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problem_set1_answers - University of Toronto Mississauga...

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Unformatted text preview: University of Toronto Mississauga Department of Economics ECO 209Y — Macroeconomic Theory and Policy Prof. Kambourov Problem Set 1 Deadline: Friday, October 1, 2010. 1. In year 1 and year 2, there are two products produced in a given economy, computers and bread. Suppose that there are no intermediate goods. In year 1, 20 computers are produced and sold at $1,000 each, and in year 2, 25 computers are sold at $1,500 each. In year 1, 10,000 loaves of bread are sold for $1.00 each, and in year 2, 12,000 loaves of bread are sold for $1.10 each. (a) Calculate nominal GDP in each year. (b) Calculate real GDP in each year, and the percentage increase in real GDP from year 1 to year 2 using year 1 as the base year. Next, do the same calculations using the chain—weighting method. (0) Calculate the implicit GDP price deflator and the percentage inflation rate from year Ito year 2 using year 1 as the base year. Next, do the same calculations using the chain-weighting method. (d) Suppose that computers in year 2 are twice as productive as computers in year 1. How does this change your calculations in parts (a)—(c)? Explain any differences. Williamson, Third Canadian Edition, Chapter 4, Problem 1. Use a diagram to show that if the consumer prefers more to less, then indifference curves cannot cross. Williamson, Third Canadian Edition, Chapter 4, Problem 2. Suppose that leisure and consumption goods are perfect substitutes. In this case, an indifference curve is described by the equation u = al + bC, _ (1) where a and b are positive constants, and u is the level of utility. That is, a given indifference curve has a particular value for u, with higher indifference curves having higher values for u. (a) Show what the consumer’s indifference curves look like when consumption and leisure are perfect substitutes, and determine graphically and algebraically what consumption bundle the consumer will choose. Show that the consump— tion bundle the consumer chooses depends on the relationship between % and w, and explain why. (b) Do you think it likely that any consumer would treat consumption goods and leisure as perfect substitutes? (0) Given perfect substitutes, is more preferred to less? Do preferences satisfy the diminishing—rate-of substitution property? 4. Williamson, Third Canadian Edition, Chapter 4, Problem 4. Suppose that the government imposes a proportional income tax on the representative consumer’s wage income. That is, the consumers wage income is w(1 — t)(h —— l), where t is the tax rate. What effect does the income tax have on consumption and labor supply? Explain your results in terms of income and substitution effects. 5. Williamson, Third Canadian Edition, Chapter 4, Problem 11. Suppose that the government subsidizes employment. That is, the government pays the firm sunits of consumption goods for each unit of labor that the firm hires. Determine the effect of the subsidy on the firm’s demand for labor. 6. Consider a representative firm with the following technology: Y = zKO‘Nl"a, (2) where K = capital and N : labor. Assume the firm behaves competitively, that is it takes prices as given. (a) Show that the marginal product of labor is equal to Y MPN: (1—a)fi. (3) (b) Find the optimal demand for labor N if the stock of capital available to the firm is exogenously given and the firm takes as given the price of output p and the price of labor w. (c) What happens to N if 2, K, or w increase? University of Toronto Mississauga Department of Economics ECO 209Y — Macroeconomic Theory and Policy Prof. Kambourov Problem Set 1 Suggested Answers 1. Question 1 a) Year 1 nominal GDP : 20($1,000) + 10,000($1.00) = $30,000. Year 2 nominal GDP = 25($1,500) + 12,000($1.10) : $50,700. b) With year 1 as the base year, we need to value both years7 production at year 1 prices. In the base year, year 1, real GDP equals nominal GDP equals $30,000. In year 2, we need to value year 2’s output at year 1 prices. Year 2 real GDP 2 25($1,000) + 12,000($1.00) = $37,000 . The percentage change in real GDP equals ($37,000 — $30,000)/$30,000 : 22.3%. We next calculate chain—weighted real GDP. To perform this calculation, we first compute average prices. The average price for computers equals ($1,000 + $1,500) / 2 = $1,250. The average price for bread equals ($1.00 + $1.10)/2 = $1.05. Year 1 output valued at average prices equals 20($1,250) + 10,000($1.05) = $35,500. Year 2 output valued at average prices equals 25($1,250) + 12,000($1.05) : $43,850. The percentage change in chain—weighted GDP is therefore equal to ($43,850—$35,500) /$35,500 = 23.5%. c) To calculate the implicit GDP deflator, we divide nominal GDP by real GDP, and then multiply by 100 to express it as an index number. With year 1 as the base year, base year nominal GDP equals base year real GDP, so the base year implicit GDP deflator is 100. For the year 2, the implicit GDP deflator is ($50,700/$37,000)X100 = 137.0. The percentage change in the deflator is equal to 37.0%. With chain weighting, the base year is now the midpoint between the two years. The year 1 GDP deflator equals ($30,000/$35,500)X100 : 84.5. The year 2 GDP defiator equals ($50,700/$43,850)X100 = 115.6. The percentage change in the chain— weighted deflator equals (115.6—84.5)/84.5 : 36.8%. d) We next consider the possibility that year 2 computers are twice as productive as year 1 computers. As one possibility, let us define a ’computer’ as a year 1 computer. In this case, the 25 computers produced in year 2 are the equivalent of 50 year 1 computers. Year 2 real GDP, in year 1 prices is now 50($1,000) + 12,000($1.00) = $62,000. The percentage change in real GDP is equal to ($62,000-$30,000)/$30,000 = 106.7%. We next revise the value of output at average prices. Chain—weighted year 1 real GDP is equal to 20($875) + 10,000($1.05) : $28,000. Chain—weighted year 2 real GDP is equal to 50($875) + 12,000($1.05) = $56,350. The percentage change in chain—weighted real GDP is therefore equal to ($56,350—$28,000)/$28,000 : 101.25%. With year 1 as the base year, the year 2 real GDP deflator equals ($50,700/$62,000)x100 : 81.8. The percentage rate of change of the deflator equals 18.2%. The chain-weighted deflator for year 1 is now equal to ($30,000/$28,000)x100 : 107.1. The chain—weighted defiator for year 2 is now equal to ($50,700/$56,350)x100 : 89.97. The percentage change in the GDP deflator is equal to (89.97v107.1)/107.1 : ~15.9%. When there is no quality change. the difference between using year 1 as the base year and using chain weighting is relatively small. In the case of increased quality, the production of computers rises dramatically while its relative price falls. Chain weighting provides a smaller estimate of the increase in production and a smaller estimate of the reduction in prices. This difference is due to the fact that the relative price of the good that increases most in quantity (computers) is much higher in year 1. Therefore, the use of historical prices puts more weight on the increase in quality— adjusted computer output. 2. Question 6 (a) The marginal product of labor (MPN) is equal to MPN = 13% 2 (1 - a)zKo‘N"o‘ = (1 w a)zK"‘N‘O‘ ~% = (1 —— 05)};— (b) The firm chooses N to maximize profits. This optimization problem is given by Max{szaN1_“ — wN}. The first order conditions for a maximum imply:(1 — oz) pZKaN‘O‘ — w = 0. Solving for N we obtain N : [(1 ~ 04)sz11/0‘ : [(1— a)pz]l/aK w w (c) Given that or > 0, the above formula implies that N is an increasing function of z and a decreasing function of 'w. Moreover, N is an increasing function of K. Are these results intuitive? 1. Consider the two hypothetical indifference curves in Figure 4.1. Point A is on both indifference curves, 1; and 12, By construction, the consumer is indifferent between A and B, as both points are on 12, In like fashion, the consumer is indifferent between A and C, as both points are on I 1. But at point C, the consumer has more consumption and more leisure than at point B. As long as the consumer prefers more to less, he or she must strictly prefer C to A. We therefore contradict the hypothesis that two indifference curves can cross. C Figure 4.1 2. u = a] + bC a.) To specify an indifference curve, we hold utility constant at 71. Next rearrange in the form: C = 1 _ fl 1 b b Indifference