这个作业是完成微观经济学相关的问题

EC2066 ZA Microeconomics

回答本节中的所有8个问题（每个5分）。

1.下图显示了跨时间预算约束。赋

点是（Y0，Y1）。随着利率的增加，预算约束会不断变化

如图所示，在捐赠点附近。使用该图，说明一个保护程序

如果利率上升，就不能成为借款人。

Y1

Y + Y /（1 + r）0 1

Y（1 + r）+ Y10

00

消费

在期间0

消费

在第一期捐赠中

2.假设以下游戏无限重复。玩家有共同点

折现因子δ∈（0，1）。假设δ非常接近零。在这种情况下，确定重复博弈的均衡。 （只需确定将要采取的措施

每段播放一次-无需进行任何计算。）

2

D

1 C 2,2 0,3

D 3,0 1,1

3.假设基因检测技术的改进使保险公司能够高精度地预测任何人都会患的疾病

在以后的生活中发展。这样的测试对效率的可能影响是什么？

健康保险市场？简要说明。

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4.吉尔只购买两种商品，苹果（A）和橙子（R）。苹果的价格

是2，橙子的价格是4。吉尔的收入为56，其效用函数

是

u（A，R）= min [A，5R]

吉尔应该购买哪些苹果和橙子组合以最大化效用？

5.公共物品有3个消费者。消费者的需求功能如下：

P1 = 20 − Q（消费者1）

P2 = 50 − Q（消费者2）

P3 = 90 − Q（消费者3）

Q是公共物品的数量，Pi是我愿意的价格消费者

支付。公共物品的边际成本恒定为70，并且

没有固定成本。寻找公共产品的有效生产水平。

6.假设排放交易市场上的许可证价格降至零。什么

这是否表明该市场在减少排放方面的有效性？

简要说明。

7.如果垄断者的边际收益曲线与市场需求一致

曲线上，没有垄断带来的无谓损失。这是对还是错？说明

你的答案。

8.假设需求逆曲线由P = 10 − Q给出。

下图。在图中所示的点A是需求弹性

或无弹性（关于价格变化）？解释你的答案。

数量

价钱

P = 10-Q

5

0

10

5 10

一个

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B部分

回答本节中的三个问题（每个20分）。

9.（a）假设一个行业中有两个完全相同的公司通过设定数量进行竞争。公司1的输出用q1表示，公司2的输出用q1表示。

用q2表示。每个公司都面临30的不变边际成本。让Q表示

总输出，即Q = q1 + q2。市场逆需求曲线为

由

P = 300-2Q

一世。找出每个公司产生的古诺-纳什均衡量

和市场价格。 [5分]

ii。每个公司生产的数量和市场价格是多少

在Stackelberg双寡头垄断下，如果公司1先走？ [5分]

（b）假设两家公司（1和2）出售差异化产品并通过以下方式竞争

设定价格。需求函数是

q1 = 2-P1 +

P2

2

和

q2 = 2-P2 +

P1

2

企业的生产成本为零。

一世。在同时移动游戏中找到纳什均衡。也找到

每个公司出售的数量。 [5分]

ii。如果1在2之前移动，则找到子博弈完美平衡。

每个公司出售的数量。 [5分]

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10.（a）考虑下面的战略形式博弈，其中有两个参与者1和2。

玩家2

A2 B2 C2

A1 2,2 4,2 0,4

玩家1 B1 4,0 6,8 2,2

C1 6,4 4,0 X，6

一世。设置X =0。通过迭代消除严格控制的问题来解决游戏

策略。 [6分]

ii。设置X =8。消除所有严格控制的策略，然后找到

游戏的纯策略和混合策略纳什均衡。

[6分]

(b) Suppose two firms (1 and 2) sell differentiated products and compete by

setting prices. The demand functions are

q1 = 2 − P1 +

P2

2

and

q2 = 2 − P2 +

P1

2

Firms have a zero cost of production.

i. Find the Nash equilibrium in the simultaneous-move game. Also find

the quantities sold by each firm. [5 marks]

ii. Find the subgame-perfect equilibrium if 1 moves before 2. Also find

the quantities sold by each firm. [5 marks]

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10. (a) Consider the strategic-form game below with two players, 1 and 2.

Player 2

A2 B2 C2

A1 2,2 4,2 0,4

Player 1 B1 4,0 6,8 2,2

C1 6,4 4,0 X,6

i. Set X = 0. Solve the game by iteratively eliminating strictly dominated

strategies. [6 marks]

ii. Set X = 8. Eliminate any strictly dominated strategies and then find

the pure strategy and mixed strategy Nash equilibria of the game.

[6 marks]

(b) Consider the following extensive-form game with two players, 1 and 2.

a3

1

a1 a2

b2

(1,3)

b1

(0,1)

b2

(0,1)

b1

(2,2)

2

c2

(2,0)

c1

(-1,1)

2

i. Find the pure-strategy Nash equilibria of the game. [4 marks]

ii. Find the pure-strategy subgame-perfect equilibria of the game.

[4 marks]

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11. A seller sells a good of quality q at a price t. The cost of producing at quality

level q is given by q2. There is a buyer who receives a utility of θq − t by consuming the unit of quality q at price t. If he decides not to buy, he gets a utility

of zero. θ can take two values θ1 = 2 and θ2 = 4.

(a) Suppose the seller can observe θ. Derive the profit maximizing price-quality

pairs offered when θ = 2 and when θ = 4. [6 marks]

(b) Prove that the full information price-quality pairs are not incentive compatible if the seller cannot observe θ. [7 marks]

(c) Suppose the seller cannot observe θ, and suppose he decides to set q1 =

1/2 and q2 = 2. Calculate the optimal values of t1 and t2 such that both

types participate, type θ = 1 takes the contract (q1, t1) and type θ = 2 takes

the contract (q2, t2). [7 marks]

[Hint: write down the participation constraint of type θ1 and the incentive

constraint of type θ2 and solve for t1 and t2.]

12. Consider a competitive industry with several identical firms. Entry and exit are

possible in the long-run. The long-run cost function of a firm and the market

demand function are as follows:

Q = 127 − P (Market demand)

C(q) = 10q − 2q2 + q3

3 (Long-run cost function of a firm)

(a) Derive the long-run supply function of the representative firm, paying

proper attention to the shut-down decision of the firm. [5 marks]

(b) Derive the long-run market supply function. [5 marks]

(c) How many firms would operate in the market in the long-run? [5 marks]

(d) To boost consumer demand, the government announces a per-unit subsidy

of 6 to be given to consumers. Calculate the social cost of this policy and

also show it in a diagram. [5 marks]

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13. Suppose u(W) = ln W for an agent. The agent faces the following gamble:

with probability 1/2 wealth is 10 and with probability 1/2 a loss occurs so that

wealth becomes 4. The agent can buy any amount of insurance: a coverage of

X (to be paid if the loss occurs) can be purchased by paying premium rX.

(a) Calculate the Arrow-Pratt measure of risk aversion for the agent.[5 marks]

(b) Write down the expected utility of the agent after purchasing insurance

coverage X. [5 marks]

(c) How much coverage X would the agent choose if r is set so that the insurer’s payoff is zero? [5 marks]

(d) How much coverage X would the agent choose if r = 2/3? [5 marks]

14. Rai spends her income on fuel for heating her house (H) and a composite of all

other goods (Y). Her preferences are represented by the utility function

u(H,Y) = HαY1−α

The price of the composite good is 1, and the price of heating fuel is p. Let M

denote Rai’s income.

(a) Derive Rai’s demand for H and Y. [5 marks]

(b) Suppose M = 300 and α = 2/3. Also suppose currently the unit price of

fuel is p = 20. The energy company offers Rai the option to switch to a

different tariff. Under the new tariff, Rai must pay a fixed fee of 100 and

then she can buy fuel at a unit price of 10. Would Rai switch to the new

tariff? Explain. [8 marks]

(c) The government decides to give Rai a heating fuel subsidy of s per unit.

