Spring 2022-2023
METU, Department of Economics
ECON 204 - Problem Set 1
1.
Jonas’ utility for daily leisure hours (l) and consumption (c) is u(l, c) = l · c; she can work for a wage
rate of $8 per hour and labor is her only source of income. Suppose that the price of the consumption per
unit is $1.
2.
(a)
Write Jonas’ daily budget constraint assuming there are only 24 hours in a day.
(b)
Find Jonas’ demand for leisure and consumption. What is her labor supply hours?
(c)
Suppose that wage rate raises to $10. How does wage rate increase affect her labor supply amount?
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Consider Noah’s preference for leisure (l) and consumption goods (c), u(l, c) = l 2 + c 2 . Assume that
Noah’s only source of income is labor, and the price for consumption good is $1 per unit. Suppose the
maximum number of available hours is l. Is Noah’s labor supply backward bending? That is, after a certain
level of wage rate, does his labor supply start to decrease as wage is getting higher?
3.
Ulrich’s utility function over leisure (l) and consumption goods (c) is u(l, c) = c + lc. Assume that
Ulrich’s only source of income is labor. Find the optimal working hours and show that, no matter what the
wage rate is, the optimal number of labor hours is always the same.
4.
Marta’s utility function is u(l, c) = c − (12 − l)2 , where l is the amount of leisure she gets per day, and
she has 16 hours a day to divide between labor and leisure. She has an income of $m a day from non-labor
source. The price of consumption goods is $1 per unit.
(a)
If wage rate is zero, how many hours of leisure will she choose? How many hours will she work?
(b)
If wage rate is $10 per hour, how many hours of leisure will she choose? How many hours will she
work?
(c)
If Marta’s non-labor income decreased to $ m
2 a day while wage rate remained at $10, will her
optimal working hours be different than part (b)?
(d)
Suppose that Marta has to pay an income tax of 20 percent on all of her income, and suppose that
her before-tax wage is $10 per hour and her before-tax non-labor income is $20 per day. How many hours
would she choose to work? What is the optimal consumption level?
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5.
(Williamson, Ch.4, Q.2) Suppose that leisure and consumption goods are perfect substitutes. In this
case, an indifference curve is described by the equation:
u = al + bC
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 chooses.
Show that the consumption bundle the consumer chooses depends on the relationship between a/b and w,
and explain why.
(b)
Do you think it likely that any consumer would treat consumption goods and leisure as perfect
substitutes?
(c)
Given perfect substitutes, is more preferred to less? Do preferences satisfy the diminishing marginal
rate of substitution property?
6.
(Williamson, Ch.4, Q.3) Consider the consumer choice example in this chapter, where consumption and
leisure are perfect complements. Assume that the consumer always desires a consumption bundle where the
quantities of consumption and leisure are equal, that is, a = 1.
(a)
Suppose that w = 0.75, π = 0.8, and T = 6. Determine the consumer’s optimal choice of consump-
tion and leisure, and show this in a diagram.
(b)
Now suppose that w = 1.5, π = 0.8, and T = 6. Again, determine the consumer’s optimal choice
of consumption and leisure, and show this in your diagram. Explain how and why the consumer’s optimal
consumption bundle changes, with reference to income and substitution effects.
7.
(Williamson, Ch.4, Q.4) Suppose that the government imposes a proportional income tax on the rep-
resentative consumer’s wage income. That is, the consumer’s 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.
8.
(Williamson, Ch.4, Q.6) Suppose that the representative consumer’s dividend income increases, and his
or her wage rate falls at the same time. Determine the effects on consumption and labor supply, and explain
your results in terms of income and substitution effects.
9.
(Williamson, Ch.4, Q.7) Suppose that a consumer can earn a higher wage rate for working overtime.
That is, for the first q hours the consumer works, he or she receives a real wage rate of w1 , and for hours
worked more than q he or she receives w2 , where w2 > w1 . Suppose that the consumer pays no taxes and
receives no nonwage income, and he or she is free to choose hours of work.
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(a)
Draw the consumer’s budget constraint, and show his or her optimal choice of consumption and
leisure.
(b)
Show that the consumer would never work q hours, or anything very close to q hours. Explain the
intuition behind this.
(c)
Determine what happens if the overtime wage rate w2 increases. Explain your results in terms of
income and substitution effects. You must consider the case of a worker who initially works overtime, and a
worker who initially does not work overtime.
10.
(Williamson, Ch.4, Q.11) Suppose that the government imposes a producer tax. That is, the firm pays
t units of consumption goods to the government for each unit of output it produces. Determine the effect of
this tax on the firm’s demand for labor.
11.
(Williamson, Ch.4, Q.12) Suppose that the government subsidizes employment. That is, the govern-
ment pays the firm s units 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.
12.
(Williamson, Ch.4, Q.15) In the course of producing its output, a firm causes pollution. The government
passes a law that requires the firm to stop polluting, and the firm discovers that it can prevent the pollution
by hiring x workers for every worker that is producing output. That is, if the firm hires N workers, then xN
workers are required to clean up the pollution caused by the N workers who are actually producing output.
Determine the effect of the pollution regulation on the firm’s profit-maximizing choice of labor input, and
on the firm’s labor demand curve.
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