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 1 / 24

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. 1 This 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 2 / 24

Introduction Overview Reference-Dependent Preferences Example 1: Shopping Example 2: Labour Supply Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 3 / 24

Reference-Dependent Preferences The utility function Let c = ( c 1 , c 2 ) 2 R 2 be consumption and r = ( r 1 , r 2 ) 2 R 2 a reference point. For example c 1 2 f 0 , 1 g denotes whether the consumer purchases a good or not, and c 2 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 Z Z U ( F j G ) = u ( c j r ) dG ( r ) dF ( c ) . (1) Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 4 / 24

Reference-Dependent Preferences We consider the following speci…cation of the utility function: u ( c j r ) = c 1 + c 2 + µ ( c 1 � r 1 ) + µ ( c 2 � r 2 ) . (2) c 1 + c 2 is “consumption utility” and µ ( c 1 � r 1 ) + µ ( c 2 � r 2 ) is “gain-loss utility”. Both components are additively separable across dimensions. Call µ a “universal gain-loss function”. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 5 / 24

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 µ 00 ( x ) � 0 for all x > 0, and µ 00 ( x ) � 0 for all x < 0. A4 lim x ! 0 µ 0 ( � j x j ) / lim x ! 0 µ 0 ( j x j ) = 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’ µ 00 ( x ) = 0 for all x 6 = 0. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 6 / 24

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 f D l g l 2 R be a continuum of choice sets and F ( l ) the distribution over f D l g l 2 R which de…nes the consumer’s expectations. We introduce an equilibrium concept that combines rational expectations and reference-dependence. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 7 / 24

Reference-Dependent Preferences De…nition 1. A selection f σ l 2 D l g l 2 R is a personal equilibrium (PE) if � � R � R � � � � U � � σ 0 U σ l σ l dF ( l ) σ l dF ( l ) l for all l 2 R and alternative selection σ 0 l 2 D l . De…nition 2. A selection f σ l 2 D l g l 2 R is a preferred personal equilibrium (PPE) if it is a PE, and � � R � R � � � � U � � σ 0 σ 0 U σ l σ l dF ( l ) l dF ( l ) l for all PEs f σ 0 l 2 D l g l 2 R . Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 8 / 24

Reference-Dependent Preferences A selection f σ l 2 D l g l 2 R is a map that tells us what consumption bundle σ l will be realized in situation l when the choice set is D l . 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 σ l 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. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 9 / 24

Example 1: Shopping Consider a consumer who has to decide whether to purchase a good or not, c 1 2 f 0 , 1 g . Denote by c 2 the money left after the transaction. Her endowment is given by ( 0 , 0 ) . Her utility from consumption is given by c 1 + c 2 . We assume that µ satis…es A3’ such that µ ( x ) = η x for all x > 0 and µ ( x ) = λη x for all x < 0, where λ > 1. The price of the good is denoted by p and (in all interesting cases) uncertain. Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 10 / 24

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 � q ηλ + η p � . (4) Her net gain from purchasing the good is then 1 + η ( 1 � q + λ q ) � ( 1 + η ) p . (5) Note that this term is maximal if q = 1! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 11 / 24

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 � ηλ p . (6) If she does not purchase the good, her total utility is given by � q ηλ . (7) Her net gain from purchasing the good is then 1 + η ( 1 � q + λ q ) � ( 1 + λη ) p . (8) Note that this term is minimal if q = 0! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 12 / 24

Example 1: Shopping From (5) we get that the consumer never purchases the good if p > p max = 1 + ηλ 1 + η . (9) From (8) we get that the consumer always purchases the good if p < p min = 1 + η 1 + ηλ . (10) We now conduct the following thought experiment. Suppose that with probability q L the price is p L < p min and with probability q H = 1 � q L the price is p H > p max . What will the consumer do, if she - unexpectedly - …nds that the real price is p M 2 ( p min , p max ) ? Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 13 / 24

Example 1: Shopping Note that the consumer expects to get the good with probability q L and her expected payment is q L p L . If she purchases the good at price p M , her total utility is given by 1 � p M + q H η � q L ηλ ( p M � p L ) � q H ηλ p M . (11) If she does not purchase the good at price p M , her total utility is given by � q L λη + q L η p L . (12) Make sure that you can derive these terms! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 14 / 24

Example 1: Shopping Let us …rst consider the case where p L = 0. From (11) and (12) we get that the consumer purchases the good at price p M if and only if p M � 1 + η ( λ � 1 ) η 1 + ηλ + q L 1 + ηλ . (13) Interpretation: The higher is q L , the more attached is the consumer to the good so that she is ready to pay higher prices! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 15 / 24

Example 1: Shopping Next, consider the case where p L � 0 and q L = 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 p M if and only if ( λ � 1 ) η p M � 1 + p L 1 + ηλ . (14) Interpretation: The higher is p L , 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! Prof. Dr. Heiner Schumacher (KU Leuven) Incentives and Behavior 4. Risk Preferences II 16 / 24