TEXTBOOK QUESTION SOLUTIONS Problems amazonaws com
Instructor’s Manual for Macroeconomics, Fourth Canadian Edition
TEXTBOOK QUESTION SOLUTIONS Problems 1. Although we often think about the negative externalities of congestion and pollution in cities, there may also be some positive externalities. A concentrated population is better able to support the arts, professional sports, etc; cities typically have a greater variety of good restaurants and so forth. Perhaps a more basic issue is that there may be some increasing returns to scale at low output levels that make industrial production more costly in small towns. There may also be externalities in production in being located close to other producers. One example would be the financial industry in financial centres like New York, London, and Tokyo. Another example would be large city medical centres that enhance coordination between primary physicians and specialists. One market test of whether productivity is higher in cities would be to look at the wages in cities versus the wages in smaller towns and rural areas. Wages are often higher in cities for individuals of comparable skills. Market efficiency suggests that the higher wages are reflective of a higher marginal product of labour, and that the higher wages compensate those choosing to live in cities for the negative externalities they face. 2. In a one period model, taxes must be exactly equal to government spending. A reduction in taxes is therefore equivalent to a reduction in government spending. This result is exactly the opposite of the case of an increase in government spending presented in the textbook. A reduction in government spending induces a pure income effect that induces the consumer to consume more and work less. At lower employment, the equilibrium real wage is higher because the marginal product of labour rises when employment falls. Output falls, consumption rises, employment falls, and the real wage rises. 3. A reduction in the capital stock. a) This works exactly the same as for the increase in total factor productivity in Figures 5.9 and 5.10, except in reverse, that is, the PPF shifts down. Output falls, employment may rise or fall (because of offsetting income and substitution effects), leisure may rise or fall (likewise), consumption falls, and the real wage falls. b) Shocks to the capital stock qualitatively could replicate key business cycle facts, provided that the substitution effect on labour supply dominates. However, for Canada over the last 50 years, for example, the kinds of natural disasters or other capital-stock-destroying events that have occurred have typically had very small effects on the aggregate capital stock. This is not to say that such large disasters are impossible, or could not happen in the future.
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Chapter 5: A Closed-Economy One-Period Macroeconomic Model 4. Government productivity. First consider the benchmark case in which z = 1 , and there is no effect of changes in z on government activities. Now suppose that z increases. This case of an increase in z is depicted in Figure 5.1 below. The original production possibilities frontier is labelled PPF1 and the competitive equilibrium is at point A. If the increase in z only affects the economy through the change in zF ( K , N ) , then the new production possibilities frontier is PPF2. The diagram shows a case in which the income and substitution effects on leisure exactly cancel out, and the economy moves to point B. The equation for the production possibilities frontier is C = zF (K , h − l ) − T . In the benchmark case, T = G and so we have C = zF (K , h − l ) − G . For this problem, T = G / z ,
and so the production possibilities frontier is given by C = zF (K , h − l ) −
G . When z = 1 , z
the two PPFs coincide. When z increases, the vertical intercept of the PPF increases by
G . Therefore, the new PPF is PPF3 in Figure 5.1. The competitive equilibrium is Δz
at point C. There is an additional income effect that provides an additional increase in equilibrium consumption and a reinforced income effect that tends to make leisure increase. Therefore, relative to the benchmark case, there is a larger increase in consumption and either a smaller decrease in leisure or a larger increase in leisure.
Figure 5.1
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Instructor’s Manual for Macroeconomics, Fourth Canadian Edition 5. Change in preferences. a) At the margin, the consumer decides that leisure is preferred to consumption. That is, the consumer now requires a bigger increase in consumption to willingly work more (consume less leisure). In more intuitive language, the consumer is lazier. b) To work out the effects of this change in tastes, we refer to Figure 5.2. The production possibility frontier in this example is unchanged. The consumer now picks a new point at which one of the flatter indifference curves is tangent to the production possibilities frontier. That is, equilibrium will shift from point A to point B. Consumption falls and leisure rises. Therefore, the consumer works less and produces less. Because employment has fallen, it also must be the case that the real wage increases. c) This disturbance, which some might characterize as a contagious outbreak of laziness, would have the appearance of a recession as output and employment both fall. The consequent reduction in consumption is also consistent with a typical recession. However, in this case the real wage would rise, which is inconsistent with business cycle facts. Therefore, this type of preference change is not a cause of recessions.
