Reference-Dependent Preferences with...

Motivation & Introduction Model

Reference-Dependent Preferences withExpectations as the Reference Point

(t = T )

April 16, 2009

The Koszegi/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 begained from modeling RD prefs, how we can apply them tostandard economic situations.

Featuring:

What is the (reference) point?

TK(1991):

A treatment of reference-dependent choice raises twoquestions: what is the reference state, and how does itaffect preferences? The present analysis focuses on thesecond question. We assume that the decision maker hasa definite reference state X , and we investigate itsimpact on the choice between options. The question ofthe origin and the determinants of the reference state liesbeyond the scope of the present article. Although thereference state usually corresponds to the decisionmaker’s current position, it can also be influenced byaspirations, expectations, norms, and social comparisons.

What is the (reference) point?

Candidates:

1 Aspirations/goals

2 Your neighbors

3 Recent4 Status quo

rt = (1− γ)rt−1 + γct−1 (most common, convenient)

rt = maxτ<t cτ ;∑t−1

j=11j cj∑t−1

;∑t−1

j=11j ct−j∑t−1

;∑∞

j=1 γjct−j

5 Expectations

Koszegi & Rabin argue that (5) is often most appropriate.

Expectations as the Reference Point: Why?

KR: reference point = probabilistic beliefs held in recent pastabout outcomes

In most cases where evidence is interpreted w/ status quo asr , people plausibly expect to maintain status quo.

When expectations 6= status quo, expectations generallymakes better predictions:

Endowment effect in mug experiments: people expect to keepmug, no predisposition to tradeNo endowment effect among card traders: Buyers & sellers inreal-world markets who expect to tradeSalary of $50k to someone who expected $60k feels like a $10kloss, not a $50k gain

Any theory of expectation formation could be plugged into themodel, but KR assume rational expectations

(Realistic) assumption that people can predict their ownbehaviorCan pinpoint results due to RD

(Old) Example

Suppose you get instrumental and anticipatory utility from eatingeither a muffin or a smoothie. (No RD here)

x , e ∈ {m, s}

U(x , e) is given by

e\x m s

m 3 2s 0 1

Self-fulfilling expectations: ff you expect m, m is the optimalchoice; if you expect s, s is optimal

Multiple equilibria, but (m,m) yields higher utility

Refinement: preferred personal equilibrium. Based on theassumption that you should be able to make any credible planfor your own behavior, choose the best plan.

Example: Stochastic Reference Point

Suppose Oprah is considering buying a shoe today. She went tobed last night believing that the price is equally likely to bepL = 100 or pH = 150. She forms her plan tonight, but onlyobserves the price tomorrow, before making the decision to buy.

Consumption utility: m(s, d) = vs + d , where s ∈ {0, 1} is thenumber of shoes, and d is the number of dollars, at the end ofthe day, and v > 0 is a taste parameter.

Gain/loss utility: µ(∆m) = ∆m for ∆m ≥ 0 andµ(∆m) = λ∆m for ∆m ≤ 0, where λ ≥ 1.

Find all personal equilibria (PE) and preferred personal equilibria(PPE) as a function of v .

Suppose Oprah is considering buying a shoe today. She went tobed last night believing that the price is equally likely to bepL = 100 or pH = 150. She forms her plan tonight, but onlyobserves the price tomorrow, before making the decision to buy.

Consumption utility: m(s, d) = vs + d , where s ∈ {0, 1} is thenumber of shoes, and d is the number of dollars, at the end ofthe day, and v > 0 is a taste parameter.

Gain/loss utility: µ(∆m) = ∆m for ∆m ≥ 0 andµ(∆m) = λ∆m for ∆m ≤ 0, where λ ≥ 1.

Find all personal equilibria (PE) and preferred personal equilibria(PPE) as a function of v .

Break the problem down into parts: consider ‘always buy’, ‘buy ifpL’ and ‘never buy’ seperately.

First, when is the strategy of always buying the shoes (no matterthe price) a PE?

Given r , if it’s worth buying at pH , it will always be worthbuying at pL.

So when buy at pH?

UBUY |pH= v − 150− 1

23(150− 100) = v − 225

UNO |pH= 0− 3v + 1

2(100) + 12(150) = 125− 3v

So buy iff v > 87.5

So for v > 87.5, ‘always buy’ is a PE.

So when buy at pH?

UBUY |pH= v − 150− 1

23(150− 100) = v − 225

UNO |pH= 0− 3v + 1

2(100) + 12(150) = 125− 3v

So buy iff v > 87.5

So when buy at pH?

