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第2章——技术题 3.a. Since depreciation D = Ig - In = 800 - 200 = 600 ==> NDP = GDP - D = 6,000 - 600 = 5,400 3.b. From GDP = C + I + G + NX ==> NX = GDP - C - I - G ==> NX = 6,000 - 4,000 - 800 - 1,100 = 100. 3.c. BS = TA - G - TR ==> (TA - TR) = BS + G ==> (TA - TR) = 30 + 1,100 = 1,130 3.d. YD = Y - (TA - TR) = 5,400 - 1,130 = 4,270 3.e. S = YD - C = 4,270 - 4,000 = 270 4.a. S = YD - C = 5,100 - 3,800 = 1,300 4.b. From S - I = (G + TR - TA) + NX ==> I = S - (G + TR - TA) - NX = 1,300 - 200 - (-100) = 1,200. 4.c. From Y = C + I + G + NX ==> G = Y - C - I - NX ==> G = 6,000 - 3,800 - 1,200 - (-100) = 1,100. Also: YD = Y - TA + TR ==> TA - TR = Y - YD = 6,000 - 5,100 ==> TA - TR = 900 From BS = TA - TR - G ==> G = (TA - TR) - BS = 900 - (-200) ==> G = 1,100 第三章——技术题 1.a. According to Equation (2), the growth of output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the rate of technical progress, that is, DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A, where 1 - q is the share of labor (N) and q is the share of capital (K). Thus if we assume that the rate of technological progress (DA/A) is zero, then output grows at an annual rate of 3.6 percent, since DY/Y = (0.6)(2%) + (0.4)(6%) + 0% = 1.2% + 2.4% = + 3.6%, 1.b. The so-called "Rule of 70" suggests that the length of time it takes for output to double can be calculated by dividing 70 by the growth rate of output. Since 70/3.6 = 19.44, it will take just under 20 years for output to double at an annual growth rate of 3.6%, 1.c. Now that DA/A = 2%, we can calculate economic growth as DY/Y = (0.6)(2%) + (0.4)(6%) + 2% = 1.2% + 2.4% + 2% = + 5.6%. Thus it will take 70/5.6 = 12.5 years for output to double at this new growth rate of 5.6%. 2.a. According to Equation (2), the growth of output is equal to the growth in labor times the labor share plus the growth of capital times the capital share plus the growth rate of total factor productivity (TFP), that is, DY/Y = (1 - q)(DN/N) + q(DK/K) + DA/A, where 1 - q is the share of labor (N) and q is the share of capital (K). In this example q = 0.3; therefore, if output grows at 3% and labor and capital grow at 1% each, then we can calculate the change in TFP in the following way 3% = (0.3)(1%) + (0.7)(1%) + DA/A ==> DA/A = 3% - 1% = 2%, that is, the growth rate of total factor productivity is 2%. 2.b. If both labor and the capital stock are fixed and output grows at 3%, then all this growth has to be contributed to the growth in factor productivity, that is, DA/A = 3%. 3.a. If the capital stock grows by DK/K = 10%, the effect on output would be an additional growth rate of DY/Y = (.3)(10%) = 3%. 3.b. If labor grows by DN/N = 10%, the effect on output would be an additional growth rate of DY/Y = (.7)(10%) = 7%. 3.c. If output grows at DY/Y = 7% due to an increase in labor by DN/N = 10%, and this increase in labor is entirely due to population growth, then per capita income would decrease and people’s welfare would decrease, since Dy/y = DY/Y - DN/N = 7% - 10% = - 3%. 3.d. If this increase in labor is due to an influx of women into the labor force, the overall population does not increase and income per capita would increase by Dy/y = 7%. Therefore people's welfare would increase. 