Time & Uncertainty Aggregation Atemporal Uncertainty How to evaluate? ✱ ✱ ✱ ✱ ✱ ✦ ✦ ✦ ✦ ✦ ❛ ❛ ❛ ❛ ❛ ❧ ❧ ❧ ❧ ❧ ♣ u(x 1 ) u(x 2 ) u(x 3 ) u(x 4 ) u(x 5 ) p 1 p 2 p 3 p 4 p 5 ❢ ☞ ☞ ▲ ▲ ✧ ✧ ✦ ✦ ❩ ♣ ♣ ? ? ❇ ❇ vNM axioms ▲ ▲ additive separability ✏ ✏✶ ∃ u i s.th. ❄ E p ❄ ★ ✧ ✥ ✦ ARE Departmental September 07 Intertemporal Risk Attitude – p. 5
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Time & Uncertainty Aggregation
Atemporal Uncertainty How to evaluate?
,,
,,,
!!!!!
aaaaal
ll
ll
pttttt
u(x1)
u(x2)
u(x3)
u(x4)
u(x5)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
?Ep
?
#"
!
ARE Departmental September 07 Intertemporal Risk Attitude – p. 5
Time & Uncertainty Aggregation
Atemporal Uncertainty How to evaluate?
,,
,,,
!!!!!
aaaaal
ll
ll
pttttt
u(x1)
u(x2)
u(x3)
u(x4)
u(x5)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
?Ep
?
#"
!
ARE Departmental September 07 Intertemporal Risk Attitude – p. 5
ARE Departmental September 07 Intertemporal Risk Attitude – p. 6
Time & Uncertainty Aggregation
‘Real World’: intertemporal uncertainty
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 7
Time & Uncertainty Aggregation
‘Real World’: intertemporal uncertainty
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 7
Time & Uncertainty Aggregation
‘Real World’: intertemporal uncertainty
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 7
Time & Uncertainty Aggregation
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 7
Time & Uncertainty Aggregation
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
Standard Model: intertemporally additive expected utilityU = Ep
∑
t ut(xt)
ARE Departmental September 07 Intertemporal Risk Attitude – p. 7
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
However: In general ui
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp!!BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
However: In general ui 6= ui !
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Time & Uncertainty Aggregation KPEZ
Reconsider: an axiomatic reasoning
,,
,,,
!!!!!
aaaaal
ll
ll......
......
......
......
......
pttttt
ttttt
ttttt
ttttt
u1(x11)
u1(x21)
u1(x31)
u1(x41)
u1(x51)
u2(x12)
u2(x22)
u2(x32)
u2(x42)
u2(x52)
u3(x13)
u3(x23)
u3(x33)
u3(x43)
u3(x53)
uT(x1T)
uT(x2T)
uT(x3T)
uT(x4T)
uT(x5T)
p1
p2
p3
p4
p5
f��LL
PP ""!!ZZqpp!!BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
-
-
-
-
- P
t
P
t
P
t
P
t
P
t
? ? ? ?Ep Ep Ep Ep
?...... -
E
P U
However: In general ui 6= ui !
Thus: Intertemporally additive expected utility modelonly represents very particular preferences where ui = ui!
ARE Departmental September 07 Intertemporal Risk Attitude – p. 8
Intertemporal Risk Aversion I setup
Question: What preferences are missing in the standard model?
