative relationship between the price level and the level of national income. In Chapter 9, we looked at a simplified theory of aggregate demand based on the quantity theory. In this chapter, we explore how the IS¨CLM model provides a more complete theory of aggregate demand. We can see why the aggregate demand curve slopes downward by considering what happens in the IS¨CLM model when the price level changes. As Figure 11¨C1(A) illustrates, for a given money supply, an increase in the price level from P 1 to P 2 shifts the LM curve upward because real balances decline; this reduces income from Y 1 to Y 2 . The aggregate demand curve in Figure 11¨C1(B) summarizes this relationship between the price level and income that results from the IS¨CLM model. 91 LM (P = P 2 ) Y 2 Y 2 Y 1 P 2 P 1 LM (P = P 1 ) Y 1 A. The IS LM Model B. The Aggregate Demand Curve r Interest rate B A IS Income, output Income, output P Price level AD ¨C Figure 11¨C1 CHAPTER 11 Aggregate Demand II 2. The tax multiplier in the Keynesian-cross model tells us that, for any given interest rate, the tax increase causes income to fall by ?T × [ ¨C MPC/(1 ¨C MPC)]. This IS curve shifts to the left by this amount, as in Figure 11¨C2. The equilibrium of the economy moves from point A to point B. The tax increase reduces the interest rate from r 1 to r 2 and reduces national income from Y 1 to Y 2 . Consumption falls because disposable income falls; investment rises because the interest rate falls. Note that the decrease in income in the IS¨CLM model is smaller than in the Keynesian cross, because the IS¨CLM model takes into account the fact that investment rises when the interest rate falls. 3. Given a fixed price level, a decrease in the nominal money supply decreases real money balances. The theory of liquidity preference shows that, for any given level of income, a decrease in real money balances leads to a higher interest rate. Thus, the LM curve shifts upward, as in Figure 11¨C3. The equilibrium moves from point A to point B. The decrease in the money supply reduces income and raises the interest rate. Consumption falls because disposable income falls, whereas investment falls because the interest rate rises. LM A B Y Income, output Interest rate r ? T × 1 ¨C MPC Y 1 Y 2 r 1 r 2 IS 2 IS 1 ¨C MPC 92 Answers to Textbook Questions and Problems Figure 11¨C2 B A IS Y Income, output Interest rate r Y 1 LM 2 LM 1 r 2 r 1 Y 2 Figure 11¨C3 4. Falling prices can either increase or decrease equilibrium income. There are two ways in which falling prices can increase income. First, an increase in real money balances shifts the LM curve downward, thereby increasing income. Second, the IS curve shifts to the right because of the Pigou effect: real money balances are part of household wealth, so an increase in real money balances makes consumers feel wealthier and buy more. This shifts the IS curve to the right, also increasing income. There are two ways in which falling prices can reduce income. The first is the debt-deflation theory. An unexpected decrease in the price level redistributes wealth from debtors to creditors. If debtors have a higher propensity to consume than credi- tors, then this redistribution causes debtors to decrease their spending by more than creditors increase theirs. As a result, aggregate consumption falls, shifting the IS curve to the left and reducing income. The second way in which falling prices can reduce income is through the effects of expected deflation. Recall that the real interest rate r equals the nominal interest rate i minus the expected inflation rate ¦Ð e : r = i ¨C ¦Ð e . If everyone expects the price level to fall in the future (i.e., ¦Ð e is negative), then for any given nominal interest rate, the real interest rate is higher. A higher real interest rate depresses investment and shifts the IS curve to the left, reducing income. Problems and Applications 1. a. If the central bank increases the money supply, then the LM curve shifts down- ward, as shown in Figure 11¨C4. Income increases and the interest rate falls. The increase in disposable income causes consumption to rise; the fall in the interest rate causes investment to rise as well. Chapter 11 Aggregate Demand II 93 A B IS Y Income, output Interest rate r LM 1 LM 2 r 1 r 2 Y 1 Y 2 Figure 11¨C4 b. If government purchases increase, then the government-purchases multiplier tells us that the IS curve shifts to the right by an amount equal to [1/(1 ¨C MPC)]?G. This is shown in Figure 11¨C5. Income and the interest rate both increase. The increase in disposable income causes consumption to rise, while the increase in the interest rate causes investment to fall. c. If the government increases taxes, then the tax multiplier tells us that the IS curve shifts to the left by an amount equal to [ ¨C MPC/(1 ¨C MPC)]?T. This is shown in Figure 11¨C6. Income and the interest rate both fall. Disposable income falls because income is lower and taxes are higher; this causes consumption to fall. The fall in the interest rate causes investment to rise. 94 Answers to Textbook Questions and Problems LM A B Interest rate Y Income, output Y 2 r 2 r 1 IS 2 IS 1 ? G 1 ¨C MPC r Y 1 Figure 11¨C5 LM A B Interest rate Y Income, output r Y 1 IS 2 IS 1 r 2 r 1 Y 2 ¨C MPC 1 ¨C MPC ? T Figure 11¨C6 d. We can figure out how much the IS curve shifts in response to an equal increase in government purchases and taxes by adding together the two multiplier effects that we used in parts (b) and (c): ?Y = [(1/(1 ¨C MPC))]?G] ¨C [(MPC/(1 ¨C MPC))?T] Because government purchases and taxes increase by the same amount, we know that ?G = ?T. Therefore, we can rewrite the above equation as: ?Y = [(1/(1 ¨C MPC)) ¨C (MPC/(1 ¨C MPC))]?G ?Y = ?G. This expression tells us how output changes, holding the interest rate constant. It says that an equal increase in government purchases and taxes shifts the IS curve to the right by the amount that G increases. This shift is shown in Figure 11¨C7. Output increases, but by less than the amount that G and T increase; this means that disposable income Y ¨C T falls. As a result, consumption also falls. The interest rate rises, causing investment to fall. Chapter 11 Aggregate Demand II 95 LM B A Interest rate Y Income, output r ? G Y 2 IS 1 r 2 r 1 Y 1 IS 2 Figure 11¨C7 2. a. The invention of the new high-speed chip increases investment demand, which shifts the IS curve out. That is, at every interest rate, firms want to invest more. The increase in the demand for investment goods shifts the IS curve out, raising income and employment. Figure 11¨C8 shows the effect graphically. The increase in income from the higher investment demand also raises interest rates. This happens because the higher income raises demand for money; since the supply of money does not change, the interest rate must rise in order to restore equilibrium in the money market. The rise in interest rates partially offsets the increase in investment demand, so that output does not rise by the full amount of the rightward shift in the IS curve. Overall, income, interest rates, consumption, and investment all rise. b. The increased demand for cash shifts the LM curve up. This happens because at any given level of income and money supply, the interest rate necessary to equili- brate the money market is higher. Figure 11¨C9 shows the effect of this LM shift graphically. 96 Answers to Textbook Questions and Problems B Interest rate LM Income, output r YY 1 Y 2 IS 2 IS 1 r 1 r 2 A Figure 11¨C8 B Interest rate Income, output r YY 2 Y 1 IS r 2 r 1 A LM 2 LM 1 Figure 11¨C9 The upward shift in the LM curve lowers income and raises the interest rate. Consumption falls because income falls, and investment falls because the interest rate rises. c. At any given level of income, consumers now wish to save more and consume less. Because of this downward shift in the consumption function, the IS curve shifts inward. Figure 11¨C10 shows the effect of this IS shift graphically. Income, interest rates, and consumption all fall, while investment rises. Income falls because at every level of the interest rate, planned expenditure falls. The interest rate falls because the fall in income reduces demand for money; since the supply of money is unchanged, the interest rate must fall to restore money-market equilibrium. Consumption falls both because of the shift in the consumption func- tion and because income falls. Investment rises because of the lower interest rates and partially offsets the effect on output of the fall in consumption. Chapter 11 Aggregate Demand II 97 B Interest rate LM Income, output r YY 1 Y 2 IS 2 IS 1 r 1 r 2 A Figure 11¨C10 3. a. The IS curve is given by: Y = C(Y ¨C T) + I(r)+G. We can plug in the consumption and investment functions and values for G and T as given in the question and then rearrange to solve for the IS curve for this econ- omy: Y = 200 + 0.75(Y ¨C 100) + 200 ¨C 25r + 100 Y ¨C 0.75Y = 425 ¨C 25r (1 ¨C 0.75)Y = 425 ¨C 25r Y = (1/0.25) (425 ¨C 25r) Y = 1,700 ¨C 100r. This IS equation is graphed in Figure 11¨C11 for r ranging from 0 to 8. b. The LM curve is determined by equating the demand for and supply of real money balances. The supply of real balances is 1,000/2 = 500. Setting this equal to money demand, we find: 500 = Y ¨C 100r. Y = 500 + 100r. This LM curve is graphed in Figure 11¨C11 for r ranging from 0 to 8. c. If we take the price level as given, then the IS and the LM equations give us two equations in two unknowns, Y and r. We found the following equations in parts (a) and (b): IS: Y = 1,700 ¨C 100r. LM: Y = 500 + 100r. Equating these, we can solve for r: 1,700 ¨C 100r = 500 + 100r 1,200 = 200r r = 6. Now that we know r, we can solve for Y by substituting it into either the IS or the LM equation. We find Y = 1,100. Therefore, the equilibrium interest rate is 6 percent and the equilibrium level of output is 1,100, as depicted in Figure 11¨C11. 98 Answers to Textbook Questions and Problems LMIS Y1,7001,100500 0 6 8 r Interest rate Income, output Figure 11¨C11 d. If government purchases increase from 100 to 150, then the IS equation becomes: Y = 200 + 0.75(Y ¨C 100) + 200 ¨C 25r + 150. Simplifying, we find: Y = 1,900 ¨C 100r. This IS curve is graphed as IS 2 in Figure 11¨C12. We see that the IS curve shifts to the right by 200. By equating the new IS curve with the LM curve derived in part (b), we can solve for the new equilibrium interest rate: 1,900 ¨C 100r = 500 + 100r 1,400 = 200r 7 = r. We can now substitute r into either the IS or the LM equation to find the new level of output. We find Y = 1,200. Therefore, the increase in government purchases causes the equilibrium interest rate to rise from 6 percent to 7 percent, while output increases from 1,100 to 1,200. This is depicted in Figure 11¨C12. Chapter 11 Aggregate Demand II 99 1,200500 1,100 LM 200 Y1,9001,700 0 6 7 r Interest rate Income, output IS 2 IS 1 8 Figure 11¨C12 e. If the money supply increases from 1,000 to 1,200, then the LM equation becomes: (1,200/2) = Y ¨C 100r, or Y = 600 + 100r. This LM curve is graphed as LM 2 in Figure 11¨C13. We see that the LM curve shifts to the right by 100 because of the increase in real money balances. To determine the new equilibrium interest rate and level of output, equate the IS curve from part (a) with the new LM curve derived above: 1,700 ¨C 100r = 600 + 100r 1,100 = 200r 5.5 = r. Substituting this into either the IS or the LM equation, we find Y = 1,150. Therefore, the increase in the money supply causes the interest rate to fall from 6 percent to 5.5 percent, while output increases from 1,100 to 1,150. This is depicted in Figure 11¨C13. f. If the price level rises from 2 to 4, then real money balances fall from 500 to 1,000/4 = 250. The LM equation becomes: Y = 250 + 100r. 100 Answers to Textbook Questions and Problems IS Y1,7001,1501,100600500 0 r 6.0 5.5 Interest rate Income, output 100 LM 1 LM 2 Figure 11¨C13 As shown in Figure 11¨C14, the LM curve shifts to the left by 250 because the increase in the price level reduces real money balances. To determine the new equilibrium interest rate, equate the IS curve from part (a) with the new LM curve from above: 1,700 ¨C 100r = 250 + 100r 1,450 = 200r 7.25 = r. Substituting this interest rate into either the IS or the LM equation, we find Y = 975. Therefore, the new equilibrium interest rate is 7.25, and the new equilibrium level of output is 975, as depicted in Figure 11¨C14. g. The aggregate demand curve is a relationship between the price level and the level of income. To derive the aggregate demand curve, we want to solve the IS and the LM equations for Y as a function of P. That is, we want to substitute out for the interest rate. We can do this by solving the IS and the LM equations for the interest rate: IS: Y = 1,700 ¨C 100r 100r = 1,700 ¨C Y. LM: (M/P) = Y ¨C 100r 100r = Y ¨C (M/P). Combining these two equations, we find 1,700 ¨C Y = Y ¨C (M/P) 2Y = 1,700 + M/P Y = 850 + M/2P. Since the nominal money supply M equals 1,000, this becomes Y = 850 + 500/P. Chapter 11 Aggregate Demand II 101 250 IS r 7.25 6.0 Interest rate 0 975 1,200 1,700 Y600 250 Income, output LM 2 LM 1 1,100 Figure 11¨C14 This aggregate demand equation is graphed in Figure 11¨C15. How does the increase in fiscal policy of part (d) affect the aggregate demand curve? We can see this by deriving the aggregate demand curve using the IS equa- tion from part (d) and the LM curve from part (b): IS: Y = 1,900 ¨C 100r 100r = 1,900 ¨C Y. LM: (1,000/P) = Y ¨C 100r 100r = Y ¨C (1,000/P). Combining and solving for Y: 1,900 ¨C Y = Y ¨C (1,000/P), or Y = 950 + 500/P. By comparing this new aggregate demand equation to the one previously derived, we can see that the increase in government purchases by 50 shifts the aggregate demand curve to the right by 100. How does the increase in the money supply of part (e) affect the aggregate demand curve? Because the AD curve is Y = 850 + M/2P, the increase in the money supply from 1,000 to 1,200 causes it to become Y = 850 + 600/P. By comparing this new aggregate demand curve to the one originally derived, we see that the increase in the money supply shifts the aggregate demand curve to the right. 