Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 6. - - PowerPoint PPT Presentation

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

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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)

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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,

• thers didn’t.

1Mischel, Walter, Yuichi Shoda, and Monica L. Rodriguez (1989): “Delay of

Grati…cation in Children,” Science 244(4907), 933 - 938.

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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.

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Introduction

Overview Exponential Discounting Hyperbolic Discounting Commitment I: Deadlines Commitment II: Golden Eggs

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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

• r borrow)? If you discount future utilities by δ, your intertemporal

utility is given by V = δU(D). Suppose that today, tomorrow and the day after tomorrow you can spend D. Then your intertemporal utility is V = U(D) + δU(D) + δ2U(D).

• Prof. Dr. Heiner Schumacher (KU Leuven)

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Exponential Discounting

Suppose that you spend D on every day. Then your intertemporal utility is V = U(D) + δU(D) + δ2U(D) + δ3U(D) + δ4U(D) + ... We can write this term as V = U(D) + δ[U(D) + δU(D) + δ2U(D) + δ3U(D) + ...]. Note that the term in the brackets is identical to V . Therefore, we have V = U(D) + δV , and consequently V = U(D) 1 δ .

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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.

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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).

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Hyperbolic Discounting

We now study the impact of “(quasi-) hyperbolic discounting” on

• ptimal 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.

2Ainslie, George (1992): Picoeconomics: The Strategic Interaction of Successive

Motivational States within the Person, Cambridge University Press, Cambridge, UK.

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Hyperbolic Discounting

Under standard (exponential) discounting, the discounted utility from this consumption stream in period t is u(ct) + δu(ct+1) + δ2u(ct+2) + ... (1) Under (quasi-) hyperbolic discounting, the discounted utility from a consumption stream ct, ct+1, ct+2... in period t is given by u(ct) + βδu(ct+1) + βδ2u(ct+2) + ... (2) where β < 1.

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Hyperbolic Discounting

Consider an agent who has wealth W and can consume this wealth in the periods t 2 f1, 2, 3g. Consumption in period t is denoted by ct. His utility from consumption in period t is given by a continuous function u(ct), where u0(ct) > 0, u00(ct) < 0 and limct!0 u0(ct) = ∞. The agent can save and borrow at the interest rate R > 1. The agent’s discount rate is δ < 1.

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Hyperbolic Discounting

The agent maximizes u (c1) +

2

∑

τ=1

δτu (cτ+1) (3) subject to

3

∑

τ=1

1 Rτ1 ct W . (4) Standard arguments (for example, a Lagrangian) show that the

• ptimal consumption plan is characterized by the Euler Equation

u0 (ct) = Rδu0 (ct+1) . (5) for t 2 f1, 2g.

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Hyperbolic Discounting

Now suppose that the agent exhibits hyperbolic discounting. Her intertemporal preferences in period t are given by u(ct) + β

3t

∑

τ=1

δτu(ct+τ). (6) What is the optimal consumption plan?

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Hyperbolic Discounting

In period 1, the agent would like to execute a plan that maximizes u(c1) + β

2

∑

τ=1

δτu(cτ+1) (7) subject to the budget constraint in (4). This plan is characterized by the equations u0(c1) = βRδu0(c2), (8) u0 (c2) = Rδu0 (c3) . (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.

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Hyperbolic Discounting

However, in period 2, the agent chooses c2 (for given c

1 ) to maximize

u(c2) + βδu(c3) (10) subject to c2 + 1 R c3 R(W c

1 ).

(11) The agent’s decision in period 2 is therefore characterized by u0(c2) = βRδu0(c3). (12) Hence, in period 2, the agent will consume more than what the

• ptimal plan in period 1 would allow (“time inconsistency”)!
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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!

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Hyperbolic Discounting

Time inconsistent preferences make welfare judgments more di¢cult. It is no longer clear what the correct measure for welfare is. There seems to be a tendency in the literature to use self 1’s (expected) utility as measure for welfare, because self 1 has a long-run perspective. Time inconsistent preferences create demand for commitment: self t wishes to restrict the consumption of self t + 1, self t + 2, asf. In the next chapters, we study two di¤erent commitment mechanisms (deadlines and “golden eggs”).

