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EC2020

Summer Examinations 2021/22

Microeconomics 2

Section A: Answer ALL FOUR questions

1. Consider Rajesh whose preferences can be represented by the following function:

(a) Derive Rajesh’s coefficient of relative risk aversion. (3 marks)

(b) Rajesh is contemplating betting on Arsenal beating Tottenham. The probability that Arsenal wins is ρ and the probability Arsenal does not win is (1 − ρ). If he bets X he will receive X if Arsenal wins and zero if they do not. Rajesh’s initial wealth is W and therefore his wealth in the good state (in which Arsenal win) will be W + X and in the bad state it will be W − X. Derive an expression representing Rajesh’s bet, X. (5 marks)

(d) Maintining the assumption that Rajesh is risk averse, how does his bet depend on his level of initial wealth? (5 marks)

2. Three drivers are racing towards a narrow tunnel which is only wide enough for two cars at a time. Each driver faces a choice to go straight (Gs) or to swerve (s). If they all play Gs , they will suﬀer serious injury and get payoﬀs equal to -8. If at least one driver swerves, the payoﬀ to the driver(s) that swerved is 0 whereas the payoﬀ to those that went straight is 1. The game is illustrated in the following table:

(a) Identify all pure-strategy Nash equilibria. (3 marks)

(b) Identify all Pareto eﬃcient outcomes of the game. (3 marks)

(c) Is there a mixed strategy Nash equilibrium of the game and what is the payoﬀ to each player in this equilibrium? (6 marks)

(d) Consider the following mechanism. An ‘umpire’ uses a randomisation device to select one cell in the game matrix with the probabilities shown in the following table:

Show that this mechanism can be supported in a correlated equilibrium, and compute each driver’s payoﬀ . (6 marks)

3. Consider a 2 x 2 pure exchange economy with two consumers, Andy and Beth, or {A, B|, and two goods {1, 2|. Andy has endowment eA = (4, 2) and Beth has endowment

e会 = (2, 2). Andy’s preferences can be represented by the utility function uA = zA1 + zA2 and Beth’s preferences can be represented by the utility function u会 = z会1z会2 . Let prices be (p1 , p2 ) = (1, 2).

(a) Sketch an Edgeworth box showing the dimensions, initial endowment, both players’ indiﬀerence curves through the initial endowment and budget sets. (5 marks)

(b) Find consumers’ optimal demands at prices (p1 , p2 ) = (1, 2) and show that Walras’ Law holds at these prices. (5 marks)

(d) Without calculating the Walrasian Equilibrium, say how equilibrium prices compare to (p1 , p2 ) = (1, 2). Then using the First Welfare Theorem or otherwise ﬁnd the Walrasian Equilibrium allocation and prices. (4 marks)

4. (a) The First Welfare Theorem for pure exchange economies can be stated as “If there are no externalities and everyone has locally non-satiated preferences, then every Walrasian Equilibrium is Pareto eﬃcient.” Take the contrapositive of this statement. What can we say about an economy where everyone has locally non-satiated preferences and which has a Walrasian Equilibrium which is not Pareto eﬃcient? (3 marks)

(b) Consider a 2 x 2 pure exchange economy with two consumers, Andy and Beth, or {A, B|, and two goods {1, 2|. Andy has endowment eA = (1, 1) and Beth has endowment e会 = (1, 1). Andy’s preferences can be represented by the utility function uA = zA1zA2 and Beth’s preferences can be represented by the utility function

u会 = z会1z会2 + zA1 . Notice that Andy’s consumption of good 1 creates a positive

externality for Beth . Show that there is a Walrasian Equilibrium at the initial

endowment with prices (p1 , p2 ) = (1, 2). Show that this Walrasian Equilibrium is not Pareto eﬃcient. (7 marks)

(c) Discuss the extent to which you agree with the following satement: “The problem of externalities can be solved by the people electing a government which then intervenes in markets to restore eﬃciency.” (8 marks)

Section B: Answer ONE question.

5. Two hunters must decide separately and simultaneously whether to hunt a stag or a hare. One hunter can catch a hare alone with less eﬀort, but the only way to successfully hunt a stag is with the other’s help. The value to each hunter of a stag is much higher. The game with payoﬀs from catching a stag and a hare is illustrated in the following table:

(a) Identify all Nash equilibria of the game, and illustrate them in a diagram. (6 marks) (b) What are the payoﬀs in each Nash equilibrium? (6 marks)

(c) Players can now be of two types – starving (s) or full (F). For a starving hunter, the payoﬀs are as in the previous sub-questions. For a full hunter, however, the payoﬀ when both play stag is 5. The bimatrix game for two hunters that are both of type F is thus:

Suppose the two types are equally likely. Construct the normal form representation of this game. (8 marks)