This results in an increase in utility from u0 before the subsidy to u1 after

the subsidy. Would an equivalent variation in income cost the government

less? Explain using a suitable diagram. [7 marks]

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&/%0’1″1&3

© University of London 2017

UL17/017 Page 1 of D0

~~EC2066_ZA_2016_d0

This paper is not to be removed from the Examination Halls

UNIVERSITY OF LONDON EC2066 Z%

BSc degrees and Diplomas for Graduates in Economics, Management, Finance

and the Social Sciences, the Diplomas in Economics and Social Sciences

Microeconomics

Friday, 5 May 2017 : 10:00 to 13:00

Candidates should answer ELEVEN of the following FOURTEEN questions: all EIGHT

from Section A (5 marks each) and THREE from Section B (20 marks each).

Candidates are strongly advised to divide their time accordingly.

A calculator may be used when answering questions on this paper and it must comply

in all respects with the specification given with your Admission Notice. The make and

type of machine must be clearly stated on the front cover of the answer book.

If more questions are answered than requested, only the first answers attempted will be

counted.

PLEASE TURN OVER

SECTION A

Answer all EIGHT questions from this section (5 marks each).

1. The figure below shows the inter-temporal budget constraint. The endowment

point is (Y0,Y1). As the rate of interest increases, the budget constraint pivots

around the endowment point as shown. Using the figure, explain that a saver

cannot become a borrower if the rate of interest rises.

Y1

Y + Y /(1+r) 10

Y (1+r) + Y10

Y0

Consumption

in period 0

Consumption

in period 1 Endowment

2. Suppose the following game is repeated infinitely. The players have a common

discount factor δ ∈ (0, 1). Suppose δ is very close to zero. Identify an equilibrium of the repeated game in this case. (Simply identify what actions would be

played every period – there is no need to do any calculations.)

2

C D

1 C 2,2 0,3

D 3,0 1,1

3. Suppose improvements in genetic testing technology enables insurance companies to predict with high precision the diseases that any individual would

develop later in life. What is the likely effect of such tests on the efficiency of

the health insurance market? Explain briefly.

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4. Jill purchases only two goods, apples (A) and oranges (R). The price of apples

is 2 and the price of oranges is 4. Jill has an income of 56 and her utility function

is

u(A, R) = min[A, 5R]

What bundle of apples and oranges should Jill purchase to maximize utility?

5. The von Neumann-Morgenstern utility function of an agent is given by

u(W) = ln W

What is the Arrow-Pratt measure of risk aversion for this agent?

6. Consider a competitive industry with entry and exit in the long run. Suppose

the input prices do not change as output changes in the long run. A per-unit

tax on suppliers does not change market quantity in the long run equilibrium.

Is this true or false? Explain your answer.

7. Chen consumes two goods, x and y and his utility function is

u(x, y) = ln x + y

Suppose the price of y is 1 and let p denote the (relative) price of x. Chen has

an income M > 1. Calculate the price elasticity and the income elasticity of

demand for x.

8. In the signalling model of education, equilibria in which education acts as a signal of ability would break down if the costs of acquiring education were equal

for individuals with different abilities. Is this true or false? Explain your answer.

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SECTION B

Answer THREE questions from this section (20 marks each).

9. (a) Suppose there are two identical firms in an industry who compete by setting quantities. The output of firm 1 is denoted by q1 and that of firm 2 is

denoted by q2. Each firm faces a constant marginal cost of 30. Let Q denote

total output, i.e. Q = q1 + q2. The inverse demand curve in the market is

given by

P = 300 − 2Q

i. Find the Cournot-Nash equilibrium quantity produced by each firm

and the market price. [5 marks]

ii. What would be the quantities produced by each firm and market price

under Stackelberg duopoly if firm 1 moves first? [5 marks]

(b) Suppose two firms (1 and 2) sell differentiated products and compete by

setting prices. The demand functions are

q1 = 2 − P1 +

P2

2

and

q2 = 2 − P2 +

P1

2

Firms have a zero cost of production.

i. Find the Nash equilibrium in the simultaneous-move game. Also find

the quantities sold by each firm. [5 marks]

ii. Find the subgame-perfect equilibrium if 1 moves before 2. Also find

the quantities sold by each firm. [5 marks]

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10. (a) Consider the strategic-form game below with two players, 1 and 2.

Player 2

A2 B2 C2

A1 2,2 4,2 0,4

Player 1 B1 4,0 6,8 2,2

C1 6,4 4,0 X,6

i. Set X = 0. Solve the game by iteratively eliminating strictly dominated

strategies. [6 marks]

ii. Set X = 8. Eliminate any strictly dominated strategies and then find

the pure strategy and mixed strategy Nash equilibria of the game.

[6 marks]

(b) Consider the following extensive-form game with two players, 1 and 2.

a3

1

a1 a2

b2

(1,3)

b1

(0,1)

b2

(0,1)

b1

(2,2)

2

c2

(2,0)

c1

(-1,1)

2

i. Find the pure-strategy Nash equilibria of the game. [4 marks]

ii. Find the pure-strategy subgame-perfect equilibria of the game.

[4 marks]

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11. A seller sells a good of quality q at a price t. The cost of producing at quality

level q is given by q2. There is a buyer who receives a utility of θq − t by consuming the unit of quality q at price t. If he decides not to buy, he gets a utility

of zero. θ can take two values θ1 = 2 and θ2 = 4.

(a) Suppose the seller can observe θ. Derive the profit maximizing price-quality

pairs offered when θ = 2 and when θ = 4. [6 marks]

(b) Prove that the full information price-quality pairs are not incentive compatible if the seller cannot observe θ. [7 marks]

(c) Suppose the seller cannot observe θ, and suppose he decides to set q1 =

1/2 and q2 = 2. Calculate the optimal values of t1 and t2 such that both

types participate, type θ = 1 takes the contract (q1, t1) and type θ = 2 takes

the contract (q2, t2). [7 marks]

[Hint: write down the participation constraint of type θ1 and the incentive

constraint of type θ2 and solve for t1 and t2.]

12. Consider an exchange economy with two goods (x and y) and two consumers

(A and B). There are 10 units available of each of the two goods. Consumer A

is endowed with 6 units of x and 4 units of y. Consumer B is endowed with 4

units of x and 6 units of y. Consumer A has the following utility function:

UA(x, y) = min[x, y]

Consumer B has the following utility function:

UB(x, y) = x + y

(a) Draw an Edgeworth box and show the area of mutually beneficial trades

between the two consumers. [5 marks]

(b) Let the price of y be 1 and let p be the (relative) price of x. Derive the

equilibrium value of p. [5 marks]

(c) Derive the equilibrium allocation. [5 marks]

(d) If we pick an allocation on the contract curve that is different from the

equilibrium allocation you derived in part (d), is it possible to support this

new point as an equilibrium allocation? Explain. [5 marks]

[Hint: Make use of the Second Welfare Theorem]

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13. Suppose the demand curve for corn is Q = 60 − 5P and the supply curve is

Q = 10P.

(a) Suppose the government announces a per-unit subsidy of 3 to suppliers.

Calculate the impact of the subsidy on the equilibrium market price of corn

and the equilibrium quantity of corn. [5 marks]

(b) Calculate the deadweight loss from the subsidy policy in part (a).[5 marks]

(c) Suppose the government announces a price ceiling of 2. Calculate the

deadweight loss from this policy assuming that the quantity of corn supplied is allocated to those consumers who value it the most. [5 marks]

(d) Suppose the government can either impose a price ceiling of 2 or a quota

of 20 units of corn. Which policy would consumers prefer? Which policy

would suppliers prefer? Explain using a suitable diagram. [5 marks]

14. Consider a society with 2 agents. Agent 1 takes an action a ∈ [1/4, 1] and agent

2 is also affected by the action. Agents have quasilinear utility. Suppose good m

is money. Agent i obtains a utility of ui(a) + mi. We have

u1(a) = ln a

and

u2(a) = − 2 a2

Let a0 denote the socially optimal level of a, and let a∗ denote the individually

optimal level for agent 1.

