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Consumption Level, Exogenous, Intertemporal Preferences, Stochastic Consumption, Algebra Mechanically, Relevant Possibilities, Ambiguous, Contingent Choices, Subtlety and Insight, Utility Functions. Above mentioned points are important ones from questions.
Typology: Exams
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There are 2 (long) questions on the exam. Please answer both to the best of your ability. They count roughly equally. Do not spend too much time on any one part of either problem (especially if it is not crucial to answering the rest of that problem), and don’t stress too much if you do not get all parts of all problems.
Question #1 (Psychology and Economics, 219A) Sandro lives for three periods, 0, 1, and 2, and has utility for consumption each period. He has exogenous and already determined consumption level in period 0 (his period-0 utility function will only be relevant for the last two parts of the problem), and he has $1 to spend in the final two periods. In no part of the problem will there be any interest earned on money saved. Suppose that Sandro has intertemporal preferences given by Ut^ ≡ ut + β
S∞ τ =t δ τ (^) uτ ,
with β ∈ (0, 1], and δ = 1. These are standard present-biased preferences. Sandro might have any degree of sophistication, βe ∈ [β, 1]. In all parts, Sandro will have stochastic consumption utility each period of either ln(ct) or 2ln(ct), each with probability .5, distributed independently across periods. He learns this instantaneous utility function at the beginning of the period, but not before. Please be careful to think through the set-up and the logic of the questions, using a wise balance between solving algebra mechanically and thinking through what the relevant possibilities are. a) If he cannot commit ahead of time to how to spend his $1 split between periods 1 and 2, how much will he consume in period 1 as a function of β and βe, and as a function of what the instantaneous utility occurs in period 1. (In period 2, Sandro will of course consume whatever he has left, c 2 = 1 − c 1 ). Please use the notation c∗ 1 (1) and c∗ 1 (2) to refer to his consumption levels if his period-1 utility turns out ln(c) or 2ln(c) respectively. b) Now suppose that in period 0 Sandro can put aside y ≤ 1 dollars that must be consumed in period 2, constraining him to consume c 1 ≤ 1 − y in period 1 (and of course still consume 1 − c 1 in period 2). Sandro could still consume c 1 > 1 − y. As a function of β and βe, what c∗ 1 (1), c∗ 1 (2), and y will Sandro choose? Explain briefly both the issues and the algebra Sandro confronts and the logic of his choice, and explain some of what you know about the answer in case you don’t get all the way or make a mistake with the algebra. Now suppose that on top of present bias poor Sandro has projection bias with α ∈ [0, 1]: if his utility function is ln(c) in period t = 0 or period t = 1, he thinks that his utility function in each future period will be either ln(c) or α ln(c) + 2(1 − α)ln(c) = (2 − α) ln(c), each with probability .5, independently distributed, in all future periods; if his utility function in period t is 2ln(c) then he thinks his utility function in each future period will be 2 ln(c) or 2 α ln(c) + (1 − α)ln(c) = (1 + α) ln(c), each with probability .5, independently distributed, in all future periods. c) Suppose that Sandro cannot commit in period 0, as in part (a). As a function of β, β^ e, and α, what c∗ 1 (1) and c∗ 1 (2) will Sandro choose? d) Now suppose that, as in part (b), Sandro can commit some of his consumption in period 0 to period 2. He chooses y after he finds out his period 0 (not period 1) utility function. To fully describe Sandro’s complete contingent behavior, we need to specify more than just the three values that fully described his behavior in part (b). State what are the different components of a full description of Sandro’s contingent choices in this case, explaining any notation that is ambiguous. (As always, you don’t need to explain any period 2 behavior, since it only involves Sandro consuming whatever he has left. e) Do not spend too much time on this problem unless confident you will get to the other parts of the exam. Solve as much as you can about Sandro’s behavior in this case as a function of β, βe, and α, and give intuition for your results. With or without numerical results, comment with subtlety and insight on the truthiness of the statement: “Sandro is always better off being able to predict his future utility functions more accurately”.
