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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
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
?
#"
!
von Neumann & Morgenstern (1944):
‘vNM axioms’(weak order, continuity, independence)
exists u such that
U ≡ Ep u =∑
i pi u(xi) =∫
Xu dp
represents preferences
ARE Departmental September 07 Intertemporal Risk Attitude – p. 5
Time & Uncertainty Aggregation
‘Intertemporal Certainty’ How to evaluate?
......p t t t tu1(x1) u2(x2) u3(x3) uT(xT)
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
- P
t
?
�� �
ARE Departmental September 07 Intertemporal Risk Attitude – p. 6
Time & Uncertainty Aggregation
‘Intertemporal Certainty’ How to evaluate?
......p t t t tu1(x1) u2(x2) u3(x3) uT(xT)
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
- P
t
?
�� � Koopmans (1960), ..., (Wakker 1988):
Additive Separability (also: Certainty Additivity, Coordinate Independence,...)
exist U ≡∑
t ut(xt) represents preferences
ARE Departmental September 07 Intertemporal Risk Attitude – p. 6
Time & Uncertainty Aggregation
‘Intertemporal Certainty’ How to evaluate?
......p t t t tu1(x1) u2(x2) u3(x3) uT(xT)
f��LL
PP ""!!ZZqpp??BB
vNMaxioms
LL
additiveseparability
��1 ∃ uis.th.
......t=1 t=2 t=3 t=T
- P
t
?
�� � Koopmans (1960), ..., (Wakker 1988):
Additive Separability (also: Certainty Additivity, Coordinate Independence,...)
exist u1, ..., uT such that U ≡∑
t ut(xt) represents preferences
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
u1(x1, p2) = u1(x1)+Mf2(p2, u2)
it holds that for t ∈ {1, 2}
pt �t p′t ⇔ Mft(pt, ut) ≥ Mft(p′t, ut) ∀pt, p′t ∈ Pt .
ARE Departmental September 07 Intertemporal Risk Attitude – p. 12
Representation III AIRA
Theorem 2:
In the representation of theorem 1, a decision maker is intertemporalrisk averse, if and only if, f1 is strictly concave.
ARE Departmental September 07 Intertemporal Risk Attitude – p. 13
Representation III AIRA
Theorem 2:
In the representation of theorem 1, a decision maker is intertemporalrisk averse, if and only if, f1 is strictly concave.
Define the measure of relative intertemporal risk aversion as:
RIRAt(z) = −f ′′
t(z)
f ′
t(z)
z .
The measures RIRAt
⊲ are uniquely defined if a zero welfare level is fixed,i.e. if xzero is chosen such that ut(x
zero) = 0.
⊲ It is independent of the commodity under observationand its measure scale.
ARE Departmental September 07 Intertemporal Risk Attitude – p. 13
Representation - Epstein Zin I KP
The representation in theorem 1 uses the time additive utilityfunctions ut.
In a one commodity world and with X ⊂ IR can also choose ut = id.
ARE Departmental September 07 Intertemporal Risk Attitude – p. 14
Representation - Epstein Zin I KP
The representation in theorem 1 uses the time additive utilityfunctions ut.
In a one commodity world and with X ⊂ IR can also choose ut = id.
→ recursion relation: ut = Ngt
(
xt,Mft+1(pt+1, ut)
)
Where Ngt is a nonlinear intertemporal aggregation rule similar to
Mft parameterized by the function gt : Ut → IR. (slightly sloppy)
ARE Departmental September 07 Intertemporal Risk Attitude – p. 14
Representation - Epstein Zin I KP
The representation in theorem 1 uses the time additive utilityfunctions ut.
In a one commodity world and with X ⊂ IR can also choose ut = id.
→ recursion relation: ut = Ngt
(
xt,Mft+1(pt+1, ut)
)
Where Ngt is a nonlinear intertemporal aggregation rule similar to
Mft parameterized by the function gt : Ut → IR. (slightly sloppy)
For gt(z) = βtzρ and ft(z) = zα Epstein & Zin’s (1991) generalizedisoelastic model.