Figure 5.2
6. Production-enhancing aspects of government spending. a) The increase in government spending in this example has two separate effects on the production possibilities frontier. First, the increase in government spending from G1 to G2 implies a parallel downward shift in the production possibilities frontier. Second, the productive nature of government spending is equivalent to an increase in total factor productivity that shifts the production possibilities frontier upward and increases its slope. Figure 5.3 draws the original production Copyright © 2013 Pearson Canada Inc. - 54 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model possibilities frontier as PPF1 and the new production possibilities frontier as PPF2. If the production-enhancing aspects of the increase in government spending are large enough, representative consumer utility could rise, as in Figure 5.3.
Figure 5.3
b) There are three effects at work in this example. First, there is a negative income effect from the increase in taxes needed to pay for the increased government spending. This effect tends to lower both consumption and leisure. Second, there is a substitution effect due to the productive effect of the increase in G, which is drawn as the movement from point A to point D. This effect tends to increase both consumption and leisure. Third, there is a positive income effect from the increase in G on productivity. This effect tends to increase both consumption and leisure. In Figure 5.3, the movement from point D to point B is the net effect of the two income effects. In general, consumption may rise or fall, and leisure may rise or fall. The overall effect on output is the same as in any increase in total factor productivity. Output surely rises. 7. Increase in hours spent in education a) In the present, an increase in the quantity of hours used up for the representative consumer in education can be represented in the model as a decrease in the quantity of time available, h, from h1 to h2 in the figure. This shifts the PPF to the left by a fixed amount h2 – h1 for each quantity of consumption. This will then have a similar income effect to a reduction in government spending. Consumption and leisure must fall, but note that the quantity of leisure must fall by less than the decrease in available hours, so hours worked must fall. Therefore output falls in the present. The real wage rises, just as for a decrease in G. b) In the future, because the representative agent is more productive, we can represent this as more time available (note that this makes time more productive Copyright © 2013 Pearson Canada Inc. - 55 -
Instructor’s Manual for Macroeconomics, Fourth Canadian Edition in leisure as well, by assumption) and an increase in h. This then is just the reverse experiment from part (a). Consumption rises, leisure rises, output rises, and the real wage falls. c) Education is an investment here for the economy as a whole. Using up more time in education today implies that, as a society, we have less leisure time, and less consumption, but what we gain in terms of payoffs are more leisure and more consumption in the future.
Figure 5.4
8. Increase in government spending with a proportional wage income tax. In the Figure 5.5, the production possibilities frontier is initially AB, and it shifts to DF with an increase in G. If the economy is on the good side of the Laffer curve, then with the consumer’s budget constraint in equilibrium initially given by HJ, the consumer must face a higher tax rate in the new equilibrium, facing a budget constraint like HK. The equilibrium point shifts from L to M. If we disentangle the income and substitution effects, consumption must fall, leisure may rise or fall, Copyright © 2013 Pearson Canada Inc. - 56 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model employment may rise or fall, and output may rise or fall. The consumer must be worse off. However, suppose in Figure 5.6 that the economy is on the wrong side of the Laffer curve. Then, the tax rate must fall in equilibrium, and the consumer’s budget constraint must shift up in equilibrium from HJ to HK. Consumption must rise, leisure may rise or fall, employment may rise or fall, and output must rise. Note in this case that the representative consumer is actually better off with a higher level of government spending.