(a) Calculate a∗ and a0. [5 marks]

(b) Suppose the government imposes a proportional tax of t on a (so that the

total tax bill is t times a) on agent 1. Calculate the optimal value of t.

[5 marks]

(c) Suppose the government allocates the property right on the action to agent

1, so that 1 can now ask 2 for a payment in order to reduce a from a∗. 1 can

extract a payment of T from 2 and maximises u1(a) + T subject to 2 getting

at least as much utility as under no agreement.

Show that 1 would optimally adopt a0 in this case. [5 marks]

(d) Would the adopted level of a be different if the property right is allocated

to agent 2? Explain. [5 marks]

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Page 7 of 7

Examiners’ commentaries 2017

Examiners’ commentaries 2017

EC2066 Microeconomics

Important note

This commentary reflects the examination and assessment arrangements for this course in the

academic year 2016–17. The format and structure of the examination may change in future years,

and any such changes will be publicised on the virtual learning environment (VLE).

Information about the subject guide and the Essential reading

references

Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2016).

You should always attempt to use the most recent edition of any Essential reading textbook, even if

the commentary and/or online reading list and/or subject guide refer to an earlier edition. If

di↵erent editions of Essential reading are listed, please check the VLE for reading supplements – if

none are available, please use the contents list and index of the new edition to find the relevant

section.

General remarks

Learning outcomes

At the end of this course, and having read this guide, completed the Essential reading and activities,

you should:

• be able to define and describe:

• the determinants of consumer choice, including inter-temporal choice and choice under

uncertainty

• the behaviour of firms under di↵erent market structures

• how firms and households determine factor prices

• behaviour of agents in static as well as dynamic strategic situations

• the nature of economic interaction under asymmetric information

• be able to analyse and assess:

• eciency and welfare optimality of perfectly and imperfectly competitive markets

• the e↵ects of externalities and public goods on eciency

• the e↵ects of strategic behaviour and asymmetric information on eciency

• the nature of policies and contracts aimed at improving welfare

• be prepared for further courses which require a knowledge of microeconomics.

1

EC2066 Microeconomics

Time management

Section A comprises eight questions, all of which must be answered (accounting for 40% of the total

marks). Section B comprises six questions of which three must be answered (accounting for 60% of

the total marks). Candidates are strongly advised to divide their time accordingly. On average, only

nine minutes should be allocated to any individual Section A question. On average, only 36 minutes

should be allocated to any individual Section B question.

Key steps to improvement

You need to be able to apply relevant microeconomic theory to questions that you may not have

encountered before. To prepare for this, you need not only to gain a thorough understanding of

microeconomic models but also (and importantly) to practise using relevant models to answer

specific questions. Practice is the key, not the learning of specific answers.

You should spend time planning your answers and make sure that you respond to all parts of a

question and to key words like ‘define’, ‘explain’ and ‘compare’. Precise and concise answers are to

be preferred to vague and long-winded answers.

You should be aware that, for most answers, diagrams and/or mathematical analysis are essential.

These should be correct and diagrams should be well-labelled. In addition, you should always

accompany them with appropriate explanations. Again, ‘practice makes perfect’.

Essential reading: Important information

The subject guide refers to Nicholson & Snyder as the principal text. There are also some references

to Perlo↵. In addition to this, you should practise questions from other texts. A few ‘auxiliary’ texts

that are good sources for practice questions are listed below. Further, the auxiliary texts often

develop applications not covered in the principal text. You should study these to broaden, as well as

deepen, your understanding. In some cases, reading several treatments of the same topic might help

to clarify the basic idea. You should use the auxiliary texts for this purpose as well.

The coverage of game theory is often inadequate in texts. For this topic, you should primarily rely

on the exposition in the subject guide. You should make sure that you understand the key ideas

covered in some detail in the subject guide.

Principal text

• Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application (Cengage

Learning 2015) twelfth edition [ISBN 9781133189039].

Auxiliary texts

• Perlo↵, J.M. Microeconomics with Calculus (Pearson Education, 2014) third edition [ISBN

9780273789987].

• Besanko, D. and R. Braeutigam, Microeconomics (John Wiley & Sons, 2014) fifth edition,

international student version [ISBN 9781118716380].

• Varian, H.R. Intermediate Microeconomics: A Modern Approach (W.W. Norton, 2014)

ninth edition [ISBN 9780393920772].

• Pindyck, R.S. and D.L. Rubinfeld, Microeconomics (Prentice–Hall/Pearson, 2012) eighth

edition [ISBN 9780133041705].

2

Examiners’ commentaries 2017

Examination revision strategy

Many candidates are disappointed to find that their examination performance is poorer than they

expected. This may be due to a number of reasons, but one particular failing is ‘question

spotting’, that is, confining your examination preparation to a few questions and/or topics which

have come up in past papers for the course. This can have serious consequences.

We recognise that candidates might not cover all topics in the syllabus in the same depth, but you

need to be aware that examiners are free to set questions on any aspect of the syllabus. This

means that you need to study enough of the syllabus to enable you to answer the required number of

examination questions.

The syllabus can be found in the Course information sheet available on the VLE. You should read

the syllabus carefully and ensure that you cover sucient material in preparation for the

examination. Examiners will vary the topics and questions from year to year and may well set

questions that have not appeared in past papers. Examination papers may legitimately include

questions on any topic in the syllabus. So, although past papers can be helpful during your revision,

you cannot assume that topics or specific questions that have come up in past examinations will

occur again.

If you rely on a question-spotting strategy, it is likely you will find yourself in diculties

when you sit the examination. We strongly advise you not to adopt this strategy.

3

EC2066 Microeconomics

Examiners’ commentaries 2017

EC2066 Microeconomics

Important note

This commentary reflects the examination and assessment arrangements for this course in the

academic year 2016–17. The format and structure of the examination may change in future years,

and any such changes will be publicised on the virtual learning environment (VLE).

Information about the subject guide and the Essential reading

references

Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2016).

You should always attempt to use the most recent edition of any Essential reading textbook, even if

the commentary and/or online reading list and/or subject guide refer to an earlier edition. If

di↵erent editions of Essential reading are listed, please check the VLE for reading supplements – if

none are available, please use the contents list and index of the new edition to find the relevant

section.

Comments on specific questions – Zone A

Text: We use the following abbreviation:

• N&S – Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application

(Cengage Learning 2015) twelfth edition [ISBN9781133189039].

For each question, we point out the relevant sections from the subject guide. For most topics, there

are also references to the main text (N&S).

Candidates should answer ELEVEN of the following FOURTEEN questions: all EIGHT from

Section A (5 marks each) and THREE from Section B (20 marks each). Candidates are

strongly advised to divide their time accordingly.

If more questions are answered than requested, only the first answers attempted will be counted.

Section A

Answer all EIGHT questions from this section (5 marks each).

Question 1

The figure below shows the inter-temporal budget constraint. The endowment point

is (Y0, Y1). As the rate of interest increases, the budget constraint pivots around the

endowment point as shown. Using the figure, explain that a saver cannot become a

borrower if the rate of interest rises.

4

Examiners’ commentaries 2017

Reading for this question

N&S Chapter 14.

Subject guide, Chapter 2.

Approaching the question

A saver consumes to the left of the point (Y0, Y1) on the original budget line (the solid black

line). A rise in the rate of interest would change the budget line as shown in the figure. To

become a borrower, an agent would have to consume to the right of (Y0, Y1) on the new budget

line (the dashed line). However, any such point was initially available, and was rejected in favour

of a point to the left of (Y0, Y1). Since the point initially chosen (on the solid budget line) is still

available, any point to the right of (Y0, Y1) on the dashed budget line cannot be optimal.

Therefore, a saver cannot become a borrower.