6.a. Assume the production function is of the form Y = F(K, N, Z) = AKaNbZc ==> DY/Y = DA/A + a(DK/K) + b(DN/N) + c(DZ/Z), with a + b + c = 1. Now assume that there is no technological progress, that is, DA/A = 0, and that capital and labor grow at the same rate, that is, DK/K = DN/N = n. If we also assume that all natural resources available are fixed, such that DZ/Z = 0, then the rate of output growth will be DY/Y = an + bn = (a + b)n. In other words, output will grow at a rate less than n since a + b < 1. Therefore output per worker will fall. 6.b. If there is technological progress, that is, DA/A > 0, then output will grow faster than before, namely DY/Y = DA/A + (a + b)n. If DA/A > c, then output will grow at a rate larger than n, in which case output per worker will increase. 6.c. If the supply of natural resources is fixed, then output can only grow at a rate that is smaller than the rate of population growth and we should expect limits to growth as we run out of natural resources. However, if the rate of technological progress is sufficiently large, then output can grow at a rate faster than population, even if we have a fixed supply of natural resources. 7.a. If the production function is of the form Y = K1/2(AN)1/2, and A is normalized to 1, then we have Y = K1/2N1/2 . In this case capital's and labor's shares of income are both 50%. 7.b. This is a Cobb-Douglas production function. 7.c. A steady-state equilibrium is reached when sy = (n + d)k. From Y = K1/2N1/2 ==> Y/N = K1/2N-1/2 ==> y = k1/2 ==> sk1/2 = (n + d)k ==> k-1/2 = (n + d)/s = (0.07 + 0.03)/(.2) = 1/2 ==> k1/2 = 2 = y ==> k = 4 . 第八章——技术题 1. If actual GDP is expected to be $40 billion below the full-employment level and the size of the government spending multiplier is 2, then government spending should be increased by $20 billion over its current level. For the next period, when actual GDP is expected to be $20 billion below potential, government spending should be cut by $10 billion from its new level, that is, to $10 billion over its original level. In period three, when actual GDP is expected to be at its full-employment level, the level of government spending should again be cut by $10 billion from the last period's level to bring it back to the original level of Period 0. 第九章——技术题 1.a. AD = C + I = 100 + (0.8)Y + 50 = 150 + (0.8)Y The equilibrium condition is Y = AD ==> Y = 150 + (0.8)Y ==> (0.2)Y = 150 ==> Y = 5*150 = 750. 1.b. Since TA = TR = 0, it follows that S = YD - C = Y - C. Therefore S = Y - [100 + (0.8)Y] = - 100 + (0.2)Y ==> S = - 100 + (0.2)750 = - 100 + 150 = 50. 1.c. If the level of output is Y = 800, then AD = 150 + (0.8)800 = 150 + 640 = 790. Therefore the amount of involuntary inventory accumulation is UI = Y - AD = 800 - 790 = 10. 1.d. AD' = C + I' = 100 + (0.8)Y + 100 = 200 + (0.8)Y From Y = AD' ==> Y = 200 + (0.8)Y ==> (0.2)Y = 200 ==> Y = 5*200 = 1,000 Note: This result can also be achieved by using the multiplier formula: DY = (multiplier)(DSp) = (multiplier)(DI) ==> DY = 5*50 = 250, that is, output increases from Yo = 750 to Y1 = 1,000. 1.e. From 1.a. and 1.d. we can see that the multiplier is 5. 1.f. Sp Y = Sp AD1 = 200 = (0.8)Y ADo = 150 + (0.8)Y 200 150 0 750 1,000 Y 2.a. Since the mpc has increased from 0.8 to 0.9, the size of the multiplier is now larger and we should therefore expect a higher equilibrium income level than in 1.a. AD = C + I = 100 + (0.9)Y + 50 = 150 + (0.9)Y ==> Y = AD ==> Y = 150 + (0.9)Y ==> (0.1)Y = 150 ==> Y = 10*150 = 1,500. 