Econ 4660 1st EditionExam # 2 Study Guide Lectures1: We cannot tell without more information. On one hand, Michelle is impatient (because she has δ < 1), and this impatience will tend to make her choose c1 > c2. On the other hand, future consumption is cheaper (because r > 0%), which will tend to make her choose c1 < c2. Without more information, we cannot tell which of these two conflicting effects dominates. Question 2: Question 3: (a) Using our usual normalization that D(0 days)=1, the first indifference implies $40= D(20 days)($50) or D(20 days)=0.8. The second indifference implies D(20 days)$60 = D(40 days)$75 or D(40 days) D(20 days) = 0.8. Then D(40 days) = D(40 days) D(20 days) ∗ D(20 days)=0.8 ∗ 0.8=0.64. (b) Because D(20 days) = D(40 days) D(20 days) = 0.8, this evidence is consistent with exponential discounting (and INCONSISTENT with present bias). In words, Lexi feels the same about a 20-day delay no matter whether it is 20 days starting from now or 20 days starting 20 days from now – this evenhandedness is the hallmark of exponential discounting. 1 Econ 4660 Spring 2015 Question 4 (a) An exponential discounter with δ < 1 will accelerate good outcomes and delay bad outcomes. Hence, if z > 0 then Miguel will complete the activity in one week (sooner), whereas if z < 0 then Miguel will complete the activity in two weeks (later). (b) Utility from anticipation creates a tendency to delay good outcomes and accelerate bad outcomes. Since he has δ = 1, Miguel’s behavior will be driven entirely by this force. Hence, if z > 0 then Miguel will complete the activity in two weeks (later), whereas if z < 0 then Miguel will complete the activity in one week (sooner). Question 5: The answers are: (i) Forstandard exponential discounters, only statement (a) is true. (ii) For those with naive present bias, statements (a) and (b) are true. (iii) For those with sophisticated present bias, all three statements are true. (iv) For exponential discounters with projection bias, statements (a) and (b)are true. The explanations: Statement (a) is true for any kind of agent, because lowering the marginal price of a good will (weakly) increase quantity purchased under any reasonable model.For standard agents, though, because they are time-consistent, they would purchase the monthly contract only if they expect to and actually pay less than $10 per visit (so (b) and (c) areboth false). For naifs and people with projection bias, they both perceive themselves to be time-consistent, and thus would purchase the monthly contract only if they expect to pay less than $10 per visit (so (c) is false); but they might underestimate future use of the gym and thus end up paying more than $10 per visit (so (b) is true). Finally, for sophisticates, they might purchase the monthly contract to induce themselves to exercise more, even knowing that doing so will mean paying more than $10 per visit (so (b) and (c) are true). Question 6: (a) From a period-1 perspective, Jordan’s intertemporal utilities are: U1(Project A) = (9) + β(−6) + β(−6) + β(−6) + β(−6) = −3 U1(Project B) = β(20) + β(−10) + β(−10) + β(−10) = −5 U1(Project C) = β(−35) + β(25) + β(25) = 7.5 U1(Project D) = β(−25) + β(10) = −7.5 Hence, Jordan’s preferences are (Project C) Â (Project A) Â (Project B) Â (Project D). 2 Econ 4660 Spring 2015 (b) From a period-2 perspective, Jordan’s intertemporal utilities are: U2(Project B) = (20) + β(−10) + β(−10) + β(−10) = 5 U2(Project C) = β(−35) + β(25) + β(25) = 7.5 U2(Project D) = β(−25) + β(10) = −7.5 Hence, Jordan’s preferences are (Project C) Â (Project B) Â (Project D). (c) From a period-3 perspective, Jordan’s intertemporal utilities are: U3(Project C)=(−35) + β(25) + β(25) = −10 U3(Project D) = β(−25) + β(10) = −7.5 Hence, Jordan’s preferences are (Project D) Â (Project C). (d) If Jordan is naive, then in period 1, he’ll view Project C as best, and so he’ll wait in period 1. When period 2 arrives, he’ll still view Project C as best, and so he’ll wait again. When period 3 arrives, now he’ll view Project D as best, and so he’ll commit to that. Hence, if he is naive, Jordan will end up pursuing Project D. (e) If Jordan is sophisticated, he’ll do the backward induction. In period 3, he’d prefer Project D to Project C, and so he’d commit to that. Knowing this, in period 2, he’ll compare Project B to Project D, and from a period-2 perspective, he prefers Project B. Knowing this, in period 1, he’ll compare Project A to Project B, and from a period-1 perspective, he prefers Project A. Hence, if he is sophisticated, Jordan will pursue Project A. (f) We measure welfare losses using long-run preferences. The long-run utility for naifs is U0(Project D)=(−25) + (10) = −15, while the long-run utility for sophisticates is U0(Project A) = (9) + (−6) + (−6) + (−6) + (−6) = −15. Since they get the same long-run utility, naifs and sophisticates suffer identical welfare losses. Question 7: For both parts, Se Ri would like to implement her period-1 desired behavior (c∗ 1, c∗ 2, c∗ 3), but to do so she must constrain her period-2 self. (a) Here, becauseY2 > c∗ 2, Se Ri cannot implement period-1 desired behavior. But since Y2 < cS 2 , she canconstrain her period-2 self to consume c2 = Y2 = 1025 (by investing nothing in the liquid asset), and use the illiquid asset to allocate the remaining 2475 between periods 1 and 3. Relative to period-1 optimal behavior, she’ll cut back on both period-1 and period-3 consumption. And since she has 25 less to allocate between these two periods, c1 and c3 will each be 0 to 25 less than c∗ 1 and c∗ 3. Hence: c1 ∈ (1475, 1500), c2 = 1025, and c3 ∈ (975, 1000), where she’ll put x = 0 in the bank, and z ∈ (250, 275) in the illiquid asset. 3 Econ 4660 Spring 2015 (b) Here, because in period 2 Se Ri will be able borrow $50, her period-2 self will be able to spend up to Y2 + $50 = $1075 in period 2 even if Se Ri puts nothing in the bank. Because $1075 > c∗ 2, it is NOT possible for Se Ri to implement her period-1 desired behavior. But since $1075 < cs 2, she can constrain her period-2 self to consume c2 = Y2 = 1075 (by investing nothing in the liquid asset), and use the illiquid asset to allocate the remaining 2425 between periods 1 and 3. Relative to period-1 optimal behavior, she’ll cut back on both period-1 and period-3 consumption. And