Reference-Dependent Preferences with Expectations as the Reference Point (t = T )

Transcription

1 Reference-Dependent Preferences with Expectations as the Reference Point (t = T ) April 16, 2009

2 Today The Kőszegi/Rabin model of reference-dependent preferences... Featuring: Personal Equilibrium (PE) Preferred Personal Equilibrium (PPE)

3 Today The Kőszegi/Rabin model of reference-dependent preferences... Featuring: Personal Equilibrium (PE) Preferred Personal Equilibrium (PPE)... and many more (UPE,CPE)

4 Today The Kőszegi/Rabin model of reference-dependent preferences... Featuring: Personal Equilibrium (PE) Preferred Personal Equilibrium (PPE)... and many more (UPE,CPE) Ultimate goal: more complete understanding of the insights to be gained from modeling RD prefs, how we can apply them to standard economic situations.

5 What is the (reference) point? TK(1991): A treatment of reference-dependent choice raises two questions: what is the reference state, and how does it affect preferences? The present analysis focuses on the second question. We assume that the decision maker has a definite reference state X, and we investigate its impact on the choice between options. The question of the origin and the determinants of the reference state lies beyond the scope of the present article. Although the reference state usually corresponds to the decision maker s current position, it can also be influenced by aspirations, expectations, norms, and social comparisons.

6 What is the (reference) point? Candidates: 1 Aspirations/goals 2 Your neighbors 3 Recent 4 Status quo r t = (1 γ)r t 1 + γc t 1 (most common, convenient) r t = max τ<t c τ ; 5 Expectations t 1 1 j=1 j c j t 1 1 j=1 j ; t 1 1 j=1 j c t j t 1 1 j=1 j ; j=1 γj c t j Kőszegi & Rabin argue that (5) is often most appropriate.

7 Expectations as the Reference Point: Why? KR: reference point = probabilistic beliefs held in recent past about outcomes In most cases where evidence is interpreted w/ status quo as r, people plausibly expect to maintain status quo.

8 Expectations as the Reference Point: Why? KR: reference point = probabilistic beliefs held in recent past about outcomes In most cases where evidence is interpreted w/ status quo as r, people plausibly expect to maintain status quo. When expectations status quo, expectations generally makes better predictions: Endowment effect in mug experiments: people expect to keep mug, no predisposition to trade No endowment effect among card traders: Buyers & sellers in real-world markets who expect to trade Salary of $50k to someone who expected $60k feels like a $10k loss, not a $50k gain

9 Expectations as the Reference Point: Why? KR: reference point = probabilistic beliefs held in recent past about outcomes In most cases where evidence is interpreted w/ status quo as r, people plausibly expect to maintain status quo. When expectations status quo, expectations generally makes better predictions: Endowment effect in mug experiments: people expect to keep mug, no predisposition to trade No endowment effect among card traders: Buyers & sellers in real-world markets who expect to trade Salary of $50k to someone who expected $60k feels like a $10k loss, not a $50k gain Any theory of expectation formation could be plugged into the model, but KR assume rational expectations (Realistic) assumption that people can predict their own behavior Can pinpoint results due to RD

10 (Old) Example Suppose you get instrumental and anticipatory utility from eating either a muffin or a smoothie. (No RD here) x, e {m, s} e\x m s U(x, e) is given by m 3 2 s 0 1 Self-fulfilling expectations: ff you expect m, m is the optimal choice; if you expect s, s is optimal Multiple equilibria, but (m, m) yields higher utility Refinement: preferred personal equilibrium. Based on the assumption that you should be able to make any credible plan for your own behavior, choose the best plan.

11 Example: Stochastic Reference Point Suppose Oprah is considering buying a shoe today. She went to bed last night believing that the price is equally likely to be p L = 100 or p H = 150. She forms her plan tonight, but only observes the price tomorrow, before making the decision to buy. Consumption utility: m(s, d) = vs + d, where s {0, 1} is the number of shoes, and d is the number of dollars, at the end of the day, and v > 0 is a taste parameter. Gain/loss utility: µ( m) = m for m 0 and µ( m) = λ m for m 0, where λ 1.

12 Example: Stochastic Reference Point Suppose Oprah is considering buying a shoe today. She went to bed last night believing that the price is equally likely to be p L = 100 or p H = 150. She forms her plan tonight, but only observes the price tomorrow, before making the decision to buy. Consumption utility: m(s, d) = vs + d, where s {0, 1} is the number of shoes, and d is the number of dollars, at the end of the day, and v > 0 is a taste parameter. Gain/loss utility: µ( m) = m for m 0 and µ( m) = λ m for m 0, where λ 1. Find all personal equilibria (PE) and preferred personal equilibria (PPE) as a function of v.

13 Example: Stochastic Reference Point Break the problem down into parts: consider always buy, buy if p L and never buy seperately. First, when is the strategy of always buying the shoes (no matter the price) a PE? Given r, if it s worth buying at p H, it will always be worth buying at p L. So when buy at p H?