ARE Departmental September 07 Intertemporal Risk Attitude – p. 9
Intertemporal Risk Aversion I setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
ARE Departmental September 07 Intertemporal Risk Attitude – p. 9
Intertemporal Risk Aversion I setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
For a definition assume T = 2 and define:
⊲ X: (connected compact metric) space of goods
⊲ ∆(·): Set of Borel probability measures on space ‘·’ (Prohorov metric)
ARE Departmental September 07 Intertemporal Risk Attitude – p. 9
Intertemporal Risk Aversion I setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
For a definition assume T = 2 and define:
⊲ X: (connected compact metric) space of goods
⊲ ∆(·): Set of Borel probability measures on space ‘·’ (Prohorov metric)
⊲ P2 = ∆(X): Probability measures p2 on X
⊲ P1 = ∆(X × P2): Probability measures p1 on X × P2
⊲ �2: 2nd period preference relation on P2 with elements p2
⊲ �1: 1st period preference relation on P1 with elements p1
ARE Departmental September 07 Intertemporal Risk Attitude – p. 9
Intertemporal Risk Aversion I setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
For a definition assume T = 2 and define:
⊲ X: (connected compact metric) space of goods
⊲ ∆(·): Set of Borel probability measures on space ‘·’ (Prohorov metric)
⊲ P2 = ∆(X): Probability measures p2 on X
⊲ P1 = ∆(X × P2): Probability measures p1 on X × P2
⊲ �2: 2nd period preference relation on P2 with elements p2
⊲ �1: 1st period preference relation on P1 with elements p1
⊲ C0(·): Set of all continuous functions from space ‘·’ to IR
ARE Departmental September 07 Intertemporal Risk Attitude – p. 9
Intertemporal Risk Aversion II setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
Definition: A decision maker is called
For all x1, x1, x2, x2 ∈ X such that x2 ≻2 x2 and
x1 x2 x1 x2∼1
it holds that
intertemporal risk averse iff:
ARE Departmental September 07 Intertemporal Risk Attitude – p. 10
Intertemporal Risk Aversion II setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
Definition: A decision maker is called
For all x1, x1, x2, x2 ∈ X such that x2 ≻2 x2 and
x1 x2 x1 x2∼1
it holds that
���
QQQ
x1 x2
x1 x2
x1 x2
≻1
1
2
1
2
intertemporal risk averse iff:
ARE Departmental September 07 Intertemporal Risk Attitude – p. 10
Intertemporal Risk Aversion II setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
Definition: A decision maker is called
For all x1, x1, x2, x2 ∈ X such that x2 ≻2 x2 and
x1 x2 x1 x2∼1
it holds that
���
QQQ
x1 x2
x1 x2
x1 x2
≺1
1
2
1
2
intertemporal risk seeking iff:
ARE Departmental September 07 Intertemporal Risk Attitude – p. 10
Intertemporal Risk Aversion II setup
Question: What preferences are missing in the standard model?
Answer : Those with a non-trivial intertemporal risk attitude!
Definition: A decision maker is called
For all x1, x1, x2, x2 ∈ X such that x2 ≻2 x2 and
x1 x2 x1 x2∼1
it holds that
���
QQQ
x1 x2
x1 x2
x1 x2
∼1
1
2
1
2
interemporal risk neutral iff:
standardmodel
ARE Departmental September 07 Intertemporal Risk Attitude – p. 10
Representation I
Uncertainty Aggregation Rule
For f : IR → IR continuous and strictly increasingand some compact metric space Y define
⊲ Mf : ∆(Y ) × C0(Y ) → IR
⊲ Mf (p, u) = f−1[∫
Yf ◦ u dp
]
ARE Departmental September 07 Intertemporal Risk Attitude – p. 11
Representation I
Uncertainty Aggregation Rule
For f : IR → IR continuous and strictly increasingand some compact metric space Y define
⊲ Mf : ∆(Y ) × C0(Y ) → IR
⊲ Mf (p, u) = f−1[∫
Yf ◦ u dp
]
The uncertainty aggregation rule satisfies:
⊲ Mf (y, u) = u(y) ∀ y ∈ Y
ARE Departmental September 07 Intertemporal Risk Attitude – p. 11
Representation I
Uncertainty Aggregation Rule
For f : IR → IR continuous and strictly increasingand some compact metric space Y define
