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1.You are about to sit an online assessment
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Students must answer any TWO questions. The word limit per question is 1500 words
maximum. Anything you write once you have reached the word limit will not be marked.
Question 1
Let two dimensions of choice be apples and money, with apples being dimension 1 and money being
dimension 2. Denote outcomes in apples and money by 𝑐1 and 𝑐2, respectively, and reference points in
the two dimensions by 𝑟1 and 𝑟2, respectively. The person’s utility is given by:
4𝑐1 + 𝑐2 + 4𝑣(𝑐1 − 𝑟1) + 𝑣(𝑐2 − 𝑟2)
where 𝑣(𝑥) = 𝑥, 𝑥 ≥ 0 and 𝑣(𝑥) = 𝛼𝑥, 𝑥 < 0. a. For which values of α, does this formulation capture loss aversion? How would you change the specification of the value function to capture diminishing sensitivity? [10%] b. Suppose than an individual owns one apple and has no money. What is her reference point? Given this reference point, solve for the "selling price", i.e. the minimum price at which the individual is willing to sell the apple. [10%] c. Suppose than an individual has a one pound coin but no apples. What is her reference point? Solve for the "buying price", i.e. the maximum price at which the individual is willing to buy an apple. [10%] d. Suppose than an individual has no money or apples. What is her reference point? This individual is told she can have either one apple or money. Solve for the "choosing price", i.e. the minimum amount of money which she would be willing to accept instead of one apple. [10%] e. In an actual experiment, buying and choosing prices were found to be close to each other (although the choosing price was slightly higher) while selling prices were found to be two and half times the buying price. How well does the above model explain these findings? To the extent that it cannot explain the findings, what do you think goes wrong? [10%] [Total 50%] 4 Continued Overleaf Question 2 Assume that an individual is a hyperbolic discounter with 0 < 𝛽 < 1, 𝛿 = 1 where 𝛽 denotes the quasi- hyperbolic discount factor (present bias) and 𝛿 denotes the exponential discount factor. She cannot commit her future behaviour. The individual has to undertake a costly action in one of the time periods 𝑡 = 0,1,2,3,4,5. The utility cost of the action in period 𝑡 is 𝑐𝑡. a. For which values of 𝛽, 𝑐 will a naive individual undertake the action at 𝑡 = 5? Explain the intuition behind this result. [20%] b. For which values of 𝛽, 𝑐 will a sophisticated individual undertake the action at 𝑡 = 1? [20%] c. Based on your answers, comment on the suggestion that the naive and sophisticated models of behaviour with hyperbolic discounting are "too extreme" with the sophisticated one being too rational, and the naive one being too irrational. [10%] [Total 50%] Question 3 Consider the ultimatum bargaining game (with available bargaining surplus normalised to one) where the proposer (player 1) offers a share 𝑠 ∈ [0,1] to the responder (player 2). Let the preferences of the two players be characterized by inequality aversion (as stated in Fehr and Schmidt (1999) and referred to in the lecture notes) with parameters 0 < 𝛽1 = 1 3 < 𝛼1 and 0 < 𝛽2 < 𝛼2 ≤ 1 2 . Assume that player 2 (responder) knows the correct value of 𝛼2 but player 1 (proposer) only knows the probability distribution over 𝛼2 so that 𝛼2 = 1 2 with probability (1/2) and 𝛼2 = 1 4 with probability (1/2). Compute the sub game perfect equilibrium of this bargaining game. Explain, intuitively, the answer you obtain. [50%] 5 End of Paper Question 4 Consider the following simultaneous move game with material payoffs as shown below: C D C 3X, 3X 0,4X D 4X,0 X, X (i) Show that (D,D) is a fairness equilibrium. [10%] (ii) Under what conditions on X is (C,C) also a fairness equilibrium? Explain, intuitively, your answer. [25%] (iii) Suppose the game is altered so that the row player is forced to cooperate (i.e. the action “D” is no longer available to the row player). Compute the unique fairness equilibrium in this case and intuitively explain your answer. [15%] [Total 50%]