ABC 8e Answer Key Ch 4

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Answers to Textbook Problems

Review Questions 1. Saving is current income minus consumption. For given income, any increase in consumption means an equal decrease in saving, so consumption and saving are inversely related. The basic motivation for saving is to provide for future consumption. 2. When a consumer gets an increase in current income, both current consumption and future consumption increase. Since current consumption rises, but by less than the increase in current income, saving increases. When the consumer gets an increase in expected future income, again both current and future consumption increase. Since current income does not increase, but current consumption does, saving decreases. When the consumer gets an increase in wealth, both current and future consumption again rise. Again, there has been no increase in current income, so saving decreases. At the aggregate level, these changes in consumption and saving made by individuals are decisions that change the aggregate level of desired consumption and saving. 3. The effect on desired saving of an increase in the expected real interest rate is potentially ambiguous. An increase in the real interest rate has two effects on desired saving: (1) the substitution effect increases saving, because the amount of future consumption that can be obtained in exchange for giving up a unit of current consumption rises; and (2) the income effect may increase or reduce saving. The income effect reduces saving for a lender, because a person who saves is better off as a result of having a higher real interest rate, so he or she increases current consumption. However, for a borrower, the income effect increases saving, because the borrower is worse off having to face a higher real interest rate, and so reduces current consumption. So the income effects work in different directions depending on whether a person is a lender or a borrower. For a borrower, then, both the income and substitution effects work in the same direction, and saving definitely increases. For a lender, however, the income and substitution effects work in opposite directions, so the result on desired saving is ambiguous. 4. The expected after-tax real interest rate is the after-tax nominal interest rate, (1 − t)i, minus the expected rate of inflation, π e, and represents the real return earned by a saver when a portion, t, of interest income must be paid as taxes. If the tax rate on interest income declines (that is, t declines), then 1 − t becomes larger, so the expected after-tax real interest rate increases. 5. When government purchases increase temporarily, consumers see that higher taxes will be required in the future to pay off the deficit. They reduce both current consumption and future consumption, but current consumption declines by less than the amount of the government purchases. Since national saving is output minus desired consumption minus government purchases, and government purchases have increased more than current desired consumption has decreased, national saving declines at a given real interest rate. In the case of a lump-sum tax increase, consumers have higher taxes today, but lower taxes in the future. If consumers take this into account, current desired consumption is unchanged, and since output and government purchases didn’t change, desired national saving is unchanged as well. This is the case of Ricardian equivalence, and is controversial because consumers may not understand that higher taxes today imply lower future taxes. As a result, they may reduce desired consumption today, increasing desired national saving.

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6. The two components of the user cost of capital are the interest cost and the depreciation cost. The depreciation cost is the value lost as the capital wears out during the period. The interest cost represents the opportunity cost of not using the funds that purchased the capital in some other way; an example would be if the money was invested in bonds rather than buying capital goods. 7. The desired capital stock is the amount of capital that allows the firm to earn the largest possible profit. The higher the expected future marginal product of capital, the higher the desired capital stock, since any given amount of capital will be more productive in the future. The higher the user cost of capital, the lower the desired capital stock, since a higher user cost yields lower profits on each unit of capital. The higher the effective tax rate, the lower the desired capital stock, again because the firm gets lower profits on each unit of capital. 8. Gross investment represents the total purchase or construction of new capital goods that takes place during a period. Net investment is gross investment minus the depreciation on existing capital. Thus net investment is the overall increase in the capital stock. Yes, it is possible for gross investment to be positive when net investment is negative. This occurs whenever gross investment is less than the amount of depreciation (and, in fact, happened in the United States during World War II). 9. Equilibrium in the goods market occurs when the aggregate supply of goods (Y) equals the aggregate demand for goods (Cd + Id + G). Since desired national saving (Sd ) is Y − C d − G, an equivalent condition is Sd = Id. Equilibrium is achieved by the adjustment of the real interest rate to make the desired level of saving equal to the desired level of investment, as shown in text Figure 4.6. 10. The saving curve slopes upward because saving is assumed to increase with an increase in the expected real interest rate. The investment curve slopes downward because investment is lower the higher is the expected real interest rate. The saving curve would be shifted to the right by an increase in current output, a decrease in expected future output, a decrease in wealth, a decrease in government purchases, and possibly by a rise in taxes. The investment curve would shift to the right by a decline in the effective tax rate or a rise in expected future marginal product of capital.

