UNIVERSITY OF SOUTHAMPTON ECON6021W1 SEMESTER 1 EXAMINATIONS 2018-19 ECON6021 Microeconomics Duration: 180 mins (3 HRS) This paper contains 5 questions Answer ALL questions in Section A and ALL questions in Section B. Section A carries 55 percent of the total marks (each question 25 or 30 points) for the exam paper and you should aim to spend about 90 minutes on it. Section B carries 45 percent (each question 15 points) of the total marks for the exam paper and you should aim to spend a bit less than 90 minutes on it. An outline marking scheme is shown in brackets to the right of each question. Only University approved calculators may be used. A foreign language direct ‘Word to Word’ translation dictionary (paper version) ONLY is permitted. Provided it contains no notes, additions or annotations. Copyright 2019 v01 c© University of Southampton Page 1 of 5 2 ECON6021W1 Section A A1 (25 points) Consider the game given by the extensive form below. x0 (Pl 1) (1,−1,−1) B x1 (−1, 3, 0) L x3 (Pl 3) (−1,−2, 2) k (2,−2, 1) m R M x2 x4 (Pl 3) (−1, 3, 2) k (3,−1, 1) m R (1,−1, 0) L T Pl 2 Note that nodes connected by a dashed line indicate a non-singleton information set. (a) List the proper subgames and find all pure strategy Subgame Perfect Nash equilibria of this game. [9] (b) Which strategy will player two play in any Perfect Bayesian Equi- librium? Find all pure strategy Perfect Bayesian equilibria of this game. [7] (c) Find all pure strategy sequential equilibria of this game. [9] A2 (30 points) Consider an exchange economy with 2 consumers whose preferences and endowments are represented by: • u1(x1, x2) = (x2)2 and e1 = (5, 0) • u2(x1, x2) = (x1)23(x2)13 and e2 = (5, 15) (a) Characterize the set of all Pareto efficient allocations. [6] (b) Find all Walrasian equilibria (please report both equilibrium price vector and allocation, you can restrict attention to p >> 0). [10] Copyright 2019 v01 c© University of Southampton Page 2 of 5 3 ECON6021W1 Now assume there is, in addition to the consumers from above, a third consumer with • u3(x1, x2) = 3×1 + 2×2 and e3 = (10, 10). (c) At which prices is the endowment of the third consumer one of the bundles in her demand set? [4] (d) Is the allocation xˆ = (xˆ1, xˆ2, xˆ3) = ((0, 7.5), (10, 7.5), (10, 10)) in the core? [5] (e) Is the allocation x¯ = (x¯1, x¯2, x¯3) = ((1, 1), (9, 14), (10, 10)) in the core? [5] Section B B1 (15 points) Consider the following normal form game, where player 1’s pure strategies are D, E and F and player 2’s strategies are A, B, and C. (As usual, the first entry in every field corresponds to player 1’s payoff, the second entry to player 2.) A B C D 0,9 2,10 7,7 E 2,3 2,3 7,1 F 0,1 3,2 7,0 (a) Is strategy D iteratively undominated, i.e. survives the elimina- tion of strategies that are strictly dominated? [5] (b) Find the set of all Nash equilibria (pure or mixed). [10] B2 (15 points) Let Z = (0,∞) (interpreted as a set of non-negative monetary prizes). Suppose a decision maker is an expected-utility maximizer, described by a Bernoulli utility function u(z) = √ z. Copyright 2019 v01 c© University of Southampton TURN OVER Page 3 of 5 4 ECON6021W1 (a) Compute the Arrow-Prat measure of absolute risk aversion rA(z) for any z ∈ Z, and determine how rA(z) changes as z increases. [5] From now on, consider also the lottery p¯ which results in a prize w + h with a probability of 50 percent and yields a prize of w − h (where w > h > 0) otherwise. (b) Find z(p¯), the certainty equivalent of p¯. [5] (c) Define the risk premium (ρ¯) as the difference between the ex- pectation of the lottery p¯ and its certainty equivalent z(p¯). That is, define ρ¯ = Ep¯z − z(p¯). Verify if ρ¯ is positive. How does ρ¯ change as w increases? Discuss. [5] B3 (15 points) Consider the following grim spectacle, perhaps taking place in ancient Rome. A criminal is put into an arena and a queue of three hungry lions are lined up in a narrow corridor leading into the arena. Then the door is opened and the first of the lions is released into the arena. Once in the arena the lion can decide whether to eat the criminal or hold out. The lion, of course, prefers to eat, but also faces a problem. A lion after having eaten becomes tired and slow. So, if he goes for the criminal, the next lion in the row will be released into the arena and will eat the first lion if it wishes to. If the first lion holds out, he will remain hungry, but cannot be eaten by another lion and the spectacle ends. The procedure of releasing a new lion whenever his predecessor has eaten is repeated for all lions in the queue. Any lion freshly released would always be able to eat the previous lion, but may decide to end the spectacle by holding out. (a) Draw a game tree to model this spectacle as an extensive form game, with the three lions as players. Assume payoffs are 0 for a dead lion, 10 for a hungry lion (irrespective of if it is even Copyright 2019 v01 c© University of Southampton Page 4 of 5 5 ECON6021W1 released into the arena or not) and 15 for a surviving lion who has eaten another lion or criminal. [7] (b) Find the set off all (pure strategy ) subgame perfect Nash equi- libria of this game. What are the corresponding equilibrium payoffs? [8] END OF PAPER Copyright 2019 v01 c© University of Southampton Page 5 of 5 Social Sciences Examination Feedback 2018/2019 Module Code & Title: ECON6021 Microeconomics Module Coordinator: C. Kellner Mean Exam Score: 60/100 Percentage distribution across class marks: UG Modules 1 st (70% +) 2.2 (50-59%) 3rd (40-49%) Fail (25-39%) Uncompensatable Fail (<25%) PGT Modules 70% + 27 60-69% 18 50-59% 38 <50% 18 Overall strengths of candidates’ answers: Students can find Nash equilibria in pure strategies, and apply backwards induction. Most know how to find the Walrasian equilibrium. Concepts like the certainty equivalent/risk premium are well understood. Overall weaknesses of candidates’ answers: Some students were unable to identify the demand of a consumer who only cares about good 1, and more students found it difficult to understand the implication for Pareto efficiency. Students had difficulty identifying a partially mixed Nash equilibrium. Students have a general understanding of the core, but some issues with details. Pattern of question choice: NA Issues that arose with particular questions: In A1, almost all students failed to see that the last player has four strategies moves at two different information sets when describing the strategies of the players. Further comments not covered above: NA Discipline vetting completed By (Name): Emmanouil Mentzakis Date:18 Feb 2019