Question 2

Suppose the following game is repeated infinitely. The players have a common

discount factor 2 (0, 1). Suppose is very close to zero. Identify an equilibrium of

the repeated game in this case. (Simply identify what actions would be played every

period – there is no need to do any calculations.)

2

C D

1 C 2,2 0,3

D 3,0 1,1

Reading for this question

Subject guide, Chapter 4.

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EC2066 Microeconomics

Approaching the question

You should understand that playing (D, D) in each period (after every history) is an equilibrium

for any value of . Once you understand this, it is straightforward to answer the question.

Question 3

Suppose improvements in genetic testing technology enables insurance companies to

predict with high precision the diseases that any individual would develop later in

life. What is the likely e↵ect of such tests on the eciency of the health insurance

market? Explain briefly.

Reading for this question

N&S, Chapter 15.

Subject guide, Chapter 10.

Approaching the question

With precise information available, insurers would only want to insure people who do not have

the disease, and knowing this no-one would buy insurance. Wherever such information is

available, the corresponding insurance market would break down, making everyone (insurance

buyers and insurers) worse o↵.

Question 4

Jill purchases only two goods, apples (A) and oranges (R). The price of apples is 2

and the price of oranges is 4. Jill has an income of 56 and her utility function is

u(A, R) = min[A, 5R].

What bundle of apples and oranges should Jill purchase to maximise utility?

Reading for this question

N&S, Chapter 2.

Subject guide, Chapter 2.

Approaching the question

The budget constraint is 2A + 4R = 56. At the optimum A = 5R. Using this, 14R = 56, or

R = 4. Therefore, A = 20. The optimal bundle is 20 of A and 4 of R.

Question 5

There are 3 consumers of a public good. The demand functions of the consumers

are as follows:

P1 = 20 Q (Consumer 1)

P2 = 50 Q (Consumer 2)

P3 = 90 Q (Consumer 3)

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Examiners’ commentaries 2017

where Q is the quantity of public good and Pi is the price consumer i is willing to

pay. The public good can be produced at a constant marginal cost of 70, and there

are no fixed costs. Find the ecient level of production of the public good.

Reading for this question

N&S, Chapter 16.

Subject guide, Chapter 12.

Approaching the question

Q should be such that the total willingness to pay at Q equals 70:

20 Q + 50 Q + 90 Q = 70

which gives us 3Q = 90, implying Q = 30.

Question 6

Suppose the price of permits in an emissions trading market falls to zero. What

does this indicate about the e↵ectiveness of this market in reducing emissions?

Explain briefly.

Reading for this question

Subject guide, Chapter 12.

Approaching the question

This indicates that the overall cap is very high, so that no emission-generator demands any extra

permits beyond its initial allocation. The fact that the cap is very high implies that this market

is not e↵ective in reducing emissions.

Question 7

If a monopolist’s marginal revenue curve coincides with the market demand curve,

there is no deadweight loss from monopoly. Is this true or false? Explain your

answer.

Reading for this question

N&S, Chapter 11.

Subject guide, Chapter 8.

Approaching the question

This is true. The MR curve coincides with market demand if the monopolist is able to extract

the full willingness to pay for each unit (i.e. first-degree price discrimination). In this case, the

monopolist keeps increasing output so long as P > MC and hence stops where P = MC,

therefore optimally producing the competitive quantity. Hence there is no deadweight loss.

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Question 8

Suppose the inverse demand curve is given by P = 10 Q. This is shown in the

diagram below. At the point A shown in the diagram, is the demand elastic or

inelastic (with respect to price change)? Explain your answer.

Reading for this question

N&S, Chapter 3.

Subject guide, Chapter 2.

Approaching the question

At point A, demand is elastic. The price elasticity of demand is:

” = dQ

dP

P

Q.

The slope dQ/dP is 1 at all points, so we simply have:

” = P

Q.

At point A, P >Q, so ” > 1.

Section B

Answer THREE questions from this section (20 marks each).

Question 9

(a) Suppose there are two identical firms in an industry who compete by setting

quantities. The output of firm 1 is denoted by q1 and that of firm 2 is denoted

by q2. Each firm faces a constant marginal cost of 30. Let Q denote total

output, i.e. Q = q1 + q2. The inverse demand curve in the market is given by

P = 300 2Q.

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Examiners’ commentaries 2017

i. Find the Cournot–Nash equilibrium quantity produced by each firm and the

market price.

(5 marks)

ii. What would be the quantities produced by each firm and market price under

Stackelberg duopoly if firm 1 moves first?

(5 marks)

(b) Suppose two firms (1 and 2) sell di↵erentiated products and compete by setting

prices. The demand functions are

q1 = 2 P1 +

P2

2

and

q2 = 2 P2 +

P1

2 .

Firms have a zero cost of production.

i. Find the Nash equilibrium in the simultaneous-move game. Also find the

quantities sold by each firm.

(5 marks)

ii. Find the subgame-perfect equilibrium if 1 moves before 2. Also find the

quantities sold by each firm.

(5 marks)

Reading for this question

N&S, Chapter 12.

Subject guide, Chapter 9.

Approaching the question

(a) i. Firm 1 maximises:

⇡1 = (300 2(q1 + q2) 30)q1 = (270 2(q1 + q2))q1.

The first-order condition is:

270 4q1 2q2 = 0

which implies:

q1 = 135 q2

2 .

Imposing symmetry, q1 = q2 = q⇤ where:

q⇤ = 135 q⇤

2

which implies 3q⇤/2 = 135/2, which in turn implies:

q⇤ = 45.

Therefore, q1 = q2 = 45 and the market price is 300 2(90) = 120.

ii. Now suppose firm 1 moves first. Using firm 2’s best response function, the profit of the

Stackelberg leader firm 1 is:

⇡S

1 =

✓

300 2q1 2

✓135 q1

2

◆

30◆

q1.

This simplifies to:

⇡S

1 = (135 q1)q1.

Maximising, q1 = 135/2 = 67.5. It follows that q2 = 135/4 = 33.75.

The market price is 300 2(3/4)135 = 195/2 = 97.5.

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(b) i. Firm 1 maximises ⇡1 = P1q1. Maximising, we get the best response function:

P1 =1+

P2

4 .

Imposing symmetry and solving, P1 = P2 = 4/3.

Also, q1 = q2 = 4/3.

This is not required for the answer, but note that each firm earns a profit of 16/9 ⇡ 1.78.

ii. Firm 1 maximises:

⇡1 =

✓

2 P1 +

1

2

✓

1 +

P1

4

◆◆ P1.

This simplifies to:

⇡1 =

✓5

2 7

8

P1

◆

P1.

Maximising, P1 = 10/7, q1 = 5/4.

For Firm 2, P2 =1+ P1/4 = 19/14, and q2 = 2 P2 + P1/2 = 19/14.

Again, not required for the answer, but note that ⇡1 = 25/14 ⇡ 1.79 and

⇡2 = (19/14)2 = 361/196 ⇡ 1.84. Therefore, the follower earns a greater profit. See the

subject guide for an intuition.

Question 10

(a) Consider the strategic-form game below with two players, 1 and 2.

Player 2

A2 B2 C2

A1 2, 2 4, 2 0, 4

Player 1 B1 4, 0 6, 8 2, 2

C1 6, 4 4, 0 X, 6

i. Set X = 0. Solve the game by iteratively eliminating strictly dominated

strategies.

(6 marks)

ii. Set X = 8. Eliminate any strictly dominated strategies and then find the

pure strategy and mixed strategy Nash equilibria of the game.

(6 marks)

(b) Consider the following extensive-form game with two players, 1 and 2.

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Examiners’ commentaries 2017

i. Find the pure-strategy Nash equilibria of the game.

(4 marks)

ii. Find the pure-strategy subgame-perfect equilibria of the game.

(4 marks)

Reading for this question

Subject guide, Chapter 4.

Approaching the question

(a) i. B1 dominates A1, and C2 dominates A2. After eliminating A1 and A2, in the reduced

game, B1 dominates C1. Eliminate C1. In the reduced game, B2 dominates C2.

