Motivation & Introduction
Model
Reference-Dependent Preferences with
Expectations as the Reference Point
January 11, 2011
Risk Attitudes
Motivation & Introduction
Model
Risk Attitudes
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.
Motivation & Introduction
Model
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.
Risk Attitudes
Motivation & Introduction
Model
Risk Attitudes
What is the (reference) point?
Candidates:
1. Aspirations/goals
2. Your neighbors
3. Recent
4. Status quo
• rt = (1 − γ)rt−1 + γct−1 (most common, convenient)
Pt−1
• rt = maxτ <t cτ ;
1
j cj
1
j=1 j
j=1
Pt−1
Pt−1
;
1
j ct−j
1
j=1 j
j=1
Pt−1
;
P∞
j=1
γ j ct−j
5. Expectations
Kőszegi & Rabin argue that (5) is often most appropriate.
Motivation & Introduction
Model
Risk Attitudes
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 6= 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
Motivation & Introduction
Model
Risk Attitudes
Example illustrating personal equilibrium
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
m 3 2
s 0 1
• Self-fulfilling expectations: if you expect m, m is the optimal
choice; if you expect s, s is optimal
• U(x, e) is given by
• Multiple equilibria, but (m, m) yields higher utility
Motivation & Introduction
Model
Risk Attitudes
Personal Equilibrium
• Personal equilibrium (PE):
1. Correctly predict environment & own behavior
2. Taking (reference point generated by) expectations as given,
maximizes utility (in each contingency)
• Refinement: preferred personal equilibrium (PPE). Based on
the assumption that you should be able to make any credible
plan for your own behavior, choose the best plan.
Motivation & Introduction
Model
Risk Attitudes
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
pL = 100 or pH = 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 .
Motivation & Introduction
Model
Risk Attitudes
Example: Stochastic Reference Point
Break the problem down into parts: consider ‘always buy’, ‘buy if
pL ’ 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 pH , it will always be worth
buying at pL .
• So when buy at pH ?
• U BUY |pH = v − 150 − 12 3(150 − 100) = v − 225
• U NO |pH = 0 − 3v + 21 (100) + 12 (150) = 125 − 3v
• So buy iff v > 87.5
So for v > 87.5, ‘always buy’ is a PE.
Motivation & Introduction
Model
Risk Attitudes
Example: Stochastic Reference Point
Next, when is ‘buy if pL ’ a PE?
• Given r , utilities if the price is high are:
• U BUY |pH = v − 150 + 12 (v ) − 3[ 12 (150 − 0) + 12 (150 − 100)]
• U No |pH = 0 − 3 21 v + 12 (100)
• So buy iff v > 500
3
• Utilities if the price is low are:
• U BUY |pL = v − 100 + 21 (v ) − 3[ 12 (100 − 0) + 12 (100 − 100)]
• U No |pL = 0 − 3 21 v + 12 (100)
• So buy iff v > 200
So ‘buy if pL ’ is a PE for 100 < v <
500
3
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 ?
Risk Attitudes
Motivation & Introduction
Model
Risk Attitudes
Model
• 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:
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 each
possible realization of stochastic reference point G :
Z Z
U(F |G ) =
u(c|r )dG (r )dF (c)
• Apply PE
Motivation & Introduction
Model
Risk Attitudes
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 0 ) ≥ U(F 0 |F 0 ) =⇒ U(F |F ) > U(F 0 |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.
Motivation & Introduction
Model
Risk Attitudes
Another Shoe Example
Now suppose m(s, d) = s + d, add η > 0 is weight on gain/loss
utility.
• Deterministic price: exist pL , pH such that there is a unique
PE for p < pL , p > pH ; 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.
Motivation & Introduction
Model
Risk Attitudes
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
Motivation & Introduction
Model
Risk Attitudes
Unanticipated Risk
Thinking about low-probability situations, model in extreme form
as situations where expectations are exogenous.
Example:
• Risk: 50-50 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 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 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
Motivation & Introduction
Model
Risk Attitudes
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.