Prospect Theory
Warwick Economics Summer School
Prospect Theory
Eugenio Proto
University of Warwick, Department of Economics
July 19, 2016
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Prospect Theory
Outline
1 General Introduction
2 The Expected Utility Theory
3 Main Departures from Expected Utility
4 Prospect Theory
5 Empirical Evidence
Finance
Economic Development
Housing Markets
Labor Market
Domestic Violence
6 Summary
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Prospect Theory
General Introduction
Behavioral Economics
Economics and Psychology were not separated
Adam Smith: The theory of Moral Sentiments
Jeremy Bentham, the psychological underpinnings of Utility
Francis Edgeworth, concept of envy in Utility Functions
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Prospect Theory
General Introduction
Behavioral Economics, cont’d
Neoclassical Revolution separated clearly Psychology and
Economics
Economics like a natural (and not a social) science
Psychology has too unsteady foundations
Utility can be defined as an ordinal (and not a cardinal) object
Rejection of the hedonistic assumptions of Benthamite Utility
Neoclassical economists expunged the psychology from
economics
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Prospect Theory
General Introduction
Behavioral Economics, cont’d
Some early Criticism to the neoclassical economy from Irwing
Fisher, J.M. Keynes,, Herbert Simon
More from Allais (1953), Ellsberg (1961), Markowitz (1952),
Strotz (1955), following the Expected Utility Theory and the
discounted utility models
Kahneman and Tversky (1979) on expected utility, Thaler
(1981) and Loewenstain and Prelec (1992) (on discounted
utility), Happiness Economics on Hedonic foundations of
Utility
Neuroscience looks for the bases of Individual Behavior
Neuroscience gives steadier foundations to psychology
Toward a unique discipline?
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Prospect Theory
The Expected Utility Theory
Outline
1 General Introduction
2 The Expected Utility Theory
3 Main Departures from Expected Utility
4 Prospect Theory
5 Empirical Evidence
Finance
Economic Development
Housing Markets
Labor Market
Domestic Violence
6 Summary
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Prospect Theory
The Expected Utility Theory
Expected Utility
Individuals generally, ceteris paribus, prefer certainty to
uncertainty
a prospect is a vector of probabilities and Consequences
(q = (x1 , p1 , ..., xn , pn ))
they take decision over a prospect H, p; L, (1 − p), following a
utility function
Eu = pu(H) + (1 − p)u(L)
1 is defined Expected Utility
Individuals prefer r1 = (800, 1) to r2 = (1000, 0.85; 0, 0.15)
note that E (r1 ) < E (r2 ).
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(1)
Prospect Theory
The Expected Utility Theory
Main Tenets of EU
1
The overall Utility of a prospect is the expected Utility, Eu
2
The Utility is defined over final wealth rather than gain and
losses
3
4
Risk aversion: u is concave (u 00 < 0)
Preferences are independent on the manner the prospects are
described
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Prospect Theory
Main Departures from Expected Utility
Outline
1 General Introduction
2 The Expected Utility Theory
3 Main Departures from Expected Utility
4 Prospect Theory
5 Empirical Evidence
Finance
Economic Development
Housing Markets
Labor Market
Domestic Violence
6 Summary
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Prospect Theory
Main Departures from Expected Utility
Main Departures from EU
1
Non Linear Decision Weights
2
Individuals reason in terms of loss and gains rather than final
outcome
3
Loss aversion
4
Framing Effects
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Prospect Theory
Main Departures from Expected Utility
Non Linear Decision Weights
Expected Utility requires a linear response to variation of
probability
Experimentally: raising the probability from 0.39 to 0.40 has
much less impact than increasing the probability from 0.99 to
1, or 0 to 0.01 (certainty effect)
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Prospect Theory
Main Departures from Expected Utility
Non Linear Decision Weights: Allais Paradox
Problem 1: A = (2500, 0.33; 2400, 0.66; 0, 0.01) or
B = (2400, 1)
Problem 2: C = (2500, 0.33; 0, 0.67) or
D = (2400, 0.34; 0, 0.66)
choosing B and C is not consistent with EU theory,
u(2400) > 0.33u(2500) + 0.66u(2400) or
0.34u(2400) > 0.33u(2500)
The second inequality is inconsistent with C > D (assume
u(0) = 0)
Problem 2 is obtained from Problem 1, by eliminating .66 of
winning 2400
Eliminating a large chance of winning alters the ordering
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Prospect Theory
Main Departures from Expected Utility
Non Linear Decision Weights: Allais Paradox; cont’d
Problem 3: A = (4000, .80) or B = (3000, 1)
Problem 4: C = (4000, 0.20) or D = (3000, 0.25)
choosing B and C is not consistent with EU theory,
Problem 4 is obtained from Problem 3, by dividing all
probabilities of a positive outcome by 4
Reducing the probability of winning from 1 to .25 has a
greater effect than reducing it from .8 to .2
Certainty effect
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Prospect Theory
Main Departures from Expected Utility
Non Linear Decision Weights
Problem 7: A = (6000, 0.45) or B = (3000, 0.90)
Problem 8: C = (6000, 0.001) or D = (3000, 0.002)
choosing B and C is not consistent with EU theory,
Problem 8 is obtained from Problem 7, by dividing all
