Sunspots and Multiplicity

Sunspots and Multiplicity
Matthew Hoelle
European University Institute
Max Weber Programme
Via delle Fontanelle 10
50014 San Domenico di Fiesole, Italy
February 9, 2011
Abstract
This paper proves that, in a general …nancial model with incomplete markets, the
multiplicity of certainty equilibria is not necessary for the existence of sunspot effects. These e¤ects are present, by de…nition, when real economic variables di¤er
across realizations of extrinsic uncertainty. In a …nancial model of incomplete markets
with identical payouts across realizations of extrinsic uncertainty, the past literature
has concluded that multiplicity of certainty equilibria is necessary for sunspot e¤ects.
However, all these papers model extrinsic uncertainty using expected utility. In this
paper, I consider a generalization of expected utility, a form that Dave Cass labeled
UBU (utility-based utility). With this form, my result proves that the class of rational
expectations equilibria exhibiting sunspot e¤ects is much broader than the class of
multiple certainty equilibria.
JEL Classi…cations: D61; D81; D91
Key words: sunspots; extrinsic uncertainty; incomplete markets; determinacy; intertemporal consistency; variance aversion; Cass-Shell Immunity Theorem
1
Introduction
The idea of sunspots, far simpler than a new model or a new solution concept, has been
extensively studied by economists, because sunspots are a formal representation of seemingly
irrational behavior exhibited in …nancial markets. In these markets, traders respond to
irrelevant information that has no bearing on "fundamentals." These responses, if adopted as
I thank Masanori Kashiwagi, Laurie Anderson, and Alyson Price for helpful feedback, both in the
substance and style of the paper. I especially thank Dave Cass for the idea behind the UBU utility form
used in this result and the one or two key ideas needed to properly de…ne extrinsic uncertainty with the given
utility form. He is not responsible for the choices I have made in how to use the utility form. I acknowledge
the support received from a Max Weber Fellowship (EUI). Comments are welcome at matthew.hoelle@eui.eu.
1
the "market psychology," are self-ful…lling optimal actions of traders. In terms of economic
modeling, sunspots are realizations of extrinsic uncertainty, or uncertainty that does not
impact the fundamentals (endowments, preferences, payouts) of the economy. As was shown
thirty years ago (Cass and Shell (1983), Balasko (1983), and Azariadis (1981)), even when
the tenet of rational expectations is maintained, sunspots can a¤ect the real equilibrium
variables.
Economists, for better or worse, seek meaning in the intuition behind an economic result.
In the case of sunspot e¤ects, the intuitive explanation has limited the potential scope for
sunspot phenomena and has truly been a disservice.1 Consider a typical exchange between
a sunspot expert (economist A) and one who is in the dark about sunspots (economist
B). Economist A says that beliefs can be self-ful…lling in equilibrium. Economist B states
that this is only possible if the rational expectations assumption is removed. Economist
A counters by citing sunspot e¤ects as outcomes of rational expectations models in which
beliefs can have real e¤ects. Economist B appears interested and questions how sunspots can
have such e¤ects. Economist A responds by asking B to envision an economy with multiple
certainty equilibria in which sunspots serve to coordinate the beliefs of agents on one vector
of certainty equilibrium prices.
This intuitive explanation is certainly misleading. I prove, within a class of models
described below, that sunspots can have real e¤ects independent of the number of certainty
equilibria. To study sunspot e¤ects, I must consider a dynamic model of uncertainty and
this paper, in fact, considers the most classical version of such a model: a two-period general
…nancial model with incomplete markets. The model contains extrinsic uncertainty in the
…nal period, but not intrinsic uncertainty. The …nancial element of the model is the trading
of numeraire assets, whose payouts in the …nal period are identical across realizations of
extrinsic uncertainty. Without intrinsic uncertainty, the only independent numeraire asset
is a risk-free bond whose payouts are equal in all realizations of extrinsic uncertainty and
normalized to one.
A friction is required for sunspot e¤ects to occur (Cass-Shell Immunity Theorem, 1983).
In this paper, the friction is incomplete markets. A mature literature considers a di¤erent
friction, restricted market participation, which is often caused by imposing a demographic
structure on the model. This literature begins with the two-period model of Cass and Shell
(1983), proceeds to the more general overlapping generations (OLG) model of Azariadis
and Guesnerie (1986), and culminates with the most recent papers (particularly, Dávila,
Gottardi, and Kajii, 2007) to study sunspot equilibria in OLG economies. Each of the …rst
two papers proves, for its respective model, that multiplicity of certainty equilibria is not
necessary for sunspot e¤ects.
Within the class of two-period …nancial models with incomplete markets, this paper considers numeraire assets, which have far di¤erent implications for sunspot equilibria than other
asset types. With nominal assets, Cass (1992) proves that sunspot equilibria are generically
indeterminate when there are fewer assets than states of extrinsic uncertainty. This result
only requires the reader to count. With incomplete markets and nominal assets, equilibria
1
The reasons behind the disservice can perhaps be traced to the seminal Cass and Shell (1983) paper.
Most readers have tended to focus on the canonical example in the body of the paper, in which multiplicity of
certainty equilibria appears to be necessary for sunspot e¤ects, rather than on the example in the appendix.