Figure 5.5
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Instructor’s Manual for Macroeconomics, Fourth Canadian Edition
Figure 5.6
9. In Figure 5.7, the production possibilities frontier shifts to the right from AB to AD. Since the substitution effect dominates the income effect on labour supply, the Laffer curve shifts up for each tax rate, and so the tax rate must be lower in equilibrium, since the government is financing the same quantity of government spending, and the economy is on the good side of the Laffer curve. In Figure 5.7, the consumer’s budget constraint shifts from FH to FJ, and the equilibrium point shifts from K to L. Since the substitution effect dominates, consumption must increase, leisure decreases, labour supply increases, and real output increases.
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Chapter 5: A Closed-Economy One-Period Macroeconomic Model
Figure 5.7
10. a) With perfect substitutes preferences, indifference curves are straight lines with slope –b, where b is the marginal rate of substitution. If b > 1/q, so that the indifference curves are steeper than the PPF, then the optimal choice for the government is G=qY, so that C = 0. Thus if b is relatively large (the consumer cares relatively more about public goods relative to private goods) and q is relatively large (the government is relatively efficient), then all production should be carried on by the government. Alternatively if b < 1/q, then G = 0 and C = Y, so that government is inactive. Thus, if b increases or q increases, this makes it more likely that b > 1/q and we have the first case, where all production comes from the government. b) With perfect complements, indifference curves are as depicted in Figure 5.8, and the initial equilibrium is at point A. If a increases, then the equilibrium shifts from Copyright © 2013 Pearson Canada Inc. - 59 -
Instructor’s Manual for Macroeconomics, Fourth Canadian Edition A to B in Figure 5.9. An increase in a represents a greater preference for private goods relative to public goods, and in Figure 5.9, this results in less public goods and more private consumption in equilibrium. If q increases, this shifts the PPF out as in Figure 5.10, and the equilibrium shifts from A to B. Both C and G increase, driven by income effects.
Figure 5.8
Figure 5.9
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Chapter 5: A Closed-Economy One-Period Macroeconomic Model
Figure 5.10
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TEXTBOOK QUESTION SOLUTIONS Problems 1. Although we often think about the negative externalities of congestion and pollution in cities, there may also be some positive externalities. A concentrated population is better able to support the arts, professional sports, etc; cities typically have a greater variety of good restaurants and so forth. Perhaps a more basic issue is that there may be some increasing returns to scale at low output levels that make industrial production more costly in small towns. There may also be externalities in production in being located close to other producers. One example would be the financial industry in financial centres like New York, London, and Tokyo. Another example would be large city medical centres that enhance coordination between primary physicians and specialists. One market test of whether productivity is higher in cities would be to look at the wages in cities versus the wages in smaller towns and rural areas. Wages are often higher in cities for individuals of comparable skills. Market efficiency suggests that the higher wages are reflective of a higher marginal product of labour, and that the higher wages compensate those choosing to live in cities for the negative externalities they face. 2. In a one period model, taxes must be exactly equal to government spending. A reduction in taxes is therefore equivalent to a reduction in government spending. This result is exactly the opposite of the case of an increase in government spending presented in the textbook. A reduction in government spending induces a pure income effect that induces the consumer to consume more and work less. At lower employment, the equilibrium real wage is higher because the marginal product of labour rises when employment falls. Output falls, consumption rises, employment falls, and the real wage rises. 3. A reduction in the capital stock. a) This works exactly the same as for the increase in total factor productivity in Figures 5.9 and 5.10, except in reverse, that is, the PPF shifts down. Output falls, employment may rise or fall (because of offsetting income and substitution effects), leisure may rise or fall (likewise), consumption falls, and the real wage falls. b) Shocks to the capital stock qualitatively could replicate key business cycle facts, provided that the substitution effect on labour supply dominates. However, for Canada over the last 50 years, for example, the kinds of natural disasters or other capital-stock-destroying events that have occurred have typically had very small effects on the aggregate capital stock. This is not to say that such large disasters are impossible, or could not happen in the future.