probabilities of a positive outcome by 45
Violation of the Independence axiom
With high probability of winning individuals are risk averse
with low individuals are more risk seekers
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Prospect Theory
Main Departures from Expected Utility
Willingness to Pay vs Willingness to Accept (Kahneman,
Knetsch and Thaler 1990)
Markets for: Tokens and Mugs
Willingness to Pay is the maximum price an individuals want
to pay a good
Willingness to Accept is the minimum compensation
demanded by the owner to sell a good
Standard Assumptions imply that WTA ≈ WTP
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Prospect Theory
Main Departures from Expected Utility
Willingness to Pay vs Willingness to Accept
Market for Tokens and Mugs (6 USD)
in an experiment involving the exchange of a mug (6 USD
value), some individuals were endowed with a mug, some
other with the money to buy this mug
at a price 8.75 I will sell I will not sell
at a price 8.25 I will sell I will not sell
For the token: WTP = WTA
for the mug the Median WTP = 2.75 and the Median WTA
= 5.25
Similar Experiments were conducted with pens, folding
binoculars, lottery tickets etc.
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Prospect Theory
Main Departures from Expected Utility
Willingness to Pay vs Willingness to Accept
Why WTA > WTP?
Endowment Effect
Mugs belong to sellers’ endowment but not to the Buyers’
endowment
A manifestation of Loss aversion
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Prospect Theory
Main Departures from Expected Utility
Framing Effects and status quo
The EU theory implies that choices are invariant to the way
options are described. An outbreak of a disease is expected to kill
600 people.
Program A : 200 people will be saved (72%)
Program B : 1/3 probability to save 600 people, 2/3
probability that no people will be saved (28%)
Program A (reframed) : 400 people will die (22%)
Program B (reframed) : 1/3 probability that nobody will die,
2/3 that no people will be saved (78%)
In the first version the reference is:“everybody will die”. In the
second version is “nobody will die”.
Individuals prefer the status quo.
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Prospect Theory
Main Departures from Expected Utility
Framing of Gains and Losses
Decision frames: Individuals evaluates outcome separately
rather than jointly
Hedonic Frames: Individuals aggregate losses and segregate
gains :
who is happier someone who win 50 and 25 in two lotteries
(64%) or someone that wins 75 in one lottery (36%)?
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Prospect Theory
Prospect Theory
Outline
1 General Introduction
2 The Expected Utility Theory
3 Main Departures from Expected Utility
4 Prospect Theory
5 Empirical Evidence
Finance
Economic Development
Housing Markets
Labor Market
Domestic Violence
6 Summary
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Prospect Theory
Prospect Theory
Prospect Theory
“Transforming” the EU theory to accommodate some of the above
anomalies
V = pu(x) + (1 − p)u(y ) is the value function for the
Expected Utility
V = π(p)v (x) + (1 − π(p))v (y )
where π(p) is a decision weight which reflect the overall
impact of p on the value of the prospect
v is a function different from utility u
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Prospect Theory
Prospect Theory
Prospect Theory: the function v
Individuals reason in terms of loss and gains, they are risk averse in
gains and risk lovers in losses,
A difference between a gain of 100 to 200 appears larger than
a difference between 1100 and 1200
The difference between a loss of -100 and -200 appears larger
than difference between -1100 and -1200
The effect of a change diminishes with the distance to the
reference point: principle of diminishing sensitivity
Hence v 00 (x) < 0 for x > 0 and v 00 (x) > 0 for x < 0
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Prospect Theory
Prospect Theory
Prospect Theory : the function v cont’d
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Prospect Theory
Prospect Theory
Prospect Theory : the weighting function π(p)
the principle principle of diminishing sensitivity applies to π(p)
The natural reference for p are certainty p = 1 and
impossibility p = 0
an increase of 0.1 in the probability of winning a prize has
more impact when it changes to probability from 0.9 to 1 or
from 0 to 0.1 than from 0.5 to 0.6
diminishing sensitivity gives rise to a weighting function π(p)
concave near 0 and convex near 1
implies subadditivity for unlikely event and superadditivity
near certainty
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Prospect Theory : the weighting function π(p)
Prospect Theory
Empirical Evidence
Outline
1 General Introduction
2 The Expected Utility Theory
3 Main Departures from Expected Utility
4 Prospect Theory
5 Empirical Evidence
Finance
Economic Development
Housing Markets
Labor Market
Domestic Violence
6 Summary
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Prospect Theory
Empirical Evidence
Finance
The Equity Premium Puzzle (Benartzi and Thaler 1995)
Stocks’ returns are more volatile than bonds’ returns
The average return to stocks is 8% higher than the average
return to bonds
this would imply an unrealistically high degree of risk aversion,
i.e. lotteries (51.2K , 1) and (50K , 1/2; 100K , 1/2) should be
equivalent
Since stocks have negative returns more often than bonds,
this can be explained with the loss aversion.
