Examiners’ commentaries 2016
Examiners’ commentaries 2016
EC2066 Microeconomics
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. 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
different 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 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 different 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:
• efficiency and welfare optimality of perfectly and imperfectly competitive markets
• the effects of externalities and public goods on efficiency
• the effects of strategic behaviour and asymmetric information on efficiency
• the nature of policies and contracts aimed at improving welfare
•
be prepared for further courses which require a knowledge of microeconomics.
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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 Perloff. 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 guide.
Principal text
•
Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application
(Cengage Learning 2015) twelfth edition [ISBN 9781133189039].
Auxiliary texts
2
•
Perloff, 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].
Examiners’ commentaries 2016
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. The Examiners’ commentaries suggest ways of
addressing common problems and improving your performance. 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 may not cover all topics in the syllabus in the same depth, but you
need to be aware that the 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 in the section of the VLE dedicated to
each course. You should read the syllabus carefully and ensure that you cover sufficient 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 difficulties
when you sit the examination. We strongly advise you not to adopt this strategy.
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EC2066 Microeconomics
Examiners’ commentaries 2016
EC2066 Microeconomics
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. 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
different 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 abbreviations:
•
N&S – Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application
(Cengage Learning 2015) twelfth edition [ISBN 9781133189039].
•
Perloff – Perloff, J.M. Microeconomics with Calculus, Pearson Education, third edition
(global edition), 2014, [ISBN: 9780273789987].
For each question, we point out the relevant sections from the main text (N&S) as well as the
subject guide. Additional references from Perloff are provided for a few questions.
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.
Section A
Answer all EIGHT questions from this section (5 marks each).
Question 1
Consider the following simultaneous-move game with two players, 1 and 2. If 1
and/or 2 have any dominated strategies, eliminate them. Once you have done this,
consider the remaining game. In this remaining game, eliminate any dominated
strategies of 1 and/or 2 and so on. This method is called ‘iterated elimination of
dominated strategies’. Find the equilibrium using this method. Your answer must
show each round of elimination clearly.
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Examiners’ commentaries 2016
Player 1
A1
B1
C1
Player 2
A2
B2
2, 3 3, 1
4, 0 5, 1
3, 4 4, 2
C2
−1, 4
−2, −1
0, 3
Reading for this question
N&S Chapter 5; subject guide Chapter 4. In particular, for game theory, you should rely more
on the subject guide as the coverage in the text is not always ideal for this.
Approaching the question
C1 dominates A1 . After eliminating A1 , in the reduced game, A2 dominates C2 . After
eliminating C2 , in the reduced game, B1 dominates C1 . Eliminate C1 . In the reduced game, B2
dominates A2 . Therefore, using the method of iterated elimination of dominated strategies, we
get the equilibrium B1 , B2 .
Question 2
Consider a market for used cars. There are many sellers and even more buyers. A
seller values a high quality car at 1800 and a low quality car at 800. A buyer values
a high quality car at 2000 and a low quality car at 1000. All agents are risk-neutral.
Sellers know the quality of own car, but buyers only know that a proportion α of
the cars is low quality and the remaining proportion (1 − α) of the cars are high
quality. For what values of α do all sellers sell their used cars?
Reading for this question
N&S Chapter 15, section on ‘Market for Lemons’, pp. 508–9; subject guide Section 10.4.
Approaching the question
If all cars are offered for sale, buyers’ expected value is 1000α + 2000(1 − α) = 2000 − 1000α. We
need 2000 − 1000α ≥ 1800, which implies 1000α ≤ 200, i.e. α ≤ 1/5. For α ≤ 1/5, all cars are
sold at price 2000 − 1000α (which is at least 1800).
Question 3
u(x, y) = x2 + y 2 . The price of x is 3 and the price of y is 2. The consumer’s income
is 18. What is the optimal consumption bundle? [Hint: Compare interior bundles to
corner solutions.]
Reading for this question
N&S Chapter 2; subject guide Section 2.3.
Approaching the question
An indifference curve is given by x2 + y 2 = k, where k is any positive constant. Note that this is
the equation of a circle. Ignoring the quadrants where either x or y or both are negative, you
should see that the indifference curves are quarter-circles, as shown in the diagram below. You
should be able to see that the MRS is x/y which is increasing as x increases and y falls, giving
rise to non-convex indifference curves (normally, under decreasing MRS, the MRS should fall as
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EC2066 Microeconomics
x increases and y falls). There is no interior maximum in this case. Indeed, the interior tangency
point is in fact the point of minimum utility on the budget line – any other point on the budget
line yields a higher utility. You should see that the optimal consumption bundle lies at a corner.
Spending all 18 on x yields a utility of 36. Spending all on y yields 81. Therefore, the optimal
consumption bundle is to spend all income on y, implying buying 9 units of y.
Question 4
Suppose the market demand curve is infinitely elastic. In a diagram, show the
deadweight loss from a per-unit subsidy to producers. What causes the deadweight
loss to arise in this case?
Reading for this question
N&S Sections 9.9 and 9.10; subject guide Section 6.7.
Approaching the question
The figure below shows the deadweight loss from a subsidy. You should understand that the loss
arises because, after the subsidy, production extends to levels where the marginal benefit (given
by the demand curve) is lower than the marginal cost (given by the supply curve). Many
candidates identify a wrong area as the deadweight loss. Once you understand why the
deadweight loss arises, it should be straightforward to see what the correct area is in the diagram.
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Examiners’ commentaries 2016
Question 5
Suppose two goods x and y are perfect complements. Suppose the price of good x
falls. The change in demand for x due to the price change is entirely due to
substitution effect. Is this true or false? Explain your answer carefully.
Reading for this question
N&S Chapter 3; subject guide Section 2.4.
Approaching the question
This is false. This is one of the easiest questions in this year’s examination paper. It simply
requires you to understand that when two goods are perfect complements, you cannot substitute
one for the other. If you tried to answer this without reading the answer first, and wrote an
incorrect answer, it is likely that you are not well-prepared for the examination. The answer
requires you to realise that since the two goods are perfect complements, a substitution effect
cannot arise. It then follows that the entire price effect is due to the income effect.
Question 6
Jo consumes two goods x and y and her utility function is
u(x, y) = x + y.
Jo has an income of 8. Initially the price of x is 4 and the price of y is 2. Then price
of x falls to 1. Calculate the equivalent variation of the price change.
Reading for this question
Perloff Section 5.2; subject guide Section 2.4.4.
