Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 6. Time Preferences I Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 1 / 38
Introduction In this lecture, we focus on time preferences, i.e., how people value a good at an earlier date compared to its value at a later date (“discounting”). Time preferences are crucial for many important decisions like capital accumulation (How much should I save for retirement) or human capital accumulation (Should I work immediately or …rst obtain an university degree?). Moreover, time preference matter in the domain of health behaviors such as smoking (Should I stop smoking now? Or tomorrow?), the consumption of fat foods and exercising (Should I start with the diet/training today? Or tomorrow?). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 2 / 38
Introduction The “Stanford Marshmallow Experiment” (SME) A good example for the importance of time preferences is the Stanford Marshmallow Experiment (Mischel et al. 1989). 1 In the SME, four-year-old children get a Marshmallow which they can consume immediately (assume for the moment that all children love Marshmallows). The experimenter o¤ers the following deal: If they can wait for 15 minutes without eating the Marshmallow, they get a second one. Otherwise, they don’t. As you may expect, some children managed to wait for 15 minutes, others didn’t. 1 Mischel, Walter, Yuichi Shoda, and Monica L. Rodriguez (1989): “Delay of Grati…cation in Children,” Science 244(4907), 933 - 938. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 3 / 38
Introduction Many years later, the experimenters got data about the children’s development after the experiment. Those children who managed to wait, achieved higher SAT scores and were healthier (as measured by the BMI) than those who did not wait. They were described by their parents “as adolescents who were signi…cantly more competent”. Apparently, the ability to delay grati…cation to achieve later, greater rewards is decisive for the probability of reaching long-term goals like wealth, health or education. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 4 / 38
Introduction Overview Exponential Discounting Hyperbolic Discounting Commitment I: Deadlines Commitment II: Golden Eggs Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 5 / 38
Exponential Discounting We brie‡y recall the exponential discounting model. Suppose that you have the amount D to spend today so that your utility from consumption is V = U ( D ) . What is your intertemporal utility if today you have no money to spend and tomorrow you have D (there are no opportunities to save or borrow)? If you discount future utilities by δ , your intertemporal utility is given by V = δ U ( D ) . Suppose that today, tomorrow and the day after tomorrow you can spend D . Then your intertemporal utility is V = U ( D ) + δ U ( D ) + δ 2 U ( D ) . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 6 / 38
Exponential Discounting Suppose that you spend D on every day. Then your intertemporal utility is V = U ( D ) + δ U ( D ) + δ 2 U ( D ) + δ 3 U ( D ) + δ 4 U ( D ) + ... We can write this term as V = U ( D ) + δ [ U ( D ) + δ U ( D ) + δ 2 U ( D ) + δ 3 U ( D ) + ... ] . Note that the term in the brackets is identical to V . Therefore, we have V = U ( D ) + δ V , and consequently V = U ( D ) 1 � δ . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 7 / 38
Exponential Discounting The discount factor δ measures the weight of future payo¤s relative to current payo¤s in the intertemporal utility function. If δ is close to 1, the agent is patient and attaches almost the same weight to current and future payo¤s. If δ is close to 0, the agent is impatient and values current payo¤s much higher than future payo¤s. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 8 / 38
Hyperbolic Discounting Exponential discounting implies that an individual’s time preferences are constant: in period t , the discount rate between the periods t and t + 1 is the same as between the periods t + 10 and t + 11. An immediate consequence of exponential discounting is “time-consistent” behavior: A consumption plan that is optimal in period t , is also optimal in all subsequent periods. However, there is no reason why we should expect that this is true. In fact, there is substantial evidence for time-inconsistent behavior. Individuals frequently change their plans or choose actions in order to commit themselves to certain plans in the future. In particular, they overemphasize immediate costs relative to future gains (for example, the failure to give up smoking). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 9 / 38
Hyperbolic Discounting We now study the impact of “(quasi-) hyperbolic discounting” on optimal consumption/savings decisions. Research on animal and human behavior showed that discount functions are approximately hyperbolic (Ainslie 1992). 2 We will see that hyperbolic discounting leads to time-inconsistent behavior. This has serious economic implications. 2 Ainslie, George (1992): Picoeconomics: The Strategic Interaction of Successive Motivational States within the Person , Cambridge University Press, Cambridge, UK. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 10 / 38
Hyperbolic Discounting Under standard (exponential) discounting, the discounted utility from this consumption stream in period t is u ( c t ) + δ u ( c t + 1 ) + δ 2 u ( c t + 2 ) + ... (1) Under (quasi-) hyperbolic discounting, the discounted utility from a consumption stream c t , c t + 1 , c t + 2 ... in period t is given by u ( c t ) + βδ u ( c t + 1 ) + βδ 2 u ( c t + 2 ) + ... (2) where β < 1. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 11 / 38
Hyperbolic Discounting Consider an agent who has wealth W and can consume this wealth in the periods t 2 f 1 , 2 , 3 g . Consumption in period t is denoted by c t . His utility from consumption in period t is given by a continuous function u ( c t ) , where u 0 ( c t ) > 0, u 00 ( c t ) < 0 and lim c t ! 0 u 0 ( c t ) = ∞ . The agent can save and borrow at the interest rate R > 1. The agent’s discount rate is δ < 1. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 12 / 38
Hyperbolic Discounting The agent maximizes 2 δ τ u ( c τ + 1 ) ∑ u ( c 1 ) + (3) τ = 1 subject to 3 1 ∑ R τ � 1 c t � W . (4) τ = 1 Standard arguments (for example, a Lagrangian) show that the optimal consumption plan is characterized by the Euler Equation u 0 ( c t ) = R δ u 0 ( c t + 1 ) . (5) for t 2 f 1 , 2 g . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 13 / 38
Hyperbolic Discounting Now suppose that the agent exhibits hyperbolic discounting. Her intertemporal preferences in period t are given by 3 � t δ τ u ( c t + τ ) . ∑ u ( c t ) + β (6) τ = 1 What is the optimal consumption plan? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 14 / 38
Hyperbolic Discounting In period 1, the agent would like to execute a plan that maximizes 2 δ τ u ( c τ + 1 ) ∑ u ( c 1 ) + β (7) τ = 1 subject to the budget constraint in (4). This plan is characterized by the equations u 0 ( c 1 ) = β R δ u 0 ( c 2 ) , (8) u 0 ( c 2 ) = R δ u 0 ( c 3 ) . (9) Hence, the agent (i) would like to consume more in period 1 compared to the case with standard preferences, and (ii) would like his future selves to consume as if they had standard preferences. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 15 / 38
Hyperbolic Discounting However, in period 2, the agent chooses c 2 (for given c � 1 ) to maximize u ( c 2 ) + βδ u ( c 3 ) (10) subject to c 2 + 1 R c 3 � R ( W � c � 1 ) . (11) The agent’s decision in period 2 is therefore characterized by u 0 ( c 2 ) = β R δ u 0 ( c 3 ) . (12) Hence, in period 2, the agent will consume more than what the optimal plan in period 1 would allow (“time inconsistency”)! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 16 / 38
Hyperbolic Discounting Hyperbolic discounting implies that the agent’s intertemporal preferences change over time. We therefore have to deal with an agent’s “multiple selves”. Self t of an agent has di¤erent preferences over consumption than her self t + 1, self t + 2, asf. In particular, the agent’s self t optimizes against the actions of her future selves. The optimization problem becomes an (intrapersonal) game (with players, strategies and payo¤ functions). Note that (in general) the “optimal plan” of self 1, characterized by the equations (8) and (9), is not an “optimal strategy” against the actions of her future selves! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 6. Time Preferences I 17 / 38
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