Simon Fraser University
ECON 302
Spring 2016
S. Lu
Problem Set 1
Due on January 18, at 2:30pm
Remember to show your work on all questions.
1. The goal of this question is to review material from ECON 201, to derive the relation
between quasilinear utility and income effects, and to show that, as claimed in lecture, the
amount of money needed to compensate a consumer for a change in prices is equal to the
amount of money that the consumer is willing to pay to avoid that change in prices if there is
no income effect. Throughout the semester, you are highly encouraged to ask yourself what
the purpose of each problem set question is.
A consumer has quasilinear utility 𝑢(𝑚, 𝑥1 , 𝑥2 , … , 𝑥𝑛 ) = 𝑚 + 𝑣(𝑥1 , 𝑥2 , … , 𝑥𝑛 ), where:
𝑥𝑖 is the quantity of good 𝑖,
𝑚 is money (left over after paying for goods 1, … , 𝑛), and
𝑣 is a differentiable strictly concave function (this just means that if the optimum is
interior, you can find it by taking first-order conditions).
She has income 𝐼, and faces prices 𝑝1 , 𝑝2 , … , 𝑝𝑛 .
a) Find the first-order conditions that determine how much of each good is consumed,
assuming that the solution is interior.
b) What condition on 𝐼 is needed for an interior solution? [Hint: 𝑚 cannot be negative at the
consumer’s optimum.] Express your answer in terms of 𝐼, 𝑝1 , 𝑝2 , … , 𝑝𝑛 and the quantities
from the optimum determined in part a, which you may denote 𝑥1∗ , 𝑥2∗ , … , 𝑥𝑛∗ .
For the remainder of this question, assume that the solution is always interior, so that your
answer from part a is valid.
𝑑𝑥 ∗
c) Find 𝑑𝐼𝑖 . [Hint: Do your first-order conditions depend on 𝐼?] Are goods 1, … , 𝑛 normal,
inferior, or neither?
Suppose the price of good 1 increases from 𝑝1 to 𝑝1′, and let the quantities consumed under
the new prices be 𝑥1∗′ , 𝑥2∗′ , … , 𝑥𝑛∗′ . Consider the following claims:
i.
The amount that you need to pay the consumer to compensate her for the price
change, denoted 𝑦, is determined by the equation:
(𝐼 − 𝑝1 𝑥1∗ − ⋯ − 𝑝𝑛 𝑥𝑛∗ + 𝑣(𝑥1∗ , 𝑥2∗ , … , 𝑥𝑛∗ ))
= (𝐼 + 𝑦 − 𝑝1′ 𝑥1∗′ − ⋯ − 𝑝𝑛 𝑥𝑛∗′ + 𝑣(𝑥1∗′ , 𝑥2∗′ , … , 𝑥𝑛∗′ )).
ii.
The amount that the consumer would pay to avoid the price change, denoted 𝑧, is
determined by the equation: (𝐼 − 𝑧 − 𝑝1 𝑥1∗ − ⋯ − 𝑝𝑛 𝑥𝑛∗ + 𝑣(𝑥1∗ , 𝑥2∗ , … , 𝑥𝑛∗ )) =
(𝐼 − 𝑝1′ 𝑥1∗′ − ⋯ − 𝑝𝑛 𝑥𝑛∗′ + 𝑣(𝑥1∗′ , 𝑥2∗′ , … , 𝑥𝑛∗′ )).
d) Explain why the claims are true by stating what each side of each equation represents,
and by using your answer to part c. [Hint: Note that 𝑥1∗ , 𝑥2∗ , … , 𝑥𝑛∗ and 𝑥1∗′ , 𝑥2∗′ , … , 𝑥𝑛∗′ were
obtained for an income of 𝐼 (and not 𝐼 + 𝑦 or 𝐼 − 𝑧). This is where part c comes in.]
e) Solve for 𝑦 and 𝑧. Show that they are equal.
2. X and Y are random variables. Their joint probability distribution is as follows:
- X = 0, Y = 0 with probability 0.2
- X = 1, Y = 0 with probability 0.1
- X = 3, Y = 2 with probability 0.2
- X = 1, Y = 2 with probability 0.5
a) Find the expected value of X, denoted E[X].
b) For each possible value of Y, find the expected value of X conditional on that value. That
is, find E[X|Y=0] and E[X|Y=2].
c) Find the probability that Y=0, denoted Pr(Y=0).
d) For each possible value of X, find the probability that Y=2 conditional on that value. That
is, find Pr(Y=2|X=0), Pr(Y=2|X=1) and Pr(Y=2|X=3).
A third random variable, Z, is independent of X and has the following probability distribution:
- Z = 5 with probability 0.6
- Z = 7 with probability 0.4
e) Find the joint probability distribution of X and Z.
f) Find the expected value of ZX, denoted E[ZX].
3. Show that whether an outcome x is Pareto efficient:
a) cannot change when a strictly increasing function fi is applied to each agent i’s utility
over outcomes.
b) can change when an increasing function fi is applied to each agent i’s utility over
outcomes.
Hint: For part a, you need an argument, since your answer needs to be valid for all possible
functions. You might want to proceed by contradiction: suppose x is Pareto efficient when
each agent i’s utility is ui, but not Pareto efficient when it’s fi(ui) instead. Then show that this
assumption leads to something that is impossible.
For part b, you just need an example, since you’re only asked to show that something can
occur.
4. Consider a perfectly competitive market with demand Q=20-P and supply Q=2P-1 with a
per-unit tax of $3. Find the quantity, the price paid by buyers, the price received by sellers,
and the deadweight loss.