lOMoARcPSD|6562757
Final 2019, questions and answers
Public Economics (University of Queensland)
Studocu is not sponsored or endorsed by any college or university
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Sample Final Exam - Solutions
ECON3450 Public Finance
ECON7440 Public Economics
University of Queensland
This sample final exam is for your exercise. You do not need to submit this sample
final and this sample final will not be counted toward your course grade. The best use
of this sample final is to attempt all the questions before looking at the answers. The
final exam may contain questions related to any topics covered in lectures from Week 1
to Week 11. The format of the actual final will be similar to the format of this sample
final.
Exam Conditions and Instructions
• This is a closed book exam - no materials permitted.
• Exam started: Check UQ Exam Schedule
• Exam duration: 120 minutes
• Reading time: 10 minutes
• Exam venue: Check UQ Exam Schedule
• This exam will not be released to the library.
• No electronic aids are permitted e.g. laptops, phones.
• Any unmarked paper dictionary is permitted.
• Calculators - Casio FX82 series or UQ approved (labeled)
• Materials to be supplied to students:
– 1 × 14 page answer booklet.
– Rough paper
• Answer all questions in Part A and Part B.
1
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Part A: Multiple Choice Questions
For each question, choose one and only one answer. Copy the choice alphabets such
as (a) and the entire answers to the answer booklet.
1. Which of the following statements is correct?
(A) Besley and Burgess (2002) study how state governments in India respond to
falls in food production and crop flood damage. They find that in states with
subscribers of cable TV, governments are more responsive to economic shocks
and food shortages via public food distribution and calamity relief expenditure.
(B) Synder & Strömberg (2010) study the impact of press coverage on citizen
knowledge, politicians’ actions and policy. Voters living in areas where the
press covers their Australian House representatives less are less likely to recall
their representative’s name and less able to describe and rate him/her.
(C) McMillan and Zoido (2004) study the bribery records of Peru’s former secretpolice chief, Vladimiro Montesinos Torres. They find that the typical bribe
paid to a television-channel owner was many times larger than that paid to a
politician or a judge. X
(D) DellaVigna & Kaplan (2007) investigate whether and how the introduction of
conservative Fox News channel affected the campaign contributions to senate
candidates of different parties in the U.S.
(E) Gentzkow & Shapiro (2010) measure media bias through eliciting survey subject’s perception of different media outlets. They find that the bias of radio
stations is significantly correlated with the ideology of their audience.
2. Suppose new driver P have a 50% chance to scratch his new car next year. P has
√
an income of Y = 100 and a utility function U = C. If P does not scratch his car,
his consumption is C = Y . If P scratches his car, he will have to pay L = 36 to fix
his car. An insurance contract with a premium $18.10 and covering all the losses
in case of an accident is available. Which of the following statements is correct?
(a) The driver would not buy the insurance as it gives lower expected utility.
(b) The driver would not buy the insurance as it gives the same expected utility.
(c) The driver would not buy the insurance because it is not actuarially fair.
(d) The driver would buy the insurance despite that it is not actuarially fair.
2
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Solution:
Without insurance:
EU0 = 0.5 ∗ 10 + 0.5 ∗ 8 = 9
With insurance:
EU1 =
√
100 − 18.1 = 9.05 > EU0
Thus, the driver would buy insurance as it gives higher expected utility.
Moreover, the insurance is not actuarially fair because the premium is higher than
the expected pay-out:
18.1 > 0.5 × 36
3. Which of the following is not a social insurance program in Australia or the United
States?
(a) Age Pension
(b) Newstart Allowance
(c) Medicare
(d) Agricultural subsidies to farmers X
(e) Food Stamps Program
4. Consider the following figure plotting intergenerational elasticity in earnings (intergenerational income persistence) and income inequality in a number of countries.
3
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Which of the following country sequences correctly indicates dots A, B, and C
respectively?
