Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 7. Time Preferences II Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 1 / 26
Introduction “Saturday 31 December. New Year’s Resolutions. I WILL [...] go to the gym three times a week not merely to buy a sandwich.” (Bridget Jones’s Diary: A Novel) “Monday 28 April. [...] Gym visits 0, no. of gym visits so far this year 1, cost of gym membership per year £370; cost of single gym visit £123 (v. bad economy).” (Bridget Jones: The Edge of Reason) Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 2 / 26
Introduction There is substantial evidence that many people do not anticipate future (time-) preferences. In this case, they do not make e¤ective use of commitment devices. For example, consider all those unsuccessful attempts to quit smoking. Gallup reports 74% of smokers would like to give up smoking, but only 4% to 7% of quit attempts are successful. We use ( β , δ ) -preferences to model naiveté about future time-preferences. 1 We will discover a number of behaviors that may be quite familiar to us. 1 This extension of the ( β , δ ) -model is due to O’Donoghue, Ted, and Matthew Rabin (1999): “Doing it now or later,” American Economic Review 89(1), 103 - 124. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 3 / 26
Introduction Overview Naiveté: Doing It Now or Later Paying Not to Go to the Gym Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 4 / 26
Naiveté: Doing It Now or Later Let there be periods t 2 f 1 , ..., T g with T < ∞ . An agent’s intertemporal utility in period t is T � t U t = u t + β δ τ � t u τ , ∑ (1) τ = t + 1 where u t is her instantaneous utility in period t . For simplicity, we assume δ = 1. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 5 / 26
Naiveté: Doing It Now or Later The agent must perform a task exactly once. If she waits until period T , she must do it in period T . If the agent performs the task in period τ , then she immediately incurs costs c τ , and receives a reward of υ τ in the future (later we will discuss the case of immediate rewards). Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 6 / 26
Naiveté: Doing It Now or Later The agent’s utility in period t from performing the task in period τ > t is U t ( τ ) = � β c τ + βυ τ , (2) and her utility in period t from performing the task in period t is U t ( t ) = � c t + βυ t . (3) The cost- and reward-structure is given by c = ( c 1 , c 2 , ..., c T ) and υ = ( υ 1 , υ 2 , ..., υ T ) . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 7 / 26
Naiveté: Doing It Now or Later We analyze the behavior of three types of agents. Agents with standard exponential, time-consistent preferences ( TC ). Their intertemporal preferences are given by (1) with β = 1. Sophisticated agents ( S ). Their intertemporal preferences are given by (1) with β < 1. Naive agents ( N ). They have the same intertemporal preferences as S . However, in each period t , they assume that in periods after t they have the same preferences as TC . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 8 / 26
Naiveté: Doing It Now or Later The agent’s behavior can be described by her strategy s = ( s 1 , s 2 , ..., s T ) , (4) where s t 2 f y , n g speci…es whether the agent performs the task ( y ) or not ( n ), given she has not yet done it . By assumption, we must have s T = y . � � � � s A De…ne τ A = min t t = Y for A = f TC , N , S g . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 9 / 26
Naiveté: Doing It Now or Later A critical component of any model with naive agents is the solution concept or equilibrium de…nition. Recall that the assumption of rational expectations is at the heart of every equilibrium de…nition in game theory (Nash Equilibrium, Subgame-Perfect Equilibrium). This assumption is violated here. So let us see, what O’Donoghue and Rabin (1999) come up with! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 10 / 26
Naiveté: Doing It Now or Later De…nition 1. A strategy is perception-perfect if in all periods the agent chooses the optimal action given her current preferences and her perceptions of future behavior. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 11 / 26
Naiveté: Doing It Now or Later Example 1. Suppose that T = 4, β = 1 2 , c = ( 3 , 5 , 8 , 13 ) and υ = ( ¯ υ , ¯ υ , ¯ υ , ¯ υ ) . Compute the perception-perfect strategies for TC , N , S and determine the corresponding values τ TC , τ N , τ S . Example 2. Suppose that T = 3, β = 1 2 , c = ( 3 , 8 , 13 ) and υ = ( 12 , 18 , 18 ) . Compute the perception-perfect strategies for TC , N , S and determine the corresponding values τ TC , τ N , τ S . Provide an interpretation for these results! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 12 / 26
Naiveté: Doing It Now or Later Similarly, we can analyze the agent’s behavior if rewards are immediate and costs occur in the future. With immediate rewards, an agent’s utility in period t from performing the task in period τ > t is U t ( τ ) = � β c τ + βυ τ , (5) and her utility in period t from performing the task in period t is U t ( t ) = � β c t + υ t . (6) Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 13 / 26
Naiveté: Doing It Now or Later When rewards are immediate, then S may perform even worse than N (as they anticipate future self-control problems). As an exercise, compute the perception-perfect strategies when rewards are immediate, T = 4, β = 1 2 , υ = ( 3 , 5 , 8 , 13 ) and c = ( 0 , 0 , 0 , 0 ) . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 14 / 26
Naiveté: Doing It Now or Later People are very bad at predicting future preferences. Therefore, naiveté about future time preferences may be a reasonable assumption. Time inconsistency with naiveté (i.e., the assumption that from the next day on time preferences are exponential) explains why people may change their plans (and are surprised by that!). Next, we consider an empirical example that shows time inconsistent preferences (with sophistication/naiveté) in action. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 15 / 26
Paying Not to Go to the Gym As an application of our naiveté-model, we consider the famous study by Stefano DellaVigna and Ulrike Malmendier (2006) on health club attendance. 2 For most people, going to the gym creates immediate costs (e¤ort costs of physical activity) and long-run bene…ts (health, physical shape). How do health club members trade-o¤ immediate costs versus long-run gains? Is there evidence for hyperbolic discounting and/or naiveté? 2 DellaVigna, Stefano, and Ulrike Malmendier (2006): “Paying Not to Go to the Gym,” American Economic Review 96(3), 694 - 719. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 16 / 26
Paying Not to Go to the Gym DellaVigna and Malmendier (2006) analyze the data on day-to-day attendance of around 7.700 members of three health clubs located in New England (the sample period is 1997–2001). Health clubs o¤er a menu of contracts from which consumers can choose: pay-per-visit contracts, monthly contracts, and annual contracts. Do consumers pick the best contract for their needs? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 17 / 26
Paying Not to Go to the Gym Pay-per-visit system: A ten-visit pass is available for 100 USD. Monthly contract: The monthly contract costs 85 USD per month. Cancellation can be done in person or by sending a written note. It must take place before the 10th of the month (otherwise, the contract extends for one more month). Annual contract: The annual contract costs 850 USD per year. At the end of the year, the contract expires and members who wish to stay enrolled have to sign up again (for any type of contract). All contracts give right to the same services. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 18 / 26
Paying Not to Go to the Gym Prediction 1. For users who choose a monthly contract, we have 85 USD expected no. of visits/month < 10 USD . For users who choose an annual contract, we have 850 USD expected no. of visits/year < 10 USD . Prediction 2. The average initial attendance of annual members is higher than the average initial attendance of monthly members. Prediction 3. The average forecast of attendance equals the average actual attendance. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 7. Time Preferences II 19 / 26
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