2
are generically indeterminate, but the set of equilibria without sunspot e¤ects (namely those
with identical real variables in all states) is generically …nite. With real assets, Gottardi
and Kajii (1999) prove that for a generic subset of economies, sunspot e¤ects occur without
a multiplicity of certainty equilibria. The asset yields, in their model, are identical across
realizations of extrinsic uncertainty, but the asset payouts, as functions of endogenous commodity prices, are not. Thus, their result states that a "potential multiplicity" of certainty
equilibria is necessary for sunspot e¤ects. With numeraire assets, the sunspot equilibria are
not generically indeterminate and the asset payouts are parameters, independent of endogenous commodity prices.
Within the narrower class of two-period …nancial models with incomplete markets and
numeraire assets, Hens (2000) proves that multiple certainty equilibria are necessary for
sunspot e¤ects, except when the asset payouts di¤er across realizations of extrinsic uncertainty. Obviously, this exceptional case violates the de…nition of extrinsic uncertainty itself
as fundamental elements of the economy, the asset payouts, are impacted. Thus, to prove
my result that multiple certainty equilibria are not necessary for sunspot e¤ects, I set the
asset payouts (parameters) equal for all realizations of extrinsic uncertainty.
To explain the apparent contradiction that Hens (2000), with identical payouts across
realizations of extrinsic uncertainty, proves that multiplicity of certainty equilibria is necessary for sunspot e¤ects, while this paper proves the opposite, consider the utility form
used by Hens (2000). Not just Hens (2000), but in fact the entire literature on sunspots,
including all of the above-cited works, assume that preferences are represented by the expected utility form with identical objective probabilities for all households. In this paper,
I employ a di¤erent utility form that generalizes expected utility. This generalized form is
free from the restriction that uncertainty is simply risk characterized by the assignment of
objective probabilities to states in the …nal period, and instead is de…ned in terms of the
broader concept of a …ltration of a state space. In Cass (2008), this generalized utility form
is introduced together with the assumptions required for extrinsic uncertainty and is labeled
UBU (utility-based utility).
I introduce the UBU form for a general …nite horizon model, before detailing the speci…cs
for my two-period model. The UBU form is the representation of choice over a menu of
consumption in the current period and prospective utilities over all vaguely uncertain states
in the following period. UBU is a special case of the recursive tree-structure of utility
functions introduced by Johnsen and Donaldson (1985). Both forms are generalizations of
expected utility that satisfy intertemporal consistency, speci…cally (i) all prior contingent
consumption plans are optimal once the current state is realized and (ii) given the current
realized state, current choices are independent of contingent choices made for states that
cannot be realized. Skiadas (1998) extends the idea of Johnsen and Donaldson (1985) so as
to include choice over both contingent consumption plans and information …ltrations. These
two papers and the UBU form are in sharp contrast to the "standard" modeling of uncertain
realizations over time as epitomized by Epstein and Zin (1989). The fact that Epstein-Zin
preferences are de…ned over lotteries across realizations, rather than directly on realizations
as in the UBU form, veri…es that this "standard" modeling is still wed to probabilities.
Regarding a two-period …nancial model of uncertainty, the utility in the UBU form is
a function of the initial period consumption and the prospective utilities for all states of
uncertainty in the …nal period. For each state, the prospective utility is the value of utility
3
only for consumption in that state, not unlike the Bernoulli utility in the expected utility
form. For my purposes, in showing that multiplicity of certainty equilibria is not necessary for
sunspot e¤ects, it is essential that the initial period consumption and the prospective utilities
in the UBU form are not additively separable. With this dependence between initial period
consumption and prospective utilities, sunspots leading to di¤erent …nal period consumptions
a¤ect the initial period consumption and ultimately the asset choice. With expected utility,
the asset choice can be …xed and then sunspots have no e¤ect on initial period consumption.
This paper is organized as follows. Section 2 introduces the UBU functional form and
speci…es the assumptions required to de…ne extrinsic uncertainty. Section 3 provides a robust
example in which sunspots have e¤ects in a general …nancial model with incomplete markets
and numeraire assets. Section 4 proves that the sunspot equilibrium is not the randomization
over certainty equilibria and additionally that a unique certainty equilibrium exists for this
example. Section 5 concludes with general observations about the example.
2
UBU
I consider a general …nancial model with two time periods and extrinsic uncertainty in the
…nal period. The extrinsic uncertainty is modeled as a …nite number of states s 2 S =
f1; :::; Sg that can be realized in the …nal period. By convention, the initial period is state
s = 0: In all states, a …nite number of households h 2 H = f1; :::; Hg trade and consume a
…nite number of physical commodities l 2 f1; ::; Lg: The model is a …nancial model, because
assets may be included to allow households to transfer wealth between states.
For this paper, the uncertainty is only extrinsic. Extrinsic uncertainty means that the
fundamentals (household endowments and utility functions and asset payouts) in the states
s 2 S are identical. A sunspot, by de…nition, is the realization of extrinsic uncertainty.
Sunspot e¤ects occur if real economic variables di¤er across realizations of extrinsic uncertainty.
Let h be any household. The consumption of a bundle of L physical commodities by h
in state s = 0 is xh (0) and in state s 2 S is xh (s): Consumption over all states is de…ned
L(S+1)
by the vector xh = xh (0); xh (s) 8s2S : The consumption set is de…ned by X h = R+
and the utility function by uh : X h ! R: The UBU form de…nes uh as a function of S + 1
prospective utility functions v0h ; vsh 8s2S :
uh (xh ) = v0h xh (0); vsh (xh (s))
where vsh : RL+ ! R 8s 2 S and v0h : RL+
s2S
8s2S
vsh RL+ ! R: By appropriate restrictions on
the function v0h ; the expected utility form is obtained.