Therefore, the game is dominance solvable and the equilibrium is B1, B2.

ii. B1 dominates A1, and C2 dominates A2. After eliminating A1 and A2, no further such

elimination can be done. In the remaining 2 ⇥ 2 game, the pure Nash equilibria are

B1, B2 and C1, C2.

Mixed-strategy Nash equilibrium: Suppose 1 plays B1 with probability p and C1 with

probability (1 p). Suppose 2 plays B2 with probability q and C2 with probability

(1 q). Therefore, p should be such that 8p = 2p + 6(1 p), which implies p = 1/2.

Further, q must be such that 6q + 2(1 q)=4q + 8(1 q) which implies q = 3/4.

Therefore, the mixed-strategy Nash equilibrium is given by p = 1/2, q = 3/4.

(b) i. For player 1, the set of actions and set of strategies are the same: (a1, a2, a3).

For player 2, there are 4 actions b1, b2, c1, c2. Strategies: Note that at the information set

player 2 can only choose either b1 or b2 (cannot play di↵erent actions at di↵erent nodes

of an information set). Hence there are two places (one information set and one node) at

which player 2 can move, with player 2 actions in each case, implying 4 strategies. The

strategy set is (b1c1, b1c2, b2c1, b2c2).

To identify the pure-strategy Nash equilibria, write down the normal form:

Player 2

b1c1 b1c2 b2c1 b2c2

a1 0, 1 0, 1 1, 3 1, 3

Player 1 a2 2, 2 2, 2 0, 1 0, 1

a3 1, 1 2, 0 1, 1 2, 0

Pure-strategy Nash equilibria are (a2, b1c1), (a2, b1c2) and (a1, b2c1).

ii. Next, identify the pure-strategy subgame-perfect Nash equilibria. The whole game is

always a subgame. Apart from this, there is only 1 strict subgame: starting from player

2’s decision node after player 1 plays a3. Note that you cannot cut an information set or

start a subgame from an information set, therefore no other strict subgames exist.

In the subgame, the optimal choice is c1. Therefore, any Nash equilibrium of the whole

game that involves playing c1 after a3 is subgame perfect. It follows that the

pure-strategy subgame-perfect equilibria are (a2, b1c1) and (a1, b2c1).

Question 11

A seller sells a good of quality q at a price t. The cost of producing at quality level q

is given by q2. There is a buyer who receives a utility of ✓q t by consuming the

unit of quality q at price t. If he decides not to buy, he gets a utility of zero. ✓ can

take two values ✓1 = 2 and ✓2 = 4.

(a) Suppose the seller can observe ✓. Derive the profit maximising price-quality

pairs o↵ered when ✓ = 2 and when ✓ = 4.

(6 marks)

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(b) Prove that the full information price-quality pairs are not incentive compatible

if the seller cannot observe ✓.

(7 marks)

(c) Suppose the seller cannot observe ✓, and suppose he decides to set q1 = 1/2 and

q2 = 2. Calculate the optimal values of t1 and t2 such that both types

participate, type ✓ = 1 takes the contract (q1, t1) and type ✓ = 2 takes the

contract (q2, t2).

(7 marks)

[Hint: write down the participation constraint of type ✓1 and the incentive

constraint of type ✓2 and solve for t1 and t2.]

Reading for this question

N&S, Chapter 15.

Subject guide, Chapter 10.

Approaching the question

(a) The seller will just satisfy the participation constraint of a buyer, so that the seller

maximises t q2 subject to ✓q = t. Substituting the value of t, the seller maximises ✓q q2,

so that the optimal choice is given by 2q = ✓. So we have q = ✓/2.

Here ✓1 = 2 and ✓2 = 4. Therefore, the optimal choice is q1 = ✓1/2 = 1, and q2 = ✓2/2 = 2.

Further, t1 = ✓1q1 = 2 and t2 = ✓2q2 = 8.

(b) The incentive constraint of type ✓2 is:

✓2q2 t2 ✓2q1 t1.

Here, the left-hand side is 0 while the right-hand side is 4 2 = 2. Therefore, the incentive

constraint of type ✓2 is violated.

(c) The participation constraint of type ✓1 and the incentive constraint of type ✓2 bind. So we

have:

✓1q1 t1 = 0 PC1

✓2q2 t2 = ✓2q1 t1 IC2.

From these, t1 = ✓1q1 = 2 ⇥ (1/2) = 1, and t2 = ✓2q2 ✓2q1 + t1 = 4 ⇥ 2 4 ⇥ (1/2) + 1 = 7.

This is not required for the answer, but if you want you can check that the other two

constraints do not bind.

First, the participation constraint of type ✓2 requires ✓2q2 t2 0. Here, the left-hand side

is 8 7 = 1. Therefore, this constraint does not bind.

Second, the incentive constraint of type ✓1 requires:

✓1q1 t1 ✓1q2 t2.

Here, the left-hand side is 0 and the right-hand side is 4 7 = 3. Therefore, this does not

bind as well.

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Examiners’ commentaries 2017

Question 12

Consider a competitive industry with several identical firms. Entry and exit are

possible in the long-run. The long-run cost function of a firm and the market

demand function are as follows:

Q = 127 P (Market demand)

C(q) = 10q 2q2 + q3

3 (Long-run cost function of a firm)

(a) Derive the long-run supply function of the representative firm, paying proper

attention to the shut-down decision of the firm.

(5 marks)

(b) Derive the long-run market supply function.

(5 marks)

(c) How many firms would operate in the market in the long-run?

(5 marks)

(d) To boost consumer demand, the government announces a per-unit subsidy of 6

to be given to consumers. Calculate the social cost of this policy and also show

it in a diagram.

(5 marks)

Reading for this question

N&S, Chapters 8 and 9.

Subject guide, Chapter 6.

Approaching the question

(a) The long-run supply is given by P = MC for P LRACmin.

Here:

LRMC = 10 4q + q2

and:

LRAC = 10 2q + q2/3.

The first-order condition for a minimum is 2+2q/3 = 0 (note that the second derivative is

positive implying a minimum), which gives q = 3.

At q = 3, the AC is 10 6 + 3 = 7.

Therefore, the long-run supply curve of a firm is given by:

P =

(

10 4q + q2 for P 7

0 for P < 7.

(b) With entry and exit, the long-run price equals the minimum LRAC. Therefore, the long-run

market supply curve is simply:

P = LRACmin.

Here, the long-run market supply curve is:

P = 7.

(c) Equating supply with demand, we get:

7 = 127 Q

implying that the equilibrium quantity in the market is 120.

From part (a) we know that each firm produces 3 units. It follows that there are 40 firms

operating in the market equilibrium.

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(d) With a subsidy of 6, the demand curve becomes:

P = 127 Q + 6 = 133 Q.

Equating supply with this new demand, we get 7 = 133 Q or:

Q = 126.

As shown in the diagram, the social cost is (1/2)(126 120)6 = 18.

Question 13

Suppose u(W) = ln W for an agent. The agent faces the following gamble: with

probability 1/2 wealth is 10 and with probability 1/2 a loss occurs so that wealth

becomes 4. The agent can buy any amount of insurance: a coverage of X (to be

paid if the loss occurs) can be purchased by paying premium rX.

(a) Calculate the Arrow–Pratt measure of risk aversion for the agent.

(5 marks)

(b) Write down the expected utility of the agent after purchasing insurance

coverage X.

(5 marks)

(c) How much coverage X would the agent choose if r is set so that the insurer’s

payo↵ is zero?

(5 marks)

(d) How much coverage X would the agent choose if r = 2/3?

(5 marks)

Reading for this question

N&S, Chapter 4.

Subject guide, Chapter 3.

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Examiners’ commentaries 2017

Approaching the question

(a) The Arrow–Pratt measure of risk aversion for the agent is:

⇢ = 1/w2

1/w = 1

w .

(b) The expected utility after purchasing coverage X is:

1

2 ln[10 rX] + 1

2 ln[4 + X rX].