4. a. The IS curve represents the relationship between the interest rate and the level of income that arises from equilibrium in the market for goods and services. That is, it describes the combinations of income and the interest rate that satisfy the equa- tion Y = C(Y ¨C T) + I(r)+G. 102 Answers to Textbook Questions and Problems P 4.0 2.0 1.0 0.5 0 975 1,100 1,350 1,850 Y Price level Income, output Figure 11¨C15 If investment does not depend on the interest rate, then nothing in the IS equa- tion depends on the interest rate; income must adjust to ensure that the quantity of goods produced, Y, equals the quantity of goods demanded, C + I + G. Thus, the IS curve is vertical at this level, as shown in Figure 11¨C16. Monetary policy has no effect on output, because the IS curve determines Y. Monetary policy can affect only the interest rate. In contrast, fiscal policy is effec- tive: output increases by the full amount that the IS curve shifts. b. The LM curve represents the combinations of income and the interest rate at which the money market is in equilibrium. If money demand does not depend on the interest rate, then we can write the LM equation as M/P = L(Y). For any given level of real balances M/P, there is only one level of income at which the money market is in equilibrium. Thus, the LM curve is vertical, as shown in Figure 11¨C17. Fiscal policy now has no effect on output; it can affect only the interest rate. Monetary policy is effective: a shift in the LM curve increases output by the full amount of the shift. c. If money demand does not depend on income, then we can write the LM equation as M/P = L(r). Chapter 11 Aggregate Demand II 103 IS r Interest rate LM YY Income, output Figure 11¨C16 LM IS YY Income, output r Interest rate For any given level of real balances M/P, there is only one level of the interest rate at which the money market is in equilibrium. Hence, the LM curve is horizon- tal, as shown in Figure 11¨C18. Fiscal policy is very effective: output increases by the full amount that the IS curve shifts. Monetary policy is also effective: an increase in the money supply causes the interest rate to fall, so the LM curve shifts down, as shown in Figure 11¨C18. d. The LM curve gives the combinations of income and the interest rate at which the supply and demand for real balances are equal, so that the money market is in equilibrium. The general form of the LM equation is M/P = L(r, Y). Suppose income Y increases by $1. How much must the interest rate change to keep the money market in equilibrium? The increase in Y increases money demand. If money demand is extremely sensitive to the interest rate, then it takes a very small increase in the interest rate to reduce money demand and restore equilibrium in the money market. Hence, the LM curve is (nearly) horizontal, as shown in Figure 11¨C19. 104 Answers to Textbook Questions and Problems LM 1 LM 2 IS Y Income, output r Interest rate Figure 11¨C18 LM IS Y Income, output r Interest rate Figure 11¨C19 An example may make this clearer. Consider a linear version of the LM equation: M/P = eY ¨C fr. Note that as f gets larger, money demand becomes increasingly sensitive to the interest rate. Rearranging this equation to solve for r, we find r = (e/f)Y ¨C (1/f)(M/P). We want to focus on how changes in each of the variables are related to changes in the other variables. Hence, it is convenient to write this equation in terms of changes: ? r = (e/f)?Y ¨C (1/f)?(M/P). The slope of the LM equation tells us how much r changes when Y changes, hold- ing M fixed. If ?(M/P) = 0, then the slope is ?r/?Y = (e/f). As f gets very large, this slope gets closer and closer to zero. If money demand is very sensitive to the interest rate, then fiscal policy is very effective: with a horizontal LM curve, output increases by the full amount that the IS curve shifts. Monetary policy is now completely ineffective: an increase in the money supply does not shift the LM curve at all. We see this in our example by considering what happens if M increases. For any given Y (so that we set ?Y = 0), ?r/?(M/P) = ( ¨C 1/f); this tells us how much the LM curve shifts down. As f gets larger, this shift gets smaller and approaches zero. (This is in contrast to the hori- zontal LM curve in part (c), which does shift down.) 5. To raise investment while keeping output constant, the government should adopt a loose monetary policy and a tight fiscal policy, as shown in Figure 11¨C20. In the new equilibrium at point B, the interest rate is lower, so that investment is high- er. The tight fiscal policy¡ªreducing government purchases, for example¡ªoffsets the effect of this increase in investment on output. Chapter 11 Aggregate Demand II 105 A B Y r Interest rate IS 1 Income, output r 1 r 2 Y 1 LM 2 LM 1 IS 2 Figure 11¨C20 The policy mix in the early 1980s did exactly the opposite. Fiscal policy was expansionary, while monetary policy was contractionary. Such a policy mix shifts the IS curve to the right and the LM curve to the left, as in Figure 11¨C21. The real interest rate rises and investment falls. 6. a. An increase in the money supply shifts the LM curve to the right in the short run. This moves the economy from point A to point B in Figure 11¨C22: the interest rate falls from r 1 to r 2 , and output rises from Y to Y 2 . The increase in output occurs because the lower interest rate stimulates investment, which increases output. Since the level of output is now above its long-run level, prices begin to rise. A rising price level lowers real balances, which raises the interest rate. As indicat- ed in Figure 11¨C22, the LM curve shifts back to the left. Prices continue to rise until the economy returns to its original position at point A. The interest rate returns to r 1 , and investment returns to its original level. Thus, in the long run, there is no impact on real variables from an increase in the money supply. (This is what we called monetary neutrality in Chapter 4.) 106 Answers to Textbook Questions and Problems Income, output Y 1 r 2 r 1 IS 2 IS 1 LM 2 LM 1 A B Y r Interest rate Figure 11¨C21 Figure 10-19 A B IS Y r Interest rate r 1 LM 2 Y= Y Income, output LM 1 Y Y 2 r 2 Figure 11¨C22 b. An increase in government purchases shifts the IS curve to the right, and the economy moves from point A to point B, as shown in Figure 11¨C23. In the short run, output increases from Y to Y 2 , and the interest rate increases from r 1 to r 2 . The increase in the interest rate reduces investment and ¡°crowds out¡± part of the expansionary effect of the increase in government purchases. Initially, the LM curve is not affected because government spending does not enter the LM equa- tion. After the increase, output is above its long-run equilibrium level, so prices begin to rise. The rise in prices reduces real balances, which shifts the LM curve to the left. The interest rate rises even more than in the short run. This process continues until the long-run level of output is again reached. At the new equilibri- um, point C, interest rates have risen to r 3 , and the price level is permanently higher. Note that, like monetary policy, fiscal policy cannot change the long-run level of output. Unlike monetary policy, however, it can change the composition of output. For example, the level of investment at point C is lower than it is at point A. c. An increase in taxes reduces disposable income for consumers, shifting the IS curve to the left, as shown in Figure 11¨C24. In the short run, output and the inter- est rate decline to Y 2 to r 2 as the economy moves from point A to point B. Chapter 11 Aggregate Demand II 107 C B A Y r Interest rate r 3 r 2 r 1 IS 1 IS 2 LM 2 LM 1 Income, output Y= Y YY 2 Figure 11¨C23 A B C Y r Interest rate Y= Y LM 2 LM 1 IS 1 IS 2 Income, output YY 2 r 2 r 3 r 1 Figure 11¨C24 Initially, the LM curve is not affected. In the longer run, prices begin to decline because output is below its long-run equilibrium level, and the LM curve then shifts to the right because of the increase in real money balances. Interest rates fall even further to r 3 and, thus, further stimulate investment and increase income. In the long run, the economy moves to point C. Output returns to Y, the price level and the interest rate are lower, and the decrease in consumption has been offset by an equal increases in investment. 7. Figure 11¨C25(A) shows what the IS¨CLM model looks like for the case in which the Fed holds the money supply constant. Figure 11¨C25(B) shows what the model looks like if the Fed adjusts the money supply to hold the interest rate constant; this policy makes the effective LM curve horizontal. a. If all shocks to the economy arise from exogenous changes in the demand for goods and services, this means that all shocks are to the IS curve. Suppose a shock caus- es the IS curve to shift from IS 1 to IS 2 . Figures 11¨C26(A) and (B) show what effect this has on output under the two policies. It is clear that output fluctuates less if the Fed follows a policy of keeping the money supply constant. Thus, if all shocks are to the IS curve, then the Fed should follow a policy of keeping the money sup- ply constant. LM A. Holding the Money Supply Constant B. Holding the Interest Rate Constant r Interest rate IS Y Income, output r Interest rate Y Income, output LM IS 108 Answers to Textbook Questions and Problems Figure 11¨C25 r Interest rate LM Y r Interest rate Y LM A. Holding the Money Supply Constant B. Holding the Interest Rate Constant Y 2 Income, output IS 1 IS 2 Y 1 Y 1 Y 2 Income, output IS 1 IS 2 A. Holding the Money Supply Constant B. Holding the Interest Rate Constant Figure 11¨C26 b. If all shocks in the economy arise from exogenous changes in the demand for money, this means that all shocks are to the LM curve. If the Fed follows a policy of adjusting the money supply to keep the interest rate constant, then the LM curve does not shift in response to these shocks¡ªthe Fed immediately adjusts the money supply to keep the money market in equilibrium. Figures 11¨C27(A) and (B) show the effects of the two policies. It is clear that output fluctuates less if the Fed holds the interest rate constant, as in Figure 11¨C27(B). If the Fed holds the inter- est rate constant and offsets shocks to money demand by changing the money sup- ply, then all variability in output is eliminated. Thus, if all shocks are to the LM curve, then the Fed should adjust the money supply to hold the interest rate con- stant, thereby stabilizing output. 8. a. The analysis of changes in government purchases is unaffected by making money demand dependent on disposable income instead of total expenditure. An increase in government purchases shifts the IS curve to the right, as in the standard case. The LM curve is unaffected by this increase. Thus, the analysis is the same as it was before; this is shown in Figure 11¨C28. Chapter 11 Aggregate Demand II 109 A. Holding the Money Supply Constant r Interest rate IS Y B. Holding the Interest Rate Constant r Interest rate Y LM IS Y Income, output LM 1 LM 2 Income, output Y 1 Y 2 Figure 11¨C27 b. A tax cut causes disposable income Y ¨C T to increase at every level of income Y. This increases consumption for any given level of income as well, so the IS curve shifts to the right, as in the standard case. This is shown in Figure 11¨C29. If money demand depends on disposable income, however, then the tax cut increases money demand, so the LM curve shifts upward, as shown in the figure. Thus, the analysis of a change in taxes is altered drastically by making money demand dependent on disposable income. As shown in the figure, it is possible for a tax cut to be contractionary. 110 Answers to Textbook Questions and Problems B A Y r Interest rate Income, output Y 2 Y 1 IS 1 IS 2 LM 1 LM 2 Figure 11¨C29 LM Y r Interest rate IS 1 IS 2 Income, output Y 2 Y 1 Figure 11¨C28 Questions for Review 1. In the Mundell¨CFleming model, an increase in taxes shifts the IS* curve to the left. If the exchange rate floats freely, then the LM * curve is unaffected. As shown in Figure 12¨C1, the exchange rate falls while aggregate income remains unchanged. The fall in the exchange rate causes the trade balance to increase. 111 A B Y Income, output e Exchange rate IS 2 * * IS 1 LM * Figure 12¨C1 CHAPTER 12 Aggregate Demand in the Open Economy Now suppose there are fixed exchange rates. When the IS * curve shifts to the left in Figure 12¨C2, the money supply has to fall to keep the exchange rate constant, shift- ing the LM * curve from LM to LM . As shown in the figure, output falls while the exchange rate remains fixed. Net exports can only change if the exchange rate changes or the net exports sched- ule shifts. Neither occurs here, so net exports do not change. We conclude that in an open economy, fiscal policy is effective at influencing out- put under fixed exchange rates but ineffective under floating exchange rates. 2. In the Mundell¨CFleming model with floating exchange rates, a reduction in the money supply reduces real balances M/P, causing the LM * curve to shift to the left. As shown in Figure 12¨C3, this leads to a new equilibrium with lower income and a higher exchange rate. The increase in the exchange rate reduces the trade balance. 112 Answers to Textbook Questions and Problems * 1 * 2 AB Y e e Exchange rate Fixed exchange rate IS 2 IS 1 Income, output Y 1 LM 2 LM 1 ** Y 2 * * Figure 12¨C2 e Exchange rate B A Y Income, output LM 2 LM 1 IS * Y 1 Y 2 ** Figure 12¨C3 If exchange rates are fixed, then the upward pressure on the exchange rate forces the Fed to sell dollars and buy foreign exchange. This increases the money supply M and shifts the LM * curve back to the right until it reaches LM again, as shown in Figure 12¨C4. In equilibrium, income, the exchange rate, and the trade balance are unchanged. We conclude that in an open economy, monetary policy is effective at influencing output under floating exchange rates but impossible under fixed exchange rates. 