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EC2066 Microeconomics
Approaching the question
Initially all income is spent on y and u0 = 4. After the price change all income is spent on x and
u1 = 8. The equivalent variation (EV) is the extra income that would create the same change in
utility without changing the price. To raise utility to 8 at the initial prices, the consumer’s
income needs to rise to 16. Therefore, the EV is 16 − 8 = 8.
Question 7
Firms in a constant-cost industry (an industry in which input prices do not change
as industry output changes) do not experience diminishing returns to scale. Is this
true or false? Explain your answer carefully with the help of appropriate diagrams.
Reading for this question
N&S Chapters 6 and 7; subject guide Chapter 5.
Approaching the question
This is false. The constant-cost part refers to input prices not changing as industry output
changes. You should understand that this has nothing to do with the production technology of a
firm. It is entirely possible that the long-run average cost (AC) curve of a firm is U-shaped, so
that there is decreasing returns to scale to the right of the point of minimum AC. You should
draw appropriate diagrams (see, for example, Figure 5.4 in the subject guide) to clarify your
understanding of this point.
Question 8
A new gadget has arrived in the market. In the short run the demand for the
gadget is totally inelastic. In the long run, many competitors produce similar
gadgets so that demand becomes perfectly elastic. Suppose a tax of t per unit of
gadget purchased is imposed on consumers. What is the incidence of the tax (i.e.
who bears the tax) in the short-run and in the long-run?
Reading for this question
N&S Section 9.10; subject guide Section 6.7.1.
Approaching the question
In the short run, demand is perfectly inelastic so that the incidence is entirely on consumers. If
the short-run price without tax is ps , the price paid by consumers rises from ps to ps + t. Sellers
bear none of the tax. In the long run, demand is perfectly elastic, so that the incidence is entirely
on sellers. If the long-run price without tax is p` , the sellers receive p` − t rather than p` .
If you want to draw a diagram, note that the tax is on consumers, so does not affect the supply
curve. When demand is totally inelastic, it is a vertical line, which remains unchanged after tax
as does the market price received by sellers – just the price paid by consumers rises by the tax.
When demand is perfectly elastic, the total price consumers pay remains the same – the effective
demand curve facing suppliers now moves down by the extent of the tax.
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Examiners’ commentaries 2016
Section B
Answer THREE questions from this section (20 marks each).
Question 9
Suppose there are two identical firms in an industry. The output of firm 1 is denoted
by q1 and that of firm 2 is denoted by q2 . The cost function of firm 1 is given by
C(q1 ) = q12 /2 and the cost function of firm 2 is given by C(q2 ) = q22 /2. Let Q denote
total output, i.e. Q = q1 + q2 . The inverse demand curve in the market is given by
P = 420 − Q
(a) Find the Cournot–Nash equilibrium quantity produced by each firm and the
market price.
[5 marks]
(b) What would be the quantities produced by each firm and market price under
Stackelberg duopoly if firm 1 moves first?
[5 marks]
(c) Suppose the firms can collude and produce a total quantity Q. Suppose firm 1
produces a fraction α of Q (so q1 = αQ) and firm 2 produces a fraction (1 − α)
of Q (so q2 = (1 − α)Q). What value of α minimises total cost of producing Q?
[5 marks]
[Hint: Write the expression for C(q1 ) + C(q2 ) and minimise with respect to α.]
(d) Using the cost-minimising value of α from part (c), find the quantity produced
by each firm under collusion and the market price.
[5 marks]
Reading for this question
N&S Section 12.2; subject guide Section 9.3.
Approaching the question
(a) Firm 1 maximises profit given by P q1 − q12 /2 which is (420 − (q1 + q2 ))q1 − q12 /2. The
first-order condition for a maximum is:
420 − 2q1 − q2 − q1 = 0
from which we get the best response function of firm 1:
q1 =
q2
420 − q2
= 140 − .
3
3
At this point we can impose symmetry: q1 = q2 = q ∗ , and then solve for q ∗ . Alternatively,
we can write down firm 2’s best response function:
q2 = 140 −
q1
3
and solve the two equations in two unknowns. Solving, we get:
q1 = q2 =
3
× 140 = 105.
4
The total output is 210. Therefore, the market price is P = 420 − 210 = 210.
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EC2066 Microeconomics
(b) Firm 1 is the Stackelberg leader. Firm 1 maximises (420 − (q1 + q2 ))q1 − q12 /2 where
q2 = 140 − q1 /3. Substituting the value of q2 , firm 1 maximises:
q1 q2
420 − q1 + 140 −
q1 − 1
3
2
which simplifies to:
2
q2
280 − q1 q1 − 1 .
3
2
Maximising this, we get:
4
280 − q1 − q1 = 0
3
or:
7
q1 = 280
3
which implies:
q1 = 120.
Substituting in firm 2’s best response function, q2 = 100. The total output is 220, so that
the market price is P = 420 − 220 = 200.
(c) Firm 1 produces a fraction α of Q and firm 2 produces a fraction (1 − α) of Q. Therefore,
the total cost of production is:
(αQ)2
((1 − α)Q)2
Q2 2
+
=
(α + (1 − α)2 ).
2
2
2
We need to minimise α2 + (1 − α)2 with respect to α. The first-order condition is:
2α − 2(1 − α) = 0
which implies:
α = 1/2.
You should check that the second-order condition for a minimum holds.
(d) Using α = 1/2, the total cost of production is Q2 /4.
Therefore, the optimal profit given Q is:
(420 − Q)Q −
Q2
.
4
Maximising with respect to Q, we get:
420 − 2Q − Q/2 = 0
which implies 5Q/2 = 420 or Q = 168. Each firm produces half of this, i.e. 84.
The market price is 420 − 168 = 252.
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Examiners’ commentaries 2016
Question 10
Consider a second-hand car market with two kinds of cars: type A cars work with
probability 9/10 and break down after a while with probability 1/10, and type B
cars work with probability 1/2 and break down after a while with probability 1/2.
Other than the difference in probability of breaking down, type A and type B cars
are the same.
There are many car owners, and even more buyers. Car owners value a car that
works at 800 and buyers value a car that works at 1000. Both owners and buyers
have a zero value for a car that breaks down after a while. Both buyers and owners
are risk neutral. Finally each seller knows the type of own car, but this information
is not available to the buyers. The buyers only know that 1/2 the cars are type A
and thus break down with probability 1/10 and 1/2 are type B, which break down
with probability 1/2.
In answering the question, assume that the sellers get all the surplus.