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Numerical Problems 1.

First, a general formulation of the problem is useful. With income of Y1 in the first year and Y2 in the second year, the consumer saves Y1 − C in the first year and Y2 − C in the second year, where C is the consumption amount, which is the same in both years. Saving in the first year earns interest at rate r, where r is the real interest rate. And the consumer needs to accumulate just enough after two years to pay for college tuition, in the amount T. So the key equation is (Y1 − C)(1 + r) + (Y2 − C) = T. (a) Y1 = Y2 = $50,000, r = 10%, T = $12,600. The key equation gives ($50,000 − C)1.1 + ($50,000 − C) = $12,600. This can be simplified to $50,000 − C = $12,600/2.1 = $6000, which can be solved to get C = $44,000. Then S = Y − C = $50,000 − $44,000 = $6000. (b) Y1 = $54,200. The key equation is now ($54,200 − C)1.1 + ($50,000 − C) = $12,600. This can be simplified to ($54,200 × 1.1) + $50,000 − $12,600 = 2.1 C, or $97,020 = 2.1 C, so C = $46,200. Then S = Y1 − C = $54,200 − $46,200 = $8000. This illustrates that a rise in current income increases saving. (c) Y2 = $54,200. The key equation is now ($50,000 − C)1.1 + ($54,200 − C) = $12,600. This can be simplified to ($50,000 × 1.1) + $54,200 − $12,600 = 2.1 C, or $96,600 = 2.1 C, so C = $46,000. Then S = Y1 − C = $50,000 − $46,000 = $4000. This illustrates that a rise in future income decreases saving. (d) With the increase in wealth of W, the total amount invested for the second period is W + Y1 − C, so the key equation becomes ($1050 + $50,000 − C)1.1 + ($50,000 − C) = $12,600. This can be simplified to ($51,050 × 1.1) + $50,000 − $12,600 = 2.1 C, or $93,555 = 2.1 C, so C = $44,550. Then S = Y1 − C = $50,000 − $44,550 = $5450. This illustrates that a rise in wealth decreases saving. (e) T = $14,700. The key equation is now ($50,000 − C)1.1 + ($50,000 − C) = $14,700. This can be simplified to $50,000 − C = $14,700/2.1 = $7000, which can be solved to get C = $43,000. Then S = Y − C = $50,000 − $43,000 = $7000. The rise in targeted wealth needed in the future raises current saving. (f) r = 25%. The key equation is now ($50,000 − C)1.25 + ($50,000 − C) = $12,600. This can be simplified to $50,000 − C = $12,600/2.25 = $5600, which can be solved to get C = $44,400. Then S = Y − C = $50,000 − $44,400 = $5600. The rise in the real interest rate, with a given wealth target, reduces current saving.

2.

(a) This chart shows the MPKf as the increase in output from adding another fabricator: # Fabricators 0 1 2 3 4 5 6