The following assumptions are imposed on the prospective utility functions v0h ; vsh
8s2S
1. vsh is C 0 ; concave, and locally non-satiated 8s 2 S:
2. v0h is C 0 ; concave, the function v0h :
s2S
vsh RL+ ! R; de…ned by holding xh (0) 2 RL+
4
:
…xed at any level, is strictly increasing, and the function v^0h : RL+ ! R; de…ned by
vsh RL+ …xed at any level, is locally non-satiated.
holding vsh (xh (s)) 8s2S 2
s2S
From the continuity assumptions in 1 and 2, uh is C 0 : From the monotonicity assumptions in 1 and 2, uh is locally non-satiated. Further, the contingent consumption plans are
intertemporally consistent: (i) contingent plans xh (s) 8s2S made in the initial period are
optimal when any state s 2 S is realized and (ii) choices xh (s) are independent of choices
xh (s0 ) for s0 6= s:2 From the concavity assumptions in 1 and 2, uh is concave. To verify this,
take any xh ; y h 2 X h and consider the convex combination xh + (1
)y h for 2 [0; 1] :
uh
xh + (1
)y h
= v0h
xh (0) + (1
)y h (0); vsh ( xh (s) + (1
v0h
xh (0) + (1
)y h (0);
v0h xh (0); vsh (xh (s))
=
uh xh + (1
8s2S
)y h (s))
vsh (xh (s)) + (1
+ (1
8s2S
)vsh (y h (s))
8s2S
)v0h y h (0); vsh (y h (s))
)uh y h :
by 1
8s2S
Extrinsic uncertainty
By de…nition, for all possible realizations of extrinsic uncertainty, the fundamentals remain unchanged. Unchanging fundamentals means more than just holding household parameters (endowments and prospective utility functions) constant across states s 2 S: Restrictions need to be made so that the overall utility function uh satis…es variance aversion, a
property that is de…ned below and proven to hold for the UBU form in theorem 1.
Sunspots, the realizations of extrinsic uncertainty, have e¤ects when they lead to di¤erent
consumptions choices. If sunspots have e¤ects, they are said to matter.
De…nition 1 Sunspots have e¤ects i¤ 9h 2 H s.t. xh (s) 6= xh (s0 ) for some s; s0 2 S:
s + s0
if s + s0 S
:
s + s0 S if s + s0 > S
The key concept of circular symmetry is borrowed from Cass (2008) and de…ned below.
De…ne the permutation
:S
S ! S by s 7 !
(s; s0 ) =
De…nition 2 A function v0h satis…es circular symmetry i¤ 8s0 = 1; :::; S
v0h xh (0); vsh (xh (s))
8s2S
= v0h xh (0); vsh (xh ( (s; s0 )))
1:
8s2S
:
The assumptions for extrinsic uncertainty are given by:
3. vsh = v h 8s 2 S:
4. v h is strictly concave.
5. v0h satis…es circular symmetry.
2
The utility form of Johnsen and Donaldson (1985) is the most general form satisfying intertemporal
consistency. UBU is a special case of their form.
5
by 2
De…ne the average consumption bundle across states s 2 S as:
1X
xh =
xh (s):
s2S
S
De…nition 3 The utility function uh satis…es variance aversion i¤ uh (xh (0); xh ; :::; xh )
uh (xh (0); (xh (s))8s2S ); with strict inequality if xh (s) 6= xh (s0 ) for some s; s0 2 S:
Theorem 1 Under assumptions 1 thru 5, uh satis…es variance aversion.
X
Proof. De…ne xh = S1
xh (s): Then by assumption 4:
s2S
v h xh
1X
v h xh (s) ;
s2S
S
(1)
with strict inequality if xh (s) 6= xh (s0 ) for some s; s0 2 S: Since v0h is concave (assumption
2):
X
X
v h xh (s) ; :::; S1
v h xh (s)
v0h xh (0); S1
s2S
s2S
X
(2)
1
v0h xh (0); v h xh (s) ; v h xh ( (s; 1)) ; :::; v h xh ( (s; S 1)) :
S
s2S
For s = 1; no further steps are needed. For s > 1; de…ne s0 = s
for s~ 2 f1; ::; S 1g :
( (1; s0 ) ; s~) =
by de…nition of
(1 + s~; s0 ) =
1: Then s =
(1; s0 ) and
1 + s~ + s0
if 1 + s~ + s0 S
1 + s~ + s0 S if 1 + s~ + s0 > S
and
v h xh (s) ; v h xh ( (s; 1)) ; :::; v h xh ( (s; S 1)) =
v h xh ( (1; s0 )) ; v h xh ( (2; s0 )) ; :::; v h xh ( (S; s0 )) = v h (xh ( ( ; s0 )))
: (3)
8 2S
Thus, 8s 2 S; using equation (3) and the circular symmetry assumption 5:
v h xh (s) ; v h xh ( (s; 1)) ; :::; v h xh ( (s; S
v h (xh ( ( ; s0 ))) 8 2S = v h (xh ( )) 8 2S :
1))
=
(4)
Using equations (1); (2); and (4) and the assumption 2 that v0h is strictly increasing in
vsh 8s2S :
v0h xh (0); v h xh ; :::; v h xh
v0h xh (0); v h (xh (s))
8s2S
;
with strict inequality if xh (s) 6= xh (s0 ) for some s; s0 2 S: Therefore, uh (xh (0); xh ; :::; xh )
uh (xh (0); (xh (s))8s2S ); with strict inequality if xh (s) 6= xh (s0 ) for some s; s0 2 S:
Corollary 1 If the assumptions of the …rst basic welfare theorem are met (no frictions
present), then the Cass-Shell Immunity Theorem (Cass and Shell, 1983) is applicable and
xh (s) = xh (s0 ) 8s; s0 2 S and 8h 2 H: Thus, sunspots cannot have any e¤ects; they do not
matter.
