Reference-Dependent Preferences with...
Transcript of Reference-Dependent Preferences with...
Motivation & Introduction Model
Reference-Dependent Preferences withExpectations as the Reference Point
(t = T )
April 16, 2009
Motivation & Introduction Model
Today
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.
Motivation & Introduction Model
Today
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.
Motivation & Introduction Model
Today
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.
Motivation & Introduction Model
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.
Motivation & Introduction Model
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
j=11j
;∑t−1
j=11j ct−j∑t−1
j=11j
;∑∞
j=1 γjct−j
5 Expectations
Koszegi & Rabin argue that (5) is often most appropriate.
Motivation & Introduction Model
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
Motivation & Introduction Model
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
Motivation & Introduction Model
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
Motivation & Introduction Model
(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.
Motivation & Introduction Model
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 .
Motivation & Introduction Model
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 .
Motivation & Introduction Model
Example: Stochastic Reference Point
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.
Motivation & Introduction Model
Example: Stochastic Reference Point
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.
Motivation & Introduction Model
Example: Stochastic Reference Point
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.
Motivation & Introduction Model
Example: Stochastic Reference Point
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.
Motivation & Introduction Model
Example: Stochastic Reference Point
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.
Motivation & Introduction Model
Example: Stochastic Reference Point
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
Motivation & Introduction Model
Example: Stochastic Reference Point
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
Motivation & Introduction Model
Example: Stochastic Reference Point
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
Motivation & Introduction Model
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?
Motivation & Introduction Model
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?
Motivation & Introduction Model
Model
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
Motivation & Introduction Model
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.
Motivation & Introduction Model
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.
Motivation & Introduction Model
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.
Motivation & Introduction Model
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.
Motivation & Introduction Model
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.
Motivation & Introduction Model
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
Motivation & Introduction Model
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.
Motivation & Introduction Model
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.