Output 0 100 150 180 195 205 210

MPKf — 100 50 30 15 10 5

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(b) uc = (r + d)pK = (0.12 + 0.20)$100 = $32. HHHHC should buy two fabricators, since at two fabricators, MPK f = 50 > 32 = uc. But at three fabricators, MPK f = 30 < 32 = uc. You want to add fabricators only if the future marginal product of capital exceeds the user cost of capital. The MPK f of the third fabricator is less than its user cost, so it should not be added. (c) When r = 0.08, uc = (0.08 + 0.20)$100 = $28. Now they should buy three fabricators, since MPK f = 30 > 28 = uc for the third fabricator and MPK f = 15 < 28 = uc for the fourth fabricator. (d) With taxes, they should add additional fabricators as long as (1 − τ)MPK f > uc. Since τ = 0.4, 1 − τ = 0.6. They should buy just one fabricator, since (1 − τ)MPK f = 0.6 × 100 = 60 > 32 = uc. They shouldn’t buy two, since then (1 − τ)MPK f = 0.6 × 50 = 30 < 32 = uc. (e) When output doubles, the MPKf doubles as well. At r = 0.12, they should buy three fabricators, since then MPK f = 60 > 32 = uc; they shouldn’t buy four, since then MPK f = 30 < 32 = uc. At r = 0.08, they should buy four fabricators, since then MPK f = 30 > 28 = uc; they shouldn’t buy five, since then MPK f = 20 < 28 = uc. 3.

(a) The expected after-tax real interest rate is r = i(1 − t) − πe = 0.10 (1 − 0.30) − 0.05 = 0.07 − 0.05 = 0.02. (b) The cost of maintaining the house is depreciation. So the annual user cost of capital is uc = (r + d)pK = (0.02 + 0.06)$300,000 = $24,000. (c) You should be indifferent between buying and renting if the rent is $16,000 per year.

4.

Since the price of capital declines from 60 to 51, the depreciation rate is 9/60 = .15. (a) uc = (r + d)pK = (.10 + .15)60 = 15 units of output per year. (b) The desired capital stock is such that MPK f = uc, so 165 − 2K = 15, or 2K = 150, so K = 75. (c) The tax-adjusted user cost of capital is uc/(1 − τ), so with τ = .4, the condition for the desired capital stock is 165 − 2K = 15/0.6, or 2K = 140; the solution is K = 70. Thus taxation decreases the firm’s desired capital stock. (d) The investment tax credit basically lowers the price of capital from 60 to (1 − 0.2)60 = 48. So the tax-adjusted user cost of capital is only (.25 × 48)/0.6 = 20. Then the equation for setting the desired capital stock is 165 − 2K = 20, or 2K = 145; the solution is K = 72.5. Thus the investment tax credit increases the firm’s desired capital stock.

5.

(a) Desired consumption declines as the real interest rate rises because the higher return to saving encourages higher saving; desired investment declines as the real interest rate rises because the user cost of capital is higher, reducing the desired capital stock, and thus investment. (b) Use the following table, where Sd = Y − Cd − G = 9000 − Cd − 2000 = 7000 − Cd. Cd Id Sd Cd + Id + G r 2 6100 1500 900 9600 3 6000 1400 1000 9400 4 5900 1300 1100 9200 5 5800 1200 1200 9000 6 5700 1100 1300 8800 d d (c) Equation (4.7) says that Y = C + I + G at equilibrium. Looking at the last column of the table, with Y = 9000, this is true only at r = 5%. At this point, Sd = Id = 1200. Equation (4.8) says that Sd = Id at equilibrium. From the table, this occurs at r = 5%. ©2014 Pearson Education, Inc.

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(d) When government purchases fall by 400 to 1600, each Sd entry in the table is higher by 400, and each Cd + Id + G entry is lower by 400. Then Y = Cd + Id + G occurs at r = 3%, as does Sd = Id = 1400.

6.

r

Cd

Id

Sd

Cd + Id + G

2 3 4 5 6

6100 6000 5900 5800 5700

1500 1400 1300 1200 1100

1300 1400 1500 1600 1700

9200 9000 8800 8600 8400

(a) Sd = Y − Cd − G = Y − (3600 − 2000r + 0.1Y) − 1200 = −4800 + 2000r + 0.9Y (b) (1) Using Eq. (4.7): Y = Cd + Id + G Y = (3600 − 2000r + 0.1Y) + (1200 − 4000r) + 1200 = 6000 − 6000r + 0.1Y So 0.9Y = 6000 − 6000r At full employment, Y = 6000. Solving 0.9 × 6000 = 6000 − 6000r, we get r = 0.10. (2) Using Eq. (4.8): S d = Id −4800 + 2000r + 0.9Y = 1200 − 4000r 0.9Y = 6000 − 6000r When Y = 6000, r = 0.10. So we can use either Eq. (4.7) or (4.8) to get to the same result. (c) When G = 1440, desired saving becomes Sd = Y − Cd − G = Y − (3600 − 2000r + 0.1Y) − 1440 = −5040 + 2000r + 0.9Y. Sd is now 240 less for any given r and Y; this shows up as a shift in the Sd line from S1 to S2 in Figure 4.3.