curves are therefore linear with slope, -a/b, which represents the marginal rate of substitution. There are two main cases, according to whether 2 > M or g < M. The top panel of Figure 4.2 shows the case of ~21 < M. In this [9 case the indifference curves are flatter than the budget line and the consumer picks point A, at which I = 0 and C = wh+ 7r~T . The bottom panel of Figure 4.2 shows the case of g > M. In this case the indifference curves are steeper than the budget line, and the consumer picks point B, at which I = h and C = 1r -T . In the coincidental case in which % = M, the highest attainable indifference curve coincides with the indifference curve, and the consumer is indifferent among all possible amounts of leisure and hours worked. I Figure 4.2 b.) The utility function in this problem does not obey the property that the consumer prefers diversity, and is therefore not a likely possibility. c.) This utility function does have the property that more is preferred to less. However, the marginal rate of substitution is constant, and therefore this utility function does not satisfy the property of diminishing marginal rate of substitution. 4. When the government imposes a proportional tax on wage income, the consumer’s budget constraint is now given by: C=w(l—t)(h—l)+7t—T , where t is the tax rate on wage income. In Figure 4.3, the budget constraint for t = 0, is FGH. When t> 0, the budget constraint is EGH. The slope of the original budget line is —w, while the slope of the new budget line is —(1—t)w. Initially the consumer picks the point A on the original budget line. After the tax has been imposed, the consumer picks point B. The substitution effect of the imposition of the tax is to move the consumer from point A to point D on the original indifference curve. The point D is at the tangent point of indifference curve, 11, with a line segment that is parallel to EG. The pure substitution effect induces the consumer to reduce consumption and increase leisure (work less). The tax also makes the consumer worse off, in that he or she can no longer be on indifference curve, 11, but must move to the less preferred indifference curve, 12. This pure income effect moves the consumer to point B, which has less consumption and less leisure than point D, because both consumption and leisure are normal goods. The net effect of the tax is to reduce consumption, but the direction of the net effect on leisure is ambiguous. Figure 4.3 shows the case in which the substitution effect on leisure dominates the income effect. In this case, leisure increases and hours worked fall. Although consumption must fall, hours worked may rise, fall, or remain the same. Figure 4.3 11. The firm chooses its labor input Nd so as to maximize profits. When there is no subsidy, profits for the firm are given by n:zF(K,N“)—de That is, profits are the difference between revenue and costs. In the top panel in Figure 4.7 the revenue function is ZF ( K 7 N d) and the cost function is the straight line, de. The firm maximizes profits by choosing the quantity of labor where the slope of the revenue function equals the slope of the cost function: MPNzw. The firm’s demand for labor curve is the marginal product of labor schedule in the bottom panel of Figure 4.7. With an employment subsidy, the firm’s profits are given by: n :zF(K,Nd)—(w—S)Nd where the term 2F ( K , N d) is the unchanged revenue function, and (w—s)Nd is the cost function. The subsidy acts to reduce the cost of each unit of labor by the amount of the subsidy, s. In the top panel of Figure 4.7, the subsidy acts to shift down the cost function for the firm by reducing its slope. As before, the firm will maximize profits by choosing the quantity of labor input where the slope of the revenue function is equal to the slope of the cost function, (t—s), so the firm chooses the quantity of labor where MPN = w — s . In the bottom panel of Figure 4.7, the labor demand curve is now MPN + s, and the labor demand curve has shifted up. The subsidy acts to reduce the marginal cost of labor, and the firm will hire more labor at any given real wage. Revenues, Costs d WN (w—s)N"’ zF(K,N") MPN +5 MPN N0 Figure 4.7 ...
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