Incentives and Behavior Prof. Dr. Heiner Schumacher KU Leuven 4. - - PowerPoint PPT Presentation

Incentives and Behavior

• Prof. Dr. Heiner Schumacher

KU Leuven

• 4. Risk Preferences II
• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 4. Risk Preferences II

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Introduction

In the previous lecture, we learned that Kahneman and Tversky’s (1979) prospect theory provides a risk-preference model that avoids the problems of EUT. Its main conceptual problem is the determination of the reference point. In this lecture, we consider a risk-preference model in which the reference-point is derived endogenously from expectations.1 This model is now used in many economic applications.

1This theory is based on Köszegi, Botond, and Matthew Rabin (2006): “A Model of

Reference-Dependent Preferences,” Quarterly Journal of Economics 121(4), 1133 - 1165.

• Prof. Dr. Heiner Schumacher (KU Leuven)

Incentives and Behavior

• 4. Risk Preferences II

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Introduction

Overview Reference-Dependent Preferences Example 1: Shopping Example 2: Labour Supply

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Reference-Dependent Preferences

The utility function Let c = (c1, c2) 2 R2 be consumption and r = (r1, r2) 2 R2 a reference point. For example c1 2 f0, 1g denotes whether the consumer purchases a good or not, and c2 is the money left after the transaction. Consumption may be uncertain (e.g. the consumer may not know the exact prices of the good). Let c be drawn from the probability measure F. Also the reference point may be stochastic. Let r be drawn from the probability measure G. The consumer’s utility is then U(F jG ) =

Z Z

u(c jr )dG(r)dF(c). (1)

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Reference-Dependent Preferences

We consider the following speci…cation of the utility function: u(c jr ) = c1 + c2 + µ(c1 r1) + µ(c2 r2). (2) c1 + c2 is “consumption utility” and µ(c1 r1) + µ(c2 r2) is “gain-loss utility”. Both components are additively separable across dimensions. Call µ a “universal gain-loss function”.

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Reference-Dependent Preferences

We impose the following assumptions on µ. They correspond to Kahneman and Tversky’s description of the “value function” de…ned on c r. A0 µ is continuous, twice di¤erentiable for x 6= 0, and µ(0) = 0. A1 µ is strictly increasing. A2 If y > x > 0, then µ(y) + µ(y) < µ(x) + µ(x). A3 µ00(x) 0 for all x > 0, and µ00(x) 0 for all x < 0. A4 limx!0 µ0( jxj)/ limx!0 µ0(jxj) = λ > 1. Note that A3 captures the fact that the marginal change in µ is larger for changes that are close to the reference level. For some results we have to replace this assumption by A3’ µ00(x) = 0 for all x 6= 0.

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• 4. Risk Preferences II

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Reference-Dependent Preferences

Reference Point and Personal Equilibrium Unfortunately, it is not clear where the reference point r does come from. There is very little empirical evidence on this issue. In most models, r is just the status quo. We will assume that the reference point is given by the consumer’s expectations. Speci…cally, let fDlgl2R be a continuum of choice sets and F(l) the distribution over fDlgl2R which de…nes the consumer’s expectations. We introduce an equilibrium concept that combines rational expectations and reference-dependence.

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Reference-Dependent Preferences

De…nition 1. A selection fσl 2 Dlgl2R is a personal equilibrium (PE) if U

• σl
• R

σldF(l) U

• σ0

l

• R

σldF(l)

• for all l 2 R and alternative selection σ0

l 2 Dl.

De…nition 2. A selection fσl 2 Dlgl2R is a preferred personal equilibrium (PPE) if it is a PE, and U

• σl
• R

σldF(l) U

• σ0

l

• R

σ0

ldF(l)

• for all PEs fσ0

l 2 Dlgl2R.

