By the end of this course, and having completed the Essential readings and activities, you should:

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1 Important note This commentary reflects the examination and assessment arrangements for this course in the academic year 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 Unless otherwise stated, all cross-references will be to the latest version of the subject guide (011). 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 refers 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 By the end of this course, and having completed the Essential readings and activities, you should: be able to define and describe: the determinants of consumer choices, including inter-temporal choices and those involving risk firms behaviour how firms behaviour differs in different market structures and may help to determine those structures how firms and households determine factor prices. 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 government policies aimed at improving welfare. be prepared for further units which require a knowledge of 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. 1

2 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 Morgan, Katz and Rosen as the principal text. In addition to this, you should practise questions from other texts. Two 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. You should make sure that you understand the key ideas covered in some detail in the subject guide. Principal text Wyn Morgan, Michael Katz and Harvey Rosen, Microeconomics (Boston, Mass.: Irwin/McGraw-Hill, 009) second edition [ISBN ]. Auxiliary texts Jeffrey M. Perloff, Microeconomics with Calculus (Pearson Education, 013) third edition [ISBN ]. Robert S. Pindyck and Daniel L. Rubinfeld, Microeconomics (Upper Saddle River, New Jersey: Prentice Hall/Pearson, 01) eighth edition [ISBN ].

3 Question spotting Many candidates are disappointed to find that their examination performance is poorer than they expected. This can be due to a number of different reasons and the Examiners commentaries suggest ways of addressing common problems and improving your performance. We want to draw your attention to one particular failing question spotting, that is, confining your examination preparation to a few question topics which have come up in past papers for the course. This can have very 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 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 this course. You should read the syllabus very 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 every topic on the syllabus is a legitimate examination target. So although past papers can be helpful in 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 paper. We strongly advise you not to adopt this strategy. 3

4 Important note This commentary reflects the examination and assessment arrangements for this course in the academic year 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 Unless otherwise stated, all cross-references will be to the latest version of the subject guide (011). 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 refers 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: M,K& R - Wyn Morgan, Michael Katz and Harvey Rosen, Microeconomics, McGraw-Hill, second edition, 009, ISBN: For each question, we point out the relevant sections from the main text (M,K & R) as well as the subject guide. Candidates should answer ELEVEN of the following FOURTEEN questions: all EIGHT of Section A (5 marks each) and THREE from Section B (0 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. 1

5 Section A Answer all eight questions in this section (5 marks each). Question 1 Consider the following simultaneous-move game with two players, 1 and. If 1 and/or 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 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. Player A B C A 1, 4, 0,4 Player 1 B 1 4,0 6,8, C 1 6,4 4,0 0,6 The coverage of game theory in M,K& R (Chapter 16) is not ideal. See Chapter 9 (Game theory: an introduction) of the subject guide for a detailed discussion. If you know what a dominated strategy is, this is easy. However, many candidates seem to confuse dominated strategy calculation with best response calculation. When you say for player 1 strategy B 1 dominates strategy A 1 that means the payoff of 1 from playing B 1 is better than that from playing A 1 no matter what player does. In other words, for every box in the row B 1, the first number (1 s payoff) must be higher than the first number in the box above. As you can see, this is true here. So we can conclude that B 1 dominates A 1. A best response, on the other hand, refers to the best response against a specific strategy. For player 1, the best response to A is C 1. This sort of calculation is not useful here we want to compare all possible payoffs of one strategy against all possible payoffs of another to determine whether one of these dominates the other. Once you understand the above, the answer is straightforward. Here B 1 dominates A 1, and C dominates A. So eliminate A 1 and A. After eliminating A 1 and A, we have the following reduced game: Player B C Player 1 B 1 6,8, C 1 4,0 0,6 In this game, B 1 dominates C 1. Eliminate C 1. We now have the further reduced game: Player B C Player 1 B 1 6,8, In the game above, B dominates C. Eliminate C. We are then left with B 1, B. Therefore, using the method of iterated elimination of dominated strategies, we get the equilibrium B 1, B.

6 Question Consider an exchange economy with two goods (milk and honey) 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 milk and 4 units of honey. Consumer B is endowed with 4 units of milk and 6 units of honey. Let M denote units of milk and H denote units of honey. Consumer A has the following utility function: U A (M, H) = min[m, H] Consumer B has the following utility function: U B (M, H) = M + H Draw an Edgeworth box and show the area of mutually beneficial trades between the two consumers. M,K& R Chapter 1; subject guide Chapter 11 (General equilibrium and welfare economics) Milk and honey are perfect substitutes for consumer A and perfect complements for consumer B. You should know how to draw indifference curves for these preferences. The area of mutually beneficial trades is given by the shaded area EYZ. The question does not ask you this, but let us also understand where the equilibrium trades would lie in this box. Any equilibrium must be on the contract curve. So equilibrium trades lie somewhere on the part of the contract curve contained in the area EYZ. This is shown in the picture as the line ZA. 3