(a) Given that quality is observable to sellers but not to buyers, which type(s) of
car(s) would be traded and at what price(s)?
[8 marks]
[Hint: Note that the value of a type A car to a buyer is (9/10)1000 = 900, and
the value of a type A car to a seller is (9/10)800 = 720. Similarly derive the
value of a type B car to a buyer and a seller. Then calculate the average value
of a car to buyers, which is 1/2 times the value of a type A car and 1/2 times
the value of type B car.]
(b) Is the market outcome efficient? Explain.
[4 marks]
(c) Suppose any seller can offer a guarantee, which is a contract that promises to
pay the buyer R if the car breaks down (no payment is made if the car does not
break down).
Find the range of values of R for which there is an equilibrium in which type A
cars sell at 900 with a guarantee and type B cars sell at 500 without a guarantee.
[8 marks]
[Hint: For such an equilibrium to exist, a seller with a type A car should want
to sell with a guarantee rather than not sell. Further, a seller with a type B car
must not have the incentive to offer a guarantee. Write down these constraints
and find the required range of R.]
Reading for this question
N&S Chapter 15, section on ‘Market for Lemons’ pp. 508–9; subject guide Section 10.4.
Approaching the question
(a) The values are as follows.
Type A
Type B
Seller
720
400
Buyer
900
500
The average value of buyers is 700. However, at prices up to 700 type A sellers would not
sell, and buyers know this, so they are prepared to pay at most 500 for a car. Since there are
more buyers than sellers, type B cars would be traded at a price of 500, and it is not
possible to have any trading of type A cars in equilibrium.
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EC2066 Microeconomics
(b) You should point out that there are gains from trade from type A cars. It then follows that
the market outcome is inefficient as gains from trade of type A cars are not realised.
(c) For a separating equilibrium, we need to satisfy the following constraints. The participation
constraint of type A is:
1
900 − R ≥ 720
(PCA ).
10
The incentive constraint of type B is:
1
500 ≥ 900 − R
2
(ICB ).
(Note that the incentive constraint of type A and participation constraint of type B do not
bind. The former is 900 − R/10 ≥ 500, which is weaker than PCA , and the latter is
500 ≥ 400.)
From the two constraints above, we get 800 ≤ R ≤ 1800.
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Examiners’ commentaries 2016
Question 11
(a) Find the pure-strategy and mixed-strategy Nash equilibria of the following
game.
[6 marks]
Player 1
A1
B1
Player 2
A2
B2
2, 1 −1, −1
1, 0
0, 4
(b) Find the pure-strategy subgame-perfect equilibria of the following game.
[6 marks]
(c) Suppose the following game is repeated infinitely. The players have a common
discount factor 0 < δ < 1.
Player 1
C
D
Player 2
C
D
4, 2 0, 4
5, 0 1, 1
i. Specify strategies that players can use to sustain cooperation (which implies
playing (C, C) in each period).
[4 marks]
ii. Find conditions on the discount factor under which cooperation can be
sustained as an equilibrium of the infinitely repeated game.
[4 marks]
Reading for this question
N&S Chapter 5; subject guide Chapter 4. In particular, for game theory, you should rely more
on the subject guide as the coverage in the text is not always ideal for this.
Approaching the question
(a) • Pure-strategy NE: (A1 , A2 ), (B1 , B2 ).
• Mixed-strategy NE: Suppose player 1 plays A1 with probability p and B1 with
probability (1 − p), and player 2 plays A2 with probability q and B2 with probability
(1 − q).
p must be such that player 2 is indifferent between A2 and B2 , so we need:
p = −p + 4(1 − p)
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EC2066 Microeconomics
which implies p = 2/3.
Similarly, q must be such that player 1 is indifferent between A1 and B1 , so we need:
2q − (1 − q) = q
which implies q = 1/2.
Therefore, the mixed-strategy NE is as follows. Player 1 plays A1 with probability 2/3
and B1 with probability 1/3, and player 2 plays A2 with probability 1/2 and B2 with
probability 1/2.
(b) Solve by backwards induction. At the left node, player 2 chooses b2 ; at the right node, either
b1 or b2 .
If player 2 plays b2 b1 (which means play b2 after a1 and b1 after a2 ), player 1’s best response
is a2 . If player 2 plays b2 b2 , player 1’s best response is a1 .
Therefore, the two pure-strategy subgame-perfect Nash equilibria are a2 , b2 b1 and a1 , b2 b2 .
(c) i. Trigger strategy for each player:
∗ Start by playing C (that is, cooperate at the very first period when there is no history
yet).
∗ In any period t > 1:
· if (C, C) was played last period, play C
· if anything else was played last period, play D.
ii. Let us see if these strategies will sustain cooperation. Since the payoffs are not
symmetric, we need to consider the two players separately.
Suppose player 1 deviates in period t. We only need to consider what happens from t
onwards. The payoff starting at period t is given by:
5 + δ + δ2 + · · · = 5 +
δ
.
1−δ
If player 1 did not deviate in period t, the payoff from t onwards would be:
4 + 4δ + 4δ 2 + · · · =
4
.
1−δ
For deviation to be suboptimal for player 1, we need:
4
δ
≥5+
1−δ
1−δ
which implies:
δ≥
1
.
4
Next, consider player 2. Doing similar calculations, we see that player 2 will not deviate
if:
2
δ
≥4+
1−δ
1−δ
which implies:
2
δ≥ .
3
Therefore, δ ≥ 2/3 is sufficient to ensure that no player would want to deviate from
cooperation. In other words, playing (C, C) always can be the outcome of an equilibrium
if the discount factor δ is at least 2/3.
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Examiners’ commentaries 2016
Question 12
Consider the following price support policy. The government announces a support
price PS for wheat. Farmers then choose how much wheat to produce, then sell
everything to consumers at the price at which everything can be sold (i.e. they set
price so that market demand for wheat equals total quantity of wheat produced).
Let this price be P ∗ . The government then pays farmers a deficiency payment equal
to (PS − P ∗ ) per unit on the entire quantity produced.
Suppose the demand curve for wheat is Q = 100 − 10P and the supply curve is
Q = 10P . Suppose the government announces a support price of PS = 6 using a
deficiency payment programme.
(a) Calculate the quantity supplied by farmers under the deficiency payment
programme, the price at which this output is sold to consumers, and the price
received by farmers.
[5 marks]
(b) Calculate the total deficiency payment made by the government.
[5 marks]
(c) Calculate the effect of the policy on consumer surplus and producer surplus.
