Summer 2014 Week 4 Tutorial Questions (Ch3) Solutions
Chapter 3:
Q1: Macroeconomics P101 Numerical Problems #6
Q2: Macroeconomics P.101 Numerical Problems #7
Q3: Macroeconomics P.101 Numerical Problems #8
Q4: Macroeconomics P.102 Analytical Problems #2
Q5: Macroeconomics P.102 Analytical Problems #4
Q1: Suppose that the production function is Y = 9K
0.5
N
0.5
. With this production function, the marginal product of labour is MPN = 4.5 K
0.5
N
-0.5
. The capital stock is K = 25. The labour supply curve is NS = 100[(1- t) ω]
2 , where ω is the real wage rate, t is the tax rate on labour income, and hence (1 – t)ω is the after tax real wage rate.
Answer: Since ω = 4.5 K
0.5
25, N =506.25/ω
2
.
N
−0.5
, N
0.5
= 4.5 K
0.5
/ω, so N = 20.25 K/ω
2
. When K = a.
Assume that the tax rate on labour income, t, equals zero. Find the equation of the labour demand curve. Calculate the equilibrium levels of the real wage and employment, the level of full-employment output, and the total after-tax wag income of workers.
Answer: If t = 0, then NS = 100ω
2
.
Setting labour demand equal to labour supply gives 506.25/ω
2
= 5.0625, or ω = 1.5. Then NS = 100 (1.5)
2
= 225.
= 100ω
2
, so ω
4
Y = 9K
0.5
N
0.5
= 45N
0.5
= 45(225)
0.5
= 675.
The total after-tax wage income of workers is (1−t) ω NS = 1.5 x 225 = 337.5. b.
Repeat part (a) under the assumption that the tax rate on labour income, t, equals 0.6.
Answer: The marginal product of labour is MPN = 4.5 K
0.5
N
-0.5
Equate NS = ND, so N = 100 [(1 − 0.6) x 22.5 / N
0.5
]
= 22.5 / N
2
, so N
0.5
= ω.
2
= 8100, so N
=90.
Then Y = 9K
0.5
N
0.5
=45N
0.5
= 45(90)
0.5
= 426.91.
Then ω= 22.5 / 90
0.5
= 2.37. The total after-tax wage income of workers is
(1−t) ω NS = 0.4 x 2.37 x 90 =85.38. Note that there is a big decline in output and income, although the wage is higher.
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Summer 2014 Week 4 Tutorial Questions (Ch3) Solutions c.
Suppose that a minimum wage of ω = 2 is imposed. If the tax rate on labour income, t, equals zero, what are the resulting values of employment and the real wage? Does the introduction of the minimum wage increase the total income of workers, taken as a group?
Answer: A minimum wage of 2 is binding if the tax rate is zero. Then N = 506.25/2
2
=126.6, NS = 100 x 2
2
= 400. Unemployment is 273.4. Income of workers is
ωN = 2 x 126.6 = 253.2, which is lower than without a minimum wage, because employment has declined so much.
Q2: Consider an economy with 500 people in the labour force. At the beginning of every month, five people lose their jobs and remain unemployed for exactly one month; one month later, they find new jobs and become employed. In addition, on
January 1 of each year, 20 people lose their jobs and remain unemployed for six months before finding new jobs. Finally, on July 1 of each year, 20 people lose their jobs and remain unemployed for six months before finding new jobs. a.
What is the unemployment rate in this economy in a typical month?
Answer: At any date, 25 people are unemployed: 5 who have lost their jobs at the start of the month and 20 who have lost their jobs either on January 1 or July 1.
The unemployment rate is 25 / 500 = 5%. b.
What fraction of unemployment spells lasts for one month? What fraction lasts for six months?
Answer: Each month, 5 people have one-month spells. Every six months, 20 people have six-month spells. The total number of spells during the year is (5 × 12) +
(20 × 2) = 100. Sixty of the spells (60% of all spells) last one month, while 40 of the spells (40% of all spells) last six months. c.
What is the average duration of an unemployment spell?
Answer: The average duration of a spell is (0.60 × 1 month) + (0.40 × 6 months) = 3 months. d.
On any particular date, what fraction of the unemployed is suffering a long spell (six months) of unemployment?
Answer: On any given date, there are 25 people unemployed. Twenty of them (80%) have long spells of unemployment, while 5 of them (20%) have short spells.
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Summer 2014 Week 4 Tutorial Questions (Ch3) Solutions
Q3:
Assume Okun’s law holds exactly for Canada and that for the past 3 years,
Canada’s rate of real output growth has averaged 4% per year. What has happened to the unemployment rate during that period?
Answer: The unemployment rate has fallen by point five percentage points over that period.
Q4: How would each of the following affect the current level of full-employment output? Explain. a.
A large number of immigrants enter the country.
Answer: An increase in the number of immigrants increases the labour force, increasing employment and increasing full-employment output. b.
Energy supplies become depleted.
Answer: If energy supplies become depleted, this is likely to reduce productivity, because energy is a factor of production. So the reduction in energy supplies reduces full-employment output . c.
New teaching techniques improve the educational performance of high school students.
Answer: Better education raises future productivity and output, but has no effect on current full-employment output. d.
A new law mandates the shutdown of some unsafe forms of capital.
Answer: This reduction in the capital stock reduces full-employment output (although it may very well increase welfare).
Q5:
How would each of the following affect Helena Handbasket’s supply of labour? a.
The value of Helena’s home triples in an unexpectedly hot real estate market.
Answer: The increased value of Helena's home increases her wealth. The rise in wealth leads to an income effect that leads Helena to reduce her labour supply. b.
Originally an unskilled worker, Helena acquires skills that give her access to higher-paying job. Assume that her preferences about leisure are not affected by the change in jobs.
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Summer 2014 Week 4 Tutorial Questions (Ch3) Solutions
Answer: The permanent rise in Helena's real wage gives rise to offsetting income and substitution effects. The income effect of the higher wage reduces Helena's labour supply, but the substitution effect increases it. So the result is theoretically ambiguous. Empirically, women tend to increase labour supply in response to a permanent increase in the real wage, and men tend to reduce labour supply in response to a permanent increase in the real wage. c.
A temporary income tax surcharge raises the percentage of her income that she must pay in taxes, for the current year only. (Taxes are proportional to income in Helena’s country.)
Answer: The temporary income tax surcharge is equivalent to a temporary reduction in the real wage, which reduces current labour supply, assuming that the income effect is smaller than the substitution effect. (“Ambiguous” is also acceptable.)
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