7 Question 3 A monopolist has a well-defined supply function. Is this true or false? Explain your answer. M,K& R Chapter 13; subject guide Chapter 8 (Competition and monopoly). Many candidates answered this question by saying that the statement is false because profit maximisation by the monopolist gives rise to a deadweight loss. You should understand that the presence or absence of deadweight loss has nothing to do with presence or absence of a supply function. Consider, for example, a firm in a competitive market. We know that a competitive firm has a well-defined supply function. Of course, we also know that there is no deadweight loss in the market. But now suppose the government imposes a per-unit tax. You know that this would distort the market equilibrium and give rise to a deadweight loss. But the supply function is still well-defined. So what is the answer? The crucial point is that a monopolist s optimisation is not independent of the market demand curve. Given a market demand (which is the average revenue curve), the monopolist chooses the best point on this curve (the point where marginal revenue equals marginal cost). So if demand changes, the monopolist s optimal price-quantity bundle changes. Therefore, for a monopolist, there is no unique relationship between price and quantity supplied. Therefore a monopolist does not have a well-defined supply function. The statement is false. Question 4 A bond pays a fixed sum of 100 per year for ever (i.e. the bond is a perpetuity). The annual interest rate is 5%. What is the maximum amount an agent should pay to buy this bond? M,K& R Chapter 5.3; subject guide Chapter 5 (Saving, investment and choice over time). The discount factor is δ = 1 where r is the interest rate. The maximum price P should equal 1 + r the present value of payments received from next period onwards: P = δ100 + δ = δ 1 δ 100. Using the value of δ, P = 100/r. Since r = 0.05 = 1/0, P = 000. Question 5 In 010, price of petrol was 10 and Kara purchased 600 units over the year. In 013, price of petrol was 15 and Kara consumed 800 units over the year. It follows that Kara s demand function for petrol is upward sloping. Is this true or false? Explain your answer. 4 M,K& R Chapter 3; subject guide Chapter 3 (Consumer theory).

8 This is false. From the data it does not follow that the demand is upward sloping. Another possible explanation (and a much more likely one) is that the demand curve is downward sloping as usual, but the demand shifted up between 010 and 013. This could happen because of a variety of factors. For example, Kara s income might have increased over the years and petrol is a normal good, or Kara s preferences for driving might have changed so that at any price of petrol she used more petrol in 013, or the price of alternative means of transport might have increased forcing a substitution towards greater car use. Question 6 The compensated (Hicksian) demand curve for a Giffen good is upward sloping. Is this true or false? Explain your answer. M,K& R Chapter 4; subject guide Chapter 3 (Consumer theory). This is an easy question. You should know that only the substitution effect matters for the compensated demand curve. Since the substitution effect is always negative, the compensated (Hicksian) demand curve is always downward sloping. The statement is therefore false. Question 7 There are 3 consumers of a public good. The demand functions of the consumers are as follows: P 1 = 60 Q (Consumer 1) (.1) P = 100 Q (Consumer ) (.) P 3 = 140 Q (Consumer 3) (.3) where Q is the quantity of public good and P i is the price consumer i is willing to pay, i {1,, 3}. The public good can be produced at a constant marginal cost of 180, and there are no fixed costs. Find the efficient level of production of the public good. M,K& R Chapter 18; subject guide Chapter 13 (Externalities and public goods). All you need to do here is to calculate the total willingness to pay at any quantity, and see at which quantity this equals 180. In other words, Q should be such that the total willingness to pay at Q equals 180: which gives us 3Q = 10 implying Q = Q Q Q = 180 5

9 Question 8 Identical firms in a competitive industry use capital (K) and labour (L) to produce output (Q). The labour supply curve is upward sloping, while the supply of capital is infinitely elastic in the long run. It follows that the long run industry supply curve is upward sloping. Is this true or false? Explain your answer. M,K& R Chapter 10.1; subject guide Chapter 8 (Competition and monopoly). This is true. While the cost of capital is constant, higher demand for labour implies a higher wage rate. Therefore this is an increasing cost industry. It follows that the long run supply curve is upward sloping. In the long run, the industry can produce higher output only at a higher price, which is needed to compensate for the rise in unit labour costs. Section B Answer three questions from this section (0 marks each). Question 9 Suppose there are two identical firms in an industry. The output of firm 1 is denoted by q 1 and that of firm is denoted by q. Each firm can produce output at a constant marginal cost of 6. There are no fixed costs. Let Q denote total output, i.e. Q = q 1 + q. The inverse demand curve in the market is given by P = 30 Q (a) Find the Cournot-Nash equilibrium quantity produced by each firm and the market price. (b) What would be the quantities produced by each firm and market price under Stackelberg duopoly if firm 1 moves first? (c) Calculate the deadweight loss arising from Cournot-Nash and Stackelberg duopoly. Which market structure is more efficient? (d) Suppose, as in part (b), firm 1 moves first and decides how much to produce. Firm moves second and makes its production decision. There is then a third stage at which firm 1 can change its mind about how much to produce and makes a final decision. Find the equilibrium quantities produced by the two firms under this three-stage game. M,K& R Chapter 15; subject guide Chapter 10 (Oligopoly and strategic behaviour). 6 (a) Firm 1 maximises: π 1 = (30 (q 1 + q ))q 1 6q 1.