(a) (A, B, C) = (Sweden, Chile, Australia)
(b) (A, B, C) = (Australia, Brazil, China)
(c) (A, B, C) = (United States, Sweden, Australia) X
(d) (A, B, C) = (Australia, United States, Colombia)
√
5. Suppose Gigi has utility function U (C) = 3C + C and income Y = 81. She is also
an expected utility maximizer. With probability 0.9, no adverse event happens and
Gigi’s consumption would be C = Y = 81. With probability 0.1, an adverse event
happens, costing Gigi L = 72, and her consumption would C = Y − L. Would Gigi
buy an insurance contract with premium 7.5 and covering full lost in case of an
adverse event?
(a) Yes
(b) No X
(c) Uncertain
6. Which of the following statement is incorrect?
4
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
(a) Synder & Strömberg (2010) find that press coverage of House representatives
in the U.S. improves political accountability of elected officials to their constituents.
(b) Besley and Burgess (2002) study how state governments in India respond falls
in food production and crop flood damage.
(c) Among developed countries, Australia has the lowest level of market concentration of newspapers. X
(d) Provision of information and knowledge related to public policy is a public
good.
7. Which of the following statements is correct?
(a) In a representative democracy, voters vote directly on most policy issues to
express their preferences.
(b) Regardless whether politicians or parties could or could not credibly commit
to policy platform before elections, electoral competition drives policy convergence.
(c) The income share of the higher 1% has been constant in the U.S. but increasing
in Australia.
(d) Piketty & Saez (2011) show that top 1% income share is uncorrelated with
top marginal tax rate.
(e) None of above. X
8. Consider the following two configurations of media markets and electoral districts
from Synder and Strömberg (2010). Which of the following statements is incorrect?
5
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
(a) Configuration (2) has low congruence between media markets and electoral
districts.
(b) Relative to configuration (2), Newspaper B in configuration (1) cover more
news about political candidates of the district on the right-hand side.
(c) Synder and Strömberg (2010) find that whether congruence between media
market and electoral districts looks like configuration (1) or configuration (2)
does not affect political accountability. X
(d) Voters in areas look like configuration (1) on average are more knowledgeable
about their representatives in the Congress.
(e) Areas look like configuration (1) on average receive more federal spending.
9. Suppose a society consists of three individuals, i.e. A, B, C. There are three policies
being considered. Each policy yields utilities for each individual according to the
following table.
A
B
C
Policy 1
Policy 2
Policy 3
30
10
2
24
12
5
25
11
4
Which of the following statements is correct?
(a) A Rawlsian welfare function measures the social welfare function on the basis
of middle class, i.e. individual B in the above example.
(b) A Utilitarian welfare function justifies income redistribution even if there is no
diminishing marginal utility for anyone.
(c) A Utilitarian would prefer Policy 2 to Policy 1.
(d) A Rawlsian would prefer Policy 1 to Policy 3.
(e) Both Utilitarian and Rawlsian would prefer Policy 2 to Policy 3. X
(f) Both Utilitarian and Rawlsian would prefer Policy 1 to Policy 2.
(g) Policy 1 is most preferred among the three policies by a Rawlsian because
Policy 1 maximizes the utilities summed over all individuals.
6
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
10. Suppose a large number of voters populate a polity. Voter i has an income yi . 10%
of the population have zero income, i.e. yi = 0. yi is distributed uniformly over
[0, 1] for the rest of the population. Let’s normalize the population to one. The
government imposes a flat tax τ and transform the tax revenue into public goods.
Let y be the aggregate income and g be the level of public goods provided by the
government. Then, g = τ y. Voter i’s utility is given by
ui = c i +
1
ln(g)
20
where ci is the private consumption of voter i. Which of the following statements
is correct?
(a) 50% of the population prefer τ = 0.5 most.
(b) 100% of the population prefer τ = 0 most.
(c) By the median voter theorem, the tax rate in a voting equilibrium is 0.
(d) By the median voter theorem, the tax rate in a voting equilibrium is
9
.
80
X
(e) By the median voter theorem, the tax rate in a voting equilibrium is 0.5.
(f) By the median voter theorem, the tax rate in a voting equilibrium is 1.
