Problem Set 4 Answers
Intermediate Microeconomics (22014)
Group 13 - Fall 2011
Due on Thursday, 10/13/2011
EXERCISE 1. - Eects of a price change in an endowment economy
Consider a consumer with preferences over good 1 and good 2 consumption given by
U (x1 , x2 ) = (x1 )3/4 (x2 )1/4 .
Suppose that the initial endowment of this consumer is (ω1 , ω2 ) = (6, 6) and that the initial
prices are (p1 , p2 ) = (1, 1). Suppose also that both goods are normal.
1. What is the budget constraint of the consumer?
Answer: p1 x1 + p2 x2 = p1 ω1 + p2 ω2 .
2. What is the other condition the optimal bundle must satisfy? Show it graphically.
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Answer: M RS = − pp12 ⇔ 3x
=1
x1
3. Compute the optimal consumption bundle of the consumer, and say for which good his
net demand is negative and for which it is positive.
Answer: x∗1 = 9, x∗2 = 3; positive net demand of good 1, negative net demand of good 2.
4. Suppose that the price of good 1 increases. Qualitatively, what do you think is the sign
of the substitution eect, the ordinary income eect, and the endowment income eect
of this price change on consumption of good 1?
Answer: Negative substitution eect, negative ordinary income eect, positive endowment
income eect.
5. Which income eect do you expect to be stronger, the ordinary or the endowment income eect? What sign do you expect for the total income eect?
Answer: Ordinary income eect stronger than endowment income eect because x∗1 > ω1∗ ;
thus, total income eect has negative income eect.
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6. What do you think is the eect of the price change on the consumer's welfare?
Answer: Negative because total income eect is negative.
7. In this type of economy when, if ever, do you think it is possible to have an increase in
x1 after an increase in p1 ?
Answer: When the endowment income eect is stronger than the ordinary income eect, so
that the total income eect is positive (and large enough to compensate for the negative
substitution eect).
8. Suppose now that the new price of good 1 is 2, so the new price vector is (p01 , p02 ) = (2, 1).
What is the total eect in the consumption of good 1?
∗
Answer: T E1 = x∗∗
1 − x1 = 6.75 − 9 = 2.25.
9. What is the income level that makes it possible for the individual to consume the initial
bundle at the new prices? Calculate the substitution eect of the price change on good 1.
Answer: ms = 2 ∗ 9 + 1 ∗ 3 = 21; SE1 = xs1 − x∗1 = 7.785 − 9 = −1.125.
10. What would be the income level of the consumer if her income was exogenous? Calculate
the ordinary income eect.
Answer: mo = 12; OIE1 = xo1 − xs1 = 4.5 − 7.785 = −3.375.
11. Calculate now the endowment income eect.
o
Answer: EIE1 = x∗∗
1 − x1 = 6.75 − 4.5 = 1.25.
12. Use a graph to show all the eects.
EXERCISE 2. - Labor supply decisions
Consider a worker with preferences over leisure and consumption given by
U (R, C) = RC
where R denotes the number of hours spent on leisure and C the number of units consumed.
Let R̄ denote the total number of hours available and M the nonlabor income of the consumer.
Suppose that R̄ = 24, M = 10, w = 1 and p = 2.
1. What is the budget constraint of the worker?
Answer: 2C + R = 34.
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2. What is the optimal relation between the hours of leisure and the consumption level?
Show it graphically.
Answer: CR = 12 .
3. Find the optimal number of leisure hours R∗ and working hours R̄ − R∗ , and the optimal
amount of consumption C ∗ .
Answer: C ∗ = 8.5, R∗ = 17, R̄ − R∗ = 7.
4. Suppose now that the wage rate becomes w0 = 2. What is the new optimal choice of
leisure, labor and consumption?
Answer: C ∗∗ = 14.5, R∗ = 14.5, R̄ − R∗ = 9.5.
5. Decompose now the total eect by computing the substitution eect, ordinary income
eect, and endowment income eect of the wage rate change.
Answer: SER = −4.25, OIER = −4.25, EIER = 6.
6. What is the sign of the substitution, the ordinary income, the endowment income, and
the total income eects? Explain why.
Answer: Substitution eect negative because leisure relatively more expensive after the wage
change; ordinary income eect negative because initial optimal point more expensive after the
wage change; endowment income eect positive because now higher potential income.
EXERCISE 3. - To Borrow or to lend
Consider another consumer with preferences over current and future consumption given
by
U (c1 , c2 ) = (c1 )1/2 (c2 )1/2
where c1 denotes the amount consumed in period 1 and c2 the amount consumed in period 2.
Suppose that period 1 income expressed in units of good 1 is p1 m1 = 10 and period 2 income
expressed in units of good 2 is p2 m2 = 40. Let p1 and p2 denote the price level in period 1
and 2, and let r denote the interest rate.
1. Write the intertemporal budget constraint of the consumer.
2 c2
2 m2
Answer: p1 c1 + p1+r
= p1 m1 + p1+r
.
2. If r = 0.1 and p1 = p2 = 1, what is the present value of her intertemporal income?
Answer: P V (m1 , m2 ) = 46.36.
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3. Find the optimal consumption bundle when r = 0.1 and p1 = p2 = 1. Is the consumer
a saver or a borrower?
Answer: c∗1 = 23.18 > m1 ; therefore, she is a borrower.
4. Suppose now the interest increases. Which consumption gets relatively cheaper period
1 or period 2 consumption? What is then the sign of the substitution eect on c1 ?
Answer: c1 gets relatively more expensive and c2 gets relative cheaper; hence, the substitution
eect on good 1 is negative.
5. Given your answer in part 3, what is the sign of the total income eect?
Answer: Negative.
6. Use Slutsky's equation to explain what are the eects on c1 if the interest rate increases.
Answer: First term of Slutsky equation negative; second term also negative because she is a
borrower.
7. Show the eects graphically.
8. Suppose now that p1 = 1 and r = 0. Find the ination rates for which the consumer is
a saver and the ones for which she is a borrower.
Answer: Always borrower unless π < −0.75!
EXERCISE 4. - Do not drink and drive
Consider a young dude called Mike with preferences over the wealth level m given by
u (m) =
√
m.
Suppose his initial wealth is 100 and his probability of having a car accident is 10%, in which
case half of his wealth is lost!
1. What is the expected utility of Mike if he does not take any insurance (assuming there
is no legal cost for that)?
Answer: EU = 9.71.
2. What is the highest insurance premium Mike is willing to pay to eliminate all the risk?
Answer: x̄ = 5.77.
3. Suppose the insurance company thinks the probability of having a car accident is 1%.
What would be the fair insurance premium for in this case?
Answer: 0.01 euros for each euro of insurance chosen.
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4. What would then be the insurance level chosen by Mike (assuming that it is not possible
to get more than the value of the accident damage)?
Answer: Full insurance (so insurance premium would be 0.01*50=0.5)
5. The insurance company now nds out about Mike's actual probability of having a car
accident, and is oered a new fair insurance premium that reects his actual probability.
What is now the fair insurance premium?
Answer: 0.1 euros for each euro of insurance chosen (he would also take full insurance, so the
insurance premium would now be 5).
6. How much would Mike be willing to pay to solve his drinking problem and reduce his
probability of having a car accident?
Answer: 5-0.5=4.5 (the dierence in insurance premiums).
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