10 The first order condition is 4 4q 1 q = 0, which gives us the best response function of firm 1: q 1 = 6 q. Similarly, the best response function of firm is: q = 6 q 1. Solving, q 1 = q = 4. The market price is then P = 30 8 = 14. (b) Firm s best response is as above. Firm 1 maximises in stage 1 knowing that will use this best response function at stage. Therefore firm 1 s problem is: max q 1 (30 (q q 1 ))q 1 6q 1. This simplifies to: max q 1 (1 q 1 )q 1. This gives us q 1 = 6. It follows that q = 3. The market price is P = 30 9 = 1. (c) Using the picture below, the deadweight loss under Cournot competition is: DWL C = 1 (14 6)(1 8) = 16. The deadweight loss under the Stackelberg market structure is: DWL S = 1 (1 6)(1 9) = 9. It follows that the Stackelberg market structure is more efficient. 7

11 (d) This is a little tricky. Try to see why a Stackelberg leader gets an advantage. This is because the first mover can choose an output it is committed to, and the second mover just plays a best response to that. It is the fact that the first mover can commit to an output before the follower has a chance to choose output is the source of the first mover advantage. Here, in contrast, firm 1 can change its mind about how much to produce and makes a final decision after decides. Therefore the initial stage has no commitment value. But can commit to an output at stage knowing that 1 will readjust and choose a best response at stage 3. It follows that Firm can now act like the Stackelberg leader, and 1 becomes the follower. Therefore, now q 1 = 3 and q = 6. Question 10 (a) Consider the following simultaneous-move game with two players, 1 and. Let p denote the probability with which player 1 plays A 1, where 0 p 1. B 1 is played with the residual probability. Next, let q denote the probability with which player plays A, where 0 q 1. B is played with the residual probability. Draw a picture with p along the horizontal axis and q along the vertical axis and draw the best response functions of players 1 and. Clearly label any equilibrium points in the picture. Player A B Player 1 A 1, 3,0 B 1 3,5 1,6 (6 marks) (b) Consider the following extensive-form game with two players. Player 1 can end the game by choosing Out. If player 1 chooses In, Player then chooses between L and R. The payoffs are written as (Payoff to 1, Payoff to ). Identify any subgame perfect Nash equilibrium. (6 marks) 8 (c) Suppose the following game is repeated infinitely. Players discount the future, so that, for each player, a payoff of x received t periods from today is worth δ t x today, where 0 < δ < 1. Show that it is possible to sustain cooperation (which in this case involves each player playing C every period) in the infinitely repeated game for high enough values of δ. (8 marks)

12 Player C D Player 1 C, 0,3 D 3,0 1,1 The coverage of game theory in M,K& R (Chapter 16) is not ideal. See Chapter 9 (Game theory: an introduction) of the subject guide for a detailed discussion. (a) There are no pure NE. The mixed Nash equilibrium is p = 1/3 and q = /3. The best response function of player 1 is given by: 0 if q > /3 p = [0, 1] if q = /3 1 if q < /3 The best response function of player is given by: 0 if p < 1/3 q = [0, 1] if p = 1/3 1 if p > 1/3 The picture below draws the best response functions and shows the mixed strategy Nash equilibrium. 9

13 (b) To solve for the subgame-perfect Nash equilibrium in this game, we can simply solve backwards. At s decision node, would choose R (which gives a payoff of 3 compared to a payoff of from L). Knowing this, 1 s best choice is to stay out (which gives a payoff of 0 compared to a payoff of -1 from choosing In. ). It follows that the SPNE is (Out, R). Some of you prefer to write s strategy as R if 1 plays In. That is fine as well, although since there is exactly one node at which chooses a move, the qualifier is unnecessary here. (c) Cooperation can be sustained in equilibrium if players are patient enough, i.e. δ close enough to 1. This can be seen as follows. Suppose each player follows the following strategy: play C to start with, and continue to play C so long as there is no deviation. After any deviation, play D always. Let us compare the payoff from conforming to the payoff from deviating. Consider a deviation in period t by player 1. Note that the payoff until t 1 plays no role in comparing deviation payoff with the payoff from conforming. The payoffs only differ staring at t, so we might as well just consider the payoff starting at period t. Since this is an infinitely repeated game, the players face an infinite future starting at any period t, and the nature of calculations is exactly the same no matter when you start the calculations. Given the strategy of each player specified above, the payoff from always cooperating (starting at t) is as follows. A player gets at t, at t + 1 which is worth δ at t and so on. So the payoff starting at period t from conforming is: + δ + δ + = 1 δ By deviating at period t, 1 gets 3 in period t, but then from t + 1 onwards each player plays D and so 1 gets a payoff of 1 every period. Therefore, the payoff at t from deviating at t is: 3 + δ + δ + = 3 + δ 1 δ Cooperation can be sustained if such a deviation is not profitable. Therefore to sustain cooperation we need 1 δ δ δ which implies δ 1/. Question 11 A monopolist can vary the quality of a good he produces. The cost of producing quality q is C(q) = q There are two types of consumers. From consuming a good of quality q 1 at price p 1, type 1 consumers get a utility of u 1 (q 1, p 1 ) = q 1 p 1 Type consumers get a higher marginal benefit from quality. From consuming a good of quality q at price p, type consumers get a utility of u (q, p ) = q p Consumers buy the good so long as they get at least 0 utility. The profit of the monopolist from a quality-price pair of (q i, p i ) where i {1, } is then given by π(q i, p i ) = p i q i (a) Suppose the monopolist knows the type of any consumer. In this case the monopolist produces a quality q 1 for a type 1 consumer and charges a price p 1. Similarly, type consumers are offered quality q at price p. For any i {1, }, the optimal pair (q i, p i ) maximizes π(q i, p i ) subject to u i (q i, p i ) = 0. Find the optimal quality-price pairs (q 1, p 1 ) and (q, p ). 10