[5 marks]
(d) Calculate the deadweight loss from the policy.
[5 marks]
Reading for this question
N&S Section 9.10; subject guide Section 6.7.
Approaching the question
(a) In the figure below, the initial equilibrium is E, where P = 5 and Q = 50. At a support
price of 6, total supply is 60. Output of 60 units can be sold in the market at a price of 4.
Therefore, consumers pay a price of 4, while the sellers get a price of 6 (the government pays
2 per unit to sellers over the quantity 60).
(Note that this is exactly like a subsidy of 2 given to sellers.)
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EC2066 Microeconomics
(b) This is easy if you have answered part (a) correctly. The deficiency payment is 60 × 2 = 120.
(c) Before the policy, the consumer surplus is (10 − 5)50/2 = 125 and the producer surplus is
5 × 50/2 = 125. Total surplus is 250 and there is no deadweight loss.
After the policy, the consumer surplus is the triangle ABC, which has area
(10 − 4)60/2 = 180. Producer surplus is the triangle FGH, which has area 6 × 60/2 = 180.
The spending on deficiency payment is 60 × 2 = 120, which is the rectangle GBCH.
(d) The shaded triangle EHC is the deadweight loss. The area is 10 × 2/2 = 10.
Another way to calculate the deadweight loss is to compare the total surplus before and
after the policy. Before the policy, the total surplus is 250. After, the total surplus is
180 + 180 − 120 = 240. Therefore, the deadweight loss is the difference, 10.
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Examiners’ commentaries 2016
Question 13
Suppose a monopolist can sell to two different customers with demand curves
Q1 = 24 − P1
Q2 = 24 − 2P2
The monopolist has a constant marginal cost of C = 6.
(a) Suppose the monopolist can charge different prices across customers. Calculate
the profit-maximising price-quantity pair for each customer.
[5 marks]
(b) Now suppose regulation forces the monopolist to charge the same price to both
customers. Calculate the profit-maximising price and the quantity sold to each
customer.
[5 marks]
(c) Does the regulation in part (b) reduce deadweight loss compared to the
situation in part (a)?
[5 marks]
(d) Suppose the monopolist must charge the same price to both customers, but can
charge a two-part tariff (a fixed fee and a price per unit). Calculate the optimal
two-part tariff.
[5 marks]
Reading for this question
N&S Section 11.4; subject guide Section 8.5.
Approaching the question
(a) Set MR = MC for each. The optimal price-quantity pairs are P1 = 15, Q1 = 9 and P2 = 9,
Q2 = 6.
(b) The aggregate demand is Q = Q1 + Q2 , which is:
Q = 48 − 3P.
Hence total revenue is Q(16 − Q/3). Therefore, MR = MC implies 16 − 2Q/3 = 6 which
implies Q = 15 and P = 11.
Under this price, Q1 = 13 and Q2 = 2.
Many candidates stopped here and declared the above as the answer. This amounts to
interpreting the question as saying that both customers must be served. While this is a
possible interpretation (and this is why candidates who stopped here did not lose any
marks), the question does not really say anything like this – leaving open the possibility that
the monopolist might choose also to serve just one customer.
To find the profit-maximising price-quantity pair, we need to compare the profit from
serving both customers to that from serving just one (it should be clear to you that in the
latter case just customer 1 is served).
If both customers are served, the profit is (11 − 6)15 = 75. If only customer 1 is served, the
profit is (15 − 6)9 = 81. Therefore, the profit-maximising price is 15, and the quantities are
Q1 = 9 and Q2 = 0.
(c) The answer here depends on how you interpreted the previous part. If you derived the
maximum as serving just customer 1, the answer to the question is straightforward. Since
customer 2 is not served, the entire surplus of customer 2 is wasted, while the surplus of
customer 1 is the same across (a) and (b). Therefore, regulation increases deadweight loss.
However, if your answer stopped after deriving the optimum when both customers are
served, the answer requires more calculation.
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EC2066 Microeconomics
• First, find the deadweight loss (DWL) in the scenario in part (a):
To find the efficient quantity, set P = MC in each market. In market 1, 24 − Q1 = 6
implies Q∗1 = 18, and in market 2, 12 − Q2 /2 = 6 implies Q∗2 = 12.
It follows that DWL1 = (15 − 6)(18 − 9)/2 = 81/2 and DWL2 = (9 − 6)(12 − 6)/2 = 9.
The total DWL is 99/2.
• Next, find the DWL in the scenario in part (b):
The efficient quantity is such that P = MC, i.e. 16 − Q/3 = 6 which implies Q∗ = 30.
Therefore, the DWL is (11 − 6)(30 − 15)/2 = 75/2 which is lower than the DWL before.
Therefore, if both customers are served, regulation does reduce the DWL.
(d) The figure below shows the two demand curves. The higher demand curve is that of
customer 1. Note that at any price P , the surplus of customer 2 is given by (12 − P )Q2 /2,
and the surplus of customer 1 is given by (24 − P )Q1 /2 (which is larger).
Case 1: Both customers served
The total demand at any price P is Q = Q1 + Q2 = 48 − 3P .
The inverse demand function of customer 2 is P2 = 12 − Q2 /2.
In this case, we must have P ≤ 12, and set the fee equal to the surplus of customer 2.
For any price P ≤ 12, the surplus of customer 2 is:
1
(12 − P )(24 − 2P ) = (12 − P )2 .
2
This fee is paid by both customers. Therefore, the monopolist’s profit is:
πboth = 2(12 − P )2 + (P − 6)(48 − 3P ).
Maximising, we get P = 9 and πboth = 81.
Case 2: Only customer 1 served
In this case, the monopolist should maximise customer 1’s surplus and extract it with a fee.
So set P = 6. Therefore, profit is:
π1 = (24 − 6)(24 − 6)/2 = 162.
Hence the optimal policy is to set a price of 6 and a fixed fee of 162.
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Examiners’ commentaries 2016
Question 14
Alice and her brother Bill support their elderly parents. There is a single good (call
it money). All consumptions are measured in units of money. Alice cares about the
number of units she consumes directly (denoted by yA ) and the number of units her
parents consume (denoted by x). Her utility function is
uA (x, yA ) = x1/3 yA
Similarly, Bill’s utility function is
uB (x, yB ) = x1/3 yB
where yB is his direct consumption. The parents’ consumption x is simply the sum
of the support contributions from Alice (xA ) and Bill (xB ), i.e.
x = xA + xB
Alice and Bill have an income of 42 each.
