Modeling self-control problems I: Multi-self vs. temptation Lectures in Behavioral economics Spring 2016, Part 2 People have demand for commitment Ariely & Wertenbroch (2002) Experiment where the students in a course had to hand in three compulsory assignments before the final exam, they could choose deadlines, and were punished if the deadlines were not observed. Result: Many choose deadlines before the end of the semester, & among these, students with evenly spread deadlines did better. 08.03.2016 G.B. Asheim, ECON4260, #2 1 Present-biased preferences: (EG)-pref. U t (ct ,! , cT ) u (ct ) E ¦W T t 1 3 G.B. Asheim, ECON4260, #2 Hyperbolic discounting leads to time-inconsistent preferences (Strotz, 1956), procrastination of tasks with immediate cost, and makes commitment desirable, given awareness of the self-contr. problems. Other ways to model the demand for commitment? 08.03.2016 G W t u (cW ) Yields time-inconsistent preferences. Behavior with time-inconsistent preferences Naive behavior: Choosing the best plan under the presumption that it will be followed. Sophisticated behavior: Choosing the best plan among those that will actually be followed. Multi-self model of sophisticated behavior: Let every decision node correspond to a different “self ”. 08.03.2016 G.B. Asheim, ECON4260, #2 2 Outline ”Do it now or later”-article O’Donoghue & Rabin (1999). Interesting application of the multi-self approach showing that sophisticates need not realize better outcomes than naifs. Problems with the multi-self approach Alternative to the multi-self approach Gul & Pesendorfer (2001, 2004). Direct modeling of temptation. Soft paternalism (”Nudge”) 08.03.2016 4 G.B. Asheim, ECON4260, #2 “Do it now or later” O’Donoghue & Rabin (1999) Model: ʀ Must perform an activity exactly once. “Do it now or later” (2) ʀ T f periods in which to perform it. Two cases: ʀ Each period, choose to “do it” or “wait”. ʀ If wait until period T, must do it then. If activity is done in period t, incur cost ct t 0 and receive reward t 0. ʀ Immediate costs: incur cost when you do it, receive reward after some delay. ʀ Immediate rewards: receive reward when you do it, incur cost after some delay. Assume (EG)-preferences with G (for simplicity): Period-t utility for “do it” in period W t t: For immedi ate costs : U t vt Reward schedule : v { (v1 , ! , vT ) For immedi ate rewards : U t Cost schedule : c { (c1 , ! , cT ) 08.03.2016 G.B. Asheim, ECON4260, #2 5 4 T 08.03.2016 Reward schedule : v { (0, 0, 0, 0) Cost schedule : c { (3, 5, 8, 13) Period-t utility for “do it” in period W t t: W t 3 0 (12) 1 W 2 5 2 W 3 4 ( 2) t 1 (23) t 2 W (3) (3) (1) 4 5 2 4 13 2 ( 4) 13 2 ( 4) (3) (1) ( 2) 132 4 5 (1) ( 2) 13 2 8 7 G.B. Asheim, ECON4260, #2 ­E vW cW if W t ® ¯E vW E cW if W ! t ­vW E cW if W t ® ¯E vW E cW if W ! t G.B. Asheim, ECON4260, #2 6 “Do it now or later”: Ex. with immediate costs (2) “Do it now or later”: Ex. with immediate costs 1 2 E Naifs do it in period 4. Sophisticates do t 3 it in period 2. 08.03.2016 Welfare comparisons of naive & sophistic. behavior: W 2 is better than W 4 at both t 0, t 1 and t 2. W 2 cannot be compared with W 4 at t 3. Period-t utility for “do it” in period W t t: Naifs do it in period 4. t W 3 0 (12) W 1 2 5 2 ( 2) t 1 (23) t 5 2 (1) 5 2 ( 2) Sophisticates do t 3 it in period 2. 08.03.2016 W 3 4 W 4 132 (3) 4 ( 4) 132 (3) 4 ( 4) 132 (1) 8 (3) 132 ( 2) (1) 8 G.B. Asheim, ECON4260, #2 “Do it now or later”: Ex. with immediate rewards 1 2 E 4 T Naifs do it in period 3. Period-t utility for “do it” in period W t t: Reward schedule : v { (3, 5, 8, 13) “Do it now or later”: Ex. with immediate rewards (2) Cost schedule : c { (0, 0, 0, 0) W t 0 t 1 t 1 W 2 W 3 W 4 13 2 ( 2) ( 4) (3) 4 5 2 3 (3) ( 4) 5 2 3 2 2 Sophisticates do t 3 it in period 1. 08.03.2016 4 ( 2) (1) 13 2 (1) (1) (3) ( 2) 13 2 4 5 ( 2) (1) 13 2 8 9 G.B. Asheim, ECON4260, #2 General lesson: Sophistication about future self-control problems can mitigate or exacerbate misbehavior 11 G.B. Asheim, ECON4260, #2 Welfare comparisons of naive & sophistic. behavior: W 3 is better than W 1 at both t 0 and t 1. W 3 cannot be compared with W 1 at t 2 and t 3. Period-t utility for “do it” in period W t t: Naifs do it in period 3. t W 1 W 3 2 0 2 W 5 2 ( 4) (3) 3 W ( 2) ( 2) (1) 13 2 4 ( 2) 4 (1) 13 2 4 ( 4) 5 2 13 2 4 (3) 5 2 3 t 1 t Sophisticates do t 3 it in period 1. 08.03.2016 Proposition 2. For both immediate costs and immediate rewards, Ws d Wn. ʀ Why? The future is always more promising from the point of view of a naif, since a sophisticate removes some future possibilities as unattainable without commitment. Hence, for a sophisticate the present is relatively more attractive, leading to the task being performed earlier. ʀ With im. costs: Naifs may procrastinate due to presentbiased preferences. With im. rewards: Sophisticates may preproperate since they realize their future misbehavior. 