14 (b) Suppose a consumer s own type is known only to the consumer. The monopolist cannot identify the type of any consumer. In this case, suppose the monopolist still offers the quality-price pairs (q1, p 1 ) and (q, p ) from part (a). Which quality-price pair would a type 1 consumer choose? Which pair would a type consumer choose? (c) Suppose the monopolist sets q 1 = 1/, p 1 = 1/ and q =. The monopolist wants to set p such that type consumers would have the incentive to choose (q, p ) rather than (q 1, p 1 ). What is the highest value of p that satisfies the monopolist s objective? (d) Given the values of q 1, q, p 1 and p from part (c), would a type 1 consumer have the incentive to choose (q 1, p 1 ) rather than (q, p )? M,K& R Chapter 17; subject guide Chapter 1 (Asymmetric information). Before trying to answer the question, it is useful to have a general understanding of its structure. Once you understand the idea behind the questions, it is very easy to do the calculations. In questions on asymmetric information, it is typical to first set the full information benchmark. We then compare the outcome under asymmetric information to the benchmark, allowing us to quantify the impact of asymmetric information. The first part of the question asks you to solve the monopolist s profit maximisation problem when there is no asymmetric information. The next part demonstrates that asymmetric information is a non-trivial problem: the price-quantity pairs offered under full information fail to be incentive compatible under asymmetric information. As the calculations below show, if the monopolist simply offers the full-information contracts under asymmetric information, type consumers gain by pretending to be type 1 consumers. If the monopolist wants to induce different types to accept different contracts, he must design these contracts so that no type has an incentive to accept the contract designed for the other type. In other words, contracts must be incentive compatible. Parts (c) and (d) are simple exercises based on this idea. (a) Maximise p 1 q 1 / subject to q 1 p 1 = 0. Using the constraint to substitute the value of p 1, the maximisation problem becomes: max q 1 q 1 q 1 which implies q 1 = 1 which in turn implies p 1 = 1. Similarly, maximise p q / subject to q p = 0. Using the constraint to substitute the value of p, the maximisation problem becomes: max q q q which implies q = which in turn implies p = 4. (b) A type 1 consumer gets 0 from (q1, p 1 ). The utility from (q, p ) is 4 < 0. Therefore a type 1 consumer chooses (q1, p 1 ). A type consumer gets 0 from (q, p ). The utility from choosing (q 1, p 1 ). is 1 > 0. Therefore a type consumer would switch to buying the quality q1 at price p 1 rather than choose the quality-price pair (q, p ). In other words, if the monopolist cannot observe the type of a consumer, but still offers the same price-quality pairs as in part (a), both types will choose (q 1, p 1 ). 11

15 (c) We need: Using the values given: q p q 1 p 1. 4 p 1 1 which implies: p 7. So the maximum value of p for which type would choose (q, p ) rather than (q 1, p 1 ) is given by 7/. (d) A type 1 consumer gets a utility of 0 from (q 1 = 1/, p 1 = 1/). The utility from (q =, p = 7/) is 7/ < 0. Therefore a type 1 consumer would indeed choose (q 1, p 1 ) rather than (q, p ). Question 1 Maya spends her income on chocolate (X) and a composite of all other goods (Y). Her preferences are represented by the utility function u(x, Y) = X + Y The price of the composite good is 1, and the price of chocolate is p. Let M denote Maya s income. (a) Derive Maya s demand for X and Y. (b) Suppose M = 10. Suppose initially the price of X is 1/, and then the price falls to 1/5. How much does Maya s demand for X change? How much of this change in demand can be attributed to income effect and how much to substitution effect? (c) Consider the change in price of X from 1/ to 1/5. Calculate the compensating variation of the price change. (d) For the change in price noted in part (c), is the equivalent variation greater than, equal to or less than the compensating variation? M,K& R Chapters 3 and 4; subject guide Chapter 3 (Consumer theory). (a) Here MU X = 1 X, and MU Y = 1. So at the optimum: 1 X = p which gives us: X = 1 p. 1