2.b. From DY = (multiplier)(DI) = 10*50 = 500 ==> Y1 = Yo + DY = 1,500 + 500 = 2,000. 2.c. Since the size of the multiplier has doubled from 5 to 10, the change in output (Y) that results from a change in investment (I) now has also doubled from 250 to 500. 2.d. Sp Y = Sp AD1 = 200 = (0.9)Y ADo = 150 + (0.9)Y 200 150 0 1,500 2,000 Y 3.a. AD = C + I + G + NX = 50 + (0.8)YD + 70 + 200 = 320 + (0.8)[Y - (0.2)Y + 100] = 400 + (0.8)(0.8)Y = 400 + (0.64)Y From Y = AD ==> Y = 400 + (0.64)Y ==> (0.36)Y = 400 ==> Y = (1/0.36)400 = (2.78)400 = 1,111.11 The size of the multiplier is (1/0.36) = 2.78. 3.b. BS = tY - TR - G = (0.2)(1,111.11) - 100 - 200 = 222.22 - 300 = - 77.78 3.c. AD' = 320 + (0.8)[Y - (0.25)Y + 100] = 400 + (0.8)(0.75)Y = 400 + (0.6)Y From Y = AD' ==> Y = 400 + (0.6)Y ==> (0.4)Y = 400 ==> Y = (2.5)400 = 1,000 The size of the multiplier is now reduced to 2.5. 3.d. BS' = (0.25)(1,000) - 100 - 200 = - 50 BS' - BS = - 50 - (-77.78) = + 27.78 The size of the multiplier and equilibrium output will both increase with an increase in the marginal propensity to consume. Therefore income tax revenue will also go up and the budget surplus should increase. 3.e. If the income tax rate is t = 1, then all income is taxed. There is no induced spending and equilibrium income only increases by the change in autonomous spending, that is, the size of the multiplier is 1. From Y = C + I + G ==> Y = Co + c(Y - 1Y + TRo) + Io + Go ==> Y = Co + cTRo + Io + Go = Ao 4. In Problem 3.d. we had a situation where the following was given: Y = 1,000, t = 0.25, G = 200 and BS = - 50. Assume now that t = 0.3 and G = 250 ==> AD' = 50 + (0.8)[Y - (0.3)Y + 100] + 70 + 250 = 370 + (0.8)(0.7)Y + 80 = 450 + (0.56)Y. From Y = AD' ==> Y = 450 + (0.56)Y ==> (0.44)Y = 450 ==> Y = (1/0.44)450 = 1,022.73 BS' = (0.3)(1,022.73) - 100 - 250 = 306.82 - 350 = - 43.18 BS' - BS = -43.18 - (-50) = + 6.82 The budget surplus has increased, since the increase in tax revenue is larger than the increase in government purchases. 5.a. While an increase in government purchases by DG = 10 will change intended spending by DSp = 10, a decrease in government transfers by DTR = -10 will change intended spending by a smaller amount, that is, by only DSp = c(DTR) = c(-10). The change in intended spending equals DSp = (1 - c)(10) and equilibrium income should therefore increase by DY = (multiplier)(1 - c)10. 5.b. If c = 0.8 and t = 0.25, then the size of the multiplier is a = 1/[1 - c(1 - t)] = 1/[1 - (0.8)(1 - 0.25)] = 1/[1 - (0.6)] = 1/(0.4) = 2.5. The change in equilibrium income is DY = a(DAo) = a[DG + c(DTR)] = (2.5)[10 + (0.8)(-10)] = (2.5)2 = 5 5.c. DBS = t(DY) - DTR - DG = (0.25)(5) - (-10) - 10 = 1.25 第十章——技术题 1.a. Each point on the IS-curve represents an equilibrium in the expenditure sector. Therefore the IS-curve can be derived by setting Y = C + I + G = (0.8)[1 - (0.25)]Y + 900 - 50i + 800 = 1,700 + (0.6)Y - 50i ==> (0.4)Y = 1,700 - 50i ==> Y = (2.5)(1,700 - 50i) ==> Y = 4,250 - 125i. 1.b. The IS-curve shows all combinations of the interest rate and the level of output such that the expenditure sector (the goods market) is in equilibrium, that is, intended spending is equal to actual output. A decrease in the interest rate stimulates investment spending, making intended spending greater than actual output. The resulting unintended inventory decrease leads firms to increase their production to the point where actual output is again equal to intended spending. This means that the IS-curve is downward sloping. 