Copyright © 2013 Pearson Canada Inc. - 52 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model 4. Government productivity. First consider the benchmark case in which z = 1 , and there is no effect of changes in z on government activities. Now suppose that z increases. This case of an increase in z is depicted in Figure 5.1 below. The original production possibilities frontier is labelled PPF1 and the competitive equilibrium is at point A. If the increase in z only affects the economy through the change in zF ( K , N ) , then the new production possibilities frontier is PPF2. The diagram shows a case in which the income and substitution effects on leisure exactly cancel out, and the economy moves to point B. The equation for the production possibilities frontier is C = zF (K , h − l ) − T . In the benchmark case, T = G and so we have C = zF (K , h − l ) − G . For this problem, T = G / z ,
and so the production possibilities frontier is given by C = zF (K , h − l ) −
G . When z = 1 , z
the two PPFs coincide. When z increases, the vertical intercept of the PPF increases by
G . Therefore, the new PPF is PPF3 in Figure 5.1. The competitive equilibrium is Δz
at point C. There is an additional income effect that provides an additional increase in equilibrium consumption and a reinforced income effect that tends to make leisure increase. Therefore, relative to the benchmark case, there is a larger increase in consumption and either a smaller decrease in leisure or a larger increase in leisure.
Figure 5.1
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Instructor’s Manual for Macroeconomics, Fourth Canadian Edition 5. Change in preferences. a) At the margin, the consumer decides that leisure is preferred to consumption. That is, the consumer now requires a bigger increase in consumption to willingly work more (consume less leisure). In more intuitive language, the consumer is lazier. b) To work out the effects of this change in tastes, we refer to Figure 5.2. The production possibility frontier in this example is unchanged. The consumer now picks a new point at which one of the flatter indifference curves is tangent to the production possibilities frontier. That is, equilibrium will shift from point A to point B. Consumption falls and leisure rises. Therefore, the consumer works less and produces less. Because employment has fallen, it also must be the case that the real wage increases. c) This disturbance, which some might characterize as a contagious outbreak of laziness, would have the appearance of a recession as output and employment both fall. The consequent reduction in consumption is also consistent with a typical recession. However, in this case the real wage would rise, which is inconsistent with business cycle facts. Therefore, this type of preference change is not a cause of recessions.
Figure 5.2
6. Production-enhancing aspects of government spending. a) The increase in government spending in this example has two separate effects on the production possibilities frontier. First, the increase in government spending from G1 to G2 implies a parallel downward shift in the production possibilities frontier. Second, the productive nature of government spending is equivalent to an increase in total factor productivity that shifts the production possibilities frontier upward and increases its slope. Figure 5.3 draws the original production Copyright © 2013 Pearson Canada Inc. - 54 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model possibilities frontier as PPF1 and the new production possibilities frontier as PPF2. If the production-enhancing aspects of the increase in government spending are large enough, representative consumer utility could rise, as in Figure 5.3.
Figure 5.3
b) There are three effects at work in this example. First, there is a negative income effect from the increase in taxes needed to pay for the increased government spending. This effect tends to lower both consumption and leisure. Second, there is a substitution effect due to the productive effect of the increase in G, which is drawn as the movement from point A to point D. This effect tends to increase both consumption and leisure. Third, there is a positive income effect from the increase in G on productivity. This effect tends to increase both consumption and leisure. In Figure 5.3, the movement from point D to point B is the net effect of the two income effects. In general, consumption may rise or fall, and leisure may rise or fall. The overall effect on output is the same as in any increase in total factor productivity. Output surely rises. 7. Increase in hours spent in education a) In the present, an increase in the quantity of hours used up for the representative consumer in education can be represented in the model as a decrease in the quantity of time available, h, from h1 to h2 in the figure. This shifts the PPF to the left by a fixed amount h2 – h1 for each quantity of consumption. This will then have a similar income effect to a reduction in government spending. Consumption and leisure must fall, but note that the quantity of leisure must fall by less than the decrease in available hours, so hours worked must fall. Therefore output falls in the present. The real wage rises, just as for a decrease in G. b) In the future, because the representative agent is more productive, we can represent this as more time available (note that this makes time more productive Copyright © 2013 Pearson Canada Inc. - 55 -
Instructor’s Manual for Macroeconomics, Fourth Canadian Edition in leisure as well, by assumption) and an increase in h. This then is just the reverse experiment from part (a). Consumption rises, leisure rises, output rises, and the real wage falls. c) Education is an investment here for the economy as a whole. Using up more time in education today implies that, as a society, we have less leisure time, and less consumption, but what we gain in terms of payoffs are more leisure and more consumption in the future.
