temporal resolution of uncertainty and recursive models of ... · applications to macroeconomics...
TRANSCRIPT
![Page 1: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/1.jpg)
Temporal Resolution of Uncertaintyand Recursive Models of Ambiguity Aversion
Tomasz StrzaleckiHarvard University
![Page 2: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/2.jpg)
Preference for Earlier Resolution of Uncertainty
• instrumental value of information Spence Zeckhauser (1972)
• intrinsic value of information Kreps Porteus (1978)
• hidden actions Ergin Sarver (2012)
![Page 3: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/3.jpg)
Aversion to Persistence/Long Term Risk
Duffie Epstein (1992)
![Page 4: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/4.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 5: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/5.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 6: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/6.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 7: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/7.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 8: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/8.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 9: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/9.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 10: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/10.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 11: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/11.jpg)
Ambiguity
Kenes (1921), Ellsberg (1961)
![Page 12: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/12.jpg)
Main Message: modeling tradeoffs
The way we choose to model ambiguity aversion will impact:
• preference for earlier resolution of uncertainty
• aversion to long-run risk,
so the model ties these dimensions of preference together
![Page 13: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/13.jpg)
Similar to the modeling tradeoff in Epstein Zin (1989), where
• risk aversion
• intertemporal elasticity of substitution
• preference for earlier resolution of uncertainty and aversion tolong-run risk
are tied together
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Preferences
Vt = u(ct)+βE (Vt+1) Discounted Expected Utility
Vt = W(u(ct), E (Vt+1)
)Kreps–Porteus, Epstein–Zin
Vt = u(ct)+βI (Vt+1) Discounted Ambiguity Aversion
![Page 15: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/15.jpg)
Preferences
Vt = u(ct)+βE (Vt+1) Discounted Expected Utility
Vt = W(u(ct), E (Vt+1)
)Kreps–Porteus, Epstein–Zin
Vt = u(ct)+βI (Vt+1) Discounted Ambiguity Aversion
![Page 16: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/16.jpg)
Preferences
Vt = u(ct)+βE (Vt+1) Discounted Expected Utility
Vt = W(u(ct), E (Vt+1)
)Kreps–Porteus, Epstein–Zin
Vt = u(ct)+βI (Vt+1) Discounted Ambiguity Aversion
![Page 17: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/17.jpg)
Preferences
Vt = u(ct)+βE (Vt+1) Discounted Expected Utility
this % is indifferent to timing and to long run risks
![Page 18: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/18.jpg)
Preferences
Vt = W(u(ct), E (Vt+1)
)Kreps–Porteus, Epstein–Zin
this % is
PERU
PLRU
IERU
iff W (u, ·) is
convex
concave
linear
![Page 19: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/19.jpg)
Preferences
Vt = u(ct)+βI (Vt+1) Discounted Ambiguity Aversion
• here the operator I replaces expectation
• captures ambiguity aversion
• question: how does PERU depend on I
![Page 20: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/20.jpg)
Setting
![Page 21: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/21.jpg)
Setting
T = 0, 1, . . . ,T — discrete time, T <∞
(S ,Σ) — shocks, measurable space
Ω = ST — states of nature
X — consequences, convex subset of a real vector space
h = (h0, h1, . . . , hT ) — consumption plan, ht : S t → X
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Consumption Plan
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IID uncertainty with EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1 |st)
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
• what is IID is the underlying state process st
• consumption can have correlation
![Page 24: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/24.jpg)
IID uncertainty with EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1 |st)
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
• what is IID is the underlying state process st
• consumption can have correlation
![Page 25: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/25.jpg)