Proof. The proof is immediate given the property of variance aversion, but a proof is
provided in Cass (2008).
6
3
A Sunspot Example
I de…ne the remaining variables and parameters and introduce the …nal assumptions required
for extrinsic uncertainty.
The household endowments are de…ned by eh = eh (0); eh (s) 8s2S and are assumed to
be strictly positive: eh >> 0:3 The spot commodity prices are de…ned by p = p(0); (p(s))8s2S
where p(0) 2 RL+ nf0g and p(s) 2 RL+ nf0g 8s 2 S by assumptions 1 and 2. The prices are
pl (s)
pL (s)
normalized so that p(s) =
;1 :
8l<L
With J < S numeraire assets, markets are incomplete. The payout of asset j in state
s 2 S in terms of the commodity l = L is given by rj (s): In 2
terms of the real 3
economic
r1 (1) ::: rJ (1)
:
: 5 has full
variables, it is innocuous to assume that the payout matrix R = 4 :
r1 (S) ::: rJ (S)
column rank.
The assumptions for extrinsic uncertainty include:
6. eh (s) = eh (1) 8s 2 S:
7. rj (s) = rj (1) 8s 2 S:
Assumption 7 together with R full column rank implies that only one numeraire asset
exists (J = 1) and it has a risk-free payout normalized to 1 in all states. The asset choice
by household h is denoted by z h 2 R and the price of the asset is q 2 R+ :
I de…ne a …nancial equilibrium with extrinsic uncertainty, referred to as a sunspot equilibrium.
De…nition 4 A …nancial equilibrium with extrinsic uncertainty is a vector
such that
1. given (p; q) ; 8h 2 H :
xh ; z h
8h2H
; p; q
2 arg max uh xh
subj: to
2. markets clear:
xh ; z h
X
Xh2H
Xh2H
h2H
p(0) xh (0) eh (0) + qz h 0
:
p(s) xh (s) eh (1)
z h 0 8s2S
xhl (0)
xhl (s)
zh = 0
ehl (0) = 0 8l
ehl (1) = 0 8l; 8s 2 S :
Given assumptions 1 thru 7 and eh >> 0 8h 2 H; a sunspot equilibrium always exists.
3
By convention, for x; y 2 Rn ; (i) x >> y if xi > yi 8i; (ii) x
and x 6= y:
7
y if xi
yi 8i; and (iii) x > y if x
y
The following example is one in which sunspot e¤ects occur. Let H = L = S = 2: The
utility function uh : R6+ ! R is de…ned in terms of two utility functions v0h ; v h :
uh xh = v0h xh (0); v h xh (1) ; v h xh (2)
where v h : R2+ ! R is de…ned by:
1
8
v 1 (x1 (s)) = 1 +
(x11 (s))
3
(
2
7
8
v 2 (x2 (s)) = 1 +
3
)
x21 (s)
2
2
+
2
+
7
8
3
(x12 (s))
3
(
2
2
1
8
8s 2 S:
2
)
x22 (s)
8s 2 S:
2
These prospective utility functions satisfy assumptions 1 and 4. For use later, all equilibria
(and any convex combination of equilibria using concavity) satisfy v h xh (s) > 0 8h; s:
Thus, v h : R2+ ! R++ 8h:
The initial period utility function v0h : R2+ R2++ ! R is de…ned by:
v0h
h
1
If
h
x (0); v
h
2
+
h
h
x (s)
=
8s2S
xh1 (0)
+
xh2 (0)
h
1
v
h
h
2
2
h
x (1)
v
h
h
x (2)
h
2
2
8h:
1; then v0h is concave. Set:
(
(
1
1;
2
1;
1
2)
2
2)
= (0:3; 0:08) :
= (0:3; 0:5756) :
With these values, plus given v h : R2+ ! R++ 8h; then v0h satis…es assumptions 2 and 5.4
The endowments for both households are given by:
eh1 (0) eh2 (0) eh1 (1) eh2 (1)
:
h = 1 16
32
47
2
h = 2 32
16
1
46
Recall from assumption 6 that eh (s) = eh (1) 8s 2 S:
As the prospective utility functions are di¤erentiable, concave, and strictly monotonic,
then (a) the …rst order conditions with respect to the variables xh and z h are necessary and
su¢ cient conditions for an optimal solution to the household problem given in de…nition 4
and (b) the budget constraints in the household problem hold with equality. Solving these
system of equations together with the market-clearing conditions yields the following sunspot
equilibrium:
(0)
initial pp21 (0)
= 33:55
q = 0:50
1
2
period x (0) = (16:67; 26:26) x (0) = (31:33; 21:74) z 1 = 1
p1 (1)
…nal
= 81
p2 (1)
period x1 (1) = (35; 2:5)
x2 (1) = (13; 45:5)
4
X
h
l
l
The
utility function for
h
h
Q
h
xhl (0) l
xh (s) l
s v
+S
h
l
p1 (2)
p2 (2)
1
=8
x (2) = (45:5; 13) v 1 (x1 (2)) = 0:872
x2 (2) = (2:5; 35) v 2 (x2 (2)) = 0:672
v 1 (x1 (1)) = 0:672
v 2 (x2 (1)) = 0:872
v0h
with
is
of
h
=
the
general
h
l ; ::
::;
< 1 8h; 8l:
8
z2 = 1
> 0
v0h xh (0); v h xh (s)
form:
8h;
h
=
::;
h
l ; ::
> 0
8s
=
8h; and
:
In this example, the sunspots have real e¤ects since:
xh (1) 6= xh (2) 8h:
4
Multiple Certainty Equilibria Not Necessary
A …nancial equilibrium without extrinsic uncertainty (a certainty equilibrium, for short) is
~ h = R2L
de…ned by setting S = 1: The consumption set is X
+ ; consumption over all states is
h
h
h
de…ned by a vector x~ = x~ (0); x~ (1) ; endowments are de…ned by eh = eh (0); eh (1) >> 0;
~ h ! R is de…ned in terms of two functions v~h ; v h :
and the utility function u~h : X
0
xh (1))
u~h (~
xh ) = v~0h x~h (0); v h (~
where v h : RL+ ! R satis…es assumptions 1 and 4 and v~0h : RL+ v h RL+ ! R satis…es assumptions 2 and 5. The function v h remain unchanged, but I use the notation u~h and v~0h as
the dimension of the domain for these functions is strictly less when compared to their counterparts uh and v0h : The economy with the utility form used with extrinsic uncertainty (the
previous section) and the economy with the utility form used without extrinsic uncertainty
(this section) are only comparable if the utility forms satisfy the following de…nition.