16 From the budget constraint, Y + px = M, so: Y = M 1 p. Clearly, X is independent of M. This is quasi-linear utility. There is no income effect on the demand for X. (b) Initially price is 1/ and X = 1 (1/) = 4. When price falls to 1/5, demand for X rises to 1 (1/5) = 5. As noted above, in this case there is no income effect on the demand for X. The entire price effect of 1 constitutes simply of substitution effect. (c) Given the optimal demands, the utility is u = /p + M 1/p = M + 1/p. At M = 10, p = 1/, u0 = 1. With a price fall, u 1 = = 15. If we now take away 3, utility falls back to u0. So CV is 3. (d) As noted above, there is no income effect. It follows that CV and EV are equal. You can also check this directly. If the price is not allowed to fall, utility is M + 1/p = 10 + = 1. If we now increase M by 3, we get 15, which is the utility after the price fall. Therefore EV = 3. Question 13 You are advising the minister responsible for housing. The minister is concerned that rents are too high and wants to reduce the rent payments by tenants without causing too much inefficiency. (a) Using appropriate diagrams explain the circumstances under which you would advise the minister to impose rent control. (7 marks) (b) Using appropriate diagrams explain the circumstances under which you would advise the minister to give a rent subsidy to tenants. (7 marks) (c) Using appropriate diagrams describe a case in which issuing more building permits would not be an effective policy. (6 marks) M,K& R Chapter 11; subject guide Chapter 8 (Competition and Monopoly). 13

17 (a) If supply is inelastic, rent control works well. As you can see from the picture, in this case the deadweight loss is small. 14

18 (b) If supply is elastic and demand is not very elastic, any subsidy would mostly go to tenants and the deadweight loss would be relatively small. 15

19 (c) If demand is very elastic (indicating consumers already have a lot of choice), raising supply would not reduce rents much. 16 Question 14 Consider a society with agents. Agent 1 takes an action a where Action a generates an utility of 0 < a 1 u 1 (a) = 1 a for agent 1. The action also affects agent. The utility of agent if 1 takes action a is given by u (a) = 4 a (a) Let a denote the individually optimal level of a for agent 1. Calculate a. (b) Let a 0 denote the socially optimal level of a. Calculate a 0 and compare to a from part (a). (c) Suppose the government imposes a proportional tax of t on a on agent 1. The utility function of agent 1 is now u 1 (a, t) = 1 a t a Let a t denote the after-tax individually optimal choice of a by 1. Calculate the value of t for which a t is equal to a 0. (d) Suppose the utility functions of both agents are known to each agent, but the government does not know the utility functions. In this case the tax policy of part (c) will not work. Can you suggest an alternative policy that the government can adopt that will result in a 0 being chosen by agent 1? (You only need to suggest an appropriate policy informally no calculations are needed.)

20 M,K& R Chapter 18; subject guide Chapter 13 (Externalities and public goods). (a) Note that u (a) = 1/(a ) > 0, i.e. u(a) is increasing in a. Therefore agent 1 s optimal choice is to choose the highest possible level of a. Thus a = 1. (b) Let a 0 denote the socially optimal level. To get the value of a 0, you need to maximise the u 1 (a) + u (a) = (1/a) 4a. The first order condition for maximum is 1/(a ) = 8a, and thus a 0 = 1/. (You can check that the second order condition is satisfied at this point.) Clearly, a 0 < a. You should, of course, expect this even before you do any calculations. Since agent s utility is decreasing in a, it is clear that 1 exerts a negative externality on. Therefore, the individual optimum calculated in part (a) must be higher than the social optimum. The calculations in parts (a) and (b) simply reflect this. (c) Agent 1 now maximises (1/a) ta. The first order condition is 1/(a ) = t. You need to set t so that this first order condition is satisfied at a = a 0. In other words, t must be such that 1/(a 0 ) = t. It follows that the optimal choice of t, denoted by t 0, is given by t 0 = 4. (d) The magic phrase here is Coase theorem. Hopefully, as soon as you read the question, this is the phrase that pops up in your mind. To answer the question, say that the government should ensure that property rights are well defined. From the Coase theorem, we know that once property rights are well defined, so long as there are no transactions costs, bargaining would lead to the socially optimal outcome. In this case, once property rights are defined, bargaining among the two agents would result in a 0 being chosen. This completes the answer. Note that it does not matter whether 1 or is given the property right. Agent 1 might have the right to choose a, or might have the right to prevent 1 from taking action a. In either case, the outcome would be a 0. The question does not ask you to show this formally, but let us do it here. First, suppose 1 has the right to choose a. In this case, if there is no bargaining between 1 and, 1 chooses a = 1. If wants 1 to lower the level of a, must pay 1. How much can 1 extract from? Suppose 1 charges T 1, and agrees to choose a. Then s utility is u (a) T 1. In the absence of any agreement, gets u (1). Therefore, the maximum value of T 1 is given by: u (a) T 1 = u (1). Now 1 maximises u 1 (a) + T 1 with respect to a subject to the above constraint. Substituting the value of T 1, 1 s problem is: max u 1 (a) + u (a) u (1). a Now, u (1) is just a constant. So we are simply maximising the sum of utilities. From part (b), you already know that the above maximisation problem leads to the socially optimal level a 0 being chosen. Next, suppose has the right to choose a. The question simply states a 1 > 0. But to do the calculations here, we need to define a minimum feasible level of a. Suppose the minimum feasible level of a is given by some ε > 0. In this case, in the absence of bargaining, would prevent 1 from the externality-generating activity by choosing a = ε. So the maximum payment T that can extract from 1 under bargaining is given by: u 1 (a) T = u 1 (ε). Agent maximises u (a) + T with respect to a subject to the above constraint. Substituting the value of T, s problem is: max u (a) + u 1 (a) u 1 (ε). a Now, u 1 (ε) is just a constant. As before, from part (b) you already know that the above maximisation problem leads to the socially optimal level a 0 being chosen. 17