(a) Suppose Alice is unable to contribute anything toward the parents’ support, so
that Bill must provide for both his own consumption yB , and his parents’
consumption x. Determine Bill’s optimal choice of x and yB .
[5 marks]
(b) Now suppose Alice is also going to contribute toward the parents’ support.
Given any level of contribution from Bill, Alice chooses her optimal
contribution. Similarly, for any level of contribution by Alice, Bill chooses his
optimal contribution. What are their equilibrium contributions to support their
parents?
[10 marks]
(c) Does the equilibrium contributions in part (b) add up to the Pareto optimal
level of support? Explain informally (you do not need to derive the Pareto
optimal level of support).
[5 marks]
Reading for this question
For Cournot equilibrium, see N&S Section 12.2; subject guide Section 9.3. For public goods, see
N&S Section 16.5; subject guide Section 12.4.
Approaching the question
(a) This is a straightforward constrained optimisation problem. Maximise x1/3 yB subject to
x + yB = 42. At the optimum:
1 −2/3
x
(42 − x) = x1/3
3
∗
which implies 3x = 42 − x. Therefore, Bill’s optimal choice is x∗ = 10.5 and yB
= 31.5.
(b) You should realise that this is like solving for a Cournot equilibrium. The best response
function of Bill is given by:
max(xA + xB )1/3 (42 − xB )
xB
which implies:
1
1
(xA + xB )−2/3 (42 − xB ) = (xA + xB ) / 3
3
which simplifies to:
3(xA + xB ) = 42 − xB .
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EC2066 Microeconomics
Similarly, Alice’s best response function is:
3(xA + xB ) = 42 − xA .
Adding the two best response functions, 7(xA + xB ) = 84, i.e. xA + xB = 12. Using this in
each best response function gives us xA = xB = 6. This can be seen also from the diagram
below.
(c) The key point is that a contribution by each agent also benefits the other agent. In other
words, these are contributions to a public good, and therefore private provision leads to an
inefficient level of contribution. Alternatively, you might say that the contribution by one
agent generates a positive externality for the other, implying that the total equilibrium level
of contribution would be too low relative to the social optimum.
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Examiners’ commentaries 2016
Examiners’ commentaries 2016
EC2066 Microeconomics
Important note
This commentary reflects the examination and assessment arrangements for this course in the
academic year 2015–16. 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
different 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 abbreviations:
•
N&S – Nicholson, W. and C. Snyder, Intermediate Microeconomics and its Application
(Cengage Learning 2015) twelfth edition [ISBN 9781133189039].
•
Perloff – Perloff, J.M. Microeconomics with Calculus, Pearson Education, third edition
(global edition), 2014, [ISBN: 9780273789987].
For each question, we point out the relevant sections from the main text (N&S) as well as the
subject guide. Additional references from Perloff are provided for a few questions.
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.
Section A
Answer all EIGHT questions from this section (5 marks each).
Question 1
Consider the following simultaneous-move game with two players, 1 and 2. If 1
and/or 2 have any dominated strategies, eliminate them. Once you have done this,
consider the remaining game. In this remaining game, eliminate any dominated
strategies of 1 and/or 2 and so on. This method is called ‘iterated elimination of
dominated strategies’. Find the equilibrium using this method. Your answer must
show each round of elimination clearly.
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EC2066 Microeconomics
Player 1
A1
B1
C1
Player 2
A2
B2
2, 3 3, 1
4, 0 5, 1
3, 4 4, 2
C2
−1, 4
−2, −1
0, 3
Reading for this question
N&S Chapter 5; subject guide Chapter 4. In particular, for game theory, you should rely more
on the subject guide as the coverage in the text is not always ideal for this.
Approaching the question
C1 dominates A1 . After eliminating A1 , in the reduced game, A2 dominates C2 . After
eliminating C2 , in the reduced game, B1 dominates C1 . Eliminate C1 . In the reduced game, B2
dominates A2 . Therefore, using the method of iterated elimination of dominated strategies, we
get the equilibrium B1 , B2 .
Question 2
Consider a market for used cars. There are many sellers and even more buyers. A
seller values a high quality car at 1800 and a low quality car at 800. A buyer values
a high quality car at 2000 and a low quality car at 1000. All agents are risk-neutral.
Sellers know the quality of own car, but buyers only know that a proportion α of
the cars is low quality and the remaining proportion (1 − α) of the cars are high
quality. For what values of α do all sellers sell their used cars?
Reading for this question
N&S Chapter 15, section on ‘Market for Lemons’, pp. 508–9; subject guide Section 10.4.
Approaching the question
If all cars are offered for sale, buyers’ expected value is 1000α + 2000(1 − α) = 2000 − 1000α. We
need 2000 − 1000α ≥ 1800, which implies 1000α ≤ 200, i.e. α ≤ 1/5. For α ≤ 1/5, all cars are
sold at price 2000 − 1000α (which is at least 1800).
Question 3
u(x, y) = x2 + y 2 . The price of x is 3 and the price of y is 2. The consumer’s income
is 18. What is the optimal consumption bundle? [Hint: Compare interior bundles to
corner solutions.]
Reading for this question
N&S Chapter 2; subject guide Section 2.3.
Approaching the question
An indifference curve is given by x2 + y 2 = k, where k is any positive constant. Note that this is
the equation of a circle. Ignoring the quadrants where either x or y or both are negative, you
should see that the indifference curves are quarter-circles, as shown in the diagram below. You
should be able to see that the MRS is x/y which is increasing as x increases and y falls, giving
rise to non-convex indifference curves (normally, under decreasing MRS, the MRS should fall as
22
Examiners’ commentaries 2016
x increases and y falls). There is no interior maximum in this case. Indeed, the interior tangency
point is in fact the point of minimum utility on the budget line – any other point on the budget
line yields a higher utility. You should see that the optimal consumption bundle lies at a corner.
Spending all 18 on x yields a utility of 36. Spending all on y yields 81. Therefore, the optimal
consumption bundle is to spend all income on y, implying buying 9 units of y.
Question 4
Suppose the market demand curve is infinitely elastic. In a diagram, show the
deadweight loss from a per-unit subsidy to producers. What causes the deadweight
loss to arise in this case?
Reading for this question
N&S Sections 9.9 and 9.10; subject guide Section 6.7.
Approaching the question
The figure below shows the deadweight loss from a subsidy. You should understand that the loss
arises because, after the subsidy, production extends to levels where the marginal benefit (given
by the demand curve) is lower than the marginal cost (given by the supply curve). Many
candidates identify a wrong area as the deadweight loss. Once you understand why the
deadweight loss arises, it should be straightforward to see what the correct area is in the diagram.
