(Time Inconsistency) A student has to do a problem set in one of T periods 0, 1, ..., 9. The cost of doing the problem set in period t is (3/2). that the student has = 0.5, = 1 and there is no commitment. The following steps will help you prove that a naive student will do the problem set in period 9. i. When would self 0 want to do the problem set? Conclude from your answer that self 0 does not do the problem set. ii. When would self 1 want to do the problem set? Conclude from your answer that self 1 does not do it, either. iii. Argue that the same logic applies to periods t = 1 -8, so that the student does the problem set in period 9. Explain the intuition behind this result. b. The following steps will help you prove that a sophisticated student does the problem set in period 1. For this, we work backwards as in class. i. Show that if the problem set is not done by period 8, self 8 will not do it, either, ii. Based on the previous part, argue that self 7 is effectively choosing between doing the problem set att = 7 and doing it at t = 9. iii. Then show that self 7 will do the problem set if it is not done by then. iv. Now show that self 6 will not do the problem set. Repeat the same logic that selves 1,3, and 5 will, but selves 0,2, and 4 will not, do the problem set if it is not done by that time. Explain the intuition. V. (Time Inconsistency) A student has to do a problem set in one of T periods 0, 1, ..., 9. The cost of doing the problem set in period t is (3/2). that the student has = 0.5, = 1 and there is no commitment. The following steps will help you prove that a naive student will do the problem set in period 9. i. When would self 0 want to do the problem set? Conclude from your answer that self 0 does not do the problem set. ii. When would self 1 want to do the problem set? Conclude from your answer that self 1 does not do it, either. iii. Argue that the same logic applies to periods t = 1 -8, so that the student does the problem set in period 9. Explain the intuition behind this result. b. The following steps will help you prove that a sophisticated student does the problem set in period 1. For this, we work backwards as in class. i. Show that if the problem set is not done by period 8, self 8 will not do it, either, ii. Based on the previous part, argue that self 7 is effectively choosing between doing the problem set att = 7 and doing it at t = 9. iii. Then show that self 7 will do the problem set if it is not done by then. iv. Now show that self 6 will not do the problem set. Repeat the same logic that selves 1,3, and 5 will, but selves 0,2, and 4 will not, do the problem set if it is not done by that time. Explain the intuition. V