3. In the Mundell¨CFleming model under floating exchange rates, removing a quota on imported cars shifts the net exports schedule inward, as shown in Figure 12¨C5. As in the figure, for any given exchange rate, such as e, net exports fall. This is because it now becomes possible for Americans to buy more Toyotas, Volkswagens, and other for- eign cars than they could when there was a quota. Chapter 12 Aggregate Demand in the Open Economy 113 * 1 e Exchange rate Fixed exchange rate A Y Income, output LM 1 IS * * e Figure 12¨C4 e e Exchange rate NX Net exports NX 1 (e) NX 2 (e) NX 1 NX 2 Figure 12¨C5 This inward shift in the net-exports schedule causes the IS * schedule to shift inward as well, as shown in Figure 12¨C6. The exchange rate falls while income remains unchanged. The trade balance is also unchanged. We know this since NX (e) = Y ¨C C(Y ¨C T) ¨C I(r) ¨C G. Removing the quota has no effect on Y, C, I, or G, so it also has no effect on the trade balance. If there are fixed exchange rates, then the shift in the IS * curve puts downward pressure on the exchange rate, as above. In order to keep the exchange rate fixed, the Fed is forced to buy dollars and sell foreign exchange. This shifts the LM * curve to the left, as shown in Figure 12¨C7. In equilibrium, income is lower and the exchange rate is unchanged. The trade balance falls; we know this because net exports are lower at any level of the exchange rate. 114 Answers to Textbook Questions and Problems e Exchange rate A B YY Income, output IS 2 IS 1 e 1 e 2 LM * * * Figure 12¨C6 e e Exchange rate BA e Y Income, output LM 2 LM 1 Y 1 ** IS 1 IS 2 * * Y 2 Figure 12¨C7 4. The following table lists some of the advantages and disadvantages of floating versus fixed exchange rates. Table 12¨C1 Floating Exchange Rates Advantages: Allows monetary policy to pursue goals other than just exchange-rate stabilization, for example, the stability of prices and employment. Disadvantages: Exchange-rate uncertainty is higher, and this might make international trade more difficult. Fixed Exchange Rates Advantages: Makes international trade easier by reducing exchange rate uncertainty. It disciplines the monetary authority, preventing excessive growth in M. As a monetary rule, it is easy to implement. Disadvantages: Monetary policy cannot be used to pursue policy goals other than maintaining the exchange rate. As a way to discipline the monetary authority, it may lead to greater instability in income and employment. Problems and Applications 1. The following three equations describe the Mundell¨CFleming model: Y = C(Y ¨C T) + I(r) + G + NX(e). (IS) M/P = L(r, Y). (LM) r = r * . In addition, we assume that the price level is fixed in the short run, both at home and abroad. This means that the nominal exchange rate e equals the real exchange rate . a. If consumers decide to spend less and save more, then the IS * curve shifts to the left. Figure 12¨C8 shows the case of floating exchange rates. Since the money sup- ply does not adjust, the LM * curve does not shift. Since the LM * curve is unchanged, output Y is also unchanged. The exchange rate falls (depreciates), which causes an increase in the trade balance equal to the fall in consumption. Chapter 12 Aggregate Demand in the Open Economy 115 e Exchange rate A B Y Income, output e 1 e 2 LM * Y 1 IS 1 * IS 2 * ? Figure 12¨C8 Figure 12¨C9 shows the case of fixed exchange rates. The IS * curve shifts to the left, but the exchange rate cannot fall. Instead, output falls. Since the exchange rate does not change, we know that the trade balance does not change either. In essence, the fall in desired spending puts downward pressure on the inter- est rate and, hence, on the exchange rate. If there are fixed exchange rates, then the central bank buys the domestic currency that investors seek to exchange, and provides foreign currency. As a result, the exchange rate does not change, so the trade balance does not change. Hence, there is nothing to offset the fall in con- sumption, and output falls. b. If some consumers decide they prefer stylish Toyotas to Fords and Chryslers, then the net-exports schedule, shown in Figure 12¨C10, shifts to the left. That is, at any level of the exchange rate, net exports are lower than they were before. 116 Answers to Textbook Questions and Problems e e Exchange rate BA Y Income, output Y 1 IS 2 IS 1 LM 2 LM 1 ** * * Y 2 Figure 12¨C9 e Exchange rate NX Net exports NX 1 NX 2 Figure 12¨C10 This shifts the IS * curve to the left as well, as shown in Figure 12¨C11 for the case of floating exchange rates. Since the LM * curve is fixed, output does not change, while the exchange rate falls (depreciates). The trade balance does not change either, despite the fall in the exchange rate. We know this since NX = S ¨C I, and both saving and investment remain unchanged. Figure 12¨C12 shows the case of fixed exchange rates. The leftward shift in the IS * curve puts downward pressure on the exchange rate. The central bank buys dollars and sells foreign exchange to keep e fixed: this reduces M and shifts the LM * curve to the left. As a result, output falls. The trade balance falls, because the shift in the net exports schedule means that net exports are lower for any given level of the exchange rate. c. The introduction of ATM machines reduces the demand for money. We know that equilibrium in the money market requires that the supply of real balances M/P must equal demand: M/P = L(r * , Y). A fall in money demand means that for unchanged income and interest rates, the right-hand side of this equation falls. Since M and P are both fixed, we know that Chapter 12 Aggregate Demand in the Open Economy 117 e Exchange rate A B Y Income, output LM * e 1 e 2 IS 2 IS 1 * * Figure 12¨C11 e e Exchange rate BA Y Income, output IS 2 IS 1 Y 1 LM 1 LM 2 ** * * Y 2 Figure 12¨C12 the left-hand side of this equation cannot adjust to restore equilibrium. We also know that the interest rate is fixed at the level of the world interest rate. This means that income¡ªthe only variable that can adjust¡ªmust rise in order to increase the demand for money. That is, the LM * curve shifts to the right. Figure 12¨C13 shows the case with floating exchange rates. Income rises, the exchange rate falls (depreciates), and the trade balance rises. Figure 12¨C14 shows the case of fixed exchange rates. The LM * schedule shifts to the right; as before, this tends to push domestic interest rates down and cause the currency to depreciate. However, the central bank buys dollars and sells foreign currency in order to keep the exchange rate from falling. This reduces the money supply and shifts the LM * schedule back to the left. The LM * curve contin- ues to shift back until the original equilibrium is restored. In the end, income, the exchange rate, and the trade balance are unchanged. 118 Answers to Textbook Questions and Problems e Exchange rate A B Y Income, output LM 1 LM 2 e 2 e 1 IS * ** Y 1 Y 2 Figure 12¨C13 e e Exchange rate A Y Income, output LM * IS * Y 1 Figure 12¨C14 2. a. The Mundell¨CFleming model takes the world interest rate r * as an exogenous variable. However, there is no reason to expect the world interest rate to be con- stant. In the closed-economy model of Chapter 3, the equilibrium of saving and investment determines the real interest rate. In an open economy in the long run, the world real interest rate is the rate that equilibrates world saving and world investment demand. Anything that reduces world saving or increases world investment demand increases the world interest rate. In addition, in the short run with fixed prices, anything that increases the worldwide demand for goods or reduces the worldwide supply of money causes the world interest rate to rise. b. Figure 12¨C15 shows the effect of an increase in the world interest rate under float- ing exchange rates. Both the IS * and the LM * curves shift. The IS * curve shifts to the left, because the higher interest rate causes investment I(r * ) to fall. The LM * curve shifts to the right because the higher interest rate reduces money demand. Since the supply of real balances M/P is fixed, the higher interest rate leads to an excess supply of real balances. To restore equilibrium in the money market, income must rise; this increases the demand for money until there is no longer an excess supply. We see from the figure that output rises and the exchange rate falls (depreciates). Hence, the trade balance increases. Chapter 12 Aggregate Demand in the Open Economy 119 e Exchange rate B A Y Income, output LM 2 LM 1 IS 2 IS 1 Y 2 e 2 e 1 ** * * Y 1 Figure 12¨C15 c. Figure 12¨C16 shows the effect of an increase in the world interest rate if exchange rates are fixed. Both the IS * and LM * curves shift. As in part (b), the IS * curve shifts to the left since the higher interest rate causes investment demand to fall. The LM * schedule, however, shifts to the left instead of to the right. This is because the downward pressure on the exchange rate causes the central bank to buy dollars and sell foreign exchange. This reduces the supply of money M and shifts the LM * schedule to the left. The LM * curve must shift all the way back to LM in the figure, where the fixed-exchange-rate line crosses the new IS * curve. In equilibrium, output falls while the exchange rate remains unchanged. Since the exchange rate does not change, neither does the trade balance. 3. a. A depreciation of the currency makes American goods more competitive. This is because a depreciation means that the same price in dollars translates into fewer units of foreign currency. That is, in terms of foreign currency, American goods become cheaper so that foreigners buy more of them. For example, suppose the exchange rate between yen and dollars falls from 200 yen/dollar to 100 yen/dollar. If an American can of tennis balls costs $2.50, its price in yen falls from 500 yen to 250 yen. This fall in price increases the quantity of American-made tennis balls demanded in Japan. That is, American tennis balls are more competitive. 120 Answers to Textbook Questions and Problems * 2 e e Exchange rate BA Y Income, output LM 2 LM 1 IS 1 IS 2 ** * * Y 2 Y 1 Figure 12¨C16 b. Consider first the case of floating exchange rates. We know that the position of the LM * curve determines output. Hence, we know that we want to keep the money supply fixed. As shown in Figure 12¨C17, we want to use fiscal policy to shift the IS * curve to the left to cause the exchange rate to fall (depreciate). We can do this by reducing government spending or increasing taxes. Now suppose that the exchange rate is fixed at some level. If we want to increase competitiveness, we need to reduce the exchange rate; that is, we need to fix it at a lower level. The first step is to devalue the dollar, fixing the exchange rate at the desired lower level. This increases net exports and tends to increase output, as shown in Figure 12¨C18. We can offset this rise in output with contrac- tionary fiscal policy that shifts the IS * curve to the left, as shown in the figure. Chapter 12 Aggregate Demand in the Open Economy 121 e Exchange rate A B Y Income, output IS 1 IS 2 e 1 e 2 LM * * * Y 1 Income, output Figure 12¨C17 e Exchange rate A CB YY Income, output IS 2 IS 1 e 2 e 1 LM * * * Figure 12¨C18 4. In the text, we assumed that net exports depend only on the exchange rate. This is analogous to the usual story in microeconomics in which the demand for any good (in this case, net exports) depends on the price of that good. The ¡°price¡± of net exports is the exchange rate. However, we also expect that the demand for any good depends on income, and this may be true here as well: as income rises, we want to buy more of all goods, both domestic and imported. Hence, as income rises, imports increase, so net exports fall. Thus, we can write net exports as a function of both the exchange rate and income: NX = NX(e, Y). Figure 12¨C19 shows the net exports schedule as a function of the exchange rate. As before, the net exports schedule is downward sloping, so an increase in the exchange rate reduces net exports. We have drawn this schedule for a given level of income. If income increases from Y 1 to Y 2 , the net exports schedule shifts inward from NX(Y 1 ) to NX(Y 2 ). a. Figure 12¨C20 shows the effect of a fiscal expansion under floating exchange rates. The fiscal expansion (an increase in government expenditure or a cut in taxes) shifts the IS * schedule to the right. But with floating exchange rates, if the LM * curve does not change, neither does income. Since income does not change, the net-exports schedule remains at its original level NX(Y 1 ). 122 Answers to Textbook Questions and Problems e Exchange rate NX Net exports NX (Y 1 ) NX (Y 2 ) Figure 12¨C19 e Exchange rate B A Income, output Income, output IS 1 * IS 2 * LM * e 1 e 2 Y 1 Figure 12¨C20 The final result is that income does not change, and the exchange rate appreciates from e 1 to e 2 . Net exports fall because of the appreciation of the currency. Thus, our answer is the same as that given in Table 12¨C1. b. Figure 12¨C21 shows the effect of a fiscal expansion under fixed exchange rates. The fiscal expansion shifts the IS * curve to the right, from IS to IS . As in part (a), for unchanged real balances, this tends to push the exchange rate up. To pre- vent this appreciation, however, the central bank intervenes in currency markets, selling dollars and buying foreign exchange. This increases the money supply and shifts the LM * curve to the right, from LM to LM . Output rises while the exchange rate remains fixed. Despite the unchanged exchange rate, the higher level of income reduces net exports because the net- exports schedule shifts inward. Thus, our answer differs from the answer in Table 12¨C1 only in that under fixed exchange rates, a fiscal expansion reduces the trade balance. Chapter 12 Aggregate Demand in the Open Economy 123 e Exchange rate AB Y Income, output IS 1 * IS 2 * Y 2 LM 2 LM 1 ** Y 1 e * 2 * 2 * 1 Figure 12¨C21 * 1 5. [Note the similarity to question 7 in Chapter 11.] We want to consider the effects of a tax cut when the LM * curve depends on disposable income instead of income: M/P = L[r, Y ¨C T]. A tax cut now shifts both the IS * and the LM * curves. Figure 12¨C22 shows the case of floating exchange rates. The IS * curve shifts to the right, from IS to IS . The LM * curve shifts to the left, however, from LM to LM . We know that real balances M/P are fixed in the short run, while the interest rate is fixed at the level of the world interest rate r * . Disposable income is the only variable that can adjust to bring the money market into equilibrium: hence, the LM * equation determines the level of disposable income. If taxes T fall, then income Y must also fall to keep disposable income fixed. In Figure 12¨C22, we move from an original equilibrium at point A to a new equi- librium at point B. Income falls by the amount of the tax cut, and the exchange rate appreciates. 