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EC2066 Microeconomics
Question 5
Consider an oligopoly where firms compete by setting quantities. Can a Stackelberg
leader get a lower profit in equilibrium compared to its profit under a
Cournot–Nash equilibrium? Explain your answer carefully.
Reading for this question
N&S Section 12.2; subject guide Section 9.3.
Approaching the question
The crucial point that you must understand is that the Stackelberg leader is free to choose the
Cournot–Nash quantity, in which case the other firm’s best response is also to choose the
Cournot–Nash quantity, so that the Stackelberg leader would get the Cournot–Nash profit.
Therefore, the Stackelberg leader cannot possibly get a profit lower than the Cournot–Nash
profit – in fact, can only improve on this profit. Since a Stackelberg leader chooses a quantity
different from the Cournot–Nash quantity, it must be earning a higher profit.
Question 6
Data from a third-world country shows that as the price of potatoes rises,
consumption of rice falls. From this we can conclude that rice might be a Giffen
good. Is this true or false? Explain your answer carefully.
Reading for this question
N& S Section 3.3; subject guide Section 2.4.
Approaching the question
This is false. The cross-price substitution effect implies more rice consumption when potatoes
become relatively more expensive. However, here rice consumption falls, implying that this must
be driven by the income effect. The fall in income from the rise in price of potatoes induces the
24
Examiners’ commentaries 2016
consumption of rice to fall. This implies that rice is a normal good. Therefore, we can rule out
the possibility that rice is a Giffen good.
Question 7
A monopolist can only charge a single price to all customers. In the presence of
positive externalities from output, the inefficiency under such a monopoly must be
greater than that under a competitive market structure. Is this true or false?
Explain your answer carefully.
Reading for this question
For monopoly, see N&S Chapter 11; subject guide Chapter 8. For externalities, see N& S
Chapter 16; subject guide Section 12.3.
Approaching the question
This is true. Under positive externalities, even the competitive output is too low compared to
the social optimum. Since a monopolist produces less than a competitive industry, the statement
is true. Note that if the monopolist could capture consumer surplus by some price-discrimination
mechanism, the monopoly output may equal the competitive output. The question puts in the
statement restricting the monopolist to a single price to rule out such a possibility and to make
the answer unambiguous.
Question 8
The demand function in a market is given by Q = aP b where Q is quantity
demanded, P is price, a and b are constants where a > 0 and b < 0. Calculate the
price elasticity of demand.
Reading for this question
N& S Section 3.12; subject guide Section 2.4.2.
Approaching the question
If you know the definition of the price elasticity of demand, the answer is straightforward:
ε=
P
dQ P
= (abP b−1 ) b = b.
dP Q
aP
Note that ε is negative, as expected, since b < 0.
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EC2066 Microeconomics
Section B
Answer THREE questions from this section (20 marks each).
Question 9
Suppose there are two identical firms in an industry. The output of firm 1 is denoted
by q1 and that of firm 2 is denoted by q2 . The cost function of firm 1 is given by
C(q1 ) = q12 /2 and the cost function of firm 2 is given by C(q2 ) = q22 /2. Let Q denote
total output, i.e. Q = q1 + q2 . The inverse demand curve in the market is given by
P = 420 − Q
(a) Find the Cournot–Nash equilibrium quantity produced by each firm and the
market price.
[5 marks]
(b) What would be the quantities produced by each firm and market price under
Stackelberg duopoly if firm 1 moves first?
[5 marks]
(c) Suppose the firms can collude and produce a total quantity Q. Suppose firm 1
produces a fraction α of Q (so q1 = αQ) and firm 2 produces a fraction (1 − α)
of Q (so q2 = (1 − α)Q). What value of α minimises total cost of producing Q?
[5 marks]
[Hint: Write the expression for C(q1 ) + C(q2 ) and minimise with respect to α.]
(d) Using the cost-minimising value of α from part (c), find the quantity produced
by each firm under collusion and the market price.
[5 marks]
Reading for this question
N&S Section 12.2; subject guide Section 9.3.
Approaching the question
(a) Firm 1 maximises profit given by P q1 − q12 /2 which is (420 − (q1 + q2 ))q1 − q12 /2. The
first-order condition for a maximum is:
420 − 2q1 − q2 − q1 = 0
from which we get the best response function of firm 1:
q1 =
q2
420 − q2
= 140 − .
3
3
At this point we can impose symmetry: q1 = q2 = q ∗ , and then solve for q ∗ . Alternatively,
we can write down firm 2’s best response function:
q2 = 140 −
q1
3
and solve the two equations in two unknowns. Solving, we get:
q1 = q2 =
3
× 140 = 105.
4
The total output is 210. Therefore, the market price is P = 420 − 210 = 210.
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Examiners’ commentaries 2016
(b) Firm 1 is the Stackelberg leader. Firm 1 maximises (420 − (q1 + q2 ))q1 − q12 /2 where
q2 = 140 − q1 /3. Substituting the value of q2 , firm 1 maximises:
q1 q2
420 − q1 + 140 −
q1 − 1
3
2
which simplifies to:
2
q2
280 − q1 q1 − 1 .
3
2
Maximising this, we get:
4
280 − q1 − q1 = 0
3
or:
7
q1 = 280
3
which implies:
q1 = 120.
Substituting in firm 2’s best response function, q2 = 100. The total output is 220, so that
the market price is P = 420 − 220 = 200.
(c) Firm 1 produces a fraction α of Q and firm 2 produces a fraction (1 − α) of Q. Therefore,
the total cost of production is:
(αQ)2
((1 − α)Q)2
Q2 2
+
=
(α + (1 − α)2 ).
2
2
2
We need to minimise α2 + (1 − α)2 with respect to α. The first-order condition is:
2α − 2(1 − α) = 0
which implies:
α = 1/2.
You should check that the second-order condition for a minimum holds.
(d) Using α = 1/2, the total cost of production is Q2 /4.
Therefore, the optimal profit given Q is:
(420 − Q)Q −
Q2
.
4
Maximising with respect to Q, we get:
420 − 2Q − Q/2 = 0
which implies 5Q/2 = 420 or Q = 168. Each firm produces half of this, i.e. 84.
The market price is 420 − 168 = 252.
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EC2066 Microeconomics
Question 10
Consider a second-hand car market with two kinds of cars: type A cars work with
probability 9/10 and break down after a while with probability 1/10, and type B
cars work with probability 1/2 and break down after a while with probability 1/2.