(c) If r is set so that the insurer’s payo↵ is zero, the insurance is fair. Since the agent is

risk-averse, he would buy full insurance. In this case, this means X = 6.

(d) Given any r, the optimal choice of X is given by the first-order condition:

1

2

1

10 rX (r) + 1

2

1

4 + (1 r)X (1 r)=0

which can be simplified to: r

10 rX = 1 r

4 + (1 r)X .

Using r = 2/3 and simplifying:

2

30 2X = 1

12 + X

which implies 24 + 2X = 30 2X, or 4X = 6, or X = 3/2.

Question 14

Rai spends her income on fuel for heating her house (H) and a composite of all

other goods (Y ). Her preferences are represented by the utility function

u(H, Y ) = H↵Y 1↵.

The price of the composite good is 1, and the price of heating fuel is p. Let M

denote Rai’s income.

(a) Derive Rai’s demand for H and Y .

(5 marks)

(b) Suppose M = 300 and ↵ = 2/3. Also suppose currently the unit price of fuel is

p = 20. The energy company o↵ers Rai the option to switch to a di↵erent tari↵.

Under the new tari↵, Rai must pay a fixed fee of 100 and then she can buy fuel

at a unit price of 10. Would Rai switch to the new tari↵? Explain.

(8 marks)

(c) The government decides to give Rai a heating fuel subsidy of s per unit. This

results in an increase in utility from u0 before the subsidy to u1 after the

subsidy. Would an equivalent variation in income cost the government less?

Explain using a suitable diagram.

(7 marks)

Reading for this question

N&S, Chapters 2 and 3.

Subject guide, Chapter 2.

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EC2066 Microeconomics

Approaching the question

(a) MUH

MUY

= p, which implies:

↵

1 ↵

Y

H = p.

Using this in pH + Y = M, we get:

H = ↵M

p

and:

Y = (1 ↵)M.

(b) At M = 300, ↵ = 2/3, p = 20, H = (2/3)(300/20) = 10. Therefore,

Y = 300 10 ⇥ 20 = 100. The basket consumed is 10 of H and 100 of Y .

Under the new tari↵, Rai would have 200 to spend on H and Y after paying the fixed fee. At

a unit price of 10 for H, the previous basket costs 10 ⇥ 10 + 100 = 200. So the new budget

line passes through the old basket. Since the new budget line is flatter as shown in the

diagram below, Rai can achieve a higher utility under the new tari↵. So she would switch.

Algebraically, at the old basket, MUH

MUY

is equal to p = 20, but price of H is now pb = 10, so

we have MUH

MUY

> pb. This implies that starting from the old basket, more H and fewer Y

would improve utility.

(c) An equivalent income boost would be less costly. Equivalent variation of a price fall is lower

than the expenditure on H after the price fall. The intuition is that the per-unit subsidy

distorts choice in favour of H, raising the total cost of the subsidy.

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EC2066 Microeconomics

Examiners’ commentaries 2017

EC2066 Microeconomics

Important note

This commentary reflects the examination and assessment arrangements for this course in the

academic year 2016–17. The format and structure of the examination may change in future years,

and any such changes will be publicised on the virtual learning environment (VLE).

Information about the subject guide and the Essential reading

references

Unless otherwise stated, all cross-references will be to the latest version of the subject guide (2016).

You should always attempt to use the most recent edition of any Essential reading textbook, even if

the commentary and/or online reading list and/or subject guide refer to an earlier edition. If

di↵erent editions of Essential reading are listed, please check the VLE for reading supplements – if

none are available, please use the contents list and index of the new edition to find the relevant

section.

Comments on specific questions – Zone B

Text: We use the following abbreviation:

• N&S – Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application

(Cengage Learning 2015) twelfth edition [ISBN9781133189039].

For each question, we point out the relevant sections from the subject guide. For most topics, there

are also references to the main text (N&S).

Candidates should answer ELEVEN of the following FOURTEEN questions: all EIGHT from

Section A (5 marks each) and THREE from Section B (20 marks each). Candidates are

strongly advised to divide their time accordingly.

If more questions are answered than requested, only the first answers attempted will be counted.

Section A

Answer all EIGHT questions from this section (5 marks each).

Question 1

The figure below shows the inter-temporal budget constraint. The endowment point

is (Y0, Y1). As the rate of interest increases, the budget constraint pivots around the

endowment point as shown. Using the figure, explain that a saver cannot become a

borrower if the rate of interest rises.

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Examiners’ commentaries 2017

Reading for this question

N&S Chapter 14.

Subject guide, Chapter 2.

Approaching the question

A saver consumes to the left of the point (Y0, Y1) on the original budget line (the solid black

line). A rise in the rate of interest would change the budget line as shown in the figure. To

become a borrower, an agent would have to consume to the right of (Y0, Y1) on the new budget

line (the dashed line). However, any such point was initially available, and was rejected in favour

of a point to the left of (Y0, Y1). Since the point initially chosen (on the solid budget line) is still

available, any point to the right of (Y0, Y1) on the dashed budget line cannot be optimal.

Therefore, a saver cannot become a borrower.

Question 2

Suppose the following game is repeated infinitely. The players have a common

discount factor 2 (0, 1). Suppose is very close to zero. Identify an equilibrium of

the repeated game in this case. (Simply identify what actions would be played every

period – there is no need to do any calculations.)

2

C D

1 C 2,2 0,3

D 3,0 1,1

Reading for this question

Subject guide, Chapter 4.

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EC2066 Microeconomics

Approaching the question

You should understand that playing (D, D) in each period (after every history) is an equilibrium

for any value of . Once you understand this, it is straightforward to answer the question.

Question 3

Suppose improvements in genetic testing technology enables insurance companies to

predict with high precision the diseases that any individual would develop later in

life. What is the likely e↵ect of such tests on the eciency of the health insurance

market? Explain briefly.

Reading for this question

N&S, Chapter 15.

Subject guide, Chapter 10.

Approaching the question

With precise information available, insurers would only want to insure people who do not have

the disease, and knowing this no-one would buy insurance. Wherever such information is

available, the corresponding insurance market would break down, making everyone (insurance

buyers and insurers) worse o↵.

Question 4

Jill purchases only two goods, apples (A) and oranges (R). The price of apples is 2

and the price of oranges is 4. Jill has an income of 56 and her utility function is

u(A, R) = min[A, 5R].

What bundle of apples and oranges should Jill purchase to maximise utility?

Reading for this question

N&S, Chapter 2.

Subject guide, Chapter 2.

Approaching the question

The budget constraint is 2A + 4R = 56. At the optimum A = 5R. Using this, 14R = 56, or

R = 4. Therefore, A = 20. The optimal bundle is 20 of A and 4 of R.

Question 5

The von Neumann–Morgenstern utility function of an agent is given by

u(W) = ln W.

What is the Arrow–Pratt measure of risk aversion for this agent?

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Examiners’ commentaries 2017

Reading for this question

Subject guide, Chapter 3.

Approaching the question

Since u(W) = ln W, then u0

(W)=1/W and u00(W) = 1/W2. It follows that:

⇢ = 1

W .

Question 6

Consider a competitive industry with entry and exit in the long run. Suppose the

input prices do not change as output changes in the long run. A per-unit tax on

suppliers does not change market quantity in the long run equilibrium. Is this true

or false? Explain your answer.

Reading for this question

N&S, Chapter 9.

Subject guide, Chapter 6.

Approaching the question

False. The long-run supply curve is horizontal. A per-unit tax would shift the long-run supply up

by the extent of the tax. So long as demand is not perfectly inelastic, this reduces the quantity in

market equilibrium and creates a deadweight loss.

Question 7

Chen consumes two goods, x and y and his utility function is

u(x, y) = ln x + y.

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EC2066 Microeconomics

Suppose the price of y is 1 and let p denote the (relative) price of x. Chen has an

income M > 1. Calculate the price elasticity and the income elasticity of demand

for x.