124 Answers to Textbook Questions and Problems * 1 * 1 * 2 * 2 e Exchange rate B A Y Income, output IS 2 * IS 1 * Y 1 e 2 e 1 LM 2 LM 1 ** Y 2 Figure 12¨C22 If there are fixed exchange rates, the IS * curve still shifts to the right; but the ini- tial shift in the LM * curve no longer matters. That is, the upward pressure on the exchange rate causes the central bank to sell dollars and buy foreign exchange; this increases the money supply and shifts the LM * curve to the right, as shown in Figure 12¨C23. The new equilibrium, at point B, is at the intersection of the new IS * curve, IS , and the horizontal line at the level of the fixed exchange rate. There is no difference between this case and the standard case where money demand depends on income. 6. Since people demand money balances in order to buy goods and services, it makes sense to think that the price level that is relevant is the price level of the goods and services they buy. This includes both domestic and foreign goods. But the dollar price of foreign goods depends on the exchange rate. For example, if the dollar rises from 100 yen/dollar to 150 yen/dollar, then a Japanese good that costs 300 yen falls in price from $3 to $2. Hence, we can write the condition for equilibrium in the money market as M/P = L(r, Y), where P = ¦ËP d + (1 ¨C ¦Ë)P f /e. Chapter 12 Aggregate Demand in the Open Economy 125 e e Exchange rate A B Y Income, output LM 1 LM 2 IS 2 * IS 1 * Y 1 Y 2 ** * 2 Figure 12¨C23 a. A higher exchange rate makes foreign goods cheaper. To the extent that people consume foreign goods (a fraction 1 ¨C ¦Ë), this lowers the price level P that is rele- vant for the money market. This lower price level increases the supply of real bal- ances M/P. To keep the money market in equilibrium, we require income to rise to increase money demand as well. Hence, the LM * curve is upward sloping. b. In the standard Mundell¨CFleming model, expansionary fiscal policy has no effect on output under floating exchange rates. As shown in Figure 12¨C24, this is no longer true here. A cut in taxes or an increase in government spending shifts the IS * curve to the right, from IS to IS . Since the LM * curve is upward sloping, the result is an increase in output. c. A central assumption in this chapter is that the price level is fixed in the short run. That is, we assumed that the short-run aggregate supply curve is horizontal at P = P, as shown in Figure 12¨C25. 126 Answers to Textbook Questions and Problems e Exchange rate A B Y Income, output IS 1 * IS 2 * e 2 e 1 LM * Y 1 Y 2 * 2 * 1 Figure 12¨C24 P P Price level Y Income, output Figure 12¨C25 A supply shock is something that shifts the AS curve. If the price level P depends on the exchange rate, then as shown in Figure 12¨C26, an appreciation of the exchange rate e causes the price level P to fall¡ªthat is, the aggregate supply curve shifts down from AS 1 to AS 2 . In other words, it looks exactly like a supply shock, except that the ¡°shock¡± is endogenous, not exogenous. 7. a. California is a small open economy, and we assume that it can print dollar bills. Its exchange rate, however, is fixed with the rest of the United States: one dollar can be exchanged for one dollar. In the Mundell¨CFleming model with fixed exchange rates, California cannot use monetary policy to affect output, because this policy is already used to control the exchange rate. Hence, if California wishes to stimulate employment, it should use fiscal policy. Chapter 12 Aggregate Demand in the Open Economy 127 P Price level Y Income, output AS 1 AS 2 Figure 12¨C26 b. In the short run, the import prohibition shifts the IS * curve out. This increases demand for Californian goods and puts upward pressure on the exchange rate. To counteract this, the Californian money supply increases, so the LM * curve shifts out as well. The new short-run equilibrium is at point K in Figures 12¨C27(A) and (B). Assuming that we started with the economy producing at its natural rate, the increase in demand for Californian goods tends to raise their prices. This rise in the price level lowers real money balances, shifting the short-run AS curve upward and the LM * curve inward. Eventually, the Californian economy ends up at point C, with no change in output or the trade balance, but with a higher real exchange rate relative to Washington. 128 Answers to Textbook Questions and Problems A. The Mundell-Fleming Model B. The Model of Aggregate Supply and Aggregate Demand ? Exchange rate Price level A C P P 2 P 1 AD 1 AD 2 K Y K SRAS A C Income, output Y Income, output LM* IS* Figure 12¨C27 More Problems and Applications to Chapter 12 1. a. Higher taxes shift the IS curve inward. To keep output unchanged, the central bank must increase the money supply, shifting the LM curve to the right. At the new equilibrium (point C in Figure 12¨C28), the interest rate is lower, the exchange rate has depreciated, and the trade balance has risen. Chapter 12 Aggregate Demand in the Open Economy 129 A. The IS¨CLM Model B. Net Foreign Investment LM 1 LM 2 IS 2 IS 1 NFI(r) Income, output Y NFI Real interest rate Exchange rate A C rr C. The Market for Foreign Exchange Net exports NX(e) NFI Net foreign investment r 2 r 1 e 2 NX 2 NX 1 e 1 e Figure 12¨C28 (e) b. Restricting the import of foreign cars shifts the NX(e) schedule outward [see panel (C)]. This has no effect on either the IS curve or the LM curve, however, because the NFI schedule is unaffected. Hence, output doesn¡¯t change and there is no need for any change in monetary policy. As shown in Figure 12¨C29, interest rates and the trade balance don¡¯t change, but the exchange rate appreciates. 130 Answers to Textbook Questions and Problems A. The IS-LM Model B. Net Foreign Investment LM IS NFI(r) Income, output YNFI Real interest rate Exchange rate rr C. The Market for Foreign Exchange Net exports NX(e) 2 NX(e) 1 NFI Net foreign investment e 1 NX e 2 e Figure 12¨C29 A. The IS¨CLM Model 2. a. The NFI curve becomes flatter, because a small change in the interest rate now has a larger effect on capital flows. b. As argued in the text, a flatter NFI curve makes the IS curve flatter, as well. c. Figure 12¨C30 shows the effect of a shift in the LM curve for both a steep and a flat IS curve. It is clear that the flatter the IS curve is, the less effect any change in the money supply has on interest rates. Hence, the Fed has less control over the interest rate when investors are more willing to substitute foreign and domestic assets. d. It is clear from Figure 12¨C30 that the flatter the IS curve is, the greater effect any change in the money supply has on output. Hence, the Fed has more control over output. Chapter 12 Aggregate Demand in the Open Economy 131 Real interest rate Y LM 1 IS steep IS flat LM 2 C flat C steep r Income, output Figure 12¨C30 3. a. No. It is impossible to raise investment without affecting income or the exchange rate just by using monetary and fiscal policies. Investment can only be increased through a lower interest rate. Regardless of what policy is used to lower the inter- est rate (e.g., expansionary monetary policy and contractionary fiscal policy), net foreign investment will increase, lowering the exchange rate. b. Yes. Policymakers can raise investment without affecting income or the exchange rate with a combination of expansionary monetary policy and contractionary fiscal policy, and protection against imports can raise investment without affecting the other variables. Both the monetary expansion and the fiscal contraction would put downward pressure on interest rates and stimulate investment. It is necessary to combine these two policies so that their effects on income exactly offset each other. The lower interest rates will, as in part (a), increase net foreign investment, which would normally put downward pressure on the exchange rate. The protectionist policies, however, shift the net-exports curve out; this puts countervailing upward pressure on the exchange rate and offsets the effect of the fall in interest rates. Figure 12¨C31 shows this combination of policies. 132 Answers to Textbook Questions and Problems LM NX(e) NX Income, output NFIY NFI(r) I S Net exports v rr NFI LM I S Y 1, Y 2 r 1 r 2 e NX 1 NX 2 e 1, e 2 NFI 1 NFI 2 Net foreign in estment Figure 12¨C31 c. Yes. Policymakers can raise investment without affecting income or the exchange rate through a home monetary expansion and fiscal contraction, combined with a lower foreign interest rate either through a foreign monetary expansion or fiscal contraction. The domestic policy lowers the interest rate, stimulating investment. The foreign policy shifts the NFI curve inward. Even with lower interest rates, the quantity of NFI would be unchanged and there would be no pressure on the exchange rate. This combination of policies is shown in Figure 12¨C32. 4. a. Figure 12¨C33 shows the effect of a fiscal contraction on a large open economy with a fixed exchange rate. The fiscal contraction shifts the IS curve to the left in panel Chapter 12 Aggregate Demand in the Open Economy 133 LM NX(e) NX Income, output NFI Y NFI(r) I S Net exports rr NFI Y 1, Y 2 r 1 r 2 e e 1, e 2 NX 1, NX 2 Net foreign investment NFI 1, NFI 2 Figure 12¨C32 (A), which puts downward pressure on the interest rate. This tends to increase NFI and cause the exchange rate to depreciate [see panels (B) and (C)]. To avoid this, the central bank intervenes and buys dollars. This keeps the exchange rate from depreciating; it also shifts the LM curve to the left. The new equilibrium, at point C, has an unchanged interest rate and exchange rate, but lower output. This effect is the same as in a small open economy. 134 Answers to Textbook Questions and Problems Income, output Y C NFI NFI A. The IS-LM Model IS 2 IS 1 LM 2 LM 1 Real interest rate r B. Net Foreign Investment Exchange rate r Net exports NX NX(e) e Net foreign investment C. The Market for Foreign Exchange A Figure 12¨C33 b. A monetary expansion tends to shift the LM curve to the right, lowering the inter- est rate [panel (A) in Figure 12¨C34]. This tends to increase NFI and cause the exchange rate to depreciate [see panels (B) and (C)]. To avoid this depreciation, the central bank must buy its currency and sell foreign exchange. This reduces the money supply and shifts the LM curve back to its original position. As in the model of a small open economy, monetary policy is ineffectual under a fixed exchange rate. Chapter 12 Aggregate Demand in the Open Economy 135 Income, output Y NFI NFI(r) A. The IS-LM Model IS LM 1 LM 2 Real interest rate r B. Net Foreign Investment Exchange rate r Net exports NX NX(e) e Net foreign investment C. The Market for Foreign Exchange Figure 12¨C34 Questions for Review 1. In this chapter we looked at three models of the short-run aggregate supply curve. All three models attempt to explain why, in the short run, output might deviate from its long-run ¡°natural rate¡±¡ªthe level of output that is consistent with the full employment of labor and capital. All three models result in an aggregate supply function in which output deviates from its natural rate Y when the price level deviates from the expected price level: Y = Y + ¦Á(P ¨C P e ). The first model is the sticky-wage model. The market failure is in the labor mar- ket, since nominal wages do not adjust immediately to changes in labor demand or sup- ply¡ªthat is, the labor market does not clear instantaneously. Hence, an unexpected increase in the price level causes a fall in the real wage (W/P). The lower real wage induces firms to hire more labor, and this increases the amount of output they produce. The second model is the imperfect-information model. As in the worker-mispercep- tion model, this model assumes that there is imperfect information about prices. Here, though, it is not workers in the labor market who are fooled: it is suppliers of goods who confuse changes in the price level with changes in relative prices. If a producer observes the nominal price of the firm¡¯s good rising, the producer attributes some of the rise to an increase in relative price, even if it is purely a general price increase. As a result, the producer increases production. The third model is the sticky-price model. The market imperfection in this model is that prices in the goods market do not adjust immediately to changes in demand con- ditions¡ªthe goods market does not clear instantaneously. If the demand for a firm¡¯s goods falls, it responds by reducing output, not prices. 2. In this chapter, we argued that in the short run, the supply of output depends on the natural rate of output and on the difference between the price level and the expected price level. This relationship is expressed in the aggregate-supply equation: Y = Y + ¦Á(P ¨C P e ). The Phillips curve is an alternative way to express aggregate supply. It provides a sim- ple way to express the tradeoff between inflation and unemployment implied by the short-run aggregate supply curve. The Phillips curve posits that inflation ¦Ð depends on the expected inflation rate ¦Ð e , on cyclical unemployment u ¨C u n , and on supply shocks : ¦Ð = ¦Ð e ¨C ¦Â(u ¨C u n ) + . Both equations tell us the same information in a different way: both imply a con- nection between real economic activity and unexpected changes in prices. 3. Inflation is inertial because of the way people form expectations. It is plausible to assume that people¡¯s expectations of inflation depend on recently observed inflation. These expectations then influence the wages and prices that people set. For example, if prices have been rising quickly, people will expect them to continue to rise quickly. These expectations will be built into the contracts people set, so that actual wages and prices will rise quickly. 