Department of Economics
Trent University
ECON4000H – Advanced Microeconomics
Winter Semester, 2021
Pete
orough Campus
Final Exam
Due April 18, 2021 at 11:59pm EST
Instructions:
This exam has 4 questions for a total of 60 marks, and is worth 25% of the course grade. Marks for
each question are as indicated, and are evenly divided among the components of the question,
except if indicated otherwise. Answers to the questions must be word-processed or hand-written
and scanned, and be legible and concise; written explanations, where requested, should have no
more than a
ief paragraph of content. While students may work together on the exam, they must
develop, write up and submit answers independently. Answers must be submitted as a PDF file via
the assignments page of the course website.
Questions:
1. (30 marks) To examine A
ow’s Impossibility Theorem, consider the social choice function
?: ≿?=1
? ⟶ ≿ representing a public choice process over a set of social outcomes ? for a
population of ? agents.
a. State the restrictions on the domain of ?, the size of the set ? and the size of the set
? required by A
ow’s Theorem.
. State A
ow’s five axioms, and identify which of them characterize democratic
public choice and which of them characterize coherent public choice.
c. Define what it means for a coalition ? ⊆ ? to be (i) decisive over some pair of social
outcomes (?, ?) and (ii) fully decisive over ?.
d. Prove the following intermediate results used to construct the proof of A
ow’s
Theorem: (i) that a decisive coalition ? exists, (ii) that any decisive coalition ? is
also fully decisive and (iii) that any decisive coalition ? includes a sub-coalition that
is also decisive.
e. Combine the results identified in Part (d) to prove A
ow’s Theorem, which can be
done even without answering Part (d).
2. (6 Marks) For each of the following scenarios, determine whether the described behaviour
is best explained by (i) present bias, (ii) reference-based preferences, (iii) loss aversion, (iv)
probability weighting, (v) framing effect or (vi) bounded rationality.
a. A gambler on a streak of bad luck believes he will eventually recover his losses if he
continues to gamble.
. A consumer purchases product wa
anty extensions and lottery tickets at the same
time.
c. Through a New Year resolution, a person plans to quit smoking but then chooses not
follow through with the plan once the New Year a
ives.
d. A consumer enters into a detailed contract for a cell phone plan, and then regrets
this decision when he eventually learns his monthly bills are considerably higher
than he previously estimated.
e. A city manager publicly announces his projection of next year’s tax increase
annually. Over time, he finds a positive co
elation between his projection and the
actual tax increase he is able to get approved by city council.
f. A charity finds it has limited success when asking prospective donors to contribute
$600 per year. With a cup of coffee costing $1.65, the charity finds greater success
when it instead asks prospective donors to register for a contribution plan that costs
only a cup of coffee per day.
3. (8 marks) Consider application of the self-control (i.e ?-?) model to the matter of saving for
etirement while employed. Consider an agent whose life proceeds through three
oad
periods: the present, the intermediate future and the distant future. These co
espond
oughly to the lifetime phases of pre-employment (25 years), employment (35 years) and
etirement (40 years), respectively. The agent will enjoy an adequate standard of living
while retired in the distant future only if he saves sufficient income while employed in the
intermediate future. In particular, suppose sufficient saving in the intermediate future costs
the agent $700,000 while the retirement benefit enjoyed by the agent as a result of such
saving is $4,000,000 in the distant future. The agent has (?, ?) preferences where 0 < ? ≤ 1
is the self-control parameter and 0 < ? ≤ 1 is the discount rate between periods. Assume
? = 0.2.
a. State the values of ? that co
espond to naïve and sophisticated behaviour,
espectively. Identify the type of bias afflicting the naïve agent.
. Show that in the present a sufficient savings plan is desirable to the agent regardless
of the value of ?.
c. Show that when the intermediate future a
ives, and hence becomes the present, the
sufficient savings plan becomes undesirable to the agent if ? = 0.75.
d. Determine the values of ? for which a naïve agent will, despite being naïve,
nonetheless follow the sufficient savings plan.
4. (16 marks) Consider a municipality composed of identical households who earn income ?
and have utility given by ?(?, ?) = ?? where ? is composite private good expenditure and ?
is local public good expenditure. The local public good is provided by the municipality and
financed by an income tax such that the municipality’s revenue per household is given by:
?(?) = {?(? − ?)