Figure 4.3

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Setting Sd = Id, we get: −5040 + 2000r + 0.9Y = 1200 − 4000r 6000r + 0.9Y = 6240 At Y = 6000, this is 6000r = 6240 − (0.9 × 6000) = 840, so r = 0.14. The market-clearing real interest rate increases from 10% to 14%. 7.

(a) r = 0.10 uc/(1 − τ) = (r + d)pK/(1 − τ) = [(.1 + 0.2) × 1]/(1 − 0.15) = 0.35. MPK f = uc/(1 − τ), so 20 − 0.02K = 0.35; solving this gives K = 982.5. Since K − K-1 = I − dK, I = K − K-1 + dK = 982.5 − 900 + (.2 × 900) = 262.5. (b) i. Solving for this in general: uc/(1 − τ) = (r + d)pK/(1 − τ) = [(r + .2) × 1]/(1 − 0.15) = .235 + 1.176r. MPKf = uc/(1 − τ), so 20 − 0.02K = 0.235 + 1.176r; solving this gives K = 988.25 − 58.8r. I = K − K-1 + dK = 988.25 − 58.8r − 900 + (0.2 × 900) = 268.25 − 58.8r. ii. Y = C + I + G 1000 = [100 + (.5 × 1000) − 200r] + (268.25 − 58.8r) + 200 1000 = 1068.25 − 258.8r, so 258.8r = 68.25 r = 0.264 C = 100 + (0.5 × 1000) − (200 × 0.264) = 547.2 I = 268.25 − (58.8 × 0.264) = 252.7 = S uc/(1 − τ) = 0.235 + (1.176 × 0.264) = 0.545 K = 988.25 − (58.8 × 0.264) = 972.7

8.

(a) PVLR = y + [y f/(1 + r)] + a = 90 + (110/1.10) + 20 = 210. f (b) c + [c / (1 + r)] = PVLR. c + (c f / 1.10) = 210. When c = 0, c f = 231; this is the vertical intercept of the budget line, shown in Figure 4.4. When c f = 0, c = 210; this is the horizontal intercept of the budget line.

Figure 4.4

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(c) c = c f: c + (c/1.10) = 210. 1.10c + c = 210 × 1.10. 2.1c = 231. c = 110. s=y−c = 90 − 110 = −20. (d) y increases by 11, so new PVLR = 221. 2.1c = 221 × 1.1 = 243.1. c = 115.76. s = y − c = 101 − 115.76 = − 14.76. So part of the temporary increase in income is consumed and part is saved. (e) y f increases by 11, so PVLR rises by 11/1.10 = 10. New PVLR = 220. 2.1c = 220 × 1.1 = 242. c = 115.24. s = y − c = 90 − 115.24 = −25.24. So a rise in future income leads to an increase in current consumption but a decrease in saving. (f) A rise in initial wealth has the same effect on the PVLR and thus on consumption as an increase in current income of the same amount, so c = 115.76 as in part (d). s = y − c = 90 − 115.76 = −25.76. So an increase in wealth increases current consumption and decreases saving. 9.