1
. To
By the median voter theorem, the tax rate in a voting equilibrium should be 45
get the median voter’s most preferred position, we need to figure out how a voter’s
utility depends on the tax rate. We could write a voter’s utility as a function of tax
rate by substituting ci and g with expressions of tax rate τ :
1
ln(g)
20
1
ln(τ ȳ)
= (1 − τ )yi +
20
1
= (1 − τ )yi + [ln(τ ) + ln(ȳ)]
20
1
1
= y i − τ yi +
ln(τ ) +
ln(ȳ)
20
20
ui = c i +
The first order condition for maximum utility is:
∂ui
1
= −yi +
=0
∂τ
20τ
So the optimal for voter i with income yi is:
τ=
1
20yi
7
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
The tax rate can not go over one. So for yi ≤ 0.05, the most preferred tax rate is 1.
1
For others, the most preferred tax rate is 20y
. The median voter has income given
i
by:
0.1 + 0.9y ∗ = 0.5
i.e. y ∗ = 49 . Thus, the median voter’s most preferred tax rate is:
τ∗ =
1
1
=
20y ∗
20 ×
4
9
=
9
80
11. Suppose a politician is competent with a probability 0.4 and incompetent with a
probability 0.6. A policy decision can be either good or bad. A competent politician
always makes good policy (with a 100% probability). An incompetent politician
only makes a good policy with a 75% chance (and makes a bad policy with a 25%
chance). Conditional on a good policy being observed, what is the (conditional)
probability that the politician making the policy is a competent one?
(a)
1
25
(b)
1
3
(c)
8
17
X
(d) 1
12. Which of the following statements is correct?
(a) Mian, Sufi, and Trebbi (2015) study how members of U.S. Congress voted on
the two bills during the U.S. mortgage default crisis in 2008. X
(b) The American Housing Rescue and Foreclosure Prevention Act amounted to
net transfers to special interests groups formed by large firms in the real estate
and construction sectors.
(c) There were sharp increases in the mortgage delinquency rate across continental United States in 2007 before the passage of the Emergency Economic
Stabilization Act later in the same year.
(d) The Emergency Economic Stabilization Act allowed the Treasury Department
to transfers money directly to households facing foreclosure, benefiting constituencies with a high rate of mortgage default.
8
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
13. Suppose a nation has a large number of voters whose population is normalized to
one. The government 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. However, voters differ in their
preferences for public goods. Voter i’s utility function is
ui = c + αi ln(3g) − αi
where c = 1 − τ is voter i’s private consumption; and parameter αi ∈ [0, 1] captures
the voter’s preference for public goods. 15% of the population have αi = 0; 15%
of the population has αi = 1. For the rest of the population, αi is distributed
uniformly over interval [0, 1]. Voters cast their votes to decide the tax rate τ for
the nation. Two parties propose a tax rate τ to compete for election in a Hotelling
framework. What would be the equilibrium tax rate?
(a) 0
(b) 0.3
(c) 0.5 X
(d) 1
Solution:
ui = c + αi ln(3g) − αi
= 1 − τ + αi ln(3τ ) − αi
= 1 − τ + αi ln(τ ) + αi ln(3) − αi
To figure out for each voter what tax rate is most preferred, we consider the first
order condition (FOC) that maximize utility:
∂ui
=0
∂τ
i.e.
αi
∂ui
= −1 +
=0
∂τ
τ
The FOC gives τ ∗ = αi . Everything is symmetry so the median voter’s αi is:
1
αM = ,
2
9
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
which according to the median voter theorem would be proposed by both parties
and become the political-economic equilibrium tax rate.
10
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Part B: Short Answers
Please answer all of the following questions. There are five questions and each question
may have several parts. For a successful attempt to a question, clearly show your reasoning and related calculation. Please also circle or highlight your final answers and
start a question on a new page.
1. 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 nation’s
income.
(a) For p = 20, what is the Gini coefficient for the country?