De…nition 5 The economy with and without extrinsic uncertainty are comparable i¤, for
S
h
h h
y h = x~h (1); :::; x~h (1) 2 RLS
x (1)); :::; v h (~
xh (1)) 2 v h RL+
; the utility
+ and w = v (~
h
h h h
h
h
h
h
h
h
h
h
h
forms u ; u~ ; v0 ; v~0 satisfy u~ x~ (0); x~ (1) = u x~ (0); y and v~0 x~ (0); v x~h (1) =
v0h x~h (0); wh for any x~h (0); x~h (1) :
The spot commodity prices are de…ned by p~ = (~
p(0); p~(1)) where p~(0); p~(1) 2 RL+ nf0g by
assumptions 1 and 2. As above, the prices are normalized so that p~(s) =
p~l (s)
p~L (s)
The household asset choice is denoted by z~h 2 R and the asset price by q~ 2 R+ :
De…nition 6 A …nancial equilibrium without extrinsic uncertainty is a vector
such that
1. given (~
p; q~) ; 8h 2 H :
x~h ; z~h
2 arg max u~h x~h
subj: to
2. markets clear:
X
Xh2H
Xh2H
h2H
p~(0) x~h (0) eh (0) + q~z~h
p~(1) x~h (1) eh (1)
z~h
x~hl (0)
ehl (0) = 0 8l
x~hl (1)
ehl (1) = 0 8l :
z~h = 0
9
0
:
0
;1 :
8l<L
x~h ; z~h
8h2H
; p~; q~
Returning to the speci…c example, H = L = 2, in order for the two economies to be
comparable, the function v~0h must be given by:
v~0h x~h (0); v h x~h (1)
= x~h1 (0) + x~h2 (0)
h
1
v h x~h (1)
h
2
8h;
where the parameter values ( 11 ; 12 ) = (0:3; 0:08) and ( 21 ; 22 ) = (0:3; 0:5756) remain unchanged.
I show in theorems 2 and 3 that the sunspot equilibrium found in section 3 was not
obtained as the randomization over multiple certainty equilibria. De…nition 7 and theorem
3 are valid for any number of states of extrinsic uncertainty, but for expositional simplicity,
consider S = 2:
De…nition 7 Given two certainty equilibria (~
xA (0); p~A (0); z~A ; q~A ; x~A (1); p~A (1)) and (~
xB (0); p~B (0);
z~B ; q~B ; x~B (1); p~B (1)); a sunspot equilibrium (x(0); p(0); z; q; x(1); p(1); x(2); p(2)) is a randomization of the two certainty equilibria i¤:
(x(0); p(0); z; q) =
(~
xA (0); p~A (0); z~A ; q~A )
+(1
) (~
xB (0); p~B (0); z~B ; q~B )
(x(1); p(1)) = (~
xA (1); p~A (1))
(x(2); p(2)) = (~
xB (1); p~B (1))
for some
2 [0; 1] where the designation of states s = 1; 2 is arbitrary.
For the sunspot equilibrium (x(0); p(0); z; q; x(1); p(1); x(2); p(2)) from section 3, there
are two possible randomizations: (i) (~
xA (0); p~A (0); z~A ; q~A ) = (~
xB (0); p~B (0); z~B ; q~B ) and (ii)
(~
xA (0); p~A (0); z~A ; q~A ) 6= (~
xB (0); p~B (0); z~B ; q~B ) : Suppose case (i) holds, then
(x(0); p(0); z; q) = (~
xA (0); p~A (0); z~A ; q~A ) = (~
xB (0); p~B (0); z~B ; q~B )
and sunspots have e¤ects as x~A (1) 6= x~B (1): The sunspots s = 1; 2 serve to coordinate price
beliefs on one of the certainty equilibria that arises in the …nal period.
Required for case (i) to hold is that a certainty equilibrium (drop the subscript A and
B) exists satisfying x~h2 (0) = xh2 (0) 8h; z~h = z h 8h; and q~ = q: Theorem 2 proves that this is
not possible.
