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Department of Economics, University of Oslo ECON 3210 DECISIONS, MARKETS AND INCENTIVES Seminar – autumn 2005 Note: exercise 1 will be discussed at the first seminar. Exercise 1 A. Consider a utility-maximising consumer with a given set of preferences and a budget who faces given market prices. Make assumptions on preferences that will ensure a unique solution to the consumer’s choice problem. Discuss how the choice of the consumer depends on the levels of (i) the budget, (ii) prices. B. Consider next a profit-maximising firm facing a price sensitive demand for its product. Characterise the optimal output decision of the firm. Discuss how the choice of the firm depends on (i) the level of demand, (ii) the slope of the demand curve, (iii) costs. Exercise 2 A. Explain what we mean by ‘an act’, ‘a state of the world’, ‘the probability of a state’ and ‘an outcome’. Discuss the reasonableness of assuming that a decision maker (i) knows the set of states and (ii) is able to assign probabilities to each of the possible states. B. What is the Expected Utility Theorem? Explain the assumptions underlying this result and discuss their reasonableness. C. Explain what is meant by a decision maker being, respectively, ‘risk averse’, ‘risk loving’ and ‘risk neutral’. What is the mathematical characterisation of a utility function which represents, respectively, risk aversion, risk attraction and risk neutrality. D. What attitude towards risk is implied by the following utility functions: 1) u ( x ) = ln x 2) u ( x ) = ex 3) u ( x ) = a + bx , where a and b are positive constants. 4) u ( x ) = x2 E. Consider the utility function u ( x ) = a + bx + cx . What assumptions have to be 2 made on this function for it to represent risk averse agent who derives utility from x? Is the function valid for any value of x? Exercise 3 A farmer has at his disposal 100 acres of land. The land may be used for growing wheat or barley. The wheat yield depends on the weather (i.e. occurrence of frost), while the barley yield does not. Consequently, if the weather is good the farmer receives a yield on wheat land worth NOK 4500 per acre, whereas if the weather is bad the value of the wheat crop is NOK 1500 per acre only. The probability of bad weather is 1/3. The yield on barley is NOK 2500 in any case. Irrespective of the choice of crop, the cost of fertilisers, seeds etcetera is NOK 1000 per acre. A. Assume the farmer choose to produce wheat only. Show that in this case his expected net income equals NOK 250,000. B. If the farmer only considers the two extremes, ‘all barley’ and ‘all wheat’, should he then choose all wheat? Explain! C. Assume the farmer has preferences that satisfy the Expected Utility Theorem and that his (von Neuman Morgenstern) utility function is u ( x ) , where x is net income. What are the expressions for the farmer’s expected utility in the two cases considered in B? D. Assume now that the farmer may choose a mix between wheat and barley. Let Y denote the amount of land used for barley. Net income in good weather conditions is denoted xG while net income in bad weather conditions is xB . Show that xG = 350, 000 − 2000Y xB = 50, 000 + 1000Y E. Derive the expression for the farmer’s expected utility and show how it depends on Y. Explain how we may (in principle) find the optimal value of Y. Exercise 4 We consider a firm that may invest in a project with a risky outcome. If the firm invests an amount K, with probability p revenues are RH ( K ) and with probability 1-p revenues are RL ( K ) , where RH ( K ) > RL ( K ) and both RH and RL are twice continuously differentiable, increasing and concave. In the first part of the exercise, we assume that the unit cost of investment is 1+r, where r is the (given) interest rate. A. Consider the case in which the firm acts as if it does not care about risk. What is the expression for expected profits? Derive the first-order condition for the profit-maximising investment level. Interpret the result. B. Consider next the case in which the firm acts as if it is risk averse. Can you give reasons for this assumption? Assume that the firm acts as if it evaluates profits according to the utility function u. What is the expression for expected utility of the firm? Derive the first-order condition for the expected-utility maximising investment level. Interpret the result. How does the resulting investment level compare to that under A? Suppose that the firm has access to two different sources of finance. In addition to being able to borrow at the fixed interest rate r, the firm may also borrow at a risky rate ri . For simplicity we assume that the interest rate ri takes on the value rH when revenues are RH and the value rL when revenues are RL , where rH > rL . C. Consider again the case of risk neutrality. Is it possible that the firm may choose to borrow from both sources? Under what conditions on the interest rates will the firm choose to borrow at the fixed and risky interest rate, respectively? Does the firm’s choice depend on the riskiness of revenues? D. Consider next the case of risk aversion. Explain why the firm would never borrow at the fixed interest rate if r > rH . Explain also why the firm would never borrow at the risky interest rate if r < rL . Suppose for the rest of the analysis that rL < r < rH . Is it conceivable that the firm will borrow at the risky interest rate even prH + [1 − p ] rL > r ? Does the choice depend on the riskiness of revenues? Explain. if We return again to the case in which the firm can borrow at a fixed rate of interest only. However, we now assume that the profit of the firm is subject to a (proportional) tax t. E. In the case of risk neutrality, does the rate of taxation affect investment behaviour? Explain. F. What about in the case of risk aversion? Explain. Exercise 5 A. (Gibbons, Problem 1.1) What is a game in normal form? What is a strictly dominated strategy in a normal-form game? What is a pure-strategy Nash equilibrium in a normalform game? B. (Gibbons, Problem 1.2) In the following normal-form game, what strategies survive iterated elimination of strictly dominated strategies? What are the pure-strategy Nash equilibria? Player 2 Player 1 Left Middle Right Top 2,0 1,1 4,2 Middle 3,4 1,2 2,3 Bottom 1,3 0,2 3,0 C. Clothes manufacturer A is reputed to be a trend setter. A’s competitor B aims at customers who want the types of clothes sold by A but are willing to accept cheaper copies if they exist. A has two different choices for the next collection. If A and B choose different styles, A earns a payoff of 40 and B a payoff of 20. If both choose the same style, A earns 20 and B 40. Formulate the normal-form representation of this game and find any Nash equilibria. D. (Varian, ch 28, Review question 2) Are dominant-strategy equilibria always Nash equilibria? Are Nash equilibria alwas dominant-strategy equilibria? E. (Varian, ch 28, Review question 3) Suppose your opponent is not playing his or her Nash equilibrium strategy. Should you play your Nash equilibrium strategy? F. (Varian, ch 28, Review question 4) We know that the single-shot prisoner’s dilemma game results in a dominant Nash equilibrium strategy that is Pareto inefficient. Suppose we allow the two prisoners to retaliate after their respective prison terms. Formally, what aspect of the game would this affect? Could a Pareto efficient outcome result? Exercise 6 A. (Cabral, Review and Practice Exercise 4.1) What are the assumptions regarding player rationality implicit in solving a game by iterated elimination of dominated strategies? Contrast this with the case of dominant-strategy solutions. B. Cabral, Review and Practice Exercise 4.3. C. (Gibbons, Problem 1.3) Players 1 and 2 are bargaining over how to split one dollar. Both players simultaneously name shares they would like to have, s 1 and s 2 , where 0 ≤ s 1 , s 2 ≤ 1 . If s 1 + s 2 ≤ 1 , players receive the shares they named; if s 1 + s 2 > 1 , both receive zero. What are the pure-strategy Nash equilibria of this game? D. (Gibbons, Problem 1.8) Consider a population of voters uniformly distributed along the ideological spectrum from left ( x = 0 ) to right ( x = 1 ). Each of the candidates for a single office simultaneously choose a campaign platform (i.e., a point on the line between x = 0 and x = 1 ). The voters observe the candidates’ choices, and then each voter votes for the candidate whose platform is closest to the voter’s position on the spectrum. If there are two candidates and choose platforms x1 = 0.3 and x2 = 0.6 , for example, then all voters to the left of x = 0.45 vote for candidate 1, all those to the right vote for candidate 2, and candidate 2 wins the election with 55 per cent of the vote. Suppose that the candidates care only about being elected – they do not really care about their platforms at all! If there are two candidates, what is the pure-strategy Nash equilibrium? If there are three candidates, exhibit a pure-strategy Nash equilibrium. (Assume that any candidates who choose the same platform split the votes case for that platform, and that ties among the leading vote-getters are resolved by coin flips). See H Hotelling, Stability in Competition, Economic Journal, 39, 41-57, 1929, for an early model along these lines. Exercise 7 A. Review and Practice Exercises 4.4, 4.5, 4.6 and 4.7 in Cabral. B. Review Question 6 in Chapter 28 in Varian. C. Problems 2.1 and 2.2 in Gibbons. Exercise 8 A. Problems 2.10, 2.11 and 2.12 in Gibbons. B. Review Question 1 in Chapter 28 in Varian. C. Review and Practice Exercises 4.2 in Cabral. D. Review Question 5 in Chapter 28 in Varian. Exercise 9 A. Review and Practice Exercises 7.1-5 in Cabral. B. Review and Practice Exercises 8.1 and 8.2 in Cabral. C. Exam Autumn 2003 Exercise 10 (Exam Autumn 2003) A. Define and explain the following terms: game, strategy, player, normal-form representation, extensive-form representation, dominant-strategy equilibrium, Nash equilibrium. B. Two firms are competing on a market. Each firm may choose one out of two possible strategies; Aggressive and Accommodating. If both firms choose the Aggressive strategy each earn zero profits; if both choose the Accommodating strategy each earn a profit of 100; if one firm choose the Aggressive strategy and the other the Accommodating strategy, the first earns a profit of 200 and the second a profit of -100. Express the game in normal form and solve it. C. Consider a repeated version of the above game. Explain under what conditions both firms always playing the Accommodating strategy may be supported as an equilibrium outcome. D. Adam Smith wrote that “People of the same trade seldom meet together, even for merriment and diversion, but the conversation ends in a conspiracy against the public, or in some contrivance to raise prices”. Discuss conditions under which such a “conspiracy” may be successful.