21 Important note This commentary reflects the examination and assessment arrangements for this course in the academic year 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 Unless otherwise stated, all cross-references will be to the latest version of the subject guide (011). 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 refers 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: M,K& R - Wyn Morgan, Michael Katz and Harvey Rosen, Microeconomics, McGraw-Hill, second edition, 009, ISBN: For each question, we point out the relevant sections from the main text (M,K & R) as well as the subject guide. Candidates should answer ELEVEN of the following FOURTEEN questions: all EIGHT of Section A (5 marks each) and THREE from Section B (0 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. 1

22 Section A Answer all eight questions in this section (5 marks each). Question 1 Consider the following simultaneous-move game with two players, 1 and. If 1 and/or 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 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. Player A B C A 1, 4, 0,4 Player 1 B 1 4,0 6,8, C 1 6,4 4,0 0,6 The coverage of game theory in M,K& R (Chapter 16) is not ideal. See Chapter 9 (Game theory: an introduction) of the subject guide for a detailed discussion. If you know what a dominated strategy is, this is easy. However, many candidates seem to confuse dominated strategy calculation with best response calculation. When you say for player 1 strategy B 1 dominates strategy A 1 that means the payoff of 1 from playing B 1 is better than that from playing A 1 no matter what player does. In other words, for every box in the row B 1, the first number (1 s payoff) must be higher than the first number in the box above. As you can see, this is true here. So we can conclude that B 1 dominates A 1. A best response, on the other hand, refers to the best response against a specific strategy. For player 1, the best response to A is C 1. This sort of calculation is not useful here we want to compare all possible payoffs of one strategy against all possible payoffs of another to determine whether one of these dominates the other. Once you understand the above, the answer is straightforward. Here B 1 dominates A 1, and C dominates A. So eliminate A 1 and A. After eliminating A 1 and A, we have the following reduced game: Player B C Player 1 B 1 6,8, C 1 4,0 0,6 In this game, B 1 dominates C 1. Eliminate C 1. We now have the further reduced game: Player B C Player 1 B 1 6,8, In the game above, B dominates C. Eliminate C. We are then left with B 1, B. Therefore, using the method of iterated elimination of dominated strategies, we get the equilibrium B 1, B.

23 Question Consider an exchange economy with two goods (milk and honey) 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 milk and 4 units of honey. Consumer B is endowed with 4 units of milk and 6 units of honey. Let M denote units of milk and H denote units of honey. Consumer A has the following utility function: U A (M, H) = min[m, H] Consumer B has the following utility function: U B (M, H) = M + H Draw an Edgeworth box and show the area of mutually beneficial trades between the two consumers. M,K& R Chapter 1; subject guide Chapter 11 (General equilibrium and welfare economics). Milk and honey are perfect substitutes for consumer A and perfect complements for consumer B. You should know how to draw indifference curves for these preferences. The area of mutually beneficial trades is given by the shaded area EYZ. The question does not ask you this, but let us also understand where the equilibrium trades would lie in this box. Any equilibrium must be on the contract curve. So equilibrium trades lie somewhere on the part of the contract curve contained in the area EYZ. This is shown in the picture as the line ZA. 3