1.c. Each point on the LM-curve represents an equilibrium in the money sector. Therefore the LM-curve can be derived by setting real money supply equal to real money demand, that is, M/P = L ==> 500 = (0.25)Y - 62.5i ==> Y = 4(500 + 62.5i) ==> Y = 2,000 + 250i. 1.d. The LM-curve shows all combinations of the interest rate and level of output such that the money sector is in equilibrium, that is, the demand for real money balances is equal to the supply of real money balances. An increase in income will increase the demand for real money balances. Given a fixed real money supply, this will lead to an increase in interest rates, which will then reduce the quantity of real money balances demanded until the money market clears. In other words, the LM-curve is upward sloping. 1.e. The level of income (Y) and the interest rate (i) at the equilibrium are determined by the intersection of the IS-curve with the LM-curve. At this point, the expenditure sector and the money sector are both in equilibrium simultaneously. From IS = LM ==> 4,250 - 125i = 2,000 + 250i ==> 2,250 = 375I ==> i = 6 ==> Y = 4,250 - 125*6 = 4,250 - 750 ==> Y = 3,500 Check: Y = 2,000 + 250*6 = 2,000 + 1,500 = 3,500 第十三章——技术题 1.a. If income remains constant over time, permanent income equals current income. Your permanent income this year is YP0 = (1/5)(5*20,000) = 20,000. 1.b. Your permanent income next year is YP1 = (1/5)(15,000 + 4*20,000) = 19,000. 1.c. Since C = 0.9YP, your consumption this year is C0 = 0.9*20,000 = 18,000. Your consumption next year is C1 = 0.9*19,000 = 17,100. 1.d. In the short run, the mpc = (0.9)(1/5) = 0.18; but in the long run, the mpc = 0.9. 1.e. We have already calculated this and next year's permanent income. In each of the coming years you add $30,000 and subtract $20,000, and therefore your permanent income (which is your average over a five year period) will increase by $2,000 each year until it reaches $30,000 after 5 years. YPo = (1/5)(5*20,000) = 20,000 YP1 = (1/5)(1*30,000 + 4*20,000) = 22,000 YP2 = (1/5)(2*30,000 + 3*20,000) = 24,000 YP3 = (1/5)(3*30,000 + 2*20,000) = 26,000 YP4 = (1/5)(4*30,000 + 1*20,000) = 28,000 YP5 = (1/5)(5*30,000) = 30,000 Y 30,000 28,000 26,000 24,000 22,000 20,000 0 1 2 3 4 5 time 第十五章——技术题 4.a. The person tries to minimize the cost of managing the portfolio. This cost is: C = ntc + (iY/2n) n = 1 ==> C = 1*1 + (0.005)(1,600)/(2*1) = 1 + 4 = 5 n = 2 ==> C = 2*1 + (0.005)(1,600)/(2*2) = 2 + 2 = 4 n = 3 ==> C = 3*1 + (0.005)(1,600)/(2*3) = 3 + 4/3 = 4.34 n = 4 ==> C = 4*1 + (0.005)(1,600)/(2*4) = 4 + 1 = 5 Therefore 2 transactions are optimal. 4.b. With two transactions, the optimal cash holding is: md = Y/(2n) = 1,600/(2*2) = $400. 4.c. n = 1 ==> C = 1*1 + (0.005)(1,800)/(2*1) = 1 + 4.5 = 5.5 n = 2 ==> C = 2*1 + (0.005)(1,800)/(2*2) = 2 + 2.25 = 4.25 n = 3 ==> C = 3*1 + (0.005)(1,800)/(2*3) = 3 + 1.5 = 4.5 n = 4 ==> C = 4*1 + (0.005)(1,800)/(2*4) = 4 + 1.125 = 5.125 Therefore it is still optimal to make two transactions, but the optimal cash holding is now md = 1,800/(2*2) = $450. This is a 12.5% increase, since (450 - 400)/400 = .125.