(a) PVLR = a + yl + yw + yr = 1500. (1) No borrowing constraint: cl + cw + cr = 1500. cl = cw = cr = c = 1500/3 = 500. sl = 200 − 500 = −300; sw = 800 − 500 = 300; sr = 200 − 500 = −300. (2) A borrowing constraint is nonbinding, since a + y l = 500 = cl, and cw = 500 < 800 = yw. So consumption and saving are the same in each period as in part (1) above. (b) PVLR = 1200. (1) No borrowing constraint: c = 1200/3 = 400. sl = 200 − 400 = −200; sw = 800 − 400 = 400; sr = 200 − 400 = −200. (2) The borrowing constraint is now binding, since cl = 400 > a + yl = 200. So cl is constrained to be 200. That leaves PVLR of 1000 for cw + cr, so they both equal 500. cw = 500 < 800 = yw, so the borrowing constraint is not binding in working age. sl = 200 − 200 = 0; sw = 800 − 500 = 300; sr = 200 − 500 = −300. Consumption can’t be lower in all periods due to a binding borrowing constraint, because the present value of lifetime consumption must be the same with and without borrowing constraints.

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Analytical Problems 1.

(a) As Figure 4.5 shows, the shift to the right in the saving curve from S1 to S2 causes saving and investment to increase and the real interest rate to decrease.

Figure 4.5 (b) This is really just a transfer from the general population to veterans. The effect on saving depends on whether the marginal propensity to consume (MPC) of veterans differs from that of the general population. If there is no difference in MPCs, there will be no shift of the saving curve; neither investment nor the real interest rate is affected. If the MPC of veterans is higher than the MPC of the general population, then desired national saving declines and the saving curve shifts to the left; the real interest rate rises and investment declines. If the MPC of veterans is lower than that of the general population, the saving curve shifts to the right; the real interest rate declines and investment rises. (c) The investment tax credit encourages investment, shifting the investment curve from I1 to I2 in Figure 4.6. Saving and investment increase, as does the real interest rate.

Figure 4.6

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(d) The increase in expected future income decreases current desired saving, as people increase desired consumption immediately. The rise of the future marginal productivity of capital shifts the investment curve to the right. The result, as shown in Figure 4.7, is that the real interest rate rises, with ambiguous effects on saving and investment.

Figure 4.7 2.

(a) With a lower capital stock, the marginal product of labor is reduced, so the labor demand curve shifts to the left from ND1 to ND2 in Figure 4.8. Then the new equilibrium point is one with lower employment and a lower real wage. With lower employment and a lower capital stock, fullemployment output will be lower.

Figure 4.8 (b) Because the capital stock is lower, the marginal product of capital will be higher, so desired investment will increase. (c) Since current output declines, desired saving declines, because people do not want to reduce their consumption. On the other hand, since future output is also lower, people desire to save more today to make up for the loss of future income.

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(d) The increase in desired investment shows up as a shift to the right in the Id curve, from I1 to I2 in Figure 4.9. Then the new equilibrium (assuming no change in desired saving) is at a higher level of investment and a higher real interest rate.

Figure 4.9 3.

(a) The temporary increase in the price of oil reduces the marginal product of labor, causing the labor demand curve to shift to the left from ND1 to ND2 in Figure 4.10. At equilibrium, there is a reduced real wage and lower employment.

Figure 4.10

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The productivity shock results in a reduction of output. Because the shock is temporary, the only effect on desired saving or investment is due to the reduction in current output, causing desired national saving to fall. This shifts the saving curve to the left, raising the real interest rate and reducing the level of desired investment, as well as desired national saving, as shown in Figure 4.11.

Figure 4.11 (b) The permanent increase in the price of oil reduces the marginal product of labor, causing the labor demand curve to shift to the left, again as in Figure 4.10. (Also, the decline in future income means the labor-supply curve will shift to the right; but we’ll assume that this shift is less than the shift to the left of the labor-demand curve.) At equilibrium, there is a reduced real wage and lower employment. The productivity shock results in a reduction of current output. Because the shock is permanent, it reduces future output as well, and reduces the future marginal product of capital. The desired investment curve shifts to the left, from I1 to I2 in Figure 4.12, because the future marginal product of capital is lower. The effect on desired saving is ambiguous—the reduction in current income reduces desired saving, but the reduction in expected future income increases desired saving. Let’s assume that the former effect outweighs the latter, so that the desired saving curve shifts to the left from S1 to S2. Then national saving and investment both decline. Again, the effect on the real interest rate is ambiguous. (Alternatively, if the effects on desired saving of the reductions in current income and future income offset each other exactly, the desired saving curve does not shift. In this case, the leftward shift of the investment curve along an unchanged saving curve reduces the real interest rate, saving, and investment.)