Solution:
Gini =
0.5 −
h
0.2×0.5
2
+
0.5
(1+0.2)×0.5
2
i
= 0.3
(b) For p = 40, what is the 90-10 earning gap, i.e. the ratio of the 90 percentile
of earnings to the 10 percentile of earnings? (10 marks)
Solution:
90 percentile = 0.6/0.5 = 56
10 percentile = 0.4/0.5 = 54
90-10 ratio = 23
2. Consider the following Hotelling-Downs model. Suppose a large number of people
whose policy preferences are distributed between 0 and 1. There are three groups
of people, L, M, and R. Group L accounts for 10% of the population and their most
preferred policy is 0. Group R accounts for 20% of the population and their most
preferred policy is 1. Group M accounts for the rest of the population and their
most preferred policy is uniformly distributed between 0 and 1:
L
M
0
R
1
Two purely office-seeking candidates, A and B, each propose a policy platform to
compete for election. The one with the largest vote share would win and a coin
flip decide in case of a tie. If policy p ∈ [0, 1] is implemented, a voter with most
preferred position i has utility
ui = −|p − i|
11
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
What policies would be proposed by candidate A and candidate B? Which policy
would be implemented?
AN SWER:
The median voter theorem applies here. The position most preferred by the median
voter would be proposed by both candidates and implemented.
But since there is already a mass of voters at 0, which is different in size as compared
to the mass of voters at 1. The median voter is no longer one that prefers i = 0.5
most.
To solve the problem. Suppose the median voter’s ideal policy is m. By definition,
it means that half of the voters (0.5) would have an ideal policy position that is
less than or equal to m.
Within Group M, the share of voters with an ideal policy position less than or
equal to m is just m since the ideal policy position is uniformly distributed between
zero and one. But this group only accounts for 0.5 of the population because
0.7 = 1 − 0.1 − 0.2. So in the overall population, the share of voters with an ideal
policy position less than or equal to m is 0.7m
But we know the Group L has an ideal policy that is zero. The voters in this group
would have an ideal position less than the median voter’s ideal position. Moreover,
this group has a population share of 0.1.
Since m by construction is the median voters, these two segments of voters must
add up to a half:
0.1 + 0.7m = 0.5
Solving this gives us m = 47 .
Applying the median voter theorem, the median voter’s most preferred position is
m = 74 , which would be proposed by both candidates and implemented.
3. A band called EconRock has three musicians. They are Keith, Mick and Charlie.
They have to decide where to play their next concert. The possible options are
Adelaide (A), Brisbane (B) and Canberra (C). Adelaide’s venue has 60,000 seats,
Brisbane’s venue 50,000, while Canberra’s stadium can hold 40,000. Keith prefers
A to B and B to C. Mick prefers C to A and A to B. Finally, Charlie prefers B to
C and C to A.
(a) Is there a location option that is a Condorcet winner? If so, which one?
AN SWER:
12
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Keith: A ≻ B ≻ C
Mick: C ≻ A ≻ B
Charlie: B ≻ C ≻ A
• A v.s. B: A wins
• C v.s. B: B wins
• A v.s. C: C wins
Thus, no Condorcet winner!
(b) Suppose that Keith, Mick and Charlie are only interested in the number of
seats of the different venues, which give rise to their preferences over different venues. Draw the preferences of the three musicians in three graphs, one
for each musician. On the horizontal axis should be the numbers of seats of
the three venues. On the vertical axis should be the utility for the musician
associated with a venue. The utility level should be reasonable to reflect the
preferences of the musicians.
AN SWER:
13
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
(c) Are the preferences of all three musicians single-peaked?
AN SWER:
No, Mick’s preference is not single-peaked.
(d) Suppose by previous agreement, the band decides their location choice through
two rounds of voting. In the first round, each musician in the bands votes for
one of the two options. In the second round, the winner in the first round by
majority vote goes against the third option. The winner of the second round
by majority vote would be the final choice. The band draws straws to decide
who become the so-called agenda setter. The role of the agenda setter is to
decide which two options are candidate in the first round of voting. For example, an agenda setting could decide that in the first round of voting, A goes
against B; and in the second round, C goes against the winner among A versus
B. Assume that there is a way to make sure each musician votes sincerely, i.e.
they vote according to their true preferences. If Charlie draws the longer straw
and becomes the agenda setter, how would Charlie set the agenda in order to
get the outcomes he most prefers? If Mick happens to be the agenda setter,
would Mick set the agenda differently?