14 Example: Stochastic Reference Point Break the problem down into parts: consider always buy, buy if p L and never buy seperately. First, when is the strategy of always buying the shoes (no matter the price) a PE? Given r, if it s worth buying at p H, it will always be worth buying at p L. So when buy at p H? U BUY ph = v ( ) = v 225

15 Example: Stochastic Reference Point Break the problem down into parts: consider always buy, buy if p L and never buy seperately. First, when is the strategy of always buying the shoes (no matter the price) a PE? Given r, if it s worth buying at p H, it will always be worth buying at p L. So when buy at p H? U BUY ph = v ( ) = v 225 U NO ph = 0 3v (100) (150) = 125 3v

16 Example: Stochastic Reference Point Break the problem down into parts: consider always buy, buy if p L and never buy seperately. First, when is the strategy of always buying the shoes (no matter the price) a PE? Given r, if it s worth buying at p H, it will always be worth buying at p L. So when buy at p H? U BUY ph = v ( ) = v 225 U NO ph = 0 3v (100) (150) = 125 3v So buy iff v > 87.5

17 Example: Stochastic Reference Point Break the problem down into parts: consider always buy, buy if p L and never buy seperately. First, when is the strategy of always buying the shoes (no matter the price) a PE? Given r, if it s worth buying at p H, it will always be worth buying at p L. So when buy at p H? U BUY ph = v ( ) = v 225 U NO ph = 0 3v (100) (150) = 125 3v So buy iff v > 87.5 So for v > 87.5, always buy is a PE.

18 Example: Stochastic Reference Point Next, when is buy if p L a PE? Given r, utilities if the price is high are: U BUY ph = v (v) 3[ 1 2 (150 0) ( )] U No ph = v (100) So buy iff v > 500 3

19 Example: Stochastic Reference Point Next, when is buy if p L a PE? Given r, utilities if the price is high are: U BUY ph = v (v) 3[ 1 2 (150 0) ( )] U No ph = v (100) So buy iff v > Utilities if the price is low are: U BUY pl = v (v) 3[ 1 2 (100 0) ( )] U No pl = v (100) So buy iff v > 200

20 Example: Stochastic Reference Point Next, when is buy if p L a PE? Given r, utilities if the price is high are: U BUY ph = v (v) 3[ 1 2 (150 0) ( )] U No ph = v (100) So buy iff v > Utilities if the price is low are: U BUY pl = v (v) 3[ 1 2 (100 0) ( )] U No pl = v (100) So buy iff v > 200 So buy if p L is a PE for 100 < v < 500 3

21 Example: Stochastic Reference Point Do the rest on your own When is never buy a PE? When is PE unique? What are the PPE, as a function of v?

22 Example: Stochastic Reference Point Do the rest on your own Really! When is never buy a PE? When is PE unique? What are the PPE, as a function of v?

23 Riskless utility u(c r) m(c) + n(c r) Consumption (m) and gain/loss (n) utilities separable across dimensions k Gain/loss utility related to consumption: n k (c k r k ) µ(m k (c k ) m k (r k )), where µ is a KT value function (A0-A4) Stochastic outcome F evaluated according to expected utility; utility of outcome is average of how it feels relative to each possible realization of stochastic reference point G: U(F G) = u(c r)dg(r)df (c) Apply PE

24 Basic Properties Lower RP makes a person happier

25 Basic Properties Lower RP makes a person happier Status quo bias: if you re willing to abandon RP for alternative, then you strictly prefer the alternative when it is the RP. U(F F ) U(F F ) = U(F F ) > U(F F )

26 Basic Properties Lower RP makes a person happier Status quo bias: if you re willing to abandon RP for alternative, then you strictly prefer the alternative when it is the RP. U(F F ) U(F F ) = U(F F ) > U(F F ) If m is linear then u(c r) exhibits some properties as µ (A0-A4). Shares properties of prospect theory for small gambles, but not for large. (DMU(w) kicks in.)

27 Basic Properties Lower RP makes a person happier Status quo bias: if you re willing to abandon RP for alternative, then you strictly prefer the alternative when it is the RP. U(F F ) U(F F ) = U(F F ) > U(F F ) If m is linear then u(c r) exhibits some properties as µ (A0-A4). Shares properties of prospect theory for small gambles, but not for large. (DMU(w) kicks in.) When choice set, choices are deterministic, PPE predictions are identical to model based solely on consumption utility. Loss aversion doesn t affect choice, welfare. Not true for PE, because if a person anticipates and option that does not maximize m, she may carry it out to avoid sense of loss.

28 Another Shoe Example Now suppose m(s, d) = s + d, add η > 0 is weight on gain/loss utility. Deterministic price: exist p L, p H such that there is a unique PE for p < p L, p > p H ; multiple eq. in between but typically unique PPE. Stochastic prices: increased likelihood of buying (e.g. higher prob of lower price) leads to attachment affect = higher willingness to pay, because not buying carries increased sense of loss. Read carefully section on driving.

29 Risk Attitudes KR apply model to settings with risk, extend it. Distinguish between surprise and anticipated risk Predicts distaste for insuring losses when risk is a surprise But first-order risk aversion when risk, possibility of insurance is anticipated Expectation of taking on risk decreases aversion to both anticipated and any additional risk For large-scale risk, consumption utility dominates

30 Unanticipated Risk Thinking about low-probability situations, model in extreme form as situations where expectations are exogenous. Example: Risk: gain 0, lose $100 Choice: pay $55 to insure? If expected status quo, prediction is same as prospect theory: don t insure because of diminishing sensitivity. If expected to get insurance, paying $55 generates no gain/loss, while gamble coded as lose $45, gain $55. With standard 2-to-1 loss aversion, wouldn t take gamble. If initially expected the risk, paying $0 can decrease expected losses, losing $100 might decrease expected gains, so gamble doesn t look so risky. Can interpret as endowment effect for risk: When ex ante expected uncertainty is large, $100 doesn t have much effect on whether outcome is coded as loss or gain, so person is

31 New Definitions Import old definition of PE, but call it UPE now, for unacclimating personal equilibrium. Reference point fixed by past expectations, taken as given. PPE is favorite UPE. New: Choice-acclimating personal equilibrium (CPE). Decision affects reference point. CPE decision maximizes expected utility given that it determines both the reference lottery and the outcome lottery.