EC202. Microeconomic Principles II. Summer 2011 Examination. 2010/2011 Syllabus ONLY

Transcription

1 Summer 2011 Examination EC202 Microeconomic Principles II 2010/2011 Syllabus ONLY Instructions to candidates Time allowed: 3 hours + 10 minutes reading time. This paper contains seven questions in three sections. Answer question one (section A) and THREE other questions, at least ONE from section B and at least ONE from section C. Question one carries 40% of the total marks; the other questions each carry 20% of the total marks. Calculators are NOT permitted clse 2011/EC202 1 of 12

2 Section A 1. Answer any FIVE questions from the following eight, (a)-(h). Each question carries eight marks. (a) A rm produces a single output from two inputs. i. If the production possibilities are given by the single technique 1 q min 3 z 1; z 2 (where q; z 1 ; z 2 0 are quantities of output and the two inputs) draw the input-requirements set for a given q. ii. Suppose there is a second technique: q min z 1 ; 1 3 z 2 : If both techniques are available to the rm, draw the inputrequirements set for a given q. iii. Now suppose there is a third technique 2 q min 5 z 1; 2 5 z 2 : If all three techniques are available to the rm, draw the inputrequirements set for a given q. (b) Which, if any, of the following statements are true? In each case brie y explain the reasoning behind your answer. i. The elasticity of supply of a monopolist is always less than that of a competitive rm. ii. A competitive rm s long-run average cost is less than or equal to its short-run average cost. iii. If the demand for apples rises when the price of bananas rises, then the demand for bananas must rise when the price of apples rises. iv. A worker s labour supply falls with a wage rise if and only if leisure is a Gi en good. clse 2011/EC202 2 of 12

3 (c) i. Brie y explain the following axioms of consumer theory: completeness, transitivity, continuity, greed. ii. Suppose all consumption bundles consist of just four goods: apples, bananas, cherries, other goods. Sophie is crazy about fruit: if she can a ord both bundles x and x 0 but x contains more apples than x 0, then she always chooses x over x 0 ; (no matter what else is in the bundles); if x and x 0, contain the same amount of apples, but x contains more bananas than x 0 then again she always chooses x; if x and x 0 contain the same amount of apples and the same amount of bananas but x contains more cherries than x 0 then once again she always chooses x. Can Sophie s choice be represented by a utility function? Explain your answer. (d) State Arrow s Impossibility Theorem. Suppose there is the following voting scheme for deciding amongst social states. Each person assigns 1 point to the worst alternative, 2 points to the next worst, 3 to the next worst... and so on. The ranking of social states is according to the aggregate number of points voted for each one. Alf, Bill and Charlie have strict preferences over four social states as follows: Alf Bill Charlie [best] s 1 s 4 s 2... s 3 s 1 s 4... s 2 s 3 s 3 [worst] s 4 s 2 s 1 i. Find the social ranking if voters choose from fs 1 ; s 2 ; s 3 ; s 4 g. ii. Find the social ranking if voters choose from fs 1 ; s 4 g. iii. Which of Arrow s social-choice axioms is violated by this voting scheme? clse 2011/EC202 3 of 12

4 (e) Consider in nite repetition of the following strategic-form game, with discount rate : 1n2 C D C 6,6 1,7 D 7,1 0,0 i. Consider a strategy that prescribes to players to choose (C,C) as long as no player ever plays D and to play (D,D) otherwise. Show that such a strategy is Nash for su ciently high (i.e. gives rise to a Nash equilibrium), but not Subgame Perfect (i.e. cannot give rise to a Subgame-Perfect equilibrium). ii. Consider a strategy that prescribes to players: to choose (C,C) so long no player ever played D; to play (D,C) forever if the rst player to choose D was 2; and to play (C,D) forever if rst player to choose D was 1. Show that such a strategy is Subgame Perfect for su ciently high. (f) Two rms compete to sell a good. Firm 1 has total costs of production C 1 (q 1 ) = (q 1 ) 2 + 2q 1 and its costs are known to Firm 2. The total costs of Firm 2 depends on its type. If Firm 2 is of type L, its costs are C L (q L ) = 2q L. If Firm 2 is of type H, its costs are C H (q H ) = 2 (q H ) 2. Firm 2 knows its type. But Firm 1 only knows that Firm 2 can have either cost structure with equal probability. The inverse demand for the output produced by the two rms in this market satis es: 10 2(q1 + q p(q 1 + q 2 ) = 2 ) if q 1 + q if q 1 + q 2 > 5 Firms choose how much output to produce in order to maximize their pro ts. Find the Bayes-Nash equilibrium of this game. Characterize the equilibrium output strategies for both rms. Find the market price for each of the two possible cost con gurations. clse 2011/EC202 4 of 12