Theorem 2 For the economy given in section 3, there does not exists a certainty equilibrium
in which x~h2 (0) = xh2 (0) 8h; z~h = z h 8h; and q~ = q:
Proof. Suppose, for contradiction, that x~h2 (0) = xh2 (0) 8h; z~h = z h 8h; and q~ = q: As the
utility functions are di¤erentiable, concave, and strictly monotonic, …rst order conditions are
h
employed to make the argument concise. Let ~ (0) be the Lagrange multiplier associated
h
with the initial period and ~ (1) the Lagrange multiplier associated with the …nal period.
The …rst order conditions for household h with respect to x~h2 (0) are given by:
h
1
x~h2 (0)
h
1
1
v h x~h (1)
10
h
2
~ h (0) = 0:
(5)
For the economy with extrinsic uncertainty, after de…ning the Lagrange multipliers as h (0)
and h (s) 8s2S ; respectively, the …rst order conditions for household h with respect to xh2 (0)
are given by:
h
1
xh2 (0)
h
1
1
h
2
2
v h xh (1)
h
2
2
v h xh (2)
h
(6)
(0) = 0:
Given (~
z 1 ; z~2 ) = (z 1 ; z 2 ) = ( 1; 1) by supposition, the prospective utility functions v h 8h
and endowments eh (1) 8h yield exactly three equilibrium prices and allocations in the …nal
period:5
(a)
p~1 (1)
p~2 (1)
= 18 ; x~1 (1) = (35; 2:5) ; v 1 (~
x1 (1)) = 0:672; x~2 (1) = (13; 45:5) ; v 2 (~
x2 (1)) = 0:872:
(b)
p~1 (1)
p~2 (1)
= 1; x~1 (1) = (42; 6) ; v 1 (~
x1 (1)) = 0:834; x~2 (1) = (6; 42) ; v 2 (~
x2 (1)) = 0:834:
(c)
p~1 (1)
p~2 (1)
= 8; x~1 (1) = (45:5; 13) ; v 1 (~
x1 (1)) = 0:872; x~2 (1) = (2:5; 35) ; v 2 (~
x2 (1)) = 0:672:
For candidate equilibrium (a), consider household h = 2; which satis…es:
v
2
2
x~ (1)
2
2
> v
2
2
2
2
2
x (1)
v
2
2
x (2)
2
2
2
(7)
:
For candidate equilibria (b) and (c), consider household h = 1; which satis…es:
v
1
1
x~ (1)
1
2
> v
1
1
2
2
1
x (1)
v
1
1
x (2)
1
2
2
(8)
:
Using either inequality (7) or (8) (for the above speci…ed household h) and comparing the
h
…rst order conditions (5) and (6) with x~h2 (0) = xh2 (0) 8h by supposition, then ~ (0) > h (0):
The …rst order conditions with respect to z~h and z h are given by:
h
h
q~~ (0) + ~ (1) = 0
:
q h (0) + h (1) + h (2) = 0
h
With q~ = q; then ~ (1) > h (1)+ h (2): However, as (7) or (8) hold and v h is strictly concave,
h
the …rst order conditions with respect to x~h (1); xh (1); xh (2) imply ~ (1) < h (1) + h (2):6
This contradiction …nishes the proof.
The following theorem holds in general, not just for the economy in section 3. Consider
the complementary condition to that assumed in theorem 2: (~
xA;L (0); z~A ; q~A ) 6= (~
xB;L (0); z~B ; q~B ) :
Theorem 3 proves that randomization with this speci…cation is not possible for the general
UBU form.
5
The inspiration for the functional forms v h
was Kubler and Schmedders (2010).
6
8h
and endowments eh (1)
8h
leading to this multiplicity
h
The …rst order conditions with respect to x
~h (1) imply that ~ (1) is proportional to Dx~h2 (1) v h x
~h (2) :
Speci…cally, they are given by:
h
2
x
~h2 (0)
h
1
vh x
~h (1)
h
2
1
Dv h x
~h (1)
h
~ h (1)
h
similar relation holds for the …rst order conditions with respect to x (1) and x (2):
11
p~1 (1)
p~2 (1) ; 1
= 01
2.
A
Theorem 3 For S = 2 and the general UBU utility form
X
h
xhl (0) l v h xh (1)
v0h xh (0); v h xh (1) ; v h xh (2) =
h
l
v h xh (2)
l
h
l
with hl
0 8 (h; l) ; hL > 0 8h; hl > 0 8 (h; l) ; hL > 0 8h; and hl + S hl < 1 8 (h; l) ; given
certainty equilibria (~
xA (0); p~A (0); z~A ; q~A ; x~A (1); p~A (1)) and (~
xB (0); p~B (0); z~B ; q~B ; x~B (1); p~B (1))
such that (~
xA;L (0); z~A ; q~A ) 6= (~
xB;L (0); z~B ; q~B ) ; then 8 2 [0; 1] ; @ a sunspot equilibrium
(x(0); p(0); z; q; x(1); p(1); x(2); p(2)) that is the randomization of the two certainty equilibria.
