Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 7. - - PowerPoint PPT Presentation

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

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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

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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.

1This 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

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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

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Naiveté: Doing It Now or Later

Let there be periods t 2 f1, ..., Tg with T < ∞. An agent’s intertemporal utility in period t is Ut = ut + β

T t

∑

τ=t+1

δτtuτ, (1) where ut 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

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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 cτ, and receives a reward of υτ in the future (later we will discuss the case of immediate rewards).

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

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Naiveté: Doing It Now or Later

The agent’s utility in period t from performing the task in period τ > t is Ut(τ) = βcτ + βυτ, (2) and her utility in period t from performing the task in period t is Ut(t) = ct + βυt. (3) The cost- and reward-structure is given by c = (c1, c2, ..., cT ) and υ = (υ1, υ2, ..., υT ).

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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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

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Naiveté: Doing It Now or Later

The agent’s behavior can be described by her strategy s = (s1, s2, ..., sT ) , (4) where st 2 fy, ng speci…es whether the agent performs the task (y)

• r not (n), given she has not yet done it.

By assumption, we must have sT = y. De…ne τA = min

• t
• sA

t = Y

• for A = fTC, N, Sg.
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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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

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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

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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 τTC , τN, τS. 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 τTC , τN, τS. Provide an interpretation for these results!

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

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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 Ut(τ) = βcτ + βυτ, (5) and her utility in period t from performing the task in period t is Ut(t) = βct + υt. (6)

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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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

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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

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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?

2DellaVigna, 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

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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

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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

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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)

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Paying Not to Go to the Gym

Price per average attendance at enrollment of consumers choosing the monthly contract Average price Average attendance Average price per month per month per average attendance M 1 55.23 3.45 16.01 M 2 80.65 5.46 14.76 M 3 70.18 4.89 14.34 M 4 81.79 4.57 17.89 M 5 81.93 4.42 18.53 M 6 81.94 4.32 18.95 M 1-6 75.26 4.36 17.27

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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Paying Not to Go to the Gym

Price per average attendance at enrollment of consumers choosing the annual contract Average price Average attendance Average price per month per month per average attendance Y 1 66.32 4.36 15.22

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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Paying Not to Go to the Gym

Average attendance in monthly and annual contracts Month 2 Month 3 Month 4 Monthly contract 5.507 5.005 4.614 Annual contract 5.805 5.629 5.193

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

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Paying Not to Go to the Gym

Finding 1. Users who choose a ‡at-rate contract pay a price per average attendance of over 17 USD in the monthly contract and over 15 USD in the annual contract. The share of users who pay ex post less than 10 USD per visit is 20 percent in the monthly contract and 24 percent in the annual contract. Finding 2. Average attendance in months 2-4 is 10 percent higher under the annual contract than under the monthly contract.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

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Paying Not to Go to the Gym

Finding 3. The average forecasted number of monthly visits, 9.50, is more than twice as large as average attendance, 4.17. Finding 4. On average, 2.31 full months elapse between the last attendance and contract termination for monthly members, with associated membership payments of 187 USD. This lag is at least four months for 20 percent of the users.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 7. Time Preferences II

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Paying Not to Go to the Gym

From these …ndings only …nding 2 is consistent with an economic standard model. Can time inconsistency with sophistication explain these patterns? Flat-rate contracts are attractive to sophisticated users. Given that health club attendance involves immediate costs and delayed bene…ts, these users may purchase a ‡at-fee membership as a commitment device that increases future attendance (explain why). However, sophisticated agents would have rational expectations (contrary to …nding 3) and would not waste much money by delaying cancellation (as …nding 4 shows). Hence, time inconsistency with sophistication cannot explain the

• bserved patterns.
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

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Paying Not to Go to the Gym

Can time inconsistency with naiveté explain the observed patterns? Naive agents with time inconsistent preferences are overoptimistic about their future self-control and may expect to go to the gym more

• ften than they actually do. This explains …ndings 1 and 3.

DellaVigna and Malmendier (2006) show that also the delayed cancellation can be explained with naive agents (they erroneously believe that they will cancel the contract tomorrow; therefore, they do not cancel it today; but when tomorrow arrives, they revise their plan and delay it to the next day, and so forth...). Hence, a model with naive agents can explain all observations.

• Prof. Dr. Heiner Schumacher (KU Leuven)

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