Other than the difference in probability of breaking down, type A and type B cars
are the same.
There are many car owners, and even more buyers. Car owners value a car that
works at 800 and buyers value a car that works at 1000. Both owners and buyers
have a zero value for a car that breaks down after a while. Both buyers and owners
are risk neutral. Finally each seller knows the type of own car, but this information
is not available to the buyers. The buyers only know that 1/2 the cars are type A
and thus break down with probability 1/10 and 1/2 are type B, which break down
with probability 1/2.
In answering the question, assume that the sellers get all the surplus.
(a) Given that quality is observable to sellers but not to buyers, which type(s) of
car(s) would be traded and at what price(s)?
[8 marks]
[Hint: Note that the value of a type A car to a buyer is (9/10)1000 = 900, and
the value of a type A car to a seller is (9/10)800 = 720. Similarly derive the
value of a type B car to a buyer and a seller. Then calculate the average value
of a car to buyers, which is 1/2 times the value of a type A car and 1/2 times
the value of type B car.]
(b) Is the market outcome efficient? Explain.
[4 marks]
(c) Suppose any seller can offer a guarantee, which is a contract that promises to
pay the buyer R if the car breaks down (no payment is made if the car does not
break down).
Find the range of values of R for which there is an equilibrium in which type A
cars sell at 900 with a guarantee and type B cars sell at 500 without a guarantee.
[8 marks]
[Hint: For such an equilibrium to exist, a seller with a type A car should want
to sell with a guarantee rather than not sell. Further, a seller with a type B car
must not have the incentive to offer a guarantee. Write down these constraints
and find the required range of R.]
Reading for this question
N&S Chapter 15, section on ‘Market for Lemons’ pp. 508–9; subject guide Section 10.4.
Approaching the question
(a) The values are as follows.
Type A
Type B
Seller
720
400
Buyer
900
500
The average value of buyers is 700. However, at prices up to 700 type A sellers would not
sell, and buyers know this, so they are prepared to pay at most 500 for a car. Since there are
more buyers than sellers, type B cars would be traded at a price of 500, and it is not
possible to have any trading of type A cars in equilibrium.
28
Examiners’ commentaries 2016
(b) You should point out that there are gains from trade from type A cars. It then follows that
the market outcome is inefficient as gains from trade of type A cars are not realised.
(c) For a separating equilibrium, we need to satisfy the following constraints. The participation
constraint of type A is:
1
900 − R ≥ 720
(PCA ).
10
The incentive constraint of type B is:
1
500 ≥ 900 − R
2
(ICB ).
(Note that the incentive constraint of type A and participation constraint of type B do not
bind. The former is 900 − R/10 ≥ 500, which is weaker than PCA , and the latter is
500 ≥ 400.)
From the two constraints above, we get 800 ≤ R ≤ 1800.
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EC2066 Microeconomics
Question 11
(a) Find the pure-strategy and mixed-strategy Nash equilibria of the following
game.
[6 marks]
Player 1
A1
B1
Player 2
A2
B2
2, 1 −1, −1
1, 0
0, 4
(b) Find the pure-strategy subgame-perfect equilibria of the following game.
[6 marks]
(c) Suppose the following game is repeated infinitely. The players have a common
discount factor 0 < δ < 1.
Player 1
C
D
Player 2
C
D
4, 2 0, 4
5, 0 1, 1
i. Specify strategies that players can use to sustain cooperation (which implies
playing (C, C) in each period).
[4 marks]
ii. Find conditions on the discount factor under which cooperation can be
sustained as an equilibrium of the infinitely repeated game.
[4 marks]
Reading for this question
N&S Chapter 5; subject guide Chapter 4. In particular, for game theory, you should rely more
on the subject guide as the coverage in the text is not always ideal for this.
Approaching the question
(a) • Pure-strategy NE: (A1 , A2 ), (B1 , B2 ).
• Mixed-strategy NE: Suppose player 1 plays A1 with probability p and B1 with
probability (1 − p), and player 2 plays A2 with probability q and B2 with probability
(1 − q).
p must be such that player 2 is indifferent between A2 and B2 , so we need:
p = −p + 4(1 − p)
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Examiners’ commentaries 2016
which implies p = 2/3.
Similarly, q must be such that player 1 is indifferent between A1 and B1 , so we need:
2q − (1 − q) = q
which implies q = 1/2.
Therefore, the mixed-strategy NE is as follows. Player 1 plays A1 with probability 2/3
and B1 with probability 1/3, and player 2 plays A2 with probability 1/2 and B2 with
probability 1/2.
(b) Solve by backwards induction. At the left node, player 2 chooses b2 ; at the right node, either
b1 or b2 .
If player 2 plays b2 b1 (which means play b2 after a1 and b1 after a2 ), player 1’s best response
is a2 . If player 2 plays b2 b2 , player 1’s best response is a1 .
Therefore, the two pure-strategy subgame-perfect Nash equilibria are a2 , b2 b1 and a1 , b2 b2 .
(c) i. Trigger strategy for each player:
∗ Start by playing C (that is, cooperate at the very first period when there is no history
yet).
∗ In any period t > 1:
· if (C, C) was played last period, play C
· if anything else was played last period, play D.
ii. Let us see if these strategies will sustain cooperation. Since the payoffs are not
symmetric, we need to consider the two players separately.
Suppose player 1 deviates in period t. We only need to consider what happens from t
onwards. The payoff starting at period t is given by:
5 + δ + δ2 + · · · = 5 +
δ
.
1−δ
If player 1 did not deviate in period t, the payoff from t onwards would be:
4 + 4δ + 4δ 2 + · · · =
4
.
1−δ
For deviation to be suboptimal for player 1, we need:
4
δ
≥5+
1−δ
1−δ
which implies:
δ≥
1
.
4
Next, consider player 2. Doing similar calculations, we see that player 2 will not deviate
if:
2
δ
≥4+
1−δ
1−δ
which implies:
2
δ≥ .
3
Therefore, δ ≥ 2/3 is sufficient to ensure that no player would want to deviate from
cooperation. In other words, playing (C, C) always can be the outcome of an equilibrium
if the discount factor δ is at least 2/3.
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EC2066 Microeconomics
Question 12
A firm belonging to a competitive industry has the following long-run cost function
C(q) = 4q 3 − 8q 2 + 10q
where q denote the firm’s output.