5 (g) Consider a Principal-Agent model. Suppose that the Agent can exert one of two e ort levels E 2 f0; 1g. The preferences of the Agent are de ned over wages W and e ort levels E by the function: u(w; E) = log(w ) E, and his reservation utility u satis es u = 0. The Principal has two possible revenues: q = 9 and q = 0. Suppose that revenues are high (q = 9) with probability p 1 = 2=3, if the Agent exerts e ort E = 1; and with probability p 0 = 1=3, if the Agent exerts e ort E = 0. Suppose the Agent is free to decide how much e ort to exert and that the Principal cannot observe the e ort exerted by the Agent, but can only observe the output produced. [Recall that the exponential approximately satis es: e = 2:7, e 2 = 7:3, e 3 = 19:7]. i. Find the optimal wages that a Principal would set in this environment to maximize its pro ts. ii. Which e ort level would be chosen by the Agent in the equilibrium? iii. Discuss how the solution would di er if the e ort were observable. (h) Consider a Spence signalling model. Explain the intuitive criterion used to reduce the set of equilibria in that game. Identify the only equilibrium that satis es the intuitive criterion, and explain why no other equilibrium can meet such a requirement. clse 2011/EC202 5 of 12

6 Section B [Answer at least ONE question] 2. A person with xed money income has the following utility function 8 < log x 1 + [1 ] log (x 2 ) if x 1 > 0; x 2 > ; U (x 1 ; x 2 ) = : 0 otherwise, where x 1 ; x 2 are quantities of two goods, and the parameters ; satisfy 0 < < 1 and > 0. (a) Draw the indi erence curves and interpret the parameters and. [2 marks] (b) Find this person s cost (expenditure) function. [5 marks] (c) Find the ordinary and compensated demand for the two goods. [5 marks] (d) Discuss how this model could be used to represent a person s consumption during youth (period 1) and retirement (period 2). [2 marks] (e) Suppose the value of is doubled. How does this a ect his rstperiod savings (i) if his exogenous income in both periods remains unchanged? (ii) if he is given an income in period 1 su cient to keep his lifetime utility unchanged? (iii) if he is given an income in period 2 su cient to keep his lifetime utility unchanged? Discuss your answer in the light of interpretation of part (d). [6 marks] clse 2011/EC202 6 of 12

7 3. In a two-commodity economy without production, there are two types of agent: each a-type agent has the utility function log (x a 1)+ 1 log 2 (xa 2), each b-type has the utility function 1 log 2 xb 1 + log x b 2, where x h i denotes an h-type s consumption of good i. There are N agents of each type. (a) If the total quantity of goods available is given by (3N; 12N) what is the set of Pareto-e cient allocations in this economy? [4 marks] (b) Which of the following allocations are e cient? (x a 1; x a 2) = 4; 2 3 x b 1; x2 b = 5 [4 marks] ; 10 3 (x a 1; x a 2) = (2; 4) x b 1; x b 2 = (1; 8) (x a 1; x a 2) = (2; 2) x b 1; x b 2 = (1; 9) (x a 1; x a 2) = (1; 3) x b 1; x b 2 = (2; 9) (c) If each a-type has the endowment (1; k) and each b-type has the endowment (2; 12 k), where 0 k 12, show that the competitive equilibrium price of good 1 (in terms of good 2) is [k + 12] =5 [5 marks] (d) For the case in part (c) nd the equilibrium allocation as a function of k. If k = 3 show that the equilibrium allocation is (1) above. [2 marks] (e) Take the case in part (d) with k = 3. Suppose that endowments of good 2 can costlessly transferred between agents, but that endowments of good 1 cannot be transferred. A social planner wants to ensure allocation (2) as a competitive equilibrium outcome rather than (1) What transfer of endowment would ensure this? How would equilibrium prices change relative to the case of part (c)? [5 marks] (1) (2) (3) (4) clse 2011/EC202 7 of 12

8 4. A taxpayer has an income y that should be reported in full to the tax authority. Tax is payable at a constant proportionate rate t so that the taxpayer keeps a proportion 1 t of reported income. The taxpayer attempts to evade taxes by concealing an amount of income z. There is a known probability that an audit is carried out. If there is an audit the true taxable income is revealed and the taxpayer has to pay all the tax on his income and, in addition, a surcharge of s times the underpaid tax. (a) Let x 0 and x 1 denote disposable income in the two cases where the person is not audited and where the person is audited, respectively. Express (x 0 ; x 1 ) as functions of y; ; s; t and z: [4 marks] (b) The taxpayer chooses z so as to maximise the utility function [1 ] u (x 0 ) + u (x 1 ) ; where u is increasing and strictly concave. Find the rst-order condition for an interior maximum. [4 marks] (c) Using the result from part (b), if s increases, will tax evasion z increase or decrease? [6 marks] (d) If income increases, will tax evasion increase or decrease? [6 marks] clse 2011/EC202 8 of 12