IID uncertainty with EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1 |st)
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
• what is IID is the underlying state process st
• consumption can have correlation
![Page 26: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/26.jpg)
IID uncertainty with EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1 |st)
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
• what is IID is the underlying state process st
• consumption can have correlation
![Page 27: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/27.jpg)
generalize beyond EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
Vt(st , h) = u(ht(st)
)+ βI
(Vt+1
((st , ·), h
))where I : RS → R
I constant over time—IID Ambiguity (Epstein and Schneider, 2003)
![Page 28: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/28.jpg)
generalize beyond EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
Vt(st , h) = u(ht(st)
)+ βI
(Vt+1
((st , ·), h
))where I : RS → R
I constant over time—IID Ambiguity (Epstein and Schneider, 2003)
![Page 29: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/29.jpg)
generalize beyond EU
Vt(st , h) = u(ht(st)
)+ β
∫S
Vt+1
((st , st+1), h
)dp(st+1)
Vt(st , h) = u(ht(st)
)+ βI
(Vt+1
((st , ·), h
))where I : RS → R
I constant over time—IID Ambiguity (Epstein and Schneider, 2003)
![Page 30: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/30.jpg)
Uncertainty Averse Preferences
I : RS → R is:
- continuous (supnorm)- monotonic (∀s∈S ξ(s) ≥ ζ(s)⇒ I (ξ) ≥ I (ζ))- normalized (∀r∈R I (r) = r)- quasiconcave
→ Uncertainty Aversion (Schmeidler, 1989)
Axiomatic foundations: Cerreia-Vioglio, Maccheroni, Marinacci,and Montrucchio (2011)
![Page 31: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/31.jpg)
special cases
1. Maxmin expected utility: I (ξ) = minp∈C∫ξ dp
2. Variational: I (ξ) = minp∈∆(Σ)
[ ∫ξ dp + c(p)
]3. Multiplier: I (ξ) = minp∈∆(Σ)
[ ∫ξ dp + θR(p‖q)
]4. Confidence: I (ξ) = minp∈∆(Σ)|ϕ(p)≥α
[1
ϕ(p)
∫ξ dp
]5. Second order expected utility: I (ξ) = φ−1
( ∫φ(ξ) dp
)6. Smooth ambiguity: I (ξ) = φ−1
( ∫∆(Σ)
φ(∫ξ dp) dµ(p)
)
![Page 32: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/32.jpg)
special cases
1. Maxmin expected utility: Gilboa and Schmeidler (1989)
2. Variational: Marinacci, Maccheroni, and Rustichini (2006)
3. Multiplier: Hansen and Sargent (2001)
4. Confidence: Chateauneuf and Faro (2009)
5. Second order expected utility: Neilson (1993)
6. Smooth ambiguity: Klibanoff, Marinacci, and Muhkerhi (2005)
![Page 33: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/33.jpg)
relations between them
• MEU ⊂ Variational
• MEU ⊂ Confidence
• Variational ∩ Confidence = MEU
• Multiplier ⊂ Variational
• SOEU ∩ Variational = Multiplier
• SOEU ⊂ KMM
![Page 34: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/34.jpg)
two key properties
• I (ξ + k) = I (ξ) + k for all ξ ∈ RS , k ∈ R
– shift invariance, Constant Absolute Ambiguity Aversion(Variational)
• I (βξ) = βI (ξ) for all ξ ∈ RS , β ∈ (0, 1)
– scale invariance, Constant Relative Ambiguity Aversion(Confidence)
• so MEU has both
![Page 35: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/35.jpg)
Discounted Uncertainty Averse Preferences
define by backward induction
VT (sT , h) := u(hT (sT ))
Vt(st , h) := u(ht(st)
)+ βI
(Vt+1
((st , ·), h
)); t = 0, . . . ,T−1
so Dynamically Consistent:
• Sarin and Wakker (1998)
• Epstein and Schneider (2003)
• Marinacci, Maccheroni, and Rustichini (2006)
• Klibanoff, Marinacci, and Muhkerhi (2009)
(without a fixed filtration, well known problems with DC)
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Applications to macroeconomics and finance
Dow and Werlang (1992), Epstein and Wang (1994), Chen andEpstein (2002), Epstein and Schneider (2007, 2008), Drechsler(2009), Ilut (2009), Mandler (2013), Condie Ganguli (2014),
Maenhout (2004), Karantounias, Hansen, and Sargent (2009),Kleshchelski and Vincent (2009), Barillas, Hansen, and Sargent(2009)
Ju and Miao (2012), Chen, Ju, and Miao (2009), Hansen (2007),Collard, Mukerji, Sheppard, and Tallon (2011), Benigno andNistico (2009)
![Page 37: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/37.jpg)
Results
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Main Message
What we assume about I will have impact on Preference for EarlierResolution of Uncertainty.
![Page 39: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/39.jpg)
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Theorem 1. A family of discounted uncertainty averse preferencessatisfies indifference to timing of resolution of uncertainty if andonly if I (ξ) = minp∈C
∫ξ dp.