(a) Suppose entry or exit of firms is not possible. In this case, derive the long-run
supply function of the firm.
[5 marks]
(b) Suppose market price of output changes from 5 to 10. Calculate the change in
the quantity supplied by the firm as a result of the price rise.
[5 marks]
(c) Now suppose entry and exit are possible. All firms in the industry and all
potential entrants are identical. Assume that the industry has constant costs:
input prices do not change as industry output changes. In this case derive the
long-run supply function of the competitive industry.
[5 marks]
(d) Consider the supply function identified in part (c). Suppose a tax of t per unit
is imposed on the firms. What is the long-run supply function of the industry in
this case? What is the incidence of the tax (i.e. who bears the tax)?
[5 marks]
Reading for this question
N& S Chapters 8 and 9; subject guide Chapter 6.
Approaching the question
(a) In this case the supply function is the part of MC above the minimum AC. AC is given by
4q 2 − 8q + 10. First find the minimum point of AC. Differentiating with respect to q, the
first-order condition is 8q = 8, implying q = 1.
At q = 1, AC = 6. Therefore, the supply function is P = MC for P ≥ 6, and supply is 0
otherwise. Hence the supply function is given by:
(
P = 12q 2 − 16q + 10 for P ≥ 6
q=0
for P < 6.
(b) At P = 5, q = 0. At P = 10, q is given by:
10 = 12q 2 − 16q + 10
which implies 4q(3q − 4) = 0, which implies q = 4/3.
Therefore, the output changes from 0 to 4/3.
(c) In a constant-cost industry, the supply function is flat at the lowest point of the AC curve.
Here, at the lowest point, AC is 6. Therefore, the long-run industry supply function is given
by:
P = 6.
(d) The lowest AC now rises to 6 + t. Therefore, the LR industry supply function is now given
by:
P = 6 + t.
Supply is perfectly elastic, so firms bear none of the tax. The incidence is entirely on the
consumers.
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Examiners’ commentaries 2016
Question 13
An agent spends her income on two goods, X and Y . Her preferences are
represented by the utility function
u(X, Y ) = min{2X, Y }.
The price of good Y is 1, and the price of X is p. Let M denote the agent’s income.
Suppose M = 100.
(a) Derive the agent’s demand for X and Y as functions of p.
[5 marks]
(b) Suppose initially p = 3, and then p rises to 8. How much of the change in
demand for X as a result of the price rise can be attributed to income effect
and how much to substitution effect?
[5 marks]
(c) Consider the rise in p from 3 to 8. Calculate the compensating variation of the
price change.
[5 marks]
(d) Consider the rise in p from 3 to 8. Calculate the equivalent variation of the
price change.
[5 marks]
Reading for this question
For demand curves, income and substitution effects, see N&S Chapter 3; subject guide Section
2.4. For compensating variation (CV) and equivalent variation (EV), see Perloff Section 5.2;
subject guide Section 2.4.4.
Approaching the question
(a) At the optimum 2X = Y . The budget constraint is pX + Y = M , which implies
pX + 2X = M , which implies:
100
M
X=
=
p+2
p+2
and:
Y =
200
2M
=
.
p+2
p+2
(b) At p = 3, X = 20. At p = 8, X = 10. Hence X decreases by 10. Since X and Y are
complements, there is no substitution possible. The substitution effect is therefore zero. The
entire change in demand is due to the income effect.
(c) CV is the extra income such that the original utility is preserved even after the price rise.
Now utility is simply 2X. At p = 3, X = 20, therefore utility is u0 = 2 × 20 = 40. Now, if
we raise income by the CV after the price rise, the demand for X becomes:
M + CV
p0 + 2
where p0 = 8. We need the utility at this X to be equal to u0 :
2(M + CV)
= u0
p0 + 2
This implies 2(100 + CV)/10 = 40, which implies CV = 100.
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EC2066 Microeconomics
(d) EV is the reduction in income needed to attain the same utility after the price rise, but
without changing the price. In other words, EV is the income reduction that has the same
impact as the price rise, but through a reduction in income rather than by a price rise.
Let u1 denote the utility after the price rise. After the price rise, X = 10. Utility is twice
this. So u1 is given by u1 = 2 × 10 = 20.
EV is taken away while price is kept at p, so:
X=
M − EV
.
p+2
Utility is twice this, so EV is such that:
2(M − EV)
= u1
p+2
which implies 2(100 − EV)/5 = 20, which implies EV = 50.
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Examiners’ commentaries 2016
Question 14
Consider a pure exchange economy with two agents, Ann and Bob, who consume
only two goods, x and y. In the economy there are 4 units of x and 2 units of y.
Ann is endowed with 1 unit of x and 2 units of y, and Bob is endowed with 3 units
of x and 0 units of y. Their utility functions are:
√
and
ub (xb , yb ) = xb + 2yb .
ua (xa , ya ) = xa ya
(a) Draw the Edgeworth box, showing the endowment point, and the indifference
curves.
[5 marks]
(b) Derive the equation of the contract curve by equating the MRS of the two
agents.
[5 marks]
(c) Calculate the equilibrium price ratio.
[5 marks]
(d) Calculate the equilibrium allocation.
[5 marks]
Reading for this question
N& S Section 10.7; subject guide Section 7.3.
Approaching the question
(a) You should understand that one of the utility functions has the Cobb–Douglas form, giving
rise to standard convex (i.e. diminishing MRS) indifference curves. The other one gives rise
to straight-line indifference curves. The diagram draws the Edgeworth box and shows the
endowment point, the indifference curves of the two agents and the contract curve. M is the
endowment point. The price line coincides with the indifference curve of B. E is the
equilibrium.
(b) The MRS of A is ya /xa , and the MRS of B is 1/2. Therefore, the contract curve is given by:
ya
1
= .
xa
2
This is shown in the diagram above.
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EC2066 Microeconomics
(c) Let p be the relative price of x (so the price of y is 1). The demand function of agent A can
be found as follows.
The budget constraint is:
pxa + ya = p + 2.
From MRS = p, we get ya /xa = p, or ya = pxa . Using this in the budget constraint, we get
the demand functions:
p+2
xa =
2p
and:
ya =
p+2
.
2
For B, MRS = 1/2. Therefore, in equilibrium we must have p = 1/2 (the price line must
coincide with the indifference curve of B through the endowment point as shown in the
diagram).
(d) The equilibrium allocation is as follows. Since p = 1/2, we get xa = 2.5 and ya = 1.25.
Market clearing implies xb = 1.5 and yb = 0.75.
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