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Prospect Theory
Empirical Evidence
Finance
The Disposition Effect (Shefrin and Statman 1985)
Individuals hold stock that have lost value too long and are
eager to sell stocks that have gained value
following the standard theory, price expectation should drive
this choice
trading volumes for stocks that have fallen in price is lower
than for stocks that have risen
in a field experiment from a brokerage firm investors held
losing stocks a median of 124 days and held winners only 104
days
a similar effect exists in the housing market: when the house
price falls the volume of the sales fall as well.
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Prospect Theory
Empirical Evidence
Finance
The Disposition Effect: Finance (Odean (JF, 1998))
Do investors sell winning stocks more than losing stocks?
Individual trade data from Discount brokerage house
(1987-1993)
Share of realized gains:
PGR =
RealizedGains
RealizedGains + PaperGains
(2)
Share of realized Losses:
PLR =
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RealizedLosses
RealizedGains + PaperGains
(3)
Prospect Theory
Empirical Evidence
Finance
The Disposition Effect, Odean (JF, 1998))
PGR and PLR for the Entire Data Set
This table compares the aggregate Proportion of Gains Realized ~PGR! to the aggregate Proportion of Losses Realized ~PLR!, where PGR is the number of realized gains divided by the
number of realized gains plus the number of paper ~unrealized! gains, and PLR is the number
of realized losses divided by the number of realized losses plus the number of paper ~unrealized!
losses. Realized gains, paper gains, losses, and paper losses are aggregated over time ~1987–
1993! and across all accounts in the data set. PGR and PLR are reported for the entire year, for
December only, and for January through November. For the entire year there are 13,883 realized gains, 79,658 paper gains, 11,930 realized losses, and 110,348 paper losses. For December
there are 866 realized gains, 7,131 paper gains, 1,555 realized losses, and 10,604 paper losses.
The t-statistics test the null hypotheses that the differences in proportions are equal to zero
assuming that all realized gains, paper gains, realized losses, and paper losses result from
independent decisions.
PLR
PGR
Difference in proportions
t-statistic
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Entire Year
December
Jan.–Nov.
0.098
0.148
-0.050
-35
0.128
0.108
0.020
4.3
0.094
0.152
-0.058
-38
Prospect Theory
Empirical Evidence
Finance
The Disposition Effect, Odean (JF, 1998))
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Prospect Theory
Empirical Evidence
Economic Development
Mental Accounting effects For the Poor (Bertrand,
Mullainathan, Shafir. 2004
Money is not fungible
Liquidity, current account , assets are perceived differently
Differential marginal propensities to consume (MPC)
current income (where MPC is high), current assets (where
MPC is intermediate), future income (where MPC is low).
Consumption functions thus end up being overly dependent
on current income,
Importance to induce poor people to open a saving account
Savings, help investments and efficiently smooth consumptions
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Prospect Theory
Empirical Evidence
Housing Markets
Loss Aversion in the Housing market (Genesove-Mayer ,
2001)
For houses sales, natural reference point is previous purchase
price, P0
Loss Aversion: Unwilling to sell house at a loss
General Prediction, when aggregate prices are low
Higer prices P relative to fundamentals
Bunching at purchase price P0
Lower probability of sale p(P)
Longer waiting on market
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Prospect Theory
Empirical Evidence
Housing Markets
Loss Aversion in the Housing market (cont’d)
Listing price Li,t
Lossi,t = P̂i.t − P0
P̂ is the real market value (estimated)
Listing price increases with the Loss
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Prospect Theory
Empirical Evidence
Housing Markets
Loss Aversion in the Housing Market (cont’d)
LOSS AVERSION AND LIST PRICES
DEPENDENT VARIABLE: LOG (ORIGINAL ASKING PRICE),
OLS equations, standard errors are in parentheses.