⊲ Mf : ∆(Y ) × C0(Y ) → IR
⊲ Mf (p, u) = f−1[∫
Yf ◦ u dp
]
The uncertainty aggregation rule satisfies:
⊲ Mf (y, u) = u(y) ∀ y ∈ Y
It includes rules corresponding to
⊲ expected value (f = id)
⊲ geometric mean (f = ln)
⊲ power mean (fα = idα) ‘CRRA type’ Example
ARE Departmental September 07 Intertemporal Risk Attitude – p. 11
Representation II
Theorem 1:
The set of preference relations (�1,�2) on (P1, P2) satisfies
i) vNM axioms
ii) additive separability for certain consumption paths
iii) time consistency
ARE Departmental September 07 Intertemporal Risk Attitude – p. 12
Representation II
Theorem 1:
The set of preference relations (�1,�2) on (P1, P2) satisfies
i) vNM axioms
ii) additive separability for certain consumption paths
iii) time consistency
if and only if, there exist continuous functions ut : X → Ut ⊂ IR anda strictly increasing and continuous function ft : Ut → IR fort ∈ {1, 2} such that with defining u2(x2) = u2(x2) and
ARE Departmental September 07 Intertemporal Risk Attitude – p. 12
Representation II
Theorem 1:
The set of preference relations (�1,�2) on (P1, P2) satisfies
i) vNM axioms
ii) additive separability for certain consumption paths
iii) time consistency
if and only if, there exist continuous functions ut : X → Ut ⊂ IR anda strictly increasing and continuous function ft : Ut → IR fort ∈ {1, 2} such that with defining u2(x2) = u2(x2) and
Instead: Linearize utility in commodity i and determine f it and gi
t.As in standard model, risk measure depends on
⊲ particular commodity under observation
⊲ measure scale
Intertemporal risk aversion in the ‘Epstein Zin form’ is characterizedby strict concavity of ft ◦ g−1
t and is independent of the commodity.
The IRA measure captures the difference betweenthe propensity to smooth certain consumption over time andthe propensity to smooth consumption between different risk states.
ARE Departmental September 07 Intertemporal Risk Attitude – p. 15
Uncertainty Aggregation Rule, Example Definition
Example: Power mean (‘CRRA type function’):
Let fα(z) = zα and p simple (finite support):
⊲ Mα(p, u) =(∑
x p(x)u(x)α)
1
α , α∈[−∞,∞],“ RRA on welf = 1 − α”
ARE Departmental September 07 Intertemporal Risk Attitude – p. 25
Uncertainty Aggregation Rule, Example Definition
Example: Power mean (‘CRRA type function’):
Let fα(z) = zα and p simple (finite support):
⊲ Mα(p, u) =(∑
x p(x)u(x)α)
1
α , α∈[−∞,∞],“ RRA on welf = 1 − α”
Consider a given, cardinal welfare function u and the lottery
⊲ u = u(x) = 100 with probability p = p(x) = .9
⊲ u = u(x) = 10 with probability p = p(x) = .1
ARE Departmental September 07 Intertemporal Risk Attitude – p. 25
Uncertainty Aggregation Rule, Example Definition
Example: Power mean (‘CRRA type function’):
Let fα(z) = zα and p simple (finite support):
⊲ Mα(p, u) =(∑
x p(x)u(x)α)
1
α , α∈[−∞,∞],“ RRA on welf = 1 − α”
Consider a given, cardinal welfare function u and the lottery
⊲ u = u(x) = 100 with probability p = p(x) = .9
⊲ u = u(x) = 10 with probability p = p(x) = .1
and find
⊲ M∞
(p, u) ≡ limα→∞Mα(p, u) = maxx u(x) = 100
⊲ M1(p, u) = Ep u = 91
⊲ M0(p, u) ≡ limα→0M
α(p, u) =
∏
x u(x)p(x) = 73.2
⊲ M−10(p, u) =
(∑
x p(x)u(x)−10)
−1
10 = 12.6
⊲ M−∞
(p, u) ≡ limα→−∞Mα(p, u) = minx u(x) = 10
ARE Departmental September 07 Intertemporal Risk Attitude – p. 25
Absolute Intertemporal Risk Aversion RIRA
In the certainty additive representation of theorem 1 (g = id),define measure of absolute intertemporal risk aversion as:
AIRA(z) = −(f◦g−1)
′′
(z)
(f◦g−1)′(z).
⊲ AIRA is uniquely defined if unit of welfare is fixed,i.e. if x1 and x2 (non-indifferent) are chosen such thatu(x1) − u(x2) = 1.
⊲ It is independent of the commodity under observationand its measure scale.
ARE Departmental September 07 Intertemporal Risk Attitude – p. 26
Good and measure scale dependence of risk measures