14
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
AN SWER:
The three musicians’ preferences are:
Keith: A ≻ B ≻ C
Mick: C ≻ A ≻ B
Charlie: B ≻ C ≻ A
Consider the following agendas as characterized by the first round voting:
• A v.s. B: A wins in the 1st round and then C wins in the 2nd round.
• C v.s. B: B wins in the 1st round and then A wins in the 2nd round.
• A v.s. C: C wins in the 1st round and then B wins in the 2nd round.
Charlie prefers B most. Thus, if Charlie becomes the agenda setter, he would
set up A v.s. C in the first round; and then B against the winner of C v.s. B
in the second round. By doing so, B would becomes the finally choice, which
is most preferred by Charlie.
Mick prefers C most. If Mick happens to be the agenda setter, he would set
up A v.s. B in the first round; and then C wins over A, which was the firstround winner. Therefore, yes Mick as the agenda setter would set the agenda
differently.
(e) Suppose the two-round voting agenda is such that A goes against B in the
first round; and C goes against the winner of A v.s. B in the second round.
Suppose all other musicians vote sincerely, would it be optimal for Keith to
vote sincerely in the first round?
AN SWER:
Keith: A ≻ B ≻ C
Mick: C ≻ A ≻ B
Charlie: B ≻ C ≻ A
Consider the following agendas as characterized by the first round voting:
• A v.s. B: A wins in the 1st round and then C wins in the 2nd round.
• C v.s. B: B wins in the 1st round and then A wins in the 2nd round.
15
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
• A v.s. C: C wins in the 1st round and then B wins in the 2nd round.
If everybody votes sincerely, then C would be the final choice, which is the
least preferred option for Keith. In the first round, Keith voting sincerely
means that A wins. But if Keith votes B instead of A, B wins in the first
round. When B goes against C in the second round, B wins and becomes
the final choice. Keith is better off with B than with C. Thus, Keith would
vote strategically instead in the first round — provided that all other votes
sincerely.
4. Suppose a society consists of two individuals, A and B. They have the same utility
function:
√
C
U (C) = C +
2
where C is the individual’s private consumption and equals to disposable income.
A new taxation policy is being considered. Under the new taxation policy, their disposable incomes would be different. Their initial disposable incomes and disposable
incomes under the new taxation policy is given by the following table:
A
B
Current Policy
New Policy
121
64
100
81
Consider a Utilitarian social welfare function (with equal weight to either individual)
and a Rawlsian social welfare function, answer the following questions.
(a) Should the new policy be enacted under a Utilitarian social welfare function?
(b) Should the new policy be enacted under a Rawlsian social welfare function?
AN SWER:
Applying the utility founction, the utilities for A and B under different policy are
tabluated below:
A
B
Current Policy
New Policy
11+60.5
8+32
10+50
9+40.5
i.e.
16
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
Current Policy
New Policy
A
B
71.5
40
60
49.5
Total
Minimum
111.5
40
109.5
49.5
Therefore, under a Utilitarian welfare function, the new policy should not be enacted
(since 111.5 > 109.5). However, under a Rawlsian welfare function, the new policy
should be enacted (because 40 < 49.5).
5. Suppose in a market of good G, the demand curve and supply curve are given
respectively by
P = (20 − QD )2
P = Q2S
where P denotes price; QS denotes quantity supplied; and QD denotes quantity demanded. However, each unit of good G produced generates a negative of externality
valued at $40. (hint: you could assume that QD ≤ 20.)
(a) What is price of good G in the free market equilibrium?
AN SWER:
At the free market equilibrium, QD = QS at price P . To solve for P , solve
(20 − QD )2 = Q2S
Let Q = QD = QS be the equilibrium quantity (production level), then
(20 − Q)2 = Q2
⇒ 400 − 40Q + Q2 = Q2
Thus, Q = 10 and P = Q2 = 100 in the free market equilibrium.
(b) What is the socially optimal quantity of good G produced and consumed?