08.03.2016 (3) (1) 13 2 8 (1) ( 2) 10 G.B. Asheim, ECON4260, #2 Propositions 3 and 4. When evaluated from a prior period 0, the following holds. For immediate costs, a small self-control problem can cause servere welfare losses if and only if you are naive. For immediate rewards, a small self-control problem can cause servere welfare losses if and only if you are sophisticated. ʀ Why? Naifs with immediate costs may procrastinate repeatedly even if E is close to 1. Sophisticates with immediate rewards may preproperate repeatedly even if E is close to 1. General lesson: Even “small” self-control problems can cause servere welfare losses. 08.03.2016 12 G.B. Asheim, ECON4260, #2 Procrastination example revisited A task to be performed at time 0, 1, 2, …, or not at all. Immediate cost: 25. Benefits at the next stage: 125. (E, G)-preferences with E 50 40 1/2 and G 4/5. But not Better worthwhile to do it to never. wait for now than 2 periods Sophisticated behavior with 3 periods 32 time 8 10 25 Even better to do it at the next stage. 08.03.2016 13 G.B. Asheim, ECON4260, #2 Backward induction Let every decision node correspond to a different “self” or “agent” of the decision-maker Backward induction Self a time 1 if A sophisticate task was not does the task done at time 0 now, since else postponed for 2 periods. Self a time 0 08.03.2016 0 1 2 24 30 Payoff of 0-self 30 25 25 Payoff of 1-self 14 G.B. Asheim, ECON4260, #2 Sophisticated behavior with odd # periods Sophisticated behavior with 4 periods A sophisticate does the task with a delay of 1 period. A sophisticate does the task now, since else postponed for 2 periods. Sophisticated behavior with even # periods A sophisticate does the task with a delay of 1 period. Conclusion: Multi-self model with sophisticated behavior may not be descriptively accurate 0 1 08.03.2016 3 2 15 G.B. Asheim, ECON4260, #2 Is it “right” to apply the multi-self model? Introduce naivete (or partial naivete) (O’Donoghue & Rabin, 08.03.2016 1999, 2001) 16 G.B. Asheim, ECON4260, #2 Sophisticated behavior with f periods If we impose that behavior is the same in all periods, conditional on the task not having been done, there is a unique equilibrium, where the task is done in each period with prob. ½. It is another equilibrium (planning) to do the task in periods 1, 3, 5, …, but not in periods 0, 2, 4, … . It is an equilibrium (planning) to do the task in periods 0, 2, 4, …, but not in periods 1, 3, 5, … . Backw. induct. cannot be used since no last period Demand for commitment Utility at t of doing the task at 30 W 1 24 W 2 W 0 3 ¼ prob. ½ prob. 2 25 0 08.03.2016 1 0 1 2 3 Suppose the decision maker can purchase a commitment device costing c, ensuring that the task be done in the next period? What is the largest c? With an odd # periods, she does the task now with present payoff 25. Commitment to next period yields payoff 30 c. Hence, c cannot exceed 5. With an even # periods, she does the task in the next period anyway. Not interested in commiting. With f periods, she receives a payoff of 25. Commitment to next period yields payoff 30 c. Hence, c cannot exceed 5. 08.03.2016 17 G.B. Asheim, ECON4260, #2 Temptation (Gul & Pesendorfer 2001) a : Doing the task 1 G.B. Asheim, ECON4260, #2 18 Temptation in a dynamic setting (Gul & Pesendorfer 2004) a : Not doing the task 0 In standard consumer theory, if a1 is preferred to a0 , then she has the following preference over menus : {a1} ~ {a0 , a1} ; {a0 } No time-inconsistency; the future is discounted by G Still, temptation yields a demand for commitment Payoff when choosing a1 : 25 54 125 t In G & P' s analysis, if tempted by a0 and gives in : Payoff when choosing a0 : {a1} ; {a0 , a1} ~ {a0 } 0 4 5 W0 Maximal payoff when task has not been done : If tempted by a0 , but does not give in : G {a1} ; {a0 , a1} ; {a0 } 08.03.2016 19 G.B. Asheim, ECON4260, #2 W0 max{75 t , 54 W0 } t : Cost of temptation 4 5 08.03.2016 20 G.B. Asheim, ECON4260, #2 Temptation in a dynamic setting (2) Optimal not to give in at a low cost of temptation ( t 75 ) W0 75 t ! 0 W > 0 0 4 5 W 0 Optimal to give in at a high cost of temptation ( t t 75 ) W0 0 W0 t 75 t Temptation and the demand for commitment Assume that she can commit by paying c to doing the task in the next period (without being tempted). Wc 0c 4 5 25 54 125 60 c Optimal to commit at a low cost of temptation ( t 75 ) if 60 c ! 75 t Wc W0 c t 15 Optimal to commit at a high cost of temptation ( t t 75 ) if Wc 08.03.2016 21 G.B. Asheim, ECON4260, #2 60 c ! 0 08.03.2016 W0 c 60 22 G.B. Asheim, ECON4260, #2 Soft paternalism With procrastination, the status quo matters E.g. organ donation, savings decisions Organ donation: Opt out. Austria: 99.98 % consent Opt in. Germany: 12 % consent Active choice. US driver’s licence 08.03.2016 23 G.B. Asheim, ECON4260, #2