Theorem 3 states that multiplicity of certainty equilibria is not su¢ cient for sunspot
e¤ects, only multiplicity of certainty equilibria satisfying (~
xA;L (0); z~A ; q~A ) = (~
xB;L (0); z~B ; q~B ) :
Proof. Suppose, for contradiction, that 9 2 [0; 1] such that the equalities de…ning randomization hold:
(9)
(x(0); p(0); z; q) =
(~
xA (0); p~A (0); z~A ; q~A )
+(1
) (~
xB (0); p~B (0); z~B ; q~B )
(x(1); p(1)) = (~
xA (1); p~A (1))
(x(2); p(2)) = (~
xB (1); p~B (1))
The payout of the risk-free bond in the sunspot equilibrium must be the same as that in each
of the certainty equilibria. Thus, z h = z~Ah = z~Bh 8h:
For the certainty equilibria, consider the equations combining the …rst order conditions
with respect to the asset with the …rst order conditions with respect to the commodity
l = L :7
q~A
h
L
x~hA;L (0)
q~B
h
L
x~hB;L (0)
h
L
1
v h x~hA (1)
2
h
L
h
L
1
v h x~hB (1)
2
h
L
= 2
h
L
x~hA;L (0)
= 2
h
L
x~hB;L (0)
h
L
h
L
v h x~hA (1)
2
h
L
1
DxL (1) v h x~hA (1) :
v h x~hB (1)
2
h
L
1
DxL (1) v h x~hB (1) :
The …rst order conditions can be reduced to:
h
L
q~A
= 2
h
x~A;L (0)
h
L
q~B
h
x~B;L (0)
h
x~hA (1)
h DxL (1) v
L
v h x~hA (1)
h
x~hB (1)
h DxL (1) v
L
v h x~hB (1)
= 2
:
(10)
:
Consider the analogous …rst order conditions for the sunspot equilibrium:
q
h
h
L xL (0)
h
h
h
L
L xL (0)
h
h
h
L
x
(0)
L
L
h
L
v
h
1
h
L
v h xh (1)
h
x (1)
v h xh (1)
h
L
h
L
1
v
h
L
v h xh (2)
h
x (2)
v h xh (2)
7
h
L
h
h
L
1
=
DxL (1) v h xh (1) +
DxL (1) v h xh (2) :
To obtain these equilibrium conditions, I take advantage of the fact that the utility functions are differentiable, concave, and strictly monotonic. The …rst order conditions are then necessary and su¢ cient
conditions for an optimal solution to the household problem given in de…nition 6.
12
The …rst order conditions can be reduced to:
h
L
q
xhL (0)
DxL (1) v h xh (2)
DxL (1) v h xh (1)
+
v h (xh (1))
v h (xh (2))
h
L
=
!
(11)
:
Comparing equations (10) and (11); with equalities v h xh (1) = v h x~hA (1) and v h xh (2) =
x~hB (1) using (9); then:
vh
q~A
h
x~A;L (0)
q~B
h
x~B;L (0)
q
:
(12)
Beginning with (12) and utilizing the de…nitions q = q~A + (1
(1
)~
xhB;L (0); algebra yields:
q~B
q~A
= h
:
h
x~A;L (0)
x~B;L (0)
)~
qB and xhL (0) = x~hA;L (0) +
+
=2
xhL (0)
(13)
If q~A 6= q~B ; without loss of generality q~A > q~B ; then x~hA;L (0) > x~hB;L (0) 8h: This contradicts market clearing. If q~A = q~B ; then x~A;L (0) = x~hA;L (0) 8h = x~hB;L (0) 8h = x~B;L (0):
This contradicts the supposition in the statement of the theorem that (~
xA;L (0); z~A ; q~A ) 6=
(~
xB;L (0); z~B ; q~B ) :
Unique certainty equilibrium
The prior result proves that the sunspot equilibrium is not the randomization over multiple certainty equilibria. In fact, for the speci…c economy given in section 3, there is only
one certainty equilibrium:
(0)
= 32:50
q~ = 0:30
initial pp~~21 (0)
1
2
period x~ (0) = (16:47; 25:09) x~ (0) = (31:53; 22:91) z~1 = 0:859
p~1 (1)
…nal
= 0:271
p~2 (1)
1
period x~ (1) = (38:19; 3:53)
x~2 (1) = (9:81; 44:47)
z~2 = 0:859
:
v 1 (~
x1 (1)) = 0:765
v 2 (~
x2 (1)) = 0:863
It is straightforward to verify that the above variables satisfy the conditions characterizing
certainty equilibria.8 In fact, the equilibrium conditions can be reduced to a system of
two equations and this system of two equations only has one solution. The system of two
equilibrium equations is constructed in two steps.
I. Final period …rst order conditions, budget constraints, and market clearing
Taking
8
p~1 (1)
p~2 (1)
and z~1 as given, the …rst order conditions with respect to x~h (1) 8h yield:
1
8
3
7
8
3
x~11 (1)
3
x~21 (1)
3
3
=
7
8
1
8
3
=
See previous footnote.
13
x~12 (1)
3
p~1 (1)
p~2 (1)
x~22 (1)
3
p~1 (1)
p~2 (1)
:
Using the budget constraints pp~~12 (1)
x~h1 (1) + x~12 (1) =
(1)
demand functions for commodity l = 2 are given by:
p~1 (1)
p~2 (1)
47
x~12 (1)
=
x~22 (1)
p~1 (1)
p~2 (1)
=
2=3
+1
z~1
+ 46
:
2=3
p~1 (1)
p~2 (1)
1
7
eh1 (1) + eh2 (1) + z~h ; the
+ 2 + z~1
p~1 (1)
p~2 (1)
7
p~1 (1)
p~2 (1)
+1
1=3
De…ning = pp~~12 (1)
; the market clearing condition x~12 (1) + x~22 (1) = 48 reduces to a
(1)
polynomial of degree 3 :
96
7
3
48
2
144
7
+ 48
48 1
z~ = 0:
7
(14)
Equation (14) is the …rst of two equations characterizing certainty equilibria.