136 ? ? CHAPTER 13 Aggregate Supply 4. Demand-pull inflation results from high aggregate demand: the increase in demand ¡°pulls¡± prices and output up. Cost-push inflation comes from adverse supply shocks that push up the cost of production¡ªfor example, the increases in oil prices in the mid- and late-1970s. The Phillips curve tells us that inflation depends on expected inflation, the differ- ence between unemployment and its natural rate, and a shock : ¦Ð = ¦Ð e ¨C ¦Â(u ¨C u n ) + . The term ¡° ¨C ¦Â(u ¨C u n )¡± is the demand-pull inflation, since if unemployment is below its natural rate (u < u n ), inflation rises. The supply shock is the cost-push inflation. 5. The Phillips curve relates the inflation rate to the expected inflation rate and to the dif- ference between unemployment and its natural rate. So one way to reduce inflation is to have a recession, raising unemployment above its natural rate. It is possible to bring inflation down without a recession, however, if we can costlessly reduce expected infla- tion. According to the rational-expectations approach, people optimally use all of the information available to them in forming their expectations. So to reduce expected inflation, we require, first, that the plan to reduce inflation be announced before people form expectations (e.g., before they form wage agreements and price contracts); and second, that those setting wages and prices believe that the announced plan will be car- ried out. If both requirements are met, then expected inflation will fall immediately and without cost, and this in turn will bring down actual inflation. 6. One way in which a recession might raise the natural rate of unemployment is by affecting the process of job search, increasing the amount of frictional unemployment. For example, workers who are unemployed lose valuable job skills. This reduces their ability to find jobs after the recession ends because they are less desirable to firms. Also, after a long period of unemployment, individuals may lose some of their desire to work, and hence search less hard. Second, a recession may affect the process that determines wages, increasing wait unemployment. Wage negotiations may give a greater voice to ¡°insiders,¡± those who actually have jobs. Those who become unemployed become ¡°outsiders.¡± If the smaller group of insiders cares more about high real wages and less about high employment, then the recession may permanently push real wages above the equilibrium level and raise the amount of wait unemployment. This permanent impact of a recession on the natural rate of unemployment is called hysteresis. Problems and Applications 1. In the sticky-wage model, we assumed that the wage did not adjust immediately to changes in the labor market. This resulted in an upward-sloping aggregate supply curve with the form Y = Y + ¦Á(P ¨C P e ). In this problem, we consider the effect of allowing these contracts to be indexed for inflation. a. In the simple sticky-wage model, the nominal wage W equals the desired real wage ¦Ø times the expected price level P e : W = ¦ØP e . Chapter 13 Aggregate Supply 137 ? ? ? Full indexing, however, makes the nominal wage depend on the actual price level. That is, the contract specifies the desired real wage ¦Ø, and the nominal wage adjusts fully to changes in the price level. As a result, W = ¦ØP, or W/P = ¦Ø. This means that unexpected price changes do not affect the real wage and, hence, do not affect the amount of labor used or the amount of output produced. The aggregate supply schedule is thus vertical at Y = Y. b. If there is partial indexing, then the aggregate supply curve will be steeper than it is without indexing, although it will not be vertical. In the sticky-wage model, an unexpected increase in the price level reduces the real wage W/P, since the nomi- nal wage W is unaffected. With partial indexing, the increase in the price level causes an increase in the nominal wage. Since the indexing is only partial, the nominal wage increases by less than the price level does, so the real wage falls. This causes firms to use more labor and increase production. However, the real wage does not fall as much as it does without indexing, so output does not rise as much. In effect, this is like making the parameter ¦Á smaller in the equation for aggregate supply. That is, output fluctuations become less responsive to surprises in the price level. 2. In this question, we examine two special cases of the sticky-price model developed in this chapter. In the sticky-price model, all firms have a desired price p that depends on the overall level of prices P as well as the level of aggregate demand Y ¨C Y. We wrote this as p = P + a(Y ¨C Y). There are two types of firms. A proportion (1 ¨C s) of the firms have flexible prices and set prices using the above equation. The remaining proportion s of the firms have sticky prices¡ªthey announce their prices in advance based on the economic conditions that they expect in the future. We assume that these firms expect output to be at its natural rate, so (Y e ¨C Y e ) = 0. Hence, these firms set their prices equal to the expected price level: p = P e . The overall price level is a weighted average of the prices set by the two types of firms: P = sP e + (1 ¨C s)[P + a(Y ¨C Y)]. Rearranging: P = P e +[a(1 ¨C s)/s](Y ¨C Y). a. If no firms have flexible prices, then s = 1. The above equation tells us that P = P e . That is, the aggregate price level is fixed at the expected price level: the aggregate supply curve is horizontal in the short run, as assumed in Chapter 9. b. If desired relative prices do not depend at all on the level of output, then a = 0 in the equation for the price level. Once again, we find P = P e : the aggregate supply curve is horizontal in the short run, as assumed in Chapter 9. 3. The economy has the Phillips curve: ¦Ð = ¦Ð ¨C1 ¨C0.5(u ¨C 0.06). a. The natural rate of unemployment is the rate at which the inflation rate does not deviate from the expected inflation rate. Here, the expected inflation rate is just 138 Answers to Textbook Questions and Problems last period¡¯s actual inflation rate. Setting the inflation rate equal to last period¡¯s inflation rate, that is, ¦Ð = ¦Ð ¨C1 , we find that u = 0.06. Thus, the natural rate of unemployment is 6 percent. b. In the short run (that is, in a single period) the expected inflation rate is fixed at the level of inflation in the previous period, ¦Ð ¨C1 . Hence, the short-run relationship between inflation and unemployment is just the graph of the Phillips curve: it has a slope of ¨C0.5, and it passes through the point where ¦Ð = ¦Ð ¨C1 and u = 0.06. This is shown in Figure 13¨C1. In the long run, expected inflation equals actual inflation, so that ¦Ð = ¦Ð ¨C1 , and output and unemployment equal their natural rates. The long- run Phillips curve thus is vertical at an unemployment rate of 6 percent. c. To reduce inflation, the Phillips curve tells us that unemployment must be above its natural rate of 6 percent for some period of time. We can write the Phillips curve in the form ¦Ð ¨C ¦Ð ¨C1 = 0.5(u ¨C 0.06). Since we want inflation to fall by 5 percentage points, we want ¦Ð ¨C ¦Ð ¨C1 = ¨C0.05. Plugging this into the left-hand side of the above equation, we find ¨C0.05 = ¨C0.5(u ¨C 0.06). We can now solve this for u: u = 0.16. Hence, we need 10 percentage point-years of cyclical unemployment above the natural rate of 6 percent. Okun¡¯s law says that a change of 1 percentage point in unemployment trans- lates into a change of 2 percentage points in GDP. Hence, an increase in unem- ployment of 10 percentage points corresponds to a fall in output of 20 percentage points. The sacrifice ratio is the percentage of a year¡¯s GDP that must be forgone to reduce inflation by 1 percentage point. Dividing the 20 percentage-point decrease in GDP by the 5 percentage-point decrease in inflation, we find that the sacrifice ratio is 20/5 = 4. d. One scenario is to have very high unemployment for a short period of time. For example, we could have 16 percent unemployment for a single year. Alternatively, we could have a small amount of cyclical unemployment spread out over a long period of time. For example, we could have 8 percent unemployment for 5 years. Both of these plans would bring the inflation rate down from 10 percent to 5 per- cent, although at different speeds. Chapter 13 Aggregate Supply 139 LRPC SRPC u 0.06 Unemployment ¦Ð Inflation 0.5 ¦Ð ?1 ¦Ð Figure 13¨C1 140 Answers to Textbook Questions and Problems 4. The cost of reducing inflation comes from the cost of changing people¡¯s expectations about inflation. If expectations can be changed costlessly, then reducing inflation is also costless. Algebraically, the Phillips curve tells us that ¦Ð = ¦Ð e ¨C ¦Â(u ¨C u n ). If the government can lower expected inflation ¦Ð e to the desired level of inflation, then there is no need for unemployment to rise above its natural rate. According to the rational-expectations approach, people form expectations about inflation using all of the information that is available to them. This includes informa- tion about current policies in effect. If everyone believes that the government is commit- ted to reducing inflation, then expected inflation will immediately fall. In terms of the Phillips curve, ¦Ð e falls immediately with little or no cost to the economy. That is, the sacrifice ratio will be very small. On the other hand, if people do not believe that the government will carry out its intentions, then ¦Ð e remains high. Expectations will not adjust because people are skep- tical that the government will follow through on its plans. Thus, according to the rational-expectations approach, the cost of reducing infla- tion depends on how resolute and credible the government is. An important issue is how the government can make its commitment to reducing inflation more credible. One possibility, for example, is to appoint people who have a reputation as inflation fighters. A second possibility is to have Congress pass a law requiring the Federal Reserve to lower inflation. Of course, people might expect the Fed to ignore this law, or expect Congress to change the law later. A third possibility is to pass a constitutional amend- ment limiting monetary growth. People might rationally believe that a constitutional amendment is relatively difficult to change. 5. In this question we consider several implications of rational expectations¡ªthe assump- tion that people optimally use all of the information available to them in forming their expectations¡ªfor the models of sticky wages and sticky prices that we considered in this chapter. Both of these models imply an aggregate supply curve in which output varies from its natural rate only if the price level varies from its expected level: Y = Y + ¦Á(P ¨C P e ). This aggregate supply curve means that monetary policy can affect real GDP only by affecting (P ¨C P e )¡ªthat is, causing an unexpected change in the price level. a. Only unanticipated changes in the money supply can affect real GDP. Since peo- ple take into account all of the information available to them, they already take into account the effects of anticipated changes in money when they form their expectations of the price level P e . For example, if people expect the money supply to increase by 10 percent and it actually does increase by 10 percent, then there is no effect on output since there is no price surprise¡ª(P ¨C P e ) = 0. On the other hand, suppose the Fed increases the money supply more than expected, so that prices increase by 15 percent when people expect them to increase by only 10 per- cent. Since P > P e , output rises. But it is only the unanticipated part of money growth that increases output. b. The Fed often tries to stabilize the economy by offsetting shocks to output and unemployment. For example, it might increase the money supply during re- cessions in an attempt to stimulate the economy, and it might reduce the money supply during booms in an attempt to slow it down. The Fed can only do this by surprising people about the price level: during a recession, they want prices to be higher than expected, and during booms, they want prices to be lower than expect- ed. If people have rational expectations, however, they will expect the Fed to respond this way. So if the economy is in a boom, people expect the Fed to reduce the money supply; in a recession, people expect the Fed to increase the money supply. In either case, it is impossible for the Fed to cause (P ¨C P e ) to vary system- atically from zero. Since people take into account the systematic, anticipated movements in money, the effect on output of systematic, active policy is exactly the same as a policy of keeping the money supply constant. c. If the Fed sets the money supply after people set wages and prices, then the Fed can use monetary policy systematically to stabilize output. The assumption of rational expectations means that people use all of the information available to them in forming expectations about the price level. This includes information about the state of the economy and information about how the Fed will respond to this state. This does not mean that people know what the state of the economy will be, nor do they know exactly how the Fed will act: they simply make their best guess. As time passes, the Fed learns information about the economy that was unknown to those setting wages and prices. At this point, since contracts have already set these wages and prices, people are stuck with their expectations P e . The Fed can then use monetary policy to affect the actual price level P, and hence can affect output systematically. 