ECON7440 Public Economics
Take-home Assignment - 02
Semester 2, 2023
This take-home assignment has two parts. Part A consists of 7 short answer questions. Part B consists
of 4 questions with multiple parts. The assignment is due at 4 PM (Brisbane time) on 24-10-2023. There
would be a 10% penalty for every hour of late submission.
This is an open-book non-invigilated assignment. You may review any lecture notes, problem sets,
lecture recording, textbooks, and other course materials to assist your assignment. However, you must
complete the assignment individually without discussing the questions with anyone else. Any possible
violation of academic integrity will be investigated and may be reported to the university for disciplinary
actions.
Please submit your answers in a PDF file via Blackboard by the due time. You are strongly encouraged
to allow at least 30 minutes in case of technical issues over submission. You could either (i) type your
answers with a text editing application (such as Microsoft Words, Google Docs, Apple Pages, etc.) and
save the file as a PDF document; or (ii) handwrite your answers, then scan and collate them into a PDF
file (apps such as Adobe Scan could facilitate it if you are scanning using your smartphone).
Part A
1. Suppose that the government imposes a tax on producers of a perfectly inelastically demanded
good. In this scenario who (producer/consumer) will bear the burden of taxation? Who will bear
the burden of the tax if this is imposed on producers of a perfectly elastically demanded good?
Briefly explain. [5 marks]
2. Suppose there is a party with 50 members. They need to select a leader among three candidates;
A, B, and C. If a member prefers candidate X to candidate Y, denote X ≻ Y . Suppose there are 5
groups of members, each of which has a preference ranking over the three candidates:
|G1| = 12: B ≻ C ≻ A
|G2| = 6 : B ≻ A ≻ C
|G3| = 20: A ≻ C ≻ B
|G4| = 2 : C ≻ A ≻ B
|G5| = 10: C ≻ B ≻ A
where |Gi|= x indicates that group i has x members. Is there a Condorcet winner? If So, who?
Please show your reasoning. [5 marks]
3. Briefly explain how inequality can affect social welfare and how governments redistribute. [5 marks]
4. Briefly explain the meaning of horizontal equity. [5 marks]
5. What are the assumptions of Hotelling-Downs Models? [5 marks]
6. Consider the following Hotelling-Downs model. Assume a large number of people whose policy
preferences are distributed between 0 and 1. But the distribution of preferences is not uniform.
The most preferred positions/policies of half of the people are to the left of 0.4. The most preferred
positions/policies of the other half are to the right of 0.4. Two purely office-seeking candidates, A
and B, each propose a policy platform to compete for the election. The one with the largest vote
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share wins (ties are randomly broken). If policy p ∈ [0, 1] is implemented, a voter whose most
preferred position is i receives utility
ui = −|p− i|
What policies would be proposed by candidate A and candidate B? Why? [5 marks]
7. Consider the following configurations of media markets and electoral districts. There are three elec-
toral districts: A, B, and C. There are two newspapers, newspaper Pound and newspaper Astrum.
The readers of newspaper Pound split between two districts, A and B. The readers of newspaper
Astrum spread across all three districts. The letter “P” below indicates one thousand readers
of newspaper Pound. A letter “S” below indicates one thousand readers of newspaper Astrum.
According to Synder & Stro¨mberg (2010), what is the congruence measure for electoral district B?
[5 marks]
Electoral Distric A Electoral District B Electoral District C
P P P P S S S S S S S S
P P P P S S S S S S S S
P P P P P P P P S S S S
S S S S P P P P S S S S
Part B
8. Suppose there is a country that has two individuals. One is high-skilled (H) and the other is low-
skilled (L). The low-skilled individual earns $10 per hour and the high-skilled individual earns $20
per hour. Each individual works 1,000 hours.
(a) Suppose that the government imposes an income tax of 20% on all workers. Assuming that
each individual works 1,000 hours, calculate the tax collected from each individual and the
total amount of tax collected. [5 marks]
(b) Now, suppose that the government wishes to collect the same tax amount as in the previous
part, but only wants to impose the tax on incomes above $10,000. Assuming that H is paid
$20 per hour and L is paid $10 per hour and that each works 1,000 hours, calculate the tax
rate the government should impose on incomes above $10,000. Briefly explain your answer
and calculations. [5 marks]
(c) Calculate the average tax rate for each individual if the government imposes the tax scheme
as per part (b). [5 marks]
9. Suppose the bottom 50 percent of a population (in terms of earnings) all receive an equal share of p
percent of the nation’s income, where 0 ≤ p ≤ 50. The top 50 percent of the population all receive
an equal share of 1− p percent of the nations income.
(a) Roughly sketch the Lorenz curve for the above country and the perfect equality line on the
same graph. [10 marks]
(b) Calculate the Gini coefficient as an expression of p. [4 marks]
(c) Calculate the Gini coefficient if p = 20 [1 mark]
10. Suppose a nation has a large number of voters whose population is normalized to one. The gov-
ernment imposes a flat tax τ and transforms the tax revenue into public goods. Let g be the level
of public goods provided by the government. Then τ = g. All voters have the same income y = 1.
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However, voters differ in their preferences for private consumption related to public goods. Voter
i’s utility function is
ui = αic+ ln g + 2
where c = 1−τ is voter i’s private consumption; and parameter αi ∈ [1, 11] captures the voter’s pref-
erence for private consumption. 10% of the population has αi = 1. For the rest of the population,
the distribution of αi is characterized by the following distribution function:
F (x) =
1
100
(x− 1)2
which means that the probability that αi ≤ x is 1100 (x− 1)2 for x ∈ [1, 11]. The probability that αi
is less than 1 is zero and the probability that αi is greater than 11 is also zero.
Voters cast theirs votes to decide the tax rate τ for the nation.
(a) Express voter i’s utility function in terms of τ . [2 marks]
(b) What is voter i’s most preferred tax rate? [5 marks]
(c) What is the value of αi for the median voter? Please show your calculations. [10 marks]
(d) Is there a tax rate τ that is a Condorcet winner? If so, what value of τ is the Condorcet
winner? [3 marks]
11. Suppose that there are three groups of people in a society. They are group A, B and C. There
are three policy alternatives being considered. The after-tax income of an individual in each group
under different policies and the population (in million) of each group are given by the Table 1.
Table 1: After-tax income
Group Population Policy 1 Policy 2 Policy 3
A 20 150 110 145
B 50 100 110 95
C 10 50 45 70
(a) Suppose that each individual has a utility function U(Y ) = Y , where Y is the after-tax income.
Under a utilitarian social welfare function assigning equal weights to each individual, which
policy should be adopted? Please show your reasoning and your calculation. [5 marks]
(b) Now suppose that each individual has an utility function U(Y ) = ln(Y ), where Y is the
after-tax income. Under a utilitarian social welfare function assigning equal weights to each
individual, which policy should be adopted? Please show your reasoning and your calculation.
[5 marks]
(c) Under a Rawlsian social welfare function, which policy should be adopted if each individual has
a utility function U(Y ) = Y , where Y is the after-tax income? Please provide your reasoning
and/or calculations. [5 marks]
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