14.13 Economics and Psychology
(Lecture 4)
Xavier Gabaix
February 12, 2004
1
Prospect Theory value of the game
Consider gambles with two outcomes: x with probability p, and y with
probability 1 − p where x ≥ 0 ≥ y.
The PT value of the game is
V = π (p) u (x) + π (1 − p) u (y)
• In prospect theory the probability weighting π is concave first and then
convex, e.g.
pβ
π (p) = β
p + (1 − p)β
for some β ∈ (0, 1). In the figure below p is on the horizontal axis and
π (p) on the vertical one.
y
1
0.75
0.5
0.25
0
0
0.25
0.5
1
0.75
x
• A useful parametrization of the PT value function is a power law
function
u (x) = |x|α for x ≥ 0
u (x) = −λ |x|α for x ≤ 0
y
2.5
0
-5
-2.5
0
2.5
5
x
-2.5
-5
Meaning - Fourfold pattern of risk aversion u
• Risk aversion in the domain of likely gains
• Risk seeking in the domain of unlikely gains
• Risk seeking in the domain of likely losses
• Risk aversion in the domain of unlikely losses
See tables on next page.
Preferences between Positive and Negative Prospects
Positive Prospects
Problem 3:
N = 95
Problem 4:
N = 95
Problem 7:
N = 66
Problem 8:
N = 66
Negative Prospects
(4,000, .80) < (3000).
[20]
[80]*
(4,000, .20) > (3,000, .25).
[65]*
[35]
(3,000, .90) > (6,000, .45).
[86]*
[14]
(3,000, .002) < (6,000, .001).
[27]
[73]*
Problem 3': (-4,000, .80) > (-3000).
N = 95
[92]*
[8]
Problem 4': (-4,000, .20) < (-3,000, .25).
N = 95
[42]
[58]
Problem 7': (-3,000, .90) < (-6,000, .45).
N = 66
[8]
[92]*
Problem 8': (-3,000, .002) > (-6,000, .001).
N = 66
[70]*
[30]
Percentage of Risk-Seeking Choices
Gain
Loss
Subject
p < .1
p > .5
p < .1
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
100
85
100
71
83
100
100
87
16
83
100
100
87
100
66
60
100
100
60
100
100
100
100
71
100
38
33
10
0
0
5
10
0
0
0
26
30
100
16
0
21
0
5
15
22
10
5
0
0
31
0
0
20
0
30
20
0
30
10
80
0
0
10
10
30
30
10
20
10
60
0
0
0
0
80
10
75
93
58
100
100
86
100
100
93
100
100
94
100
100
100
100
93
63
81
100
92
100
100
87
10
2
88a
20
0
80a
Risk seeking
Risk neutral
Risk averse
78a
12
10
a Values that correspond to the fourfold pattern.
Note: The percentage of risk-seeking choices is given for low (p < . 1)
and high (p > .5) probabilities of gain and loss for each subject (riskneutral choices were excluded). The overall percentage of risk-seeking,
risk-neutral, and risk-averse choices for each type of prospect appears
at the bottom of the table.
p > .5
87a
7
6
Properties of power law PT value functions
• they are scale invariant, i.e. for any k > 0
Consider a gamble and the same gamble scaled up by k:
g=
(
x
y
kg =
(
kx
ky
with prob p
with prob 1 − p
with prob p
with prob 1 − p
then
V P T (kg) = kαV P T (g)
• if someone prefers g to g 0 then he will prefer kg to kg 0 for k > 0
• if x, y ≥ 0, V (−g) = −λV (g)
• if x0, y 0 ≥ 0 and someone prefers g to g 0 then he will prefer −g 0 to −g
2
How robust are the results?
• Very robust: loss aversion at the reference point, λ > 1
• Medium robust: convexity of u for x < 0
• Slightly robust: underweighting and overweighting of probabilities π (p) ≷
p
3
In applications we often use a simplified PT
(prospect theory):
π (p) = p
and
u (x) = x for x ≥ 0
u (x) = λx for x ≤ 0
4
Second order risk aversion of EU
• Consider a gamble σ and −σ with 50 : 50 chances.
• Question: what risk premium Π would people pay to avoid the small
risk σ?
´
2
• We will show that as σ → 0 this premium is O σ . This is called
second order risk aversion.
³
• In fact we will show that for twice continuously differentiable utilities:
ρ 2
Π (σ) ∼
= σ ,
2
00
u
where ρ is the curvature of u at 0 that is ρ = − u0 .
• Let’s generalize and consider an agent starting with wealth x. The
agent takes the gamble iff:
1
1
B (Π) = u (x + Π + σ) + u (x + Π − σ) ≥ u (x)
2
2
i.e. Π ≥ Π∗ where:
B (Π∗) = u (x)
• Assume that u is twice differentiable and take the Taylor expansion of
B(Π) for small σ and Π:
1
u(x + Π + σ) = u(x) + u0(x)(Π + σ) + u00(x)(Π + σ)2 + o(Π + σ)2
2
1 00
0
u(x + Π − σ) = u(x) + u (x)(Π − σ) + u (x)(Π − σ)2 + o(Π − σ)2
2
hence
h
i
³
´
1 00
0
2
2
2
2
B (Π) = u (x) + u (x) Π + u (x) σ + Π + o σ + Π
2
Then use the definition B(Π∗) = u(x) to get
³
´
∗2i
ρh 2
∗
2
∗2
Π =
σ +Π +o σ +Π
2
h
i
ρ
∗
2
∗2
• to solve : Π = 2 σ + Π
for small σ, call ρ0 = ρ/2.
• Barbaric way :
— find the roots of Π∗2 − ρ10 Π∗ + σ 2 = 0.
∗ compute the discriminant
1
∆ = 02 − 4σ 2
ρ
µ
1 ± 1 1 − 4σ 2
∗ the roots are Π∗ = 2ρ
0
2 ρ02
1
1
∗
Π = 0±
2ρ
2
Ã
¶1
2
1
2
−
4σ
ρ02
!1
2
∗ as when there is no risk, the risk premium should be 0, then the
relevant root is:
1
1
Π∗ = 0 −
2ρ
2
Ã
1
2
−
4σ
ρ02
!1
2
— take the Taylor expansion for small σ
³
´1
1
1
Π∗ = 0 − 0 1 − 4ρ02σ 2 2
2ρ
2ρ
µ
¶
1
1
1 02 2
= 0 − 0 1 − 4ρ σ + o(σ 2)
2ρ
2ρ
2
= ρ0σ 2
— then remember that ρ0 = ρ/2:
Π∗= ρ2 σ 2