6. In this model, the natural rate of unemployment is an average of the unemployment rates in the past two years. Hence, if a recession raises the unemployment rate in some year, then the natural rate of unemployment rises as well. This means that the model exhibits hysteresis: short-term cyclical unemployment affects the long-term natural rate of unemployment. a. The natural rate of unemployment might depend on recent unemployment for at least two reasons, suggested by theories of hysteresis. First, recent unemployment rates might affect the level of frictional unemployment. Unemployed workers lose job skills and find it harder to get jobs; also, unemployed workers might lose some of their desire to work, and hence search less hard for a job. Second, recent unem- ployment rates might affect the level of wait unemployment. If labor negotiations give a greater voice to ¡°insiders¡± than ¡°outsiders,¡± then the insiders might push for high wages at the expense of jobs. This will be especially true in industries in which negotiations take place between firms and unions. b. If the Fed seeks to reduce inflation permanently by 1 percentage point, then the Phillips curve tells us that in the first period we require ¦Ð 1 ¨C ¦Ð 0 = ¨C1 = ¨C0.5(u 1 ¨C u ), or (u 1 ¨C u ) = 2. That is, we require an unemployment rate 2 percentage points above the original natural rate u . Next period, however, the natural rate will rise as a result of the cyclical unemployment. The new natural rate u will be u = 0.5[u 1 + u 0 ] = 0.5[(u + 2) + u ] = u + 1. Hence, the natural rate of unemployment rises by 1 percentage point. If the Fed wants to keep inflation at its new level, then unemployment in period 2 must equal the new natural rate u . Hence, u 2 = u + 1. In every subsequent period, it remains true that the unemployment rate must equal the natural rate. This natural rate never returns to its original level: we can show this by deriving the sequence of unemployment rates: u 3 = (1/2)u 2 + (1/2)u 1 = u + 1-1/2 u 4 = (1/2)u 3 + (1/2)u 2 = u + 1-1/4 u 5 = (1/2)u 4 + (1/2)u 3 = u + 1-3/8. Chapter 13 Aggregate Supply 141 n 1 n 1 n 1 n 1 n 1 n 1 n 1 n 1 n 1 n 1 n 2 n 2 n 2 Unemployment always remains above its original natural rate. In fact, we can show that it is always at least 1 percent above its original natural rate. Thus, to reduce inflation by 1 percentage point, unemployment rises above its original level by 2 percentage points in the first year, and by 1 or more percentage points in every year after that. c. Because unemployment is always higher than it started, output is always lower than it would have been. Hence, the sacrifice ratio is infinite. d. Without hysteresis, we found that there was a short-run tradeoff but no long-run tradeoff between inflation and unemployment. With hysteresis, we find that there is a long-run tradeoff between inflation and unemployment: to reduce inflation, unemployment must rise permanently. 7. a. The natural rate of output is determined by the production function, Y = F(K, L). If a tax cut raises work effort, it increases L and, thus, increases the natural rate of output. b. The tax cut shifts the aggregate demand curve outward for the normal reason that disposable income and, hence, consumption rise. It shifts the long-run aggregate supply curve outward because the natural rate of output rises. The effect of the tax cut on the short-run aggregate supply (SRAS) curve depends on which model you use. The labor supply curve shifts outward because workers are willing to supply more labor at any given real wage while the labor demand curve is unchanged. In the sticky-wage or sticky-price models the quanti- ty of labor is demand-determined, so the SRAS curve does not move. By contrast, the imperfect-information model assumes that the labor market is always in equi- librium, so the greater supply of labor leads to higher employment immediately: the SRAS shifts out. c. If you are using the sticky-wage or sticky-price model, the short-run analysis is the same as the conventional model without the labor-supply effect. That is, output and prices both rise because aggregate demand rises while short-run aggregate supply is unchanged. If you use the imperfect-information model, short-run aggregate supply shifts outward, so that the tax cut is more expansionary and less inflationary than the conventional model. Figure 13¨C2 shows the effects in both models. Point A is the original equilibrium, point SW is the new equilibrium in the sticky-wage model, and point II is the new equilibrium in the imperfect-information model. 142 Answers to Textbook Questions and Problems LRAS 1 LRAS 2 SRAS 1 , SRAS SW SRAS Imp. Info AD 2 AD 1 Y A SW II P , SRAS SP Figure 13¨C2 d. In contrast to the normal model, the tax cut raises long-run output by increasing the supply of labor. The policy¡¯s long-run effect on price is indeterminate, depend- ing, in part on whether SRAS does, in fact, shift out. The change in the long-run equilibrium is shown in Figure 13¨C3. 8. In this quote, Alan Blinder argues that in low-inflation countries like the United States, the benefits of disinflation are small whereas the costs are large. That is, menu costs, shoeleather costs, and tax distortions simply do not add up to much, so eliminat- ing inflation offers only small benefits. By contrast, the costs in terms of unemployment and lost output that are associated with lowering inflation are easily quantifiable and very large. The basic policy implication of these beliefs about the relative benefits and costs of disinflation is that policymakers should not tighten policy in order to lower inflation rates that are already relatively low. The statement leaves two other issues ambiguous. First, should policymakers concern themselves with rising inflation? Second, should policymakers concern themselves with making inflation more predictable around the level it has inherited? Blinder may feel that these issues should have little weight rela- tive to output stabilization. 9. From the BLS web site (www.bls.gov), there are various ways to get the CPI data. The approach I took (in February 2002) was to follow the links to the Consumer Price Index, then I followed the links to the most recent (current) economic news release. This led me to a table of contents, and I followed the first link to Consumer Price Index Summary (the direct web address for that link was http://www.bls.gov/ news.release/cpi.nr0.htm). Midway through that release was a table showing the ¡°per- centage change [for the] 12 months ending in December¡± for recent years. For the years 1996¨C2001, the table shows: Year 1996 1997 1998 1999 2000 2001 Overall CPI 3.3 1.7 1.6 2.7 3.4 1.6 CPI excluding food and energy 2.6 2.2 2.4 1.9 2.6 2.7 The overall CPI was clearly more volatile than the CPI excluding food and energy. For example, the overall CPI rose sharply from 1998 to 1999, and fell sharply from 2000 to 2001 (patterns that were not evident in the CPI excluding food and energy). The differ- ence reflects shocks to the price of food and energy-especially energy prices, which are highly variable. Chapter 13 Aggregate Supply 143 LRAS 1 LRAS 2 SRAS 1 SRAS 2 AD 2 AD 1 Y A B P Y 2 Y 1 P 2 P 1 Figure 13¨C3 When energy prices, say, go down, the total CPI will rise less than the CPI exclud- ing food and energy. This represents a supply shock, which shifts the aggregate supply curve and Phillips curve downward. In 1997 and 1998, for example, when the overall CPI rose less than the CPI excluding food and energy, we would expect that the decline in inflation did not necessarily require unemployment to rise above its natural rate. Indeed, unemployment was falling over that period. More Problems and Applications to Chapter 13 1. a. The classical large open economy model (from the Appendix to Chapter 5) is similar to special case 2 in the text, except that it allows the interest rate to deviate from the world interest rate. That is, this is the special case where P e =P, L(i,Y)=(1/V)Y, and CF=CF(r-r*), with a non-infinitely elastic interna- tional capital flow. Because capital flows do not respond overwhelmingly to any differences between the domestic and world interest rates, these rates can, in fact, vary in this case. b. The Keynesian cross model of Chapter 10 is the special case where (i) the economy is closed, so that CF(r-r*)=0 ; (ii) I(r) = I, so that investment is given exogenously; and (iii) ¦Á is infinite, so that the short-run aggregate-supply curve is horizontal. In this special case, output depends solely on the demand for goods and services. c. The IS-LM model for the large open economy (from the appendix to Chapter 12) is the special case where ¦Á is infinite and CF=CF(r-r*) is not infinitely elastic. In this case, the short-run aggregate supply curve is horizontal, and capital flows do not respond too much to differences between the domestic and world interest rates. 144 Answers to Textbook Questions and Problems Questions for Review 1. What is unusual about U.S. fiscal policy since 1980 is that government debt increased sharply during a period of peace and prosperity. Over the course of U.S. history, the indebtedness of the federal government relative to GDP has varied substantially. Historically, the debt¨CGDP ratio generally increases sharply during major wars and falls slowly during peacetime. The 1980s and 1990s are the only instance in U.S. histo- ry of a large increase in the debt¨CGDP ratio during peacetime. 2. Many economists project increasing budget deficits and government debt over the next several decades because of changes in the age profile of the population. Life expectancy has steadily increased, and birth rates have fallen. As a result, the elderly are becom- ing a larger share of the population. As more people become eligible for ¡°entitlements¡± of Social Security and Medicare, government spending will rise automatically over time. Without changes in tax and expenditure policies, government debt will also rise sharply. 3. Standard measures of the budget deficit are imperfect measures of fiscal policy for at least four reasons. First, they do not correct for the effects of inflation. The measured deficit should equal the change in the government¡¯s real debt, not the change in the nominal debt. Second, such measures do not offset changes in government liabilities with changes in government assets. To measure the government¡¯s overall indebtedness, we should subtract government assets from government debt. Hence, the budget deficit should be measured as the change in debt minus the change in assets. Third, standard measures omit some liabilities altogether, such as the pensions of government workers and accumulated future Social Security benefits. Fourth, they do not correct for the effects of the business cycle. 4. Public saving is the difference between taxes and government purchases, so a debt- financed tax cut reduces public saving by the full amount that taxes fall. The tax cut also increases disposable income. According to the traditional view, since the marginal propensity to consume is between zero and one, both consumption and private saving increase. Because consumption rises, private saving increases by less than the amount of the tax cut. National saving is the sum of public and private saving; because public saving falls by more than private saving increases, national saving falls. 5. According to the Ricardian view, a debt-financed tax cut does not stimulate consump- tion because it does not raise permanent income¡ªforward-looking consumers under- stand that government borrowing today means higher taxes in the future. Because the tax cut does not change consumption, households save the extra disposable income to pay for the future tax liability that the tax cut implies: private saving increases by the full amount of the tax cut. This increase in private saving exactly offsets the decrease in public saving associated with the tax cut. Therefore, the tax cut has no effect on national saving. 6. Which view of government debt you hold depends on how you think consumers behave. If you hold the traditional view, then you believe that a debt-financed tax cut stimu- lates consumer spending and lowers national saving. You might believe this for several reasons. First, consumers may be shortsighted or irrational, so that they think their permanent income has increased even though it has not. Second, consumers may face binding borrowing constraints, so that they are only able to consume their current income. Third, consumers may expect that the implied tax liability will fall on future 149 CHAPTER 15 Government Debt generations, and these consumers may not care enough about their children to leave them a bequest to offset this tax liability. If you hold the Ricardian view, then you believe that the preceding objections are not important. In particular, you believe that consumers have the foresight to see that government borrowing today implies future taxes to be levied on them or their descen- dants. Hence, a debt-financed tax cut gives consumers transitory income that eventual- ly will be taken back. As a result, consumers will save the extra income they receive in order to offset that future tax liability. 