AN SWER:
The socially optimal quantity is a quantity such that social benefit of the marginal
17
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
product equals to the social cost of producing the marginal product. The supply
schedule gives the private marginal cost of producing the marginal unit of good G:
P = Q2S
But since each unit of good G generated a negative externality of 40, we need to
add 40 to the private production cost to get the social cost:
P = Q2S + 40
In the socially optimal scenario, the marginal social cost equals to the marginal
social benefit (MSC = MSB), which is given by the demand schedule P = (20 −
QD )2 . Therefore, at social optimum:
Q2S + 40 = (20 − QD )2
⇒ Q2 + 40 = 400 − 40Q + Q2
i.e.
⇒Q=9
6. Suppose a society has two individuals, Ana and Bill. In the first period, Ana and
Bill decide whether to study for an university degree at the same time. If one
decides to attend university, his/her payoff in the first period is -10. If one decides
not to attend university, his payoff in the first period is zero. In the second period,
Ana and Bill’s earnings are not only determined by their own education level, but
also by the education level of the other as well. Let (A, B) denotes the earnings of
Ana and Bill respectively in the second period. The following table lists the second
period earnings (excluding first period payoffs) in four different scenarios:
Bill attended university
Bill did not attend university
(30, 30)
(8, 10)
(10, 8)
(5, 5)
Ana attended university
Ana did not attend university
Assume that neither Ana nor Bill discounts second period payoffs, namely their
discount factors are both 1. Their education decisions in the first period are only
determined by the sum of their own payoffs from the two periods. Moreover, they
decide whether to go to university in the first period simultaneously.
18
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
(a) Does education have externality in the example above? If so, is it positive or
negative externality?
AN SWER:
Yes, education has a positive externality as it increases the earnings of other.
This could happen due to productivity spill over.
(b) If Ana knows that Bill would not attend university, would Ana attend university? Is the case that neither Ana nor Bill attends university a Nash equilibrium? (hint: in a Nash equilibrium, no one finds it profitable to deviate from
their choice unilaterally.)
AN SWER:
It might be useful to re-write the earnings table into a table with payoffs net
of costs of attending university (10 if attending):
Bill attended university
Bill did not attend university
(20, 20)
(8, 0)
(0, 8)
(5, 5)
Ana attended university
Ana did not attend university
If Bill does not attend university, Ana is better off not attending since 5 > 0
as indicated by the last column of the above table. Similarly, if Ana does not
attend university, Bill is better off not attending since 5 > 0 as indicated by
the last row of the above table. Therefore, neither attending university is a
Nash equilibrium.
(c) If Ana knows that Bill would attend university, would Ana attend university?
Is the case that both Ana and Bill attend university a Nash equilibrium?
AN SWER:
Again examine the table of net payoffs:
Bill attended univ. Bill did not attend univ.
Ana attended univ.
Ana did not attend univ.
(20, 20)
(8, 0)
↑
19
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
(0, 8)
(5, 5)
←
lOMoARcPSD|6562757
If Bill attends university, Ana is better of attending since 20 > 8 as indicated
by the middle column of the above table.
Similarly, if Ana attends university, Bill is better of attending 20 > 8 as
indicated by the middle row of the above table.
Therefore, both attending university is also a Nash equilibrium.
7. Consider the following political agency model similar to the one we introduced in
class. There are two periods, two states of the world, and two policy alternatives:
• 2 periods: t = 1, 2.
• 2 states of the world:
st ∈ {0, 1}, for t = 1, 2.
4
3
P r(st = 0) = and P r(st = 1) = , for t = 1, 2.
7
7
The states in the two periods are independent.
• 2 policy alternative: pt ∈ {0, 1}, for t = 1, 2.
• Voters do not observe the states but are affected by policy choice:
ut =
(
∆, if pt = st
, for t = 1, 2.
0, if pt 6= st
u = u1 + βu2
• Politicians get E from holding office. However, 2 types of politicians:
i ∈ {c, d},
P r(i = c) = π
– Congruent (c):
vt =
(
E + ∆, if pt = st
, for t = 1, 2.
E,
if pt 6= st
v = v1 + βv2
– Dissonant (d):
vt =
(
E,
if pt = st
, for t = 1, 2.