• Prof. Dr. Heiner Schumacher (KU Leuven)

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Commitment I: Deadlines

People sometimes attempt to control their procrastination by setting deadlines for themselves. However, do they use this mechanism

• ptimally?

To study this question, we consider the experimental evidence of Ariely and Wertenbroch (2002).3 They test 99 students who have to write three short papers within the 14 weeks of the semester. There were two treatments: (1) The deadlines to hand in those papers were given …xed and evenly spaced (a paper at the end of each third of the course); (2) Students are free to choose the date by which he or she wanted to hand in the papers.

3Ariely, Dan, and Klaus Wertenbroch (2002): “Procrastination, Deadlines, and

Performance: Self-Control by Precommitment,” Psychological Science 13(3), 219 - 224.

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Commitment I: Deadlines

Students had to hand in their papers no later than the last lecture. Students had to announce the deadlines for submission prior to the second lecture. The dates could not be changed. Each day of delay beyond the deadline would cause a 1% penalty in the paper’s grade. What would you do in the second (free-choice) treatment?

• Prof. Dr. Heiner Schumacher (KU Leuven)

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Commitment I: Deadlines

Do students in the second treatment choose the last possible day as deadline for all papers in order to maximize ‡exibility? Note that students could privately plan to hand in the papers earlier! Which students will do better (in terms of grades), those in the no-choice or in the free-choice treatment?

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Commitment I: Deadlines

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Commitment I: Deadlines

The mean deadlines were signi…cantly earlier than the last possible deadline: …rst paper 41.78 days before the end, second paper 26.07 days before the end, third paper 9.84 days before the end. The grades in the no-choice treatment (the average is 88.76) are signi…cantly better than those in the free-choice treatment (the average there is 85.67). In the …nal exam on the last day of the course, students in the no-choice treatment were signi…cantly better than students in the free-choice treatment (the average grades were 86 and 77, respectively).

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Commitment I: Deadlines

People are willing to self-impose deadlines to overcome procrastination, even when these deadlines are costly. Although students used deadlines, they did not set these deadlines

• ptimally. Greater ‡exibility might lead to lower grades.
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Commitment II: Golden Eggs

We follow Laibson (1997) and extent the previous consumption/savings problem by an commitment mechanism.4 The agent consumes in the period t 2 f1, 2, ..., Tg. In each period t, the agent receives income yt and the interest rate is given by Rt. The income process fytgT

t=1 and the interest rates

fRtgT

t=1 are deterministic.

The utility from consumption in period t is given by the concave function u(ct).

4Laibson, David (1997): “Golden Eggs and Hyperbolic Discounting,” Quarterly

Journal of Economics 112(2), 443 - 477.

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Commitment II: Golden Eggs

The agent can hold liquid assets xt 0 (money in the pocket) and illiquid assets zt 0 (“golden eggs” such as real estate property). Both assets have the same rate of return Rt. In period 1, the agent has an endowment of x0 and z0. In period t, the agent can spend only xt and yt on consumption. Hence, the budget constraint is ct yt + Rtxt1. (13) After consuming, the agent chooses the new asset allocation xt and zt such that yt + Rt (xt1 + zt1) = ct + xt + zt. (14) Thus, holding the illiquid asset serves as a commitment device!

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Commitment II: Golden Eggs

The utility of self t from a consumption stream fct+τgT t

τ=0 is

Ut = u (ct) + β

T t

∑

τ=1

δτu (ct+τ) . (15) We are interested in the subgame perfect equilibrium of the game that is played by self 1, self 2,..., self T.

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Commitment II: Golden Eggs

Theorem 1 (Laibson 1997). There exists a unique subgame perfect

• equilibrium. The consumption process satis…es the inequalities (P1)-(P4)

that are shown on the next slide.