UBUY |pH= v − 150− 1

23(150− 100) = v − 225

UNO |pH= 0− 3v + 1

2(100) + 12(150) = 125− 3v

So buy iff v > 87.5

So when buy at pH?

UBUY |pH= v − 150− 1

23(150− 100) = v − 225

UNO |pH= 0− 3v + 1

2(100) + 12(150) = 125− 3v

So buy iff v > 87.5

So when buy at pH?

UBUY |pH= v − 150− 1

23(150− 100) = v − 225

UNO |pH= 0− 3v + 1

2(100) + 12(150) = 125− 3v

So buy iff v > 87.5

Next, when is ‘buy if pL’ a PE?

Given r , utilities if the price is high are:

UBUY |pH= v − 150 + 1

2(v)− 3[12(150− 0) + 1

2(150− 100)]

UNo |pH= 0− 31

2v + 12(100)

So buy iff v > 5003

Utilities if the price is low are:

UBUY |pL= v − 100 + 1

2(v)− 3[12(100− 0) + 1

2(100− 100)]

UNo |pL= 0− 31

2v + 12(100)

So buy iff v > 200

So ‘buy if pL’ is a PE for 100 < v < 5003

UBUY |pH= v − 150 + 1

2(v)− 3[12(150− 0) + 1

2(150− 100)]

UNo |pH= 0− 31

2v + 12(100)

So buy iff v > 5003

UBUY |pL= v − 100 + 1

2(v)− 3[12(100− 0) + 1

2(100− 100)]

UNo |pL= 0− 31

2v + 12(100)

So buy iff v > 200

UBUY |pH= v − 150 + 1

2(v)− 3[12(150− 0) + 1

2(150− 100)]

UNo |pH= 0− 31

2v + 12(100)

So buy iff v > 5003

UBUY |pL= v − 100 + 1

2(v)− 3[12(100− 0) + 1

2(100− 100)]

UNo |pL= 0− 31

2v + 12(100)

So buy iff v > 200

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?

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?

Riskless utility u(c |r) ≡ m(c) + n(c |r)

Consumption (m) and gain/loss (n) utilities separable acrossdimensions k

Gain/loss utility related to consumption:

nk(ck |rk) ≡ µ(mk(ck)−mk(rk)),

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 eachpossible realization of stochastic reference point G :

U(F |G ) =

∫ ∫u(c|r)dG (r)dF (c)

Apply PE

Basic Properties

Lower RP makes a person happier

Status quo bias: if you’re willing to abandon RP foralternative, then you strictly prefer the alternative when it isthe 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 smallgambles, but not for large. (DMU(w) kicks in.)

When choice set, choices are deterministic, PPE predictionsare 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 optionthat does not maximize m, she may carry it out to avoid senseof loss.

Basic Properties

U(F |F ′) ≥ U(F ′|F ′) =⇒ U(F |F ) > U(F ′|F )

Basic Properties

U(F |F ′) ≥ U(F ′|F ′) =⇒ U(F |F ) > U(F ′|F )

Basic Properties

U(F |F ′) ≥ U(F ′|F ′) =⇒ U(F |F ) > U(F ′|F )

Another Shoe Example

Now suppose m(s, d) = s + d , add η > 0 is weight on gain/lossutility.

Deterministic price: exist pL, pH such that there is a uniquePE for p < pL, p > pH ; multiple eq. in between but typicallyunique PPE.

Stochastic prices: increased likelihood of buying (e.g. higherprob of lower price) leads to attachment affect = higherwillingness to pay, because not buying carries increased senseof loss.

Read carefully section on driving.

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 insuranceis anticipated

Expectation of taking on risk decreases aversion to bothanticipated and any additional risk

For large-scale risk, consumption utility dominates

Unanticipated Risk

Thinking about low-probability situations, model in extreme formas situations where expectations are exogenous.

Example:

Risk: 50-50 gain 0, lose $100Choice: 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 nogain/loss, while gamble coded as 50-50 lose $45, gain $55.With standard 2-to-1 loss aversion, wouldn’t take gamble.If initially expected the risk, paying $0 can decrease expectedlosses, losing $100 might decrease expected gains, so gambledoesn’t look so risky.Can interpret as endowment effect for risk: When ex anteexpected uncertainty is large, $100 doesn’t have much effecton whether outcome is coded as loss or gain, so person iscloser to risk neutral.

New Definitions

Import old definition of PE, but call it UPE now, forunacclimating personal equilibrium. Reference point fixed bypast expectations, taken as given. PPE is favorite UPE.

New: Choice-acclimating personal equilibrium (CPE).Decision affects reference point. CPE decision maximizesexpected utility given that it determines both the referencelottery and the outcome lottery.

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