24 Question 3 A monopolist has a well-defined supply function. Is this true or false? Explain your answer. M,K& R Chapter 13; subject guide Chapter 8 (Competition and monopoly). Many candidates answered this question by saying that the statement is false because profit maximisation by the monopolist gives rise to a deadweight loss. You should understand that the presence or absence of deadweight loss has nothing to do with presence or absence of a supply function. Consider, for example, a firm in a competitive market. We know that a competitive firm has a well-defined supply function. Of course, we also know that there is no deadweight loss in the market. But now suppose the government imposes a per-unit tax. You know that this would distort the market equilibrium and give rise to a deadweight loss. But the supply function is still well-defined. So what is the answer? The crucial point is that a monopolist s optimisation is not independent of the market demand curve. Given a market demand (which is the average revenue curve), the monopolist chooses the best point on this curve (the point where marginal revenue equals marginal cost). So if demand changes, the monopolist s optimal price-quantity bundle changes. Therefore, for a monopolist, there is no unique relationship between price and quantity supplied. Therefore a monopolist does not have a well-defined supply function. The statement is false. Question 4 A bond pays a fixed sum of 100 per year for ever (i.e. the bond is a perpetuity). The annual interest rate is 5%. What is the maximum amount an agent should pay to buy this bond? M,K& R Chapter 5.3; subject guide Chapter 5 (Saving, investment and choice over time). The discount factor is δ = 1 where r is the interest rate. The maximum price P should equal 1 + r the present value of payments received from next period onwards: P = δ100 + δ = δ 1 δ 100. Using the value of δ, P = 100/r. Since r = 0.05 = 1/0, P = 000. Question 5 Growth in the economy leads to a rise in the demand for labour. It follows that the equilibrium quantity of labour (measured in the number of hours worked) must rise. Is this true or false? Explain your answer. 4 M,K& R Chapter 5.1; subject guide Chapter 4 (Labour supply and the effect of taxes).

25 The important question is how you connect demand for labour to the equilibrium quantity of labour. You need to understand that the equilibrium quantity arises from the intersection of supply and demand. Therefore the change in equilibrium wage and quantity of labour as demand shifts up depends on the nature of the supply curve. If we are shifting demand along a positively sloped supply curve, clearly quantity and wage would rise. But as you know, the supply of labour is somewhat different from supply of standard goods, and could be backward bending. So we might be operating on a negatively sloped part of the labour supply curve. In this case, as demand shifts up, we would have a lower quantity of labour and a higher wage. Therefore, the statement is false. Question 6 Suppose the inverse demand curve is given by P = 10 Q. This is shown in the picture below. At the point A shown in the picture, is the demand elastic or inelastic (with respect to price change)? Explain your answer. M,K& R Chapter 3; subject guide Chapter 3 (Consumer theory). At point A, demand is elastic. To see this, note that the price elasticity of demand is: ε = dq P dp Q. Since P = 10 Q, the slope dq dp is -1 at all points, so we simply have: ε = P Q. It is clear from the picture that at point A, P > Q. It follows that at point A, ε > 1. 5

26 Question 7 The short run supply function of a competitive firm is given by { 0 if P < 10 Q = 3P 30 if P 10 where Q is the quantity supplied and P is the price of output. Derive the equation for the firm s marginal cost curve. M,K& R Chapter 10.1; subject guide Chapter 8 (Competition and monopoly). From the supply function: P = Q Since the supply function is P = MC, the marginal cost is given by: MC = Q Question 8 Too few people use public transport in London relative to the social optimum. Provide an economic argument in support of this statement. M,K& R Chapter 18; subject guide Chapter 13 (Externalities and public goods). This tests whether you understand the basic problem of externalities. When an activity generates negative externalities (e.g. activities that lead to pollution, congestion, noise, environmental degradation), the unregulated market outcome is too high relative to the social optimum. For example, everyone driving a car only takes into account own costs and benefits but does not take into account social costs of pollution, and so the aggregate amount of car use is higher than the social optimum. Similarly, when an activity generates positive externalities, individuals do not take into account the benefit of others when deciding on the level of the activity. Therefore the unregulated market outcome is too low compared to the social optimum. Here, the words too few people... relative to the social optimum should immediately trigger the phrase positive externality in your mind. Now try to think of some benefit that people using public transport confer on society. For example, assuming that commuters must travel to work, they could use public transport or drive private vehicles. The latter, of course, is worse for the environment. The presence of any such positive externality would tend to make the market outcome on public transport use too small relative to the social optimum. It should be clear to you that to provide an economic argument in support of the given statement, you simply need to suggest some positive externality from using public transport. 6