Reading for this question

N&S, Chapter 3.

Subject guide, Chapter 2.

Approaching the question

Let p be the relative price. We have:

MRS = 1

x = p

implying x = 1/p.

Here px is a constant, implying price elasticity of demand for x is 1. Further, income elasticity of

demand for x is clearly 0.

Question 8

In the signalling model of education, equilibria in which education acts as a signal of

ability would break down if the costs of acquiring education were equal for

individuals with di↵erent abilities. Is this true or false? Explain your answer.

Reading for this question

N&S, Chapter 15.

Subject guide, Chapter 10.

Approaching the question

This is true. Without a cost di↵erential, the level of education cannot be a separating device, as

a low-ability type can mimic a high-ability type implying that we cannot satisfy the incentive

constraints that must hold for separation. Therefore, any equilibrium would involve pooling,

implying that the equilibrium education level chosen by any worker has no informational value

about his type.

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Examiners’ commentaries 2017

Section B

Answer THREE questions from this section (20 marks each).

Question 9

(a) Suppose there are two identical firms in an industry who compete by setting

quantities. The output of firm 1 is denoted by q1 and that of firm 2 is denoted

by q2. Each firm faces a constant marginal cost of 30. Let Q denote total

output, i.e. Q = q1 + q2. The inverse demand curve in the market is given by

P = 300 2Q.

i. Find the Cournot–Nash equilibrium quantity produced by each firm and the

market price.

(5 marks)

ii. What would be the quantities produced by each firm and market price under

Stackelberg duopoly if firm 1 moves first?

(5 marks)

(b) Suppose two firms (1 and 2) sell di↵erentiated products and compete by setting

prices. The demand functions are

q1 = 2 P1 +

P2

2

and

q2 = 2 P2 +

P1

2 .

Firms have a zero cost of production.

i. Find the Nash equilibrium in the simultaneous-move game. Also find the

quantities sold by each firm.

(5 marks)

ii. Find the subgame-perfect equilibrium if 1 moves before 2. Also find the

quantities sold by each firm.

(5 marks)

Reading for this question

N&S, Chapter 12.

Subject guide, Chapter 9.

Approaching the question

(a) i. Firm 1 maximises:

⇡1 = (300 2(q1 + q2) 30)q1 = (270 2(q1 + q2))q1.

The first-order condition is:

270 4q1 2q2 = 0

which implies:

q1 = 135 q2

2 .

Imposing symmetry, q1 = q2 = q⇤ where:

q⇤ = 135 q⇤

2

which implies 3q⇤/2 = 135/2, which in turn implies:

q⇤ = 45.

Therefore, q1 = q2 = 45 and the market price is 300 2(90) = 120.

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EC2066 Microeconomics

ii. Now suppose firm 1 moves first. Using firm 2’s best response function, the profit of the

Stackelberg leader firm 1 is:

⇡S

1 =

✓

300 2q1 2

✓135 q1

2

◆

30◆

q1.

This simplifies to:

⇡S

1 = (135 q1)q1.

Maximising, q1 = 135/2 = 67.5. It follows that q2 = 135/4 = 33.75.

The market price is 300 2(3/4)135 = 195/2 = 97.5.

(b) i. Firm 1 maximises ⇡1 = P1q1. Maximising, we get the best response function:

P1 =1+

P2

4 .

Imposing symmetry and solving, P1 = P2 = 4/3.

Also, q1 = q2 = 4/3.

This is not required for the answer, but note that each firm earns a profit of 16/9 ⇡ 1.78.

ii. Firm 1 maximises:

⇡1 =

✓

2 P1 +

1

2

✓

1 +

P1

4

◆◆ P1.

This simplifies to:

⇡1 =

✓5

2 7

8

P1

◆

P1.

Maximising, P1 = 10/7, q1 = 5/4.

For Firm 2, P2 =1+ P1/4 = 19/14, and q2 = 2 P2 + P1/2 = 19/14.

Again, not required for the answer, but note that ⇡1 = 25/14 ⇡ 1.79 and

⇡2 = (19/14)2 = 361/196 ⇡ 1.84. Therefore, the follower earns a greater profit. See the

subject guide for an intuition.

Question 10

(a) Consider the strategic-form game below with two players, 1 and 2.

Player 2

A2 B2 C2

A1 2, 2 4, 2 0, 4

Player 1 B1 4, 0 6, 8 2, 2

C1 6, 4 4, 0 X, 6

i. Set X = 0. Solve the game by iteratively eliminating strictly dominated

strategies.

(6 marks)

ii. Set X = 8. Eliminate any strictly dominated strategies and then find the

pure strategy and mixed strategy Nash equilibria of the game.

(6 marks)

(b) Consider the following extensive-form game with two players, 1 and 2.

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Examiners’ commentaries 2017

i. Find the pure-strategy Nash equilibria of the game.

(4 marks)

ii. Find the pure-strategy subgame-perfect equilibria of the game.

(4 marks)

Reading for this question

Subject guide, Chapter 4.

Approaching the question

(a) i. B1 dominates A1, and C2 dominates A2. After eliminating A1 and A2, in the reduced

game, B1 dominates C1. Eliminate C1. In the reduced game, B2 dominates C2.

Therefore, the game is dominance solvable and the equilibrium is B1, B2.

ii. B1 dominates A1, and C2 dominates A2. After eliminating A1 and A2, no further such

elimination can be done. In the remaining 2 ⇥ 2 game, the pure Nash equilibria are

B1, B2 and C1, C2.

Mixed-strategy Nash equilibrium: Suppose 1 plays B1 with probability p and C1 with

probability (1 p). Suppose 2 plays B2 with probability q and C2 with probability

(1 q). Therefore, p should be such that 8p = 2p + 6(1 p), which implies p = 1/2.

Further, q must be such that 6q + 2(1 q)=4q + 8(1 q) which implies q = 3/4.

Therefore, the mixed-strategy Nash equilibrium is given by p = 1/2, q = 3/4.

(b) i. For player 1, the set of actions and set of strategies are the same: (a1, a2, a3).

For player 2, there are 4 actions b1, b2, c1, c2. Strategies: Note that at the information set

player 2 can only choose either b1 or b2 (cannot play di↵erent actions at di↵erent nodes

of an information set). Hence there are two places (one information set and one node) at

which player 2 can move, with player 2 actions in each case, implying 4 strategies. The

strategy set is (b1c1, b1c2, b2c1, b2c2).

To identify the pure-strategy Nash equilibria, write down the normal form:

Player 2

b1c1 b1c2 b2c1 b2c2

a1 0, 1 0, 1 1, 3 1, 3

Player 1 a2 2, 2 2, 2 0, 1 0, 1

a3 1, 1 2, 0 1, 1 2, 0

Pure-strategy Nash equilibria are (a2, b1c1), (a2, b1c2) and (a1, b2c1).

ii. Next, identify the pure-strategy subgame-perfect Nash equilibria. The whole game is

always a subgame. Apart from this, there is only 1 strict subgame: starting from player

2’s decision node after player 1 plays a3. Note that you cannot cut an information set or

start a subgame from an information set, therefore no other strict subgames exist.

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EC2066 Microeconomics

In the subgame, the optimal choice is c1. Therefore, any Nash equilibrium of the whole

game that involves playing c1 after a3 is subgame perfect. It follows that the

pure-strategy subgame-perfect equilibria are (a2, b1c1) and (a1, b2c1).

Question 11

A seller sells a good of quality q at a price t. The cost of producing at quality level q

is given by q2. There is a buyer who receives a utility of ✓q t by consuming the

unit of quality q at price t. If he decides not to buy, he gets a utility of zero. ✓ can

take two values ✓1 = 2 and ✓2 = 4.

(a) Suppose the seller can observe ✓. Derive the profit maximising price-quality

pairs o↵ered when ✓ = 2 and when ✓ = 4.

(6 marks)

(b) Prove that the full information price-quality pairs are not incentive compatible

if the seller cannot observe ✓.