?? ? ≤ ?
0 ? > ?
where ? ∈ [0,1] is the tax rate, ? ∈ (0,1] is the tax rate at and beyond which the
epresentative household is induced to flee the municipality and ? > 0 is a parameter
epresenting the distortionary effect of taxation. The municipality is benevolent in the sense
that it seeks to maximize a representative household’s utility.
a. Determine the values of ? ≤ ? for which no revenue is raised. Derive the revenue-
maximizing tax rate ?? and verify that ?? < ? for all ?.
. Use the results of Part (a) to construct an appropriate graph of ?(?), inclusive of
labels for extrema and intercepts with the axes.
c. For the remainder of the question, assume ? = 1. Derive the equili
ium tax rate ??
and verify that ?? < ?? for all ?.
d. Derive the efficient and equili
ium levels of local public good expenditure, ?∗ and
?? , respectively. Verify that ?? < ?∗ and
iefly explain why this inefficiency arises.
Department of Economics
Trent University
ECON4000H – Advanced Microeconomics
Winter Semester, 2021
Pete
orough Campus
Assignment #2
Due Fe
uary 28, 2021 at 11:59pm EST
Instructions:
This assignment has 4 questions for a total of 50 marks, and is worth 25% of the course grade.
Marks for each question are as indicated, and are evenly divided among the components of the
question, except if indicated otherwise. Answers to the questions must be word-processed or hand-
written and scanned, and be legible and concise; written explanations, where requested, should
have no more than a
ief paragraph of content. While students may work together on the exam,
they must develop, write up and submit answers independently. Answers must be submitted as a
PDF file via the assignments page of the course website.
Questions:
1. (12 marks) Consider the production function ?(?, ?) = ????? where ? > 0, ? > 0, ? > 0
and ? + ? < 1 and where ? and ? denote labour and capital, respectively. Let ?, ? and ?
denote the prices of labour, capital and output, respectively, and let ? denote output.
a. Solve the profit maximization problem to show that the factor demand and supply
functions are:
?(?, ?, ?) = [
???1−???
?1−???
]
1
1−?−?
?(?, ?, ?) = [
?????1−?
???1−?
]
1
1−?−?
?(?, ?, ?) = [
???+?????
????
]
1
1−?−?
The profit maximization problem is:
max
?,?,?
?? − ?? − ?? ?. ?. ? = ?????
max
?,?
?????? − ?? − ??
The first-order conditions are:
?????−1?? = ?
???????−1 = ?
Dividing these yields:
?
?
=
?
?
?
?
? =
??
??
?
Substituting this into the first-order condition for labour gives:
?????−1 (
??
??
?)
?
= ?
??+?−1 =
?
???
(
??
??
)
−?
?(?, ?, ?) = [
?
???
(
??
??
)
−?
]
1
?+?−1
?(?, ?, ?) = [
???
?
(
??
??
)
?
]
1
1−?−?
?(?, ?, ?) = [
???1−???
?1−???
]
1
1−?−?
Substitution of this into the equation ? =
??
??
? yields:
?(?, ?, ?) =
??
??
[
???1−???
?1−???
]
1
1−?−?
?(?, ?, ?) = [(
??
??
)
1−?−? ???1−???
?1−???
]
1
1−?−?
?(?, ?, ?) = [
?????1−?
???1−?
]
1
1−?−?
Substitution of ?(?, ?, ?) and ?(?, ?, ?) into the production function gives:
?(?, ?, ?) = ? [
???1−???
?1−???
]
?
1−?−?
[
?????1−?
???1−?
]
?
1−?−?
?(?, ?, ?) = ? [
??+???+???(1−?)+?????+(1−?)?
??(1−?)+?????+(1−?)?
]
1
1−?−?
?(?, ?, ?) = [
???+?????
????
]
1
1−?−?
. Solve the cost minimization problem to show that the conditional factor demand
functions are:
?(?, ?, ?) = (
??
??
)
?
?+?
(
?
?
)
1
?+?
?(?, ?, ?) = (
??
??
)
?
?+?
(
?
?
)
1
?+?
Show that all cost-minimizing production plans line on a common ray from the
origin in ?-? space.
The cost minimization problem is:
min
?,?
?? + ?? ?. ?. ? = ?????
min
?
??