Can interpret
⊲ 1 − α as measure of Arrow-Pratt relative risk aversion
⊲ (1 − ρ)−1 as intertemporal elasticity of substitution
ARE Departmental September 07 Intertemporal Risk Attitude – p. 14
Representation - Epstein Zin II
Epstein Zin interpretation depends crucially on u = id in
ut = Ngt
(
xt,Mft+1(pt+1, ut)
)
.
Multi-commodity usually: Directional derivative ⇒ Doesn’t work!
ARE Departmental September 07 Intertemporal Risk Attitude – p. 15
Representation - Epstein Zin II
Epstein Zin interpretation depends crucially on u = id in
ut = Ngt
(
xt,Mft+1(pt+1, ut)
)
.
Multi-commodity usually: Directional derivative ⇒ Doesn’t work!
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
ARE Departmental September 07 Intertemporal Risk Attitude – p. 15
Representation - Epstein Zin II
Epstein Zin interpretation depends crucially on u = id in
ut = Ngt
(
xt,Mft+1(pt+1, ut)
)
.
Multi-commodity usually: Directional derivative ⇒ Doesn’t work!
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
Space of goods Coordinate space(space of numbers)
IR
Robinson’s preferences �The (economist’s)
utility function u on IR
?curvature
describes risk attitude
Φ: Coordinate system
- coconut quantity.
..........................................
....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................
.................6
12
Φ
ARE Departmental September 07 Intertemporal Risk Attitude – p. 27
Good and measure scale dependence of risk measures
Space of goods Coordinate space(space of numbers)
IR
Robinson’s preferences �The (economist’s)
utility function u on IR
?curvature
describes risk attitude
Φ: Coordinate system
- coconut quantity.
..........................................
....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................
.................6
12
Φ
��
��
���7
litchi quantity. .................................................................... ................................ .............................. .............................. ............................... ............................... ................................
................................
................................~
12
4
Φ2
Φ1
IRn
?differs for litchis
ARE Departmental September 07 Intertemporal Risk Attitude – p. 27
Good and measure scale dependence of risk measures
Space of goods Coordinate space(space of numbers)
IR
Robinson’s preferences �The (economist’s)
utility function u on IR
?curvature
describes risk attitude
Φ: Coordinate system
- coconut quantity.
..........................................
....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................
.................6
12
Φ
6
coconut quality
. ............................. ........................... ........................ ..................... ................... ................ ............... ............. ..............................................................................
...................
.....................
............................................................ ....... ......... ........... .............. ................. .................... .......................z
12
?Φ3
Φ1
IRn
?arbitrary if coordinate Φ3 arbitrary
ARE Departmental September 07 Intertemporal Risk Attitude – p. 27
Good and measure scale dependence of risk measures
Space of goods Coordinate space(space of numbers)
IR
Robinson’s preferences �The (economist’s)
utility function u on IR
?curvature
describes risk attitude
Φ: Coordinate system
- coconut quantity.
..........................................
....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................
.................6
12
Φ
��
��
���7
litchi quantity. .................................................................... ................................ .............................. .............................. ............................... ............................... ................................
................................
................................~
12
4
Φ2
Φ1
IRn
?differs for litchis
6
coconut quality
. ............................. ........................... ........................ ..................... ................... ................ ............... ............. ..............................................................................
...................
.....................
............................................................ ....... ......... ........... .............. ................. .................... .......................z
12
?Φ3
Φ1
IRn
?arbitrary if coordinate Φ3 arbitrary
Is there a risk attitude measure that is independent ofcoordinate system Φ and the particular good?
I.e. that describes attitude with respect to risk rather thanwith respect to coconuts, litchis or money?
ARE Departmental September 07 Intertemporal Risk Attitude – p. 27