(d) What are the pure-strategy Bayes-Nash equilibria of the game? (8 marks)

6. Suppose there are two players: a Worker (w) and a Firm (F). w has private information about their own level of ability. F knows that with probability p = , w is high ability type (H), and with probability (1 - p) = , w is low ability type (L). After w observes their own type (a draw by nature), they decide whether to get a degree (play E) or not (play N). F observes w ’s education, but not their type. F then decides whether to employ w in a managerial job, or a clerical job (C). The payoﬀs are as follows: if F employs a worker of type H in a managerial job, it obtains a payoﬀ equal to 10; if F employs a worker of type L in a managerial job, it obtains payoﬀ equal to 0; if F employs a worker in a clerical job, it obtains a payoﬀ equal to 4 regardless of the type of w . A worker of either type obtains a payoﬀ of 10 in a managerial job, and 4 in a clerical job. The cost of education diﬀers, however. For an H-type worker the cost of doing a degree is 4, whereas for an L-type worker it is 5.

(a) Draw this game in extensive form. (4 marks)

(b) Is there a separating equilibrium in which the H-type chooses to do a degree, and the

L-type doesn’t? Carefully explain your answer. (8 marks)

(d) Now suppose that the cost of doing a degree for the L-type is 7, whereas the cost of doing a degree for the H-type is 4 as it was in previous subquestions. All other payoﬀs stay the same. Does this change your answer in subquestion (b)? Carefully explain your answer. (8 marks)

7. Consider a 2 x 2 pure exchange economy with two consumers, Andy and Beth, or {A, B|,

and two goods {1, 2| . Andy has endowment eA = (1 , 4) and Beth has endowment

e会 = (4, 1). Andy’s preferences can be represented by the utility function wA = ^zA1zA2 and Beth’s preferences can be represented by the utility function w会 = min {z会1, z会2|.

(a) Find all Walrasian Equilibria. (5 marks)

(b) Now we introduce a ﬁrm owned equally by the two agents with production set

y = {y e R2 | y1 s 0, y2 s -2y1 |. Find the proﬁt maximising output y (p) as a function of prices and argue that there are no Walrasian Equilibria when < 2 or when > 2. (5 marks)

(c) Find the Walrasian Equilibrium. (Hint: start by ﬁnding equilibrium prices and then both agents’ demand). (6 marks)

(d) Return to the economy without production in part (a). Find the Pareto Set and use this to put this scenario into the form of a bargaining problem: deﬁne a pair (s, d) where s is the set of feasible utility pairs and d is the disagreement point (no trade). Illustrate (s, d) on a diagram. (6 marks)

(e) Find: (i) the Nash Bargaining Solution and (ii) the Subgame Perfect Equilibrium of the Ultimatum game with Andy as proposer . For each, ﬁnd both the utilities and allocation . (6 marks)

8. This question has 7 parts. For each part you should say whether the statement in quotation marks is true and false and justify your answer. For each part, 1 mark is given to identifying whether it is true or false and the remaining marks are given to the justiﬁcation. (The ﬁrst part, as only 1 mark, requires no justiﬁcation.)

(a) If somebody’s preferences violate local non-satiation then the Walrasian Equilibrium will violate Pareto eﬃciency. (1 marks)

(b) In a pure exchange economy with complete markets, if there is a Pareto eﬃcient allocation that cannot be supported as a Walrasian Equilibrium, then at least one agent must have preferences violating either local non-satiation, convexity or continuity. (2 marks)

(d) When there are 3 alternatives and 3 voters each with a strict preference relation, the

following SWF satisiﬁes the Independence of Irrelevant Alternatives (IIA) axiom. SWF: Each voter votes for their most preferred alternative and we rank alternatives by how many votes they receive: if two candidates get the same number of votes then our

SWF ranks them indiﬀerent to each other; if one candidate gets more than another then the candidate with more votes is ranked above the one with fewer by our SWF. (4 marks)

(e) Consider a pure exchange economy where there are 10 units of each good and

preferences of our two agents are uA = zA1 + ln (1 + zA2) , u会 = z会1 +^z会2 . Then

every Pareto eﬃcient allocation can be supported as a Walrasian Equilibrium . (5 marks)

(f) In the following co-operative game, the Shapley Value is an element of the core. (6 marks)

(g) Consider the following public goods game, where Alice and Beth plant ﬂowers in their shared garden. The demands of Alice and Beth respectively are given by QA = max {6 - P, 0| and Q会 = max {5 - , 0} . Let the market price be P = 4. Assume that without any intervention Alice completely free-rides on Beth. Now, an omniscient social planner implements the Lindahl Equilibrium. Our statement is: the Lindahl Equilibrium makes both consumers better oﬀ compared to what would be achieved without the social planner’s intervention. (7 marks)