7. The level of government debt might affect the government¡¯s incentives regarding money creation because the government debt is specified in nominal terms. A higher price level reduces the real value of the government¡¯s debt. Hence, a high level of debt might encourage the government to print money in order to raise the price level and reduce the real value of its debt. Problems and Applications 1. The budget deficit is defined as government purchases minus government revenues. Selling the Liberty Bell to Taco Bell would raise revenue for the U.S. government and, hence, reduce the deficit. A smaller budget deficit would lead the government to borrow less, and as a result the measured national debt would fall. If the United States adopted capital budgeting, the net national debt would be defined as the assets of the government (its schools, armies, parks, and so forth) minus the liabilities of the government (principally outstanding public debt). By selling the Liberty Bell the government would be reducing its assets by the value of the Liberty Bell and reducing its liabilities by its purchase price. Assuming Taco Bell paid a fair price, these reductions would be the same amount and the net national debt would be unchanged. Before you worry too much about the Taco Liberty Bell, you might want to notice that this ad appeared on April Fools Day. 2. Here is one possible letter: Dear Senator: In my previous letter, I assumed that a tax cut financed by government borrowing would stimulate consumer spending. Many economists make this assumption because it seems sensible that if people had more current income, then they would consume more. As a result of this increase in consumption, national saving would fall. Ricardian economists argue that the seemingly sensible assumption that I made is incorrect. Although a debt-financed tax cut would increase current disposable income, it would also imply that at some point in the future, the government must raise taxes to pay off the debt and accumulated interest. As a result, the tax cut would merely give consumers a transitory increase in income that would eventually be taken back. If con- sumers understand this, then they would know that their permanent, or lifetime, resources had not changed. Hence, the tax cut would have no effect on consumption, and households would save all of their extra disposable income to pay for the future tax liability. Because there would be no effect on consumption, there would also be no effect on national saving. If national saving did not change, then as pointed out by the prominent economist you heard from yesterday, the budget deficit would not have the effects I listed. In par- ticular, output, employment, foreign debt, and interest rates would be unaffected in both the short run and the long run. The tax cut would have no effect on economic well- being. There are several reasons the Ricardian argument may fail. First, consumers might not be rational and forward-looking: they may not fully comprehend that the cur- rent tax cut means a future tax increase. Second, some people may face constraints on 150 Answers to Textbook Questions and Problems their borrowing: in essence, the tax cut would give these taxpayers a loan that they are unable to obtain now. Third, consumers may expect the implied future taxes to fall not on them, but on future generations whose consumption they do not care about. Your committee must decide how you think consumers would behave in response to this debt-financed tax cut. In particular, would they consume more, or not? Your faithful servant, CBO Economist. 3. a. We will assume that the life-cycle model of Chapter 15 holds and that people want to keep consumption as smooth as possible. This implies that the effect on con- sumption of a temporary change in income will be spread out over a person¡¯s entire remaining life. We will also assume for simplicity that the interest rate is zero. Consider a simple example. Let T be the amount of the one-time, temporary tax levied on the young, and let B be the amount of the one-time benefit paid to the old, where B = T. If a typical elderly person has 10 years left to live, then the temporary benefit increases the present consumption of the elderly by B/10. If a typical worker has 30 years left to live, then the increase in taxes decreases their present consumption by T/30. Aggregate consumption changes by an amount ?C = B/10 ¨C T/30 = B/15. The transfer of wealth to the elderly causes a net increase in consumption and, therefore, a decrease in saving. This happens because the elderly increase consumption by more than the workers decrease it, because the elderly have fewer years left to live and thus have a higher marginal propensity to consume. b. The answer to part (a) does depend on whether generations are altruistically linked. If generations are altruistically linked, then the elderly may not feel any better off because of the Social Security benefit, since the tax and benefit increase has no effect on a typical family¡¯s permanent income; it simply transfers resources from one generation of the family to another. If the elderly do not want to take advantage of this opportunity to consume at their children¡¯s expense, they may try to offset the effect of the tax increase on the young by giving them a gift or leaving a bequest. To the extent that this takes place, it mitigates the impact of the tax change on consumption and saving. 4. If generations are altruistically linked, generational accounting is not useful. Genera- tional accounting looks at the lifetime income of different generations. If these gene- rations are altruistically linked, however, what matters is the lifetime income of the entire family line, not the lifetime income of any particular generation. For example, an increase in the lifetime taxes of the young can be offset by an increase in bequests by the old. If many consumers face binding borrowing constraints, then generational account- ing can be useful, but it is incomplete. For example, consider the following two policies that help the young. First, cut current taxes, but keep future Social Security benefits unchanged. Second, keep current taxes unchanged, but raise future Social Security benefits. These two policies may be equivalent from the point of view of generational accounting, which looks at the combined effect of the policies on the generation. If the young face binding borrowing constraints, however, then the two policies are quite dif- ferent. In particular, the first policy helps the young more than the second policy does. 5. A rule requiring a cyclically adjusted balanced budget has the potential to overcome, at least partially, the first two objections to a balanced-budget rule that were raised in this chapter. First, this rule allows the government to run countercyclical fiscal policy in order to stabilize the economy. That is, the government can run deficits dur- ing recessions, when taxes automatically fall and expenditures automatically rise. These automatic stabilizers affect the deficit but not the cyclically adjusted deficit. Second, this rule allows the government to smooth tax rates across years when Chapter 15 Government Debt 151 income is especially low or high¡ªit is not necessary to raise tax rates in recessions or to cut them in booms. On the other hand, this rule only partially overcomes these two objections, since the government can only run a deficit of a certain size, which might not be big enough. Also, a cyclically adjusted balanced budget does not allow the government to smooth tax rates across years when expenditure is especially high or low, as in times of war or peace. (We might take account of this by allowing an exemption from the balanced bud- get rule in special circumstances such as war.) This rule does not allow the government to overcome the third objection raised in the chapter, since the government cannot shift the burden of expenditure from one generation to another when this is warranted. Finally, a serious problem with a rule requiring a balanced cyclically adjusted budget is that we do not directly observe this budget. That is, we need to estimate how far we are from full employment; then we need to estimate how expenditures and taxes would differ if we were at this full-employment level. None of these estimates can be made precisely. 6. The Congressional Budget Office (www.cbo.gov) regularly provides budget forecasts. For example, on February 10, 2002, the web site had prominent links to ¡°The Budget and Economic Outlook¡± as well as related studies. Based on the testimony of the Director of the CBO (from a link off the CBO home page, which took me to http://www.cbo.gov/showdoc.cfm?index=3275&sequence=0&from=7), the debt held by the public is projected to decline from 32.7 percent of GDP in 2001 to 7.4 percent by the end of 2012. Several assumptions are notable. First, the CBO assumes that so-called discre- tionary government spending (items such as defense, administration, and the like, amounting to about 1/3 of federal spending) will grow at only the rate of inflation. Since the overall economy generally grows faster than inflation, this implies that the CBO builds in a steady decline in federal spending relative to GDP. Second, the CBO assumes that the taxes in the future will be whatever legislation currently says they will be (i.e., the CBO does not take a stand on what changes legislators might pass in the future). Third, the CBO makes an educated guess about future output growth, now projected at 3.2 percent over the next decade. The CBO Director justifies these assumptions by noting that ¡°CBO's baseline pro- jections are intended to serve as a neutral benchmark against which to measure the effects of possible changes in tax and spending policies. They are designed to project federal revenues and spending under the assumption that current laws and policies remain unchanged. Thus, these projections will almost certainly differ from actual bud- get totals: the economy may not follow the path that CBO projects, and lawmakers are likely to alter the nation's tax and spending policies.¡± It is probably reasonable to assume that policymakers will increase real spending on discretionary programs as the economy itself grows over time. They may also change taxes, although the direction is harder to predict. If the United States experiences a productivity slowdown, this will reduce output growth and hence growth in tax rev- enue; future government debt will be larger than currently projected. 152 Answers to Textbook Questions and Problems Questions for Review 1. First, Keynes conjectured that the marginal propensity to consume¡ªthe amount con- sumed out of an additional dollar of income¡ªis between zero and one. This means that if an individual¡¯s income increases by a dollar, both consumption and saving increase. Second, Keynes conjectured that the ratio of consumption to income¡ªcalled the average propensity to consume¡ªfalls as income rises. This implies that the rich save a higher proportion of their income than do the poor. Third, Keynes conjectured that income is the primary determinant of consump- tion. In particular, he believed that the interest rate does not have an important effect on consumption. A consumption function that satisfies these three conjectures is C = C + cY. C is a constant level of ¡°autonomous consumption,¡± and Y is disposable income; c is the marginal propensity to consume, and is between zero and one. 2. The evidence that was consistent with Keynes¡¯s conjectures came from studies of house- hold data and short time-series. There were two observations from household data. First, households with higher income consumed more and saved more, implying that the marginal propensity to consume is between zero and one. Second, higher-income households saved a larger fraction of their income than lower-income households, implying that the average propensity to consume falls with income. There were three additional observations from short time-series. First, in years when aggregate income was low, both consumption and saving were low, implying that the marginal propensity to consume is between zero and one. Second, in years with low income, the ratio of consumption to income was high, implying that the average propen- sity to consume falls as income rises. Third, the correlation between income and con- sumption seemed so strong that no variables other than income seemed important in explaining consumption. The first piece of evidence against Keynes¡¯s three conjectures came from the fail- ure of ¡°secular stagnation¡± to occur after World War II. Based on the Keynesian con- sumption function, some economists expected that as income increased over time, the saving rate would also increase; they feared that there might not be enough profitable investment projects to absorb this saving, and the economy might enter a long depres- sion of indefinite duration. This did not happen. The second piece of evidence against Keynes¡¯s conjectures came from studies of long time-series of consumption and income. Simon Kuznets found that the ratio of con- sumption to income was stable from decade to decade; that is, the average propensity to consume did not seem to be falling over time as income increased. 3. Both the life-cycle and permanent-income hypotheses emphasize that an individual¡¯s time horizon is longer than a single year. Thus, consumption is not simply a function of current income. The life-cycle hypothesis stresses that income varies over a person¡¯s life; saving allows consumers to move income from those times in life when income is high to those times when it is low. The life-cycle hypothesis predicts that consumption should depend on both wealth and income, since these determine a person¡¯s lifetime resources. Hence, we expect the consumption function to look like C = ¦ÁW + ¦ÂY. 153 CHAPTER 16 Consumption In the short run, with wealth fixed, we get a ¡°conventional¡± Keynesian consumption function. In the long run, wealth increases, so the short-run consumption function shifts upward, as shown in Figure 16¨C1. The permanent-income hypothesis also implies that people try to smooth con- sumption, though its emphasis is slightly different. Rather than focusing on the pattern of income over a lifetime, the permanent-income hypothesis emphasizes that people experience random and temporary changes in their income from year to year. The per- manent-income hypothesis views current income as the sum of permanent income Y p and transitory income Y t . Milton Friedman hypothesized that consumption should depend primarily on permanent income: C = ¦ÁY p . The permanent-income hypothesis explains the consumption puzzle by suggesting that the standard Keynesian consumption function uses the wrong variable for income. For example, if a household has high transitory income, it will not have higher con- sumption; hence, if much of the variability in income is transitory, a researcher would find that high-income households had, on average, a lower average propensity to con- sume. This is also true in short time-series if much of the year-to-year variation in income is transitory. In long time-series, however, variations in income are largely per- manent; therefore, consumers do not save any increases in income, but consume them instead. C Y Income ¦Áw ¦Áw Consumption ' 154 Answers to Textbook Questions and Problems Figure 16¨C1 4. Fisher¡¯s model of consumption looks at how a consumer who lives two periods will make consumption choices in order to be as well off as possible. Figure 16¨C2(A) shows the effect of an increase in second-period income if the consumer does not face a binding borrowing constraint. The budget constraint shifts outward, and the consumer increas- es consumption in both the first and the second period. Figure 16¨C2(B) shows what happens if there is a binding borrowing constraint. The consumer would like to borrow to increase first-period consumption but cannot. If income increases in the second period, the consumer is unable to increase first-period consumption. Therefore, the consumer continues to consume his or her entire income in each period. That is, for those consumers who would like to borrow but cannot, con- sumption depends only on current income. Chapter 16 Consumption 155 C 2 Y 2 Y 2 + ? Y 2 Second-period consumption I 2 I 1 C 1 Y 1 First-period consumption B A Old budget constraint New budget constraint Figure 16¨C2A New budget constraint Old budget constraint B A C 2 Y 2 + ?Y 2 Y 2 Second-period consumption Y 1 C 1 First-period consumption I 1 I 2 Figure 16¨C2B 5. The permanent-income hypothesis implies that consumers try to smooth consumption over time, so that current consumption is based on current expectations about lifetime income. It follows that changes in consumption reflect ¡°surprises¡± about lifetime income. If consumers have rational expectations, then these surprises are unpre- dictable. Hence, consumption changes are also unpredictable. 