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Reference-Dependent Preferences

A selection fσl 2 Dlgl2R is a map that tells us what consumption bundle σl will be realized in situation l when the choice set is Dl. This selection also de…nes the consumer’s expectations about what she is going to consume. The expectations in turn de…ne the reference point. In a PE, it is optimal to choose σl in situation l. Since the consumer is free to choose any (feasible) plan, she should select one that maximizes ex-ante expected utility. This must be the case in a PPE.

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Example 1: Shopping

Consider a consumer who has to decide whether to purchase a good

• r not, c1 2 f0, 1g. Denote by c2 the money left after the
• transaction. Her endowment is given by (0, 0).

Her utility from consumption is given by c1 + c2. We assume that µ satis…es A3’ such that µ(x) = ηx for all x > 0 and µ(x) = ληx for all x < 0, where λ > 1. The price of the good is denoted by p and (in all interesting cases) uncertain.

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Example 1: Shopping

Suppose that the consumer’s expected payment is p p (i.e., the price of the good does never exceed the consumer’s expectation) and the consumer expects to get the good with probability q. If she purchases the good, her total utility is given by 1 p + (1 q)η + η(p p). (3) Interpret each of these terms! If she does not purchase the good, her total utility is given by qηλ + ηp. (4) Her net gain from purchasing the good is then 1 + η(1 q + λq) (1 + η)p. (5) Note that this term is maximal if q = 1!

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Example 1: Shopping

Suppose that the consumer’s expected payment is 0 (i.e., any positive payment implies that gain-loss utility is negative) and the consumer expects to get the good with probability q. If she purchases the good, her total utility is given by 1 p + (1 q)η ηλp. (6) If she does not purchase the good, her total utility is given by qηλ. (7) Her net gain from purchasing the good is then 1 + η(1 q + λq) (1 + λη)p. (8) Note that this term is minimal if q = 0!

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Example 1: Shopping

From (5) we get that the consumer never purchases the good if p > pmax = 1 + ηλ 1 + η . (9) From (8) we get that the consumer always purchases the good if p < pmin = 1 + η 1 + ηλ. (10) We now conduct the following thought experiment. Suppose that with probability qL the price is pL < pmin and with probability qH = 1 qL the price is pH > pmax. What will the consumer do, if she - unexpectedly - …nds that the real price is pM 2 (pmin, pmax)?

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Example 1: Shopping

Note that the consumer expects to get the good with probability qL and her expected payment is qLpL. If she purchases the good at price pM, her total utility is given by 1 pM + qHη qLηλ(pM pL) qHηλpM. (11) If she does not purchase the good at price pM, her total utility is given by qLλη + qLηpL. (12) Make sure that you can derive these terms!

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Example 1: Shopping

Let us …rst consider the case where pL = 0. From (11) and (12) we get that the consumer purchases the good at price pM if and only if pM 1 + η 1 + ηλ + qL (λ 1)η 1 + ηλ . (13) Interpretation: The higher is qL, the more attached is the consumer to the good so that she is ready to pay higher prices!

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Example 1: Shopping

Next, consider the case where pL 0 and qL = 1 (i.e., the consumer expects with certainty to purchase the good). From (11) and (12) we get that the consumer purchases the good at price pM if and only if pM 1 + pL (λ 1)η 1 + ηλ . (14) Interpretation: The higher is pL, the more does the consumer expect to pay, and the less she su¤ers if she …nds that the real price is even

• higher. This constitutes a violation of the law of demand!
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Example 2: Labour Supply

In a famous paper, Camerer et al. (1997) show that the labor supply

• f New York City cabdrivers responds negatively to hourly wages.2

This clearly contradicts standard models of labor supply where workers intertemporally substitute labor and leisure. Camerer et al. (1997) suggest that cabdrivers have daily income targets (i.e., they do not optimize intertemporally): after meeting the daily target, they stop working. In the following, we use our theory of reference-dependent preferences to formalize this argument. Note that our theory does not assume …xed targets, but endogenously derives targets (i.e., reference points) from expectations.

2Camerer, Colin, Linda Babcock, George Loewenstein, and Richard Thaler (1997):

“Labor Supply of New York City Cabdrivers: One Day at a Time,” Quarterly Journal of Economics 112(2), 407 - 441.