(7 marks)

(c) Suppose the seller cannot observe ✓, and suppose he decides to set q1 = 1/2 and

q2 = 2. Calculate the optimal values of t1 and t2 such that both types

participate, type ✓ = 1 takes the contract (q1, t1) and type ✓ = 2 takes the

contract (q2, t2).

(7 marks)

[Hint: write down the participation constraint of type ✓1 and the incentive

constraint of type ✓2 and solve for t1 and t2.]

Reading for this question

N&S, Chapter 15.

Subject guide, Chapter 10.

Approaching the question

(a) The seller will just satisfy the participation constraint of a buyer, so that the seller

maximises t q2 subject to ✓q = t. Substituting the value of t, the seller maximises ✓q q2,

so that the optimal choice is given by 2q = ✓. So we have q = ✓/2.

Here ✓1 = 2 and ✓2 = 4. Therefore, the optimal choice is q1 = ✓1/2 = 1, and q2 = ✓2/2 = 2.

Further, t1 = ✓1q1 = 2 and t2 = ✓2q2 = 8.

(b) The incentive constraint of type ✓2 is:

✓2q2 t2 ✓2q1 t1.

Here, the left-hand side is 0 while the right-hand side is 4 2 = 2. Therefore, the incentive

constraint of type ✓2 is violated.

(c) The participation constraint of type ✓1 and the incentive constraint of type ✓2 bind. So we

have:

✓1q1 t1 = 0 PC1

✓2q2 t2 = ✓2q1 t1 IC2.

From these, t1 = ✓1q1 = 2 ⇥ (1/2) = 1, and t2 = ✓2q2 ✓2q1 + t1 = 4 ⇥ 2 4 ⇥ (1/2) + 1 = 7.

This is not required for the answer, but if you want you can check that the other two

constraints do not bind.

26

Examiners’ commentaries 2017

First, the participation constraint of type ✓2 requires ✓2q2 t2 0. Here, the left-hand side

is 8 7 = 1. Therefore, this constraint does not bind.

Second, the incentive constraint of type ✓1 requires:

✓1q1 t1 ✓1q2 t2.

Here, the left-hand side is 0 and the right-hand side is 4 7 = 3. Therefore, this does not

bind as well.

Question 12

Consider an exchange economy with two goods (x and y) and two consumers (A and

B). There are 10 units available of each of the two goods. Consumer A is endowed

with 6 units of x and 4 units of y. Consumer B is endowed with 4 units of x and 6

units of y. Consumer A has the following utility function:

UA(x, y) = min[x, y].

Consumer B has the following utility function:

UB(x, y) = x + y.

(a) Draw an Edgeworth box and show the area of mutually beneficial trades

between the two consumers.

(5 marks)

(b) Let the price of y be 1 and let p be the (relative) price of x. Derive the

equilibrium value of p.

(5 marks)

(c) Derive the equilibrium allocation.

(5 marks)

(d) If we pick an allocation on the contract curve that is di↵erent from the

equilibrium allocation you derived in part (d), is it possible to support this new

point as an equilibrium allocation? Explain.

(5 marks)

[Hint: Make use of the Second Welfare Theorem.]

Reading for this question

N&S, Chapter 10.

Subject guide, Chapter 7.

Approaching the question

(a) The two goods are perfect substitutes for consumer A and perfect complements for

consumer B.

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EC2066 Microeconomics

(b) The equilibrium price line must coincide with an indi↵erence curve of consumer B (the

relative price of x must be the same as MRS of B), and the equilibrium point must be such

that the price line, as well as the indi↵erence curve of B, just touch the kink-point of an

indi↵erence curve of A.

Here, MRS of B is 1. Hence we must have p = 1.

(c) For A, we have xA = yA at the optimum. A’s budget constraint is:

pxA + yA = 6p + 4.

This implies:

xA = yA = 6p + 4

1 + p .

Since p = 1, we have:

xA = yA = 5.

For B, the budget constraint is pxB + yB = 4p + 6. Given p = 1, any allocation such that

xB + yB = 10 is optimal. Market clearing implies:

xB = yB = 5.

(d) If you know the second welfare theorem, you should realise that this is possible. As the

second welfare theorem notes, this is possible by suitably redistributing the initial

endowment. In this case, consider a redistribution to the new point on the contract curve.

Therefore, this point can be supported as an equilibrium (by a price of p = 1).

28

Examiners’ commentaries 2017

Question 13

Suppose the demand curve for corn is Q = 60 5P and the supply curve is

Q = 10P .

(a) Suppose the government announces a per-unit subsidy of 3 to suppliers.

Calculate the impact of the subsidy on the equilibrium market price of corn and

the equilibrium quantity of corn.

(5 marks)

(b) Calculate the deadweight loss from the subsidy policy in part (a).

(5 marks)

(c) Suppose the government announces a price ceiling of 2. Calculate the

deadweight loss from this policy assuming that the quantity of corn supplied is

allocated to those consumers who value it the most.

(5 marks)

(d) Suppose the government can either impose a price ceiling of 2 or a quota of 20

units of corn. Which policy would consumers prefer? Which policy would

suppliers prefer? Explain using a suitable diagram.

(5 marks)

Reading for this question

N&S, Chapter 9.

Subject guide, Chapter 6.

Approaching the question

(a) Before the subsidy, 60 5P = 10P implies P = 4 and Q = 40. After the subsidy of 3 per

unit, the supply curve becomes:

P = Q

10 3

which implies:

Q = 10P + 30.

Equating with demand:

10P + 30 = 60 5P

which implies P = 2 and Q = 50. Therefore, the subsidy leads to a fall in price from 4 to 2

and an increase in quantity from 40 to 50.

(b) The deadweight loss (as shown in the diagram) is (1/2)(50 40)3 = 15.

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EC2066 Microeconomics

(c) At a price ceiling of 2, demand exceeds supply and the market quantity is the amount

supplied, which is 20. As the diagram shows, the deadweight loss from a price ceiling is

(1/2)(6)(20) = 60.

(d) The price ceiling of 2 transfers the shaded area of surplus (in the diagram) to consumers,

while a quota of 20 transfers this area of surplus to suppliers.

Question 14

Consider a society with 2 agents. Agent 1 takes an action a 2 [1/4, 1] and agent 2 is

also a↵ected by the action. Agents have quasilinear utility. Suppose good m is

money. Agent i obtains a utility of ui(a) + mi. We have

u1(a) = ln a

and

u2(a) = 2 a2.

Let a0 denote the socially optimal level of a, and let a⇤ denote the individually

optimal level for agent 1.

(a) Calculate a⇤ and a0.

(5 marks)

30

Examiners’ commentaries 2017

(b) Suppose the government imposes a proportional tax of t on a (so that the total

tax bill is t times a) on agent 1. Calculate the optimal value of t.

(5 marks)

(c) Suppose the government allocates the property right on the action to agent 1,

so that 1 can now ask 2 for a payment in order to reduce a from a⇤. 1 can

extract a payment of T from 2 and maximises u1(a) + T subject to 2 getting at

least as much utility as under no agreement.

Show that 1 would optimally adopt a0 in this case.

(5 marks)

(d) Would the adopted level of a be di↵erent if the property right is allocated to

agent 2? Explain.

(5 marks)

Reading for this question

Subject guide, Chapter 12.

Approaching the question

(a) u1 is increasing in a which implies a⇤ = 1. To get a0, maximise u1 + u2 with respect to a.

We have:

1

a 4a = 0

which implies a0 = 1/2.

(b) We need:

1

a t = 0

at a = 1/2. This implies t = 2.

(c) 1 can extract T from 2: 1 maximises u1(a) + T subject to u2(a) T = u2(a⇤). Substituting

the value of T, 1 maximises u1(a) + u2(a) u2(a⇤). This is maximised at a0.

(d) The same outcome. 2 maximises u2(a) + T subject to u1(a) T = u1(1/4). Again, the

solution is a0.

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