27 Section B Answer three questions from this section (0 marks each). Question 9 Suppose there are two identical firms in an industry. The output of firm 1 is denoted by q 1 and that of firm is denoted by q. Each firm can produce output at a constant marginal cost of 6. There are no fixed costs. Let Q denote total output, i.e. Q = q 1 + q. The inverse demand curve in the market is given by P = 30 Q (a) Find the Cournot-Nash equilibrium quantity produced by each firm and the market price. (b) What would be the quantities produced by each firm and market price under Stackelberg duopoly if firm 1 moves first? (c) Calculate the deadweight loss arising from Cournot-Nash and Stackelberg duopoly. Which market structure is more efficient? (d) Suppose, as in part (b), firm 1 moves first and decides how much to produce. Firm moves second and makes its production decision. There is then a third stage at which firm 1 can change its mind about how much to produce and makes a final decision. Find the equilibrium quantities produced by the two firms under this three-stage game. M,K& R Chapter 15; subject guide Chapter 10 (Oligopoly and strategic behaviour). (a) Firm 1 maximises: π 1 = (30 (q 1 + q ))q 1 6q 1. The first order condition is 4 4q 1 q = 0, which gives us the best response function of firm 1: q 1 = 6 q. Similarly, the best response function of firm is: q = 6 q 1. Solving, q 1 = q = 4. The market price is then P = 30 8 = 14. (b) Firm s best response is as above. Firm 1 maximises in stage 1 knowing that will use this best response function at stage. Therefore firm 1 s problem is: max q 1 (30 (q q 1 ))q 1 6q 1. This simplifies to: max q 1 (1 q 1 )q 1. This gives us q 1 = 6. It follows that q = 3. The market price is P = 30 9 = 1. 7

28 (c) Using the picture below, the deadweight loss under Cournot competition is: DWL C = 1 (14 6)(1 8) = 16. The deadweight loss under the Stackelberg market structure is: DWL S = 1 (1 6)(1 9) = 9. It follows that the Stackelberg market structure is more efficient. (d) This is a little tricky. Try to see why a Stackelberg leader gets an advantage. This is because the first mover can choose an output it is committed to, and the second mover just plays a best response to that. It is the fact that the first mover can commit to an output before the follower has a chance to choose output is the source of the first mover advantage. Here, in contrast, firm 1 can change its mind about how much to produce and makes a final decision after decides. Therefore the initial stage has no commitment value. But can commit to an output at stage knowing that 1 will readjust and choose a best response at stage 3. It follows that Firm can now act like the Stackelberg leader, and 1 becomes the follower. Therefore, now q 1 = 3 and q = 6. 8 Question 10 (a) Consider the following simultaneous-move game with two players, 1 and. Let p denote the probability with which player 1 plays A 1, where 0 p 1. B 1 is played with the residual probability. Next, let q denote the probability with which player plays A, where 0 q 1. B is played with the residual probability.

29 Draw a picture with p along the horizontal axis and q along the vertical axis and draw the best response functions of players 1 and. Clearly label any equilibrium points in the picture. Player A B Player 1 A 1, 3,0 B 1 3,5 1,6 (6 marks) (b) Consider the following extensive-form game with two players. Player 1 can end the game by choosing Out. If player 1 chooses In, Player then chooses between L and R. The payoffs are written as (Payoff to 1, Payoff to ). Identify any subgame perfect Nash equilibrium. (6 marks) (c) Suppose the following game is repeated infinitely. Players discount the future, so that, for each player, a payoff of x received t periods from today is worth δ t x today, where 0 < δ < 1. Show that it is possible to sustain cooperation (which in this case involves each player playing C every period) in the infinitely repeated game for high enough values of δ. Player C D Player 1 C, 0,3 D 3,0 1,1 (8 marks) The coverage of game theory in M,K& R (Chapter 16) is not ideal. See Chapter 9 (Game theory: an introduction) of the subject guide for a detailed discussion. (a) There are no pure NE. The mixed Nash equilibrium is p = 1/3 and q = /3. The best response function of player 1 is given by: 0 if q > /3 p = [0, 1] if q = /3 1 if q < /3. 9

30 The best response function of player is given by: 0 if p < 1/3 q = [0, 1] if p = 1/3 1 if p > 1/3. The picture below draws the best response functions and shows the mixed strategy Nash equilibrium. 10 (b) To solve for the subgame-perfect Nash equilibrium in this game, we can simply solve backwards. At s decision node, would choose R (which gives a payoff of 3 compared to a payoff of from L). Knowing this, 1 s best choice is to stay out (which gives a payoff of 0 compared to a payoff of -1 from choosing In. ). It follows that the SPNE is (Out, R). Some of you prefer to write s strategy as R if 1 plays In. That is fine as well, although since there is exactly one node at which chooses a move, the qualifier is unnecessary here. (c) Cooperation can be sustained in equilibrium if players are patient enough, i.e. δ close enough to 1. This can be seen as follows. Suppose each player follows the following strategy: play C to start with, and continue to play C so long as there is no deviation. After any deviation, play D always. Let us compare the payoff from conforming to the payoff from deviating. Consider a deviation in period t by player 1. Note that the payoff until t 1 plays no role in comparing deviation payoff with the payoff from conforming. The payoffs only differ staring at t, so we might as well just consider the payoff starting at period t. Since this is an infinitely repeated game, the players face an infinite future starting at any period t, and the nature of calculations is exactly the same no matter when you start the calculations. Given the strategy of each player specified above, the payoff from always cooperating (starting at t) is as follows. A player gets at t, at t + 1 which is worth δ at t and so on. So