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Example 2: Labour Supply

A taxi driver decides whether to go to work in the morning, and, if yes, whether to continue driving in the afternoon. Let em 2 f0, 1g be her decision in the morning, and ea 2 f0, 1g her decision in the afternoon. Both income in the morning, wm, and in the afternoon, wa, are

• uncertain. If the driver works in the morning, she learns her afternoon

income. Consumption utilities are wm + wa and f (em + ea), where f is the per-unit cost of e¤ort. We assume that µ satis…es A3’ such that µ(x) = ηx for all x > 0 and µ(x) = ληx for all x < 0, where λ > 1.

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Example 2: Labour Supply

We proceed as in the last example and de…ne wages wa

min and wa max

such that the taxi driver always (never) continues working the afternoon when wa > wa

max (wa < wa min).

Suppose that the driver’s expected (total) income is ˜ w wm + wa and she expects to work in the afternoon with probability q. If she drives in the afternoon, her additional utility is given by wa f ηλ( ˜ w wm wa) (1 q)ηλf . (15) If she does not drive in the afternoon, her additional utility is given by ηλ( ˜ w wm) + ηqf . (16) Her net gain from driving in the afternoon is then wa f + ηλwa f η((1 q)λ + q). (17) Note that this term is maximal if q = 1!

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Example 2: Labour Supply

Suppose that the driver’s expected (total) income is ˜ w = 0 and she expects to work in the afternoon with probability q. If she drives in the afternoon, her additional utility is given by wa f + η(wm + wa) (1 q)ηλf . (18) If she does not drive in the afternoon, her additional utility is given by ηwm + qηf . (19) Her net gain from driving in the afternoon is then wa(1 + η) f (1 + ηλ(1 q) + ηq). (20) Note that this term is minimal if q = 0!

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Example 2: Labour Supply

From (20) we get that the driver always works in the afternoon if wa > wmax = 1 + ηλ 1 + η f . (21) From (17) we get that the driver never works in the afternoon if wa < wmin = 1 + η 1 + ηλf . (22) We now conduct the following thought experiment. Suppose that with probability qL the wage is wa

L < wmin and with probability

qH = 1 qL the price is wa

H > wmax. What will the driver do, if she -

unexpectedly - …nds that the real wage is wa

R 2 (wmin, wmax)?

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Example 2: Labour Supply

In order to capture all interesting e¤ects, we assume that wm

E wa R < wm R wm E ,

(23) where wm

E is the wage the driver expected to earn in the morning, wm R

is the morning wage that she realized, and wa

R is the realized

afternoon wage. We are interested in the driver’s behavior if she works in the morning. Her reference point is as follows: the taxi driver expects to work all day with probability qH, and she expects to earn wm

E + qHwa H.

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Example 2: Labour Supply

The drivers utility when she continues working in the afternoon is wm

R + wa R 2f + qLη(wm R + wa R wm E )

qHηλ(wm

E + wa H wm R wa R) qLηλf .

(24) Her utility when she does not continue working in the afternoon is wm

R f qLηλ(wm E wm R )

qHηλ(wm

E + wa H wm R ) + qHηf .

(25) Hence, if wm

R = wm E , the driver continues working as long as

wa

R 1 + η + qLη(λ 1)

1 + η + qHη(λ 1)f . (26)

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Example 2: Labour Supply

Interpretation If qL is small, then the driver expects to work in the afternoon and expects a high income. Being short of the income target feels like a

• loss. Hence, the driver is willing to work in the afternoon even at a

moderate wage wa

R.

If qL is large, then the driver does not expect to work in the afternoon and expects a low income. Hence, even at a substantial wage wa

R the

driver will not continue working. The model therefore makes the following prediction: if individuals expect wages to be high, then they work more (we then have intertemporal substitution); if individuals face unexpectedly high wages, the e¤ect on working hours is weaker or negative.

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