![Page 43: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/43.jpg)
Proof
• indifference to timing ⇒ shift-invariance and scale-invariance
• shift-invariance and scale-invariance ⇒ MEU
![Page 44: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/44.jpg)
Proof
![Page 45: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/45.jpg)
Proof
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Proof
![Page 47: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/47.jpg)
Proof
![Page 48: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/48.jpg)
Proof
![Page 49: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/49.jpg)
Proof
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Proof
![Page 51: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/51.jpg)
Proof
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Proof
we have:
∃β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)
need to show:
∀β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)
(details in the paper)
![Page 53: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/53.jpg)
Proof
we have:
∃β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)
need to show:
∀β∈(0,1)∀x∈R∀ξ∈RS x + βI (ξ) = I (x + βξ)
(details in the paper)
![Page 54: Temporal Resolution of Uncertainty and Recursive Models of ... · Applications to macroeconomics and nance Dow and Werlang (1992), Epstein and Wang (1994), Chen and Epstein (2002),](https://reader033.vdocuments.site/reader033/viewer/2022052002/60155393f455a87f4c305329/html5/thumbnails/54.jpg)
comments
• in some sense this argument could be used to axiomatize therecursive multiple priors model
• a related paper by Kochov (2012) axiomatizes MEU using astrong version of Stationarity, which has a flavor of IERU
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Comparison to Risk
• Chew Epstein (1989) show IERU ⇒ EU
• Grant Kajii Polak (2000) show (rank-dependent orbetweenness) + PERU ⇒ EU
• so dispensing with objective probability makes more room
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Theorem 2. A family of discounted variational preferences,I (ξ) = minp∈∆(Σ)
∫ξ dp + c(p), always satisfies preference toward
earlier resolution of uncertainty.
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Theorem 3. A family of discounted confidence preferences,I (ξ) = minp∈∆(Σ)|ϕ(p)≥α
1ϕ(p)
∫ξ dp, displays a preference for
earlier resolution of uncertainty if and only if I (ξ) = minp∈C∫ξ dp.
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Theorem 4. A family of discounted second order expected utilitypreferences I (ξ) = φ−1
( ∫φ(ξ) dp
)with φ concave, strictly
increasing and twice differentiable displays a preference for earlierresolution of uncertainty iff Condition 1 holds.
Condition 1. There exists a real number A > 0 such that−φ′′(x)φ′(x) ∈ [βA,A] for all x ∈ R
(this condition means that the curvature of φ doesn’t vary toomuch)
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Theorem 5. A family of discounted smooth ambiguity preferencesI (ξ) = φ−1
( ∫∆(Σ) φ(
∫ξ dp) dµ(p)
)with φ concave, strictly
increasing and twice differentiable displays a preference for earlierresolution of uncertainty if Condition 1 holds
Only if under additional assumption on the support of µ.
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Persistence
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Persistence
Theorem 6. A family of discounted variational preferences,I (ξ) = minp∈∆(Σ)
∫ξ dp + c(p), always satisfies preference for iid.
A family of discounted confidence preferences,I (ξ) = minp∈∆(Σ)|ϕ(p)≥α
1ϕ(p)
∫ξ dp, always satisfies preference
for iid.
In both cases, indifference to iid is satisfied if and only ifI (ξ) = minp∈C
∫ξ dp.
(I do not know how to extend this result to all of I )
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Aggregator
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Aggregator
Vt(st , h) = u(ht(st)
)+βI
(Vt+1((st , ·), h)
)
Vt(st , h) = W
(ht(st), I
(Vt+1((st , ·), h)
))where W : X × R→ R
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Aggregator
Vt(st , h) = u(ht(st)
)+βI
(Vt+1((st , ·), h)
)Vt(st , h) = W
(ht(st), I
(Vt+1((st , ·), h)
))where W : X × R→ R
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Recursive Uncertainty Averse Preferences
VT (sT , h) = v(hT (sT ))
Vt(st , h) = W
(ht(st), I
(Vt+1
((st , ·), h
))); t = 0, . . . ,T−1
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Question
(IMEU ,W disc)
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Question
(IMEU ,W disc)
(I ,W disc)
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Question
(IMEU ,W disc)
(I ,W disc)
(IMEU ,W )
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Question
(IMEU ,W disc)
(I ,W disc)
(IMEU ,W )
Are they the same?
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Question
(IMEU ,W disc)
(I ,W disc)
(IMEU ,W )
Same iff I (ξ) = minp∈C φ−1( ∫
φ(ξ) dp)
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relation between the two models
• both W and I induce PERU
• but save for the case above, they are different %
• do they fit the data in a different way?
• W —workhorse model of macrofinance (Epstein–Zin)
• what does I add?
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recent work: Epstein, Farhi, Strzalecki (2014)
• suppose you are endowed with a consumption process ht
• for what π ∈ (0, 1) are you indifferent between
[ht , gradual resolution] ∼ [(1− π)ht , early resolution]
• π ∈ (20%, 40%) for workhorse models in finance usingEpstein–Zin preferences (Bansal and Yaron 2004; Barro, 2009)
• how high is this number for models of ambiguity?
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Conclusion:
interdependence of ambiguity and timing
MEU—only case of indifference
Questions:
theoretical: is this it? can we disentangle more?
empirical: how to measure this?
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Thank you
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