E + r, if pt 6= st
v = v1 + βv2
20
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
where r is a private benefit (rent) drawn randomly from a uniform distribution over [0, R] in each period. v is the total utility of a politician over
two periods. A politician gets a utility of zero if he or she is not elected.
Suppose π = 0.6, E = 5, R = 20, and ∆ = 9.
(a) Assume that the rent (r) drawn in period 2 is independent of the rent drawn
in period 1. Moreover, β = 1. In equilibrium, what is the probability that a
dissonant politician would pick the right policy in period 1, i.e. p1 = s1 ? Show
your reasoning and calculation. (10 marks)
Solution:
The expected utility for a dissonant politician if he/she picks the right policy
and get re-elected:
R
vd = v1 + v2 = E + β(E + )
2
(3 marks)
The expected utility for a dissonant politician if he/she does not pick the right
policy and gives up on re-election:
vd = v1 + v2 = E + r
(3 marks)
For a dissonant politician to pick the right policy in period 1:
E + r ≤ E + β(E +
R
)
2
R
⇒ r ≤ β(E + )
2
R
β(E + 2 )
R
5 + 10
P r r ≤ β(E + ) =
=
= 0.75
2
R
20
(4 marks)
(b) Assume that the rent drawn (r) in period 2 will be the same as the rent drawn
in period 1. Moreover, β = 12 . A dissonant politician observes r and knows
about the fact that in period 2 the rent will be the same. Voters know about
that in period 2 the rent for a dissonant politician would be the same as the
rent drawn in period 1 but they do not observe the value of r in any period.
In equilibrium, what is the probability that the right policy is chosen in period
21
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
1, i.e. P r(p1 = s1 )? Show your reasoning and calculation. (10 marks)
Solution:
The expected utility for a dissonant politician if he/she picks the right policy
and get re-elected:
vd = v1 + v2 = E + β(E + r)
(3 marks)
The expected utility for a dissonant politician if he/she does not pick the right
policy and gives up on re-election:
vd = v1 + v2 = E + r
(3 marks)
For a dissonant politician to pick the right policy in period 1:
E + r ≤ E + β(E + r)
⇒ (1 − β)r ≤ βE
βE
⇒r≤
1−β
βE
βE
5
1−β
=
Pr r ≤
=
= 0.25
1−β
R
20
The probability that the right policy is chosen in period 1:
P r(p1 = s1 ) = π + (1 − π) × 0.25 = 0.6 + 0.4 × 0.25 = 0.7
(4 marks)
8. Suppose a nation has a large number of voters whose population is normalized to
one. The government 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. However, voters differ in their
preferences for public goods. Voter i’s utility function is
ui = c + αi ln(3g) − αi
where c = 1−τ is voter i’s private consumption; and parameter αi ∈ [0, 0.5] captures
the voter’s preference for public goods. 10% of the population have αi = 0; 25%
22
Downloaded by ZHANG Emily (mctoxic319@gmail.com)
lOMoARcPSD|6562757
of the population has αi = 0.5. For the rest of the population, αi is distributed
uniformly over interval [0, 0.5]. Voters cast their votes to decide the tax rate τ for
the nation. Is there a tax rate τ that is a Condorcet winner? If so, what value of τ
is the Condorcet winner? Show your reasoning and calculation. (10 marks)
Solution:
Subsitute c = 1 − τ and τ = g into the utility function:
ui = 1 − τ + αi ln(3τ ) − αi = (1 − τ ) + αi (ln 3 + ln τ ) − αi
Re-arranging the terms:
ui = (1 − αi + αi ln3) − τ + αi lnτ
Take the derivative with respect to τ and set it to zero for the first order condition:
u′i = −1 +
αi
=0
τ
F.O.C. gives τ ∗ = αi
Let the median voter be αM . Then, by the definition of the median voter, we have:
0.1 + (1 − 0.1 − 0.25) ×
Median voter has
αM =
αM
= 0.5
0.5
0.2
4
=
0.65
13
,
which by the median voter theorem is the Codorcet winner. (5 marks)
23
Downloaded by ZHANG Emily (mctoxic319@gmail.com)