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Econometrica, Vol. 76, No. 5 (September, 2008), 1167–1190
SUBJECTIVE BELIEFS AND EX ANTE TRADE
LUCA RIGOTTI
Fuqua School of Business, Duke University, Durham, NC 27708, U.S.A.
CHRIS SHANNON
University of California, Berkeley, Berkeley, CA 947203880, U.S.A.
TOMASZ STRZALECKI
Northwestern University, Evanston, IL 60208, U.S.A.
The copyright to this Article is held by the Econometric Society. It may be downloaded,
printed and reproduced only
http://www.econometricsociety.org/
Econometrica
, Vol. 76, No. 5 (September, 2008), 1167–1190SUBJECTIVE BELIEFS AND EX ANTE TRADEL
UCA
R
IGOTTI
Fuqua School of Business, Duke University, Durham, NC 27708, U.S.A.
C
HRIS
S
HANNON
University of California, Berkeley, Berkeley, CA 947203880, U.S.A.
T
OMASZ
S
TRZALECKI
Northwestern University, Evanston, IL 60208, U.S.A.
The copyright to this Article is held by the Econometric Society. It may be downloaded,printed and reproduced only for educational or research purposes, including use in coursepacks. No downloading or copying may be done for any commercial purpose without theexplicit permission of the Econometric Society. For such commercial purposes contactthe Ofﬁce of the Econometric Society (contact information may be found at the websitehttp://www.econometricsociety.org or in the back cover of
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). This statement mustthe included on all copies of this Article that are made available electronically or in any otherformat.
Econometrica
, Vol. 76, No. 5 (September, 2008), 1167–1190
SUBJECTIVE BELIEFS AND EX ANTE TRADEB
Y
L
UCA
R
IGOTTI
, C
HRIS
S
HANNON
,
AND
T
OMASZ
S
TRZALECKI
1
We study a deﬁnition of subjective beliefs applicable to preferences that allow forthe perception of ambiguity, and provide a characterization of such beliefs in terms of market behavior. Using this deﬁnition, we derive necessary and sufﬁcient conditions forthe efﬁciency of ex ante trade and show that these conditions follow from the fundamental welfare theorems. When aggregate uncertainty is absent, our results show thatfull insurance is efﬁcient if and only if agents share some common subjective beliefs.Our results hold for a general class of convex preferences, which contains many functional forms used in applications involving ambiguity and ambiguity aversion. We showhow our results can be articulated in the language of these functional forms, conﬁrmingresults existing in the literature, generating new results, and providing a useful tool forapplications.K
EYWORDS
: Common prior, absence of trade, ambiguity aversion, general equilibrium.
1.
INTRODUCTION
I
N A MODEL WITH RISK

AVERSE AGENTS
who maximize subjective expectedutility, betting occurs if and only if agents’ priors differ. This link between common priors and speculative trade in the absence of aggregate uncertainty is afundamental implication of expected utility for risksharing in markets. A similar relationship holds when ambiguity is allowed and agents maximize theminimum expected utility over a set of priors, as in the model of Gilboa andSchmeidler(1989). In this case, purely speculative trade occurs when agents
hold no priors in common; full insurance is Pareto optimal if and only if agentshave at least one prior in common, asBillot, Chateauneuf, Gilboa, and Tallon(2000) showed. This note develops a more general connection between subjective beliefs and speculative trade applicable to a broad class of convex preferences, which encompasses as special cases not only the previous results for expected utility and maxmin expected utility, but all the models central in studiesof ambiguity in markets, including the convex Choquet model of Schmeidler(1989), the smooth secondorder prior models of Klibanoff, Marinacci, and
Mukerji(2005) andNau(2006), the secondorder expected utility model of
Ergin and Gul(2004), the conﬁdence preferences model of Chateauneuf and
Faro(2006), the multiplier model of Hansen and Sargent(2001), and the vari
ational preferences model of Maccheroni, Marinacci, and Rustichini(2006).
1
Tomasz Strzalecki is extremely grateful to Eddie Dekel for very detailed comments on earlier versions of this note and to Itzhak Gilboa, Peter Klibanoff, and Marciano Siniscalchi for veryhelpful discussions. We thank Larry Samuelson (the coeditor) and three anonymous refereesfor very useful suggestions; we have also beneﬁted from comments from Wojciech Olszewski,Marcin Peski, Jacob Sagi, Uzi Segal, Itai Sher, JeanMarc Tallon, Jan Werner, Asher Wolinsky,and audiences at RUD 2007 and WIEM 2007.© 2008The Econometric SocietyDOI:10.3982/ECTA7660
1168
L. RIGOTTI, C. SHANNON, AND T. STRZALECKI
By casting our results in the general setting of convex preferences, we areable to focus on several simple underlying principles. We identify a notionof subjective beliefs based on market behavior and show how it is related to various notions of belief that arise from different axiomatic treatments. Wehighlight the close connection between the fundamental welfare theorems of general equilibrium and results that link common beliefs and risksharing. Finally, by establishing these links for general convex preferences, we provide aframework for studying ambiguity in markets while allowing for heterogeneityin the way ambiguity is expressed through preferences. The generality of thisapproach identiﬁes the forces underlying betting without being restricted toany one particular representation, and in so doing uniﬁes our thinking aboutmodels of ambiguity aversion in economic settings.The note is organized as follows. Section2studies subjective beliefs andbehavioral characterizations, with illustrations for various familiar representations. Section3studies trade between agents with convex preferences. AppendixAdevelops an extension of these results to inﬁnite state spaces, while AppendixBcollects some proofs omitted in the text.2.
BELIEFS AND CONVEX PREFERENCES
2.1.
Convex Preferences
Let
S
be a ﬁnite set of states of the world. The set of consequences is
R
+
, which we interpret as monetary payoffs. The set of acts is
F
=
R
S
+
with thenatural topology. Acts are denoted by
f
,
g
, and
h
, while
f(s)
denotes the monetary payoff from act
f
when state
s
obtains. For any
x
∈
R
+
we abuse notationby writing
x
∈
F
, which stands for the constant act with payoff
x
in each stateof the world.Let
be a binary relation on
F
. We say that
is a
convex preference relation
if it satisﬁes the following axioms: A
XIOM
1—Preference:
is complete and transitive
. A
XIOM
2—Continuity:
For all
f
∈
F
,
the sets
{
g
∈
F

