This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 chainweighting 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—rateof 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 chainweighted 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 ...
View
Full Document
 Spring '11
 Kambourov
 Macroeconomics

Click to edit the document details