(1)
All
listings
Variable
LOSS
LOSS-squared
LTV
Estimated
value in
1990
Estimated
price index
at quarter of
entry
Residual from
last sale
price
Months since
last sale
Dummy
variables for
quarter of
entry
Constant
R2
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0.35
(0.06)
0.06
(0.01)
1.09
(0.01)
0.86
(0.04)
-0.0002
(0.0001)
No
(2)
All
listings
(3)
All
listings
(4)
All
listings
0.25
(0.06)
0.63
(0.04)
-0.26
(0.04)
0.03
(0.01)
1.09
(0.01)
0.53
(0.04)
-0.26
(0.04)
0.03
(0.01)
1.09
(0.01)
0.05
(0.01)
1.09
(0.01)
0.80
(0.04)
0.11
(0.02)
-0.0003
(0.0001)
No
0.91
(0.03)
-0.0002
(0.0001)
No
0.85
(0.03)
0.11
(0.02)
-0.0003
(0.0001)
No
(5)
All
listings
(6)
All
listings
0.35
(0.06)
0.06
(0.01)
1.09
(0.01)
0.24
(0.06)
-0.0002
(0.0001)
Yes
0.05
(0.01)
1.09
(0.01)
0.11
(0.02)
-0.0003
(0.0001)
Yes
-0.77
(0.14)
-0.70
(0.14)
-0.84
(0.13)
-0.77
(0.14)
-0.88
(0.10)
-0.86
(0.10)
0.85
5792
0.86
5792
0.86
5792
0.86
5792
0.86
5792
0.86
5792
Prospect Theory
Empirical Evidence
Housing Markets
Effect of the experience and loss Aversion
Variable
LOSS x owner-occupant
LOSS x investor
LOSS-squared x owner-occupant
LOSS-squared x investor
LTV x owner-occupant
LTV x investor
Dummy for investor
Estimated value in 1990
Estimated price index at quarter of
entry
Residual from last sale price
Months since last sale
Constant
R2
Number of observations
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(1)
All
listings
(2)
All
listings
(3)
All
listings
(4)
All
listings
0.50
(0.09)
0.24
(0.12)
0.42
(0.09)
0.16
(0.12)
0.66
(0.08)
0.58
(0.06)
-0.16
(0.14)
-0.30
(0.02)
0.01
(0.01)
0.02
(0.02)
-0.03
(0.01)
1.09
(0.01)
0.86
(0.04)
-0.0001
(0.0001)
-0.86
(0.14)
0.58
(0.09)
0.49
(0.06)
-0.17
(0.15)
-0.29
(0.02)
0.01
(0.01)
0.02
(0.02)
-0.03
(0.01)
1.09
(0.01)
0.82
(0.04)
0.08
(0.02)
-0.0002
(0.0001)
-0.84
(0.16)
0.86
3687
0.86
3687
0.03
(0.02)
0.053
(0.027)
-0.02
(0.014)
1.09
(0.01)
0.84
(0.05)
0.03
(0.02)
0.053
(0.027)
-0.02
(0.01)
1.09
(0.01)
0.80
(0.04)
0.08
(0.02)
-0.0002 -0.0003
(0.0002) (0.00015)
-0.80
-0.76
(0.16)
(0.16)
0.85
3687
0.85
3687
Prospect Theory
Empirical Evidence
Labor Market
New York Cab Drivers (Camerer, Babcock, Loewenstein
and Thaler, 1997)
cab drivers in New York lease their cabs for a fixed fee for up
to 12 hours
They work long hours when there is low demand (sunny days)
and short hours when there is high demand (rainy days)
The standard theory would predict the opposite
This is consistent with the loss aversion of cab drivers fix a
daily target and are averse to fall short of it.
Inexperienced drivers feature this behavior more than
experienced ones
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Prospect Theory
Empirical Evidence
Labor Market
New York Cab Drivers (Camerer, Babcock, Loewenstein
and Thaler, 1997)
Prospect Theory,
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Prospect Theory
Empirical Evidence
Domestic Violence
Domestic Violence (Card and Dahl, QJE 2011))
Consider a man in conflicted relationship with the spouse
What is the effect of an event such as the local football team
losing or winning a game?
With probability h the man loses control and becomes violent
Assume h = h(u) with h0 < 0 and u the underlying utility
Denote by p the ex-ante expectation that the team wins
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Prospect Theory
Empirical Evidence
Domestic Violence
Implication from Reference dependent utility)
The more a win is expected, the more a loss is costly in terms
of utility, the more likely it is to trigger violence
The (positive) effect of a gain is higher the more unexpected
(lower p)
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Prospect Theory
Empirical Evidence
Domestic Violence
Data)
Domestic violence (NIBRS)
Football matches by State
Expected win probability from Las Vegas predicted point
spread
Separate matches into
Predicted win (+3 points of spread)
Predicted close
Predicted loss (-3 points)
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Prospect Theory
Empirical Evidence
Domestic Violence
Results
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Prospect Theory
Empirical Evidence
Domestic Violence
Results
Unexpected loss increases domestic violence
No effect of expected loss
No effect of unexpected win, if anything increases violence
Effect disappears within a few hours of game end – Emotions
are transient
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Prospect Theory
Summary
Summary
Main departures from EU theory
Prospect Theory can accommodate most of them
several applications
Finance
Labor Market
Economic Development
Housing Market
Domestic Violence
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