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Commitment II: Golden Eggs

u0(ct)

• max

τ2f1,...,T tg βδτ τ

∏

i=1

Rt+i ! u0(ct+τ) (P1) u0(ct) > max

τ2f1,...,T tg βδτ τ

∏

i=1

Rt+i ! u0(ct+τ) (P2) ) ct = yt + Rtxt1 u0(ct+1) < max

τ2f1,...,T t1g δτ τ

∏

i=1

Rt+i ! u0(ct+1+τ) (P3) ) xt = 0 u0(ct+1) > max

τ2f1,...,T t1g δτ τ

∏

i=1

Rt+i ! u0(ct+1+τ) (P4) ) zt = 0

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Commitment II: Golden Eggs

Interpretation The inequality in (P1) is a standard Euler equation for an environment with liquidity constraints. Marginal utility can be too high (consumption can be too low) relative to future marginal utilities (future consumption)5, but never be too low since the agent always can save for the future. The inequality in (P2) re‡ects that when marginal utility is too high (consumption is too low), the liquidity constraint must be binding. High marginal utility means low consumption relative to the future. Hence, it cannot be optimal for the agent to save in the current period for the future.

5For example, if current income is very low and there are few liquid assets.

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Commitment II: Golden Eggs

The inequality in (P3) implies that self t will limit the liquidity of self t + 1 as much as possible since the consumption in period t + 1 is expected to be too high relative to what self t would prefer it to be. The inequality in (P4) implies that self t will increase the liquidity of self t + 1 as much as possible since the consumption in period t + 1 is expected to be too low relative to what self t would prefer it to be.

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Commitment II: Golden Eggs

The model explains a number of phenomena that cannot be explained with exponential discounting: the violation of the Permanent Income Hypothesis, the coexistence of substantial savings and the co-movement of consumption/income, declining saving rates, and why …nancial innovation may decrease welfare. On the next slides, we explain these issues in detail.

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Commitment II: Golden Eggs

Permanent Income Hypothesis The Permanent Income Hypothesis states that individuals “smooth”

• consumption. In particular, they should not react to expected

variations in income (for example, if I expect to inherit a large amount of money next year, I start to increase consumption now; consumption is then stable over time). However, the Permanent Income Hypothesis is rejected by the data. In the golden eggs model, the consumer consumes “too” much in periods where income is very high, although the income process is deterministic.

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Commitment II: Golden Eggs

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Commitment II: Golden Eggs

Aggregate Saving In classical macroeconomic models, high discount rates (i.e., a low δ) are necessary to explain the co-movement of consumption and income. This, however, contradicts actual levels of capital accumulation. The golden eggs model can explain both consumption-income co-movement and historical levels of capital-output ratios. The reason is that when self t considers the trade-o¤ between consumption at t + 1 and consumption at periods after t + 1, then the present-bias β does not matter. Hence, we can calibrate δ to explain capital accumulation, and β to explain consumption-income co-movement.

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Commitment II: Golden Eggs

The declining savings rates in the 80’s The golden eggs model provides an explanation for why savings rates decreased in the 80’s. In the 80’s there was an increase in consumers’ instantaneous credit (while in 1970 only 16 percent of families had a third party credit card, the …gure was 54 percent in 1989). Instantaneous credit eliminates the e¢cacy of self-imposed liquidity constraints. Hence, consumers then consume too much and save too little for the future.

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Commitment II: Golden Eggs

Negative Welfare Impact of Financial Innovation In classical economics …nancial innovation is always good for consumers since it enlarges their set of opportunities to maximize utility (decreasing saving rates then re‡ect changing preferences and are not bad for welfare per se). The golden eggs model provides an explanation for why …nancial innovation (credit cards, instantaneous consumer credit) might also reduce consumer welfare. In particular, being able to borrow against illiquid assets reduces welfare when β < 1. The smaller is β, the larger is the welfare loss.

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Summary

The standard (exponential) discounting model proposes that the discount factor δ is constant over time. Under the hyperbolic discounting model, the discount factor changes

• ver time: there is a sharp di¤erence between “now” and “the

future”: all future periods are discounted by an additional parameter β. Hyperbolic discounting explains why sometimes we are patient (e.g., when planning retirement savings) and sometimes not (e.g., when we have some money available and spend it all on a shopping tour). Agents with self-control problems demand some form of commitment. Taking away commitment opportunities from consumers (e.g., by giving them access to instantaneous credit) may reduce welfare.

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