6. Section 16.6 included several examples of time-inconsistent behavior, in which con- sumers alter their decisions simply because time passes. For example, a person may legitimately want to lose weight, but decide to eat a large dinner today and eat a small dinner tomorrow and thereafter. But the next day, they may once again make the same choice¡ªeating a large dinner that day while promising to eat less on following days. Problems and Applications 1. Figure 16¨C3 shows the effect of an increase in the interest rate on a consumer who bor- rows in the first period. The increase in the real interest rate causes the budget line to rotate around the point (Y 1 , Y 2 ), becoming steeper. We can break the effect on consumption from this change into an income and sub- stitution effect. The income effect is the change in consumption that results from the movement to a different indifference curve. Because the consumer is a borrower, the increase in the interest rate makes the consumer worse off¡ªthat is, he or she cannot achieve as high an indifference curve. If consumption in each period is a normal good, this tends to reduce both C 1 and C 2 . The substitution effect is the change in consumption that results from the change in the relative price of consumption in the two periods. The increase in the interest rate makes second-period consumption relatively less expensive; this tends to make the con- sumer choose more consumption in the second period and less consumption in the first period. On net, we find that for a borrower, first-period consumption falls unambiguously when the real interest rate rises, since both the income and substitution effects push in the same direction. Second-period consumption might rise or fall, depending on which 156 Answers to Textbook Questions and Problems New budget constraint Old budget constraint C 2 Y 2 ?C 2 Second-period consumption I 1 I 2 C 1 Y 1 ?C 1 First-period consumption BA Figure 16¨C3 effect is stronger. In Figure 16¨C3, we show the case in which the substitution effect is stronger than the income effect, so that C 2 increases. 2. a. We can use Jill¡¯s intertemporal budget constraint to solve for the interest rate: C 1 + = Y 1 + $100 + = $0 + r = 10%. Jill borrowed $100 for consumption in the first period and in the second period used her $210 income to pay $110 on the loan (principal plus interest) and $100 for consumption. b. The rise in interest rates leads Jack to consume less today and more tomorrow. This is because of the substitution effect: it costs him more to consume today than tomorrow, because of the higher opportunity cost in terms of forgone interest. This is shown in Figure 16¨C4. By revealed preference we know Jack is better off: at the new interest rate he could still consume $100 in each period, so the only reason he would change his consumption pattern is if the change makes him better off. Chapter 16 Consumption 157 C 2 1 + r $100 1 + r $210 1 + r Y 2 1 + r B A 100 191 C 1 Consumption today Consumption tomorro w 100 210 C 2 Figure 16¨C4 c. Jill consumes less today, while her consumption tomorrow can either rise or fall. She faces both a substitution effect and income effect. Because consumption today is more expensive, she substitutes out of it. Also, since all her income is in the sec- ond period, the higher interest rate raises her cost of borrowing and, thus, lowers her income. Assuming consumption in period one is a normal good, this provides an additional incentive for lowering it. Her new consumption choice is at point B in Figure 16¨C5. We know Jill is worse off with the higher interest rates because she could have consumed at point B before (by not spending all of her second-period money) but chose not to because point A had higher utility. 3. a. A consumer who consumes less than his income in period one is a saver and faces an interest rate r s . His budget constraint is C 1 + C 2 /(1 + r s ) < Y 1 + Y 2 /(1 + r s ). b. A consumer who consumes more than income in period one is a borrower and faces an interest rate r b . The budget constraint is C 1 + C 2 /(1 + r b ) < Y 1 + Y 2 /(1 + r b ). 158 Answers to Textbook Questions and Problems B A 100 182 C 1 Consumption today Consumption tomorro w 100 200 C 2 Figure 16¨C5 c. Figure 16¨C6 shows the two budget constraints; they intersect at the point (Y 1 , Y 2 ), where the consumer is neither a borrower nor a lender. The shaded area repre- sents the combinations of first-period and second-period consumption that the con- sumer can choose. To the left of the point (Y 1 , Y 2 ), the interest rate is r b . d. Figure 16¨C7 shows the three cases. Figure 16¨C7(A) shows the case of a saver for whom the indifference curve is tangent to the budget constraint along the line seg- ment to the left of (Y 1 , Y 2 ). Figure 16¨C7(B) shows the case of a borrower for whom the indifference curve is tangent to the budget constraint along the line segment to the right of (Y 1 , Y 2 ). Finally, Figure 16¨C7(C) shows the case in which the con- sumer is neither a borrower nor a lender: the highest indifference curve the con- sumer can reach is the one that passes through the point (Y 1 , Y 2 ). Chapter 16 Consumption 159 C 1 Y 1 First-period consumption Y 2 C 2 Second-period consumption Interest rate of r s Interest rate of r b Figure 16¨C6 160 Answers to Textbook Questions and Problems I C 2 C 2 Y 2 Second-period consumption C 1 First-period consumption Y 1 C 1 Figure 16¨C7A I C 2 C 2 Y 2 Second-period consumption C 1 C 1 Y 1 Figure 16¨C7B I C 2 Y 2 Second-period consumption C 1 Y 1 First-period consumption Figure 16¨C7C e. If the consumer is a saver, then consumption in the first period depends on [Y 1 + Y 2 /(1 + r s )]¡ªthat is, income in both periods, Y 1 and Y 2 , and the interest rate r s . If the consumer is a borrower, then consumption in the first period depends on [Y 1 + Y 2 /(1 + r b )]¡ªthat is, income in both periods, Y 1 and Y 2 , and the interest rate r b . Note that borrowers discount future income more than savers. If the consumer is neither a borrower nor a lender, then consumption in the first period depends just on Y 1 . 4. The potency of fiscal policy to influence aggregate demand depends on the effect on con- sumption: if consumption changes a lot, then fiscal policy will have a large multiplier. If consumption changes only a little, then fiscal policy will have a small multiplier. That is, the fiscal-policy multipliers are higher if the marginal propensity to consume is higher. a. Consider a two-period Fisher diagram. A temporary tax cut means an increase in first-period disposable income Y 1 . Figure 16¨C8(A) shows the effect of this tax cut on a consumer who does not face a binding borrowing constraint, whereas Figure 16¨C8(B) shows the effect of this tax cut on a consumer who is constrained. The consumer with the constraint would have liked to get a loan to increase C 1 , but could not. The temporary tax cut increases disposable income: as shown in the figure, the consumer¡¯s consumption rises by the full amount that taxes fall. The consumer who is constrained thus increases first-period consumption C 1 by more Chapter 16 Consumption 161 A B AB C 2 Y 2 C 2 Y 2 Second-period consumption Second-period consumption Y 1 + ? Y 1 Y 1 First-period consumption First-period consumption Y 1 + ? Y 1 I 1 C 1 I 1 Y 1 C 1 I 2 I 2 Figure 16¨C8A Figure 16¨C8B than the consumer who is not constrained¡ªthat is, the marginal propensity to consume is higher for a consumer who faces a borrowing constraint. Therefore, fis- cal policy is more potent with binding borrowing constraints than it is without them. b. Again, consider a two-period Fisher diagram. The announcement of a future tax cut increases Y 2 . Figure 16¨C9(A) shows the effect of this tax cut on a consumer who does not face a binding borrowing constraint, whereas Figure 16¨C9(B) shows the effect of this tax cut on a consumer who is constrained. The consumer who is not constrained immediately increases consumption C 1 . The consumer who is constrained cannot increase C 1 , because disposable income has not changed. Therefore, the announcement of a future tax cut has no effect on con- sumption or aggregate demand if consumers face binding borrowing constraints: fiscal policy is less potent. 5. In this question, we look at how income growth affects the pattern of consumption and wealth accumulation over a person¡¯s lifetime. For simplicity, we assume that the inter- est rate is zero and that the consumer wants as smooth a consumption path as possible. a. Figure 16¨C10 shows the case in which the consumer can borrow. Income increases during the consumer¡¯s lifetime until retirement, when it falls to zero. 162 Answers to Textbook Questions and Problems B A B A I 1 C 1 Y 1 Y 2 C 2 Y 2 + ? Y 2 Second-period consumption Second-period consumption C 2 Y 2 + ? Y 2 Y 2 C 1 Y 1 I 2 New budget constraint First-period consumption I 2 I 1 New budget constraint Figure 16¨C9A Figure 16¨C9B Desired consumption is level over the lifetime. Until year T 1 , consumption is greater than income, so the consumer borrows. After T 1 , consumption is less than income, so the consumer saves. This means that until T 1 , wealth is negative and falling. After T 1 , wealth begins to increase; after T 2 , all borrowing is repaid, so wealth becomes positive. Wealth accumulation continues until retirement, when the consumer dissaves all wealth to finance consumption. b. Figure 16¨C11 shows the case in which a borrowing constraint prevents the con- sumer from having negative wealth. Before T 1 ¡ä, the consumer would like to be bor- rowing, as in part (a), but cannot. Therefore, income is consumed and is neither saved nor borrowed. After T 1 ¡ä, the consumer begins to save for retirement, and lifetime consumption remains constant at C¡ä. Chapter 16 Consumption 163 Wealth Income Saving Dissaving ConsumptionC Levels of income, consumption, and wealth End of life Retirement begins T 1 T 2 Figure 16¨C10 Wealth Income Saving Consumption C Levels of income, consumption, and wealth End of life Retirement begins C ' T 1 ' Figure 16¨C11 Note that C¡ä is greater than C, and T 1 ¡ä is greater than T 1 , as shown. This is because in part (b), the consumer has lower consumption in the first part of life, so there are more resources left when there is no constraint¡ªconsumption will be higher. 6. The life-cycle model predicts that an important source of saving is that people save while they work to finance consumption after they retire. That is, the young save, and the old dissave. If the fraction of the population that is elderly will increase over the next 20 years, the life-cycle model predicts that as these elderly retire, they will begin to dissave their accumulated wealth in order to finance their retirement consumption: thus, the national saving rate should fall over the next 20 years. 7. In this chapter, we discussed two explanations for why the elderly do not dissave as rapidly as the life-cycle model predicts. First, because of the possibility of unpredictable and costly events, they may keep some precautionary saving as a buffer in case they live longer than expected or have large medical bills. Second, they may want to leave bequests to their children, relatives, or charities, so again, they do not dissave all of their wealth during retirement. If the elderly who do not have children dissave at the same rate as the elderly who do have children, this seems to imply that the reason for low dissaving is the precau- tionary motive; the bequest motive is presumably stronger for people who have children than for those who don¡¯t. An alternative interpretation is that perhaps having children does not increase desired saving. For example, having children raises the bequest motive, but it may also lower the precautionary motive: you can rely on your children in case of financial emer- gency. Perhaps the two effects on saving cancel each other. 8. If you are a fully rational and time-consistent consumer, you would certainly prefer the saving account that lets you take the money out on demand. After all, you get the same return on that account, but in unexpected circumstances (e.g., if you suffer an unex- pected, temporary decline in income), you can use the funds in the account to finance your consumption. This is the kind of consumer in the intertemporal models of Irving Fisher, Franco Modigliani, and Milton Friedman. By contrast, if you face the ¡°pull of instant gratification,¡± you may prefer the account that requires a 30-day notification before withdrawals. In this way, you pre- commit yourself to not using the funds to satisfy a desire for instant gratification. This precommitment offers a way to overcome the time-inconsistency problem. That is, some people would like to save more, but at any particular moment, they face such a strong desire for instant gratification that they always choose to consume rather than save. This is the type of consumer in David Laibson¡¯s theory. 164 Answers to Textbook Questions and Problems