Atemporal Uncertainty How to evaluate?are.berkeley.edu/~traeger/pdf/Slides_Traeger_EZ... ·...

Time & Uncertainty Aggregation

Atemporal Uncertainty How to evaluate?

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ARE Departmental September 07 Intertemporal Risk Attitude – p. 5


Atemporal Uncertainty How to evaluate?

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vNMaxioms

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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



‘Intertemporal Certainty’ How to evaluate?

......p t t t tu1(x1) u2(x2) u3(x3) uT(xT)

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vNMaxioms

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......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





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vNMaxioms

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��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



‘Real World’: intertemporal uncertainty

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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

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vNMaxioms

LL


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-

-

-

-

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t

P

t

P

t

P

t

P

t

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?...... -

E

P U



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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

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vNMaxioms

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��1 ∃ uis.th.

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-

-

-

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t

P

t

P

t

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t

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P U



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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

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vNMaxioms

LL


��1 ∃ uis.th.

......t=1 t=2 t=3 t=T

-

-

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-

- 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)


Time & Uncertainty Aggregation KPEZ

Reconsider: an axiomatic reasoning

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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

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vNMaxioms

LL


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......t=1 t=2 t=3 t=T

-

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-

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t

P

t

P

t

P

t

P

t

? ? ? ?Ep Ep Ep Ep

?...... -

E

P U




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......

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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

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vNMaxioms

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-

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t

P

t

P

t

P

t

P

t

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P U

However: In general ui




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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

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p5

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t

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t

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t

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P U

However: In general ui 6= ui !




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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

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vNMaxioms

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......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!


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








⊲ 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


Intertemporal Risk Aversion II setup



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:







x1 x2 x1 x2∼1

it holds that

��

QQQ

x1 x2

x1 x2

x1 x2

≻1

1

2

1

2

intertemporal risk averse iff:







x1 x2 x1 x2∼1

it holds that

��

QQQ

x1 x2

x1 x2

x1 x2

≺1

1

2

1

2

intertemporal risk seeking iff:







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


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

]


Representation I



⊲ 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


Representation I



⊲ 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


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


Representation II

Theorem 1:


i) vNM axioms



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


Representation II

Theorem 1:


i) vNM axioms



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 .


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.


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.


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)





→ 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


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!




ut = Ngt

(

xt,Mft+1(pt+1, ut)

)

.


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




ut = Ngt

(

xt,Mft+1(pt+1, ut)

)

.


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.


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”





⊲ 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






⊲ 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



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


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.


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




IR



?curvature



- coconut quantity.

..........................................

....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................

.................6

12

Φ

��

��

��7

litchi quantity. .................................................................... ................................ .............................. .............................. ............................... ............................... ................................

................................

................................~

12

4

Φ2

Φ1

IRn

?differs for litchis




IR



?curvature



- coconut quantity.

..........................................

....................................... .................................... ................................. .............................. ........................... ......................... ...................... .................... .................. ................ .....................................................................................

.................6

12

Φ

6

coconut quality

. ............................. ........................... ........................ ..................... ................... ................ ............... ............. ..............................................................................

...................

.....................

............................................................ ....... ......... ........... .............. ................. .................... .......................z

12

?Φ3

Φ1

IRn

?arbitrary if coordinate Φ3 arbitrary




IR



?curvature



- 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?


Atemporal Uncertainty How to evaluate?are.berkeley.edu/~traeger/pdf/Slides_Traeger_EZ... ·...

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Transcript of Atemporal Uncertainty How to evaluate?are.berkeley.edu/~traeger/pdf/Slides_Traeger_EZ... ·...