Figure 5.4
8. Increase in government spending with a proportional wage income tax. In the Figure 5.5, the production possibilities frontier is initially AB, and it shifts to DF with an increase in G. If the economy is on the good side of the Laffer curve, then with the consumer’s budget constraint in equilibrium initially given by HJ, the consumer must face a higher tax rate in the new equilibrium, facing a budget constraint like HK. The equilibrium point shifts from L to M. If we disentangle the income and substitution effects, consumption must fall, leisure may rise or fall, Copyright © 2013 Pearson Canada Inc. - 56 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model employment may rise or fall, and output may rise or fall. The consumer must be worse off. However, suppose in Figure 5.6 that the economy is on the wrong side of the Laffer curve. Then, the tax rate must fall in equilibrium, and the consumer’s budget constraint must shift up in equilibrium from HJ to HK. Consumption must rise, leisure may rise or fall, employment may rise or fall, and output must rise. Note in this case that the representative consumer is actually better off with a higher level of government spending.
Figure 5.5
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Instructor’s Manual for Macroeconomics, Fourth Canadian Edition
Figure 5.6
9. In Figure 5.7, the production possibilities frontier shifts to the right from AB to AD. Since the substitution effect dominates the income effect on labour supply, the Laffer curve shifts up for each tax rate, and so the tax rate must be lower in equilibrium, since the government is financing the same quantity of government spending, and the economy is on the good side of the Laffer curve. In Figure 5.7, the consumer’s budget constraint shifts from FH to FJ, and the equilibrium point shifts from K to L. Since the substitution effect dominates, consumption must increase, leisure decreases, labour supply increases, and real output increases.
Copyright © 2013 Pearson Canada Inc. - 58 -
Chapter 5: A Closed-Economy One-Period Macroeconomic Model
Figure 5.7
10. a) With perfect substitutes preferences, indifference curves are straight lines with slope –b, where b is the marginal rate of substitution. If b > 1/q, so that the indifference curves are steeper than the PPF, then the optimal choice for the government is G=qY, so that C = 0. Thus if b is relatively large (the consumer cares relatively more about public goods relative to private goods) and q is relatively large (the government is relatively efficient), then all production should be carried on by the government. Alternatively if b < 1/q, then G = 0 and C = Y, so that government is inactive. Thus, if b increases or q increases, this makes it more likely that b > 1/q and we have the first case, where all production comes from the government. b) With perfect complements, indifference curves are as depicted in Figure 5.8, and the initial equilibrium is at point A. If a increases, then the equilibrium shifts from Copyright © 2013 Pearson Canada Inc. - 59 -
Instructor’s Manual for Macroeconomics, Fourth Canadian Edition A to B in Figure 5.9. An increase in a represents a greater preference for private goods relative to public goods, and in Figure 5.9, this results in less public goods and more private consumption in equilibrium. If q increases, this shifts the PPF out as in Figure 5.10, and the equilibrium shifts from A to B. Both C and G increase, driven by income effects.
Figure 5.8
Figure 5.9
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Chapter 5: A Closed-Economy One-Period Macroeconomic Model
Figure 5.10
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