Figure 4.12

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Abel/Bernanke/Croushore • Macroeconomics, Eighth Edition

A temporary increase in government spending reduces national saving. Whether the spending is financed by current taxes or by borrowing (and raising future taxes), consumption falls, but not by the full amount of the spending. Since S = Y − Cd − G, national saving declines. This is shown in Figure 4.13 as a shift to the left in the saving curve. The real interest rate must increase to get S = I, so I declines as well. It makes no difference whether the temporary increase in spending is funded by taxes or by borrowing.

Figure 4.13 In the case of infrastructure spending, MPK f rises, so investment increases. Saving shifts from S1 to S2 and investment shifts from I1 to I2 in Figure 4.14. With upward shifts in both saving and investment, the new equilibrium is one with a higher real interest rate. However, saving and investment at the new equilibrium may be higher or lower. The effect on consumption is unclear as well. The higher real interest rate reduces consumption, but future income is higher, which increases consumption. If investment actually rises, then the increase in government spending causes private investment to be “crowded in” rather than “crowded out.” In this case consumption is crowded out.

Figure 4.14 5.

When there is a temporary increase in government spending, consumers foresee future taxes. As a result, consumption declines, both currently and in the future. Thus current consumption does not fall

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by as much as the increase in G, so national saving (Sd = Y − Cd − G) declines at the initial real interest rate, and the saving curve shifts to the left from S1 to S2, as shown in Figure 4.15. Thus the real interest rate increases and consumption and investment both fall.

Figure 4.15 When there is a permanent increase in government spending, consumers foresee future taxes as well, with both current and future consumption declining. But if there is an equal increase in current and future government spending, and consumers try to smooth consumption, they will reduce their current and future consumption by about the same amount, and that amount will be about the same amount as the increase in government spending. So the saving curve in the saving-investment diagram does not shift, and there is no change in the real interest rate. Since the saving curve shifts upward more in the case of a temporary increase in government spending, the real interest rate is higher, so investment declines by more. However, consumption falls by more in the case of a permanent increase in government spending. 6.

See Figure 4.16. The consumer is originally on budget line BL1, with consumption at point D. An increase in the real interest rate shifts the budget line to BL2, with consumption at point Q. The change can be broken down into two steps. First, the substitution effect shifts the budget line from BL1 to BLint, and the consumption point changes from point D to point P. The substitution effect results in higher future consumption and lower current consumption. The income effect shifts the budget line from BLint to BL2, with the consumption point changing from point P to point Q. The income effect results in lower current and future consumption. Thus the income and substitution effects work in the same direction, reducing current consumption and increasing saving.

Figure 4.16

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The difference in interest rates between borrowing and lending means there is a kink in the budget constraint at the no-lending, no-borrowing point, as shown in Figure 4.17. Borrowing is zero when c = y + a. If current consumption is less than y + a, the person is a saver (lender), and the budget line has slope − (1 + rl). If current consumption is greater than y + a, the person is a borrower, and faces a steeper budget constraint with slope − (1 + rb), because the interest rate is higher.

Figure 4.17 An increase in either interest rate would steepen only the portion of the budget constraint for which that interest rate is relevant. An increase in the real interest rate on lending is shown as a shift in the budget line segment from BL1 to BL2 in Figure 4.18. An increase in the real interest rate on borrowing is shown as a shift in the budget line segment from BL3 to BL4. If the indifference curve hits the budget line at the no-borrowing, no-lending point, as shown, then there will be no change in current or future consumption due to a change in either interest rate.

Figure 4.18

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An increase in the consumer’s initial wealth would lead to a parallel rightward shift of both segments of the budget line, as shown in Figure 4.19.

Figure 4.19

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