g
f
}
and
{
g
∈
F

f
g
}
are closed
. A
XIOM
3—Monotonicity:
For all
fg
∈
F
,
if
f(s) > g(s)
for all
s
∈
S
,
then
f
g
. A
XIOM
4—Convexity:
For all
f
∈
F
,
the set
{
g
∈
F

g
f
}
is convex
.These axioms are standard, and wellknown results imply that a convex preference relation
is represented by a continuous, increasing, and quasiconcave
SUBJECTIVE BELIEFS AND EX ANTE TRADE
1169function
V
:
F
→
R
.
2
Convex preferences include as special cases many common models of risk aversion and ambiguity aversion. In many of these specialcases, one element of the representation identiﬁes a notion of beliefs. In whatfollows, we adopt the notion of subjective probability suggested inYaari(1969)
to deﬁne
subjective beliefs
for general convex preferences. We then study characterizations of this concept in terms of market behavior, and illustrate particular special cases including maxmin expected utility, Choquet expected utility,and variational preferences.2.2.
Supporting Hyperplanes and Beliefs
ThedecisiontheoreticapproachofdeFinetti,Ramsey,andSavageidentiﬁesa decision maker’s subjective probability with the odds at which he is willing tomake small bets. In this spirit,Yaari(1969) identiﬁed subjective probability
with a hyperplane that supports the upper contour set.
3
If this set has kinks,for example because of nondifferentiabilities often associated with ambiguity,there may be multiple supporting hyperplanes at some acts. To encompass suchpreferences, we consider the set of
all
(normalized) supporting hyperplanes.
4
D
EFINITION
1—Subjective Beliefs: The set of subjective beliefs at an act
f
is
π
(f)
:={
p
∈
S

p
·
g
≥
p
·
f
for all
g
f
}
Given the interpretation of the elements of
π
(f)
as beliefs, we will write
E
p
g
instead of
p
·
g
. For any convex preference relation,
π
(f)
is nonempty,compact, and convex, and is equivalent to the set of (normalized) supports tothe upper contour set of
at
f
. In the next section we explore behavioralimplications of this deﬁnition, including willingness or unwillingness to trade,and their market consequences.2.3.
Market Behavior and Beliefs
We begin with a motivating example, set in the maxmin expected utility(MEU) model of Gilboa and Schmeidler(1989). The agent’s preferences are
2
Axiom4captures convexity in monetary payoffs. For Choquet expected utility agents, whoevaluate an act according to the Choquet integral of its utility with respect to a nonadditive measure (capacity), the relation between payoff convexity and uncertainty aversion has been studiedbyChateauneuf and Tallon(2002).Dekel(1989) studied the relation between payoff convexity
and risk aversion.
3
In the ﬁnance literature this is commonly called a riskneutral probability or riskadjustedprobability.
4
Alternatively,Chambers and Quiggin(2002) deﬁned beliefs using superdifferentials of the
beneﬁt function
. Their deﬁnition turns out to be equivalent to ours.