9 Section C [Answer at least ONE question] 5. Suppose that two players have to split a cake of size 10 according to an I cut, you choose protocol. In particular, assume that Player 1 cuts the cake into two parts and that Player 2 gets to choose which of the two slices to consume (while the other slice is consumed by Player 1). Suppose that the cake is continuously divisible. (a) First, assume that the cake is homogeneous, so that both players value all its parts alike, and care only about the size of the slice that they get to consume. Set this problem up as an extensive form game. Find a Subgame Perfect equilibrium of this game. Write the behavioral strategy for both players, and check that no deviation is pro table. [7 marks] (b) Now suppose that 3=5 of the cake is chocolate avoured, while 2=5 of the cake is vanilla avoured. Player 1 only likes chocolate, and only cares about the amount of chocolate in the slice he consumes; while 2 likes all the parts of the cake alike, and cares only about the size of the slice consumed. Also, assume that Player 1 can cut slices with any composition of chocolate and vanilla. Find a Subgame Perfect equilibrium of this game. Write the behavioral strategy for both players, and check that no deviation is pro table. [7 marks] (c) Does the game in part (a) possess a pure-strategy Nash equilibrium in which a player receives a slice di erent in size from the one that you have characterized in part (a)? Explain. [6 marks] clse 2011/EC202 9 of 12

10 6. Four patients have to undergo surgery and rehabilitation in one of two hospitals. Hospital A specializes in surgery. But its elite surgery unit is small. The likelihood of successful surgery, p S A, depends on the number of patients treated in the surgery unit, n, as follows: 8 17=20 if n = 1 >< p S 15=20 if n = 2 A(n) = : 11=20 if n = 3 >: 7=20 if n = 4 The rehabilitation unit of hospital A is large, but conventional. The likelihood of successful rehabilitation p R A = 1=2 is independent of the number of patients treated. Hospital B specializes in rehabilitation. But its elite rehabilitation unit is small. The likelihood of successful rehabilitation, p R B, depends on the number of patients treated by the unit, n, as follows: 8 1 if n = 1 >< p R 16=20 if n = 2 B(n) = 14=20 if n = 3 >: 12=20 if n = 4 The surgery unit of hospital B is large, but conventional. The likelihood of successful surgery p S B = 1=2 is independent of the number of patients treated. Surgery outcomes are independent of rehabilitation outcomes. A patient s payo is 1 if both treatments are successful, and 0 otherwise. A patient is classi ed as recovered, only if both treatments are successful. [Hint: Recall that if events A and B are independent Pr(A \ B) = Pr(A) Pr(B)]. (a) First suppose that the existing health regulations require all patients to undergo surgery and rehabilitation at the same hospital. Patients can only choose in which of the two hospital to get both treatments. Set this problem up as a strategic-form game. Find a Nash equilibrium of this game. Check that no deviation is profitable. What is the average probability of recovery among the four patients in this equilibrium? [7 marks] [question continues on next page] clse 2011/EC of 12

11 (b) In an attempt to increase the average recovery probability regulators decide to lift the ban on having surgery and rehabilitation at di erent hospitals. Now patients are free to choose in which hospital to get either treatment. Set this problem up as a strategic-form game. Find a Nash equilibrium of this game. Check that no deviation is pro table. What is the average probability of recovery among the four patients in this equilibrium? [7 marks] (c) Still unsatis ed about the average recovery probability, regulators decide to try a third policy, in which one of the four patients is randomly selected and sent to the two elite units (surgery in A and rehabilitation in B), while the remaining three are sent to the larger conventional units (surgery in B and rehabilitation in A). What is the average probability of recovery among the four patients with this policy in place? [3 marks] (d) Compare the average probabilities of recovery under the three regulations. Give an intuitive explanation to the observed change in recovery probabilities. Has this anything to do with externalities? [3 marks] clse 2011/EC of 12

12 7. Consider an economy with two goods: money m and consumption x. All individuals are endowed with M units of money, but with no consumption. All individuals value the consumption good equally according to a utility function u(x) = x 1=2. But the utility derived from an amount of money m di ers among individuals. In particular, the preferences of a consumer of type t 2 f2; 4g satisfy: U(x; mjt) = u(x) tm In this market 3=4 of the consumers are of type t = 4, while all the remaining consumers have type t = 2. A single rm operates in the market. The rm uses money to produce consumption at a constant marginal cost c = 1=8. (a) Suppose that the rm is able to distinguish consumers of di erent types. Find the equilibrium consumption of both types of players if the rm o ers contracts in order to maximize its pro ts. [You are not required to derive the optimal contracts]. [8 marks] (b) Now suppose that the rm is not able to distinguish consumers of di erent types. Find the equilibrium consumption of both types of players if the rm o ers contracts in order to maximize its pro ts. [You are not required to derive the optimal contracts]. Is the equilibrium e cient? [12 marks] clse 2011/EC of 12