II. First order conditions with respect to asset choice
Given the solution to (14); the equilibrium prospective utilities v 1 (~
x1 (1)) and v 2 (~
x2 (1))
can be calculated.
With the quasilinear utility form for consumption in the initial period (linear in x~h1 (0)),
(0)
the demand functions for commodity l = 2 are independent of prices pp~~21 (0)
:
1
x~12 (0)
= 48 "
x~22 (0) = 48
1
2
1
(v (~
x (1)))
1
2
(v 1 (~
x1 (1))) 1
1
1
1
1
1
2
2
+ (v 2 (~
x2 (1))) 1
2
1
#
x~12 (0):
1
Also, the quasilinear utility with equal weight on x~11 (0) and x~21 (0) guarantees ~ (0) =
~ 2 (0): Thus, …rst order conditions with respect to z~h dictate ~ 1 (1) = ~ 2 (1): Using the
1
2
…rst order conditions with respect to x~h2 (1); the equality ~ (1) = ~ (1) is equivalently
written as:
1
2
x~11 (0)
1
1
v 1 x~1 (1)
1
2
1
7
8
3
x~12 (1)
3
=
2
2
x~21 (0)
2
1
v 2 x~2 (1)
2
2
1
1
8
3
x~22 (1)
(15)
Equations (14) and (15) are in terms of only two variables: and z~1 : Once these variables are found, simple linear equations are used to compute all remaining equilibrium variables: pp~~12 (0)
; q~; x~1 (0); x~2 (0); x~1 (1); x~2 (1) : Thus, for multiple certainty equilibria to exist,
(0)
there must exist multiple solutions to the system of two equations (14) and (15):
14
3
:
In particular, as (15) is a linear equation, for multiple certainty equilibria to exist, given
the equilibrium variable z~1 ; there must exist multiple values for that satisfy (14): Consider
the cubic polynomial (14) as a function f ( ; z~1 ) = 0: The function has two local extrema,
a local maximum f ( 1 ) and a local minimum f ( 2 ) with 1 < 2 : Both 1 and 2 are
independent of z~1 ; so de…ne z~i1 such that f ( i ; z~i1 ) = 0 for both i: The function f has
three solutions i¤ f ( 1 ; z~11 )
0 and f ( 2 ; z~21 )
0: Thus, the highest possible such that
multiplicity occurs in (14) is = such that f ( ; z~11 ) = 0 and f ( 1 ; z~11 ) = 0: Likewise,
the lowest possible such that multiplicity occurs in (14) is = such that f ; z~21 = 0
=
and f ( 2 ; z~21 ) = 0: The calculated values provide me with a search window 2 ;
[0:285; 2:049] :
In the range 2 ; = [0:285; 2:049] ; use (14) to compute z~1 and de…ne the function
g ( ) = LHS(15) RHS(15): An equilibrium value for is one such that g ( ) = 0: Figure
1 graphs this function over the range 2 ; = [0:285; 2:049] :
100
75
g(rho)*1000
50
25
0
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
2
2.1
2.2
-25
-50
rho
Figure 1: Unique solution to (14) and (15)
The function has a unique solution at the value = 0:647: This veri…es that the only
solution to the system (14) and (15) yields the certainty equilibrium previously given.
5
Observations
The example given in the previous two sections, which proves that multiplicity of certainty
equilibria is not a necessary condition for sunspot e¤ects, is robust to changes in the economy
(utility functions and endowments). That is, for an open set of economies around the given
one, the same qualitative result holds. Given this result, I can make the following general
observation about sunspot e¤ects in …nancial models with incomplete markets and numeraire
assets. Provided that both of the following properties are satis…ed for an economy, then
sunspots can have real e¤ects independent of the number of underlying certainty equilibria:
15
A. There exists a state of intrinsic uncertainty (in this case the …nal period) such that,
when viewed as a subgame with a …xed initial wealth distribution, the prospective
utility functions v h 8h and endowments eh (1) 8h are such that multiple subgame
Walrasian equilibria exist.
B. The UBU utility form is such that the function v0h is not additively separable in
the vector of initial consumption xh (0) and the vector of future prospective utility
v h xh (s) 8s2S :
References
[1] Azariadis, Costas. 1981. "Self-ful…lling Prophecies." J. Econ. Theory 25 (December):
380-396.
[2] Azariadis, Costas and Roger Guesnerie. 1986. "Sunspots and Cycles." Rev. Econ. Studies 53 (October): 725-737.
[3] Balasko, Yves. 1983. "Extrinsic Uncertainty Revealed." J. Econ. Theory 31 (December):
203-210.
[4] Cass, David. 1992. "Sunspots and Incomplete Financial Markets: The General Case."
Econ Theory 2 (September): 341-358.
[5] Cass, David. 2008. "Utility-Based Utility: The Johnsen-Donaldson Hypothesis." Unpublished mimeo, University of Pennsylvania (February).
[6] Cass, David and Karl Shell. 1983. "Do Sunspots Matter?" J. Pol. Economy 91 (January): 193-227.
[7] Dávila, Julio, Piero Gottardi, and Atsushi Kajii. 2007. "Local Sunspot Equilibria Reconsidered." Econ Theory 31 (June): 401-425.
[8] Epstein, Larry G. and Stanley E. Zin. 1989. "Substitution, Risk Aversion, and the
Temporal Behavior of Consumption and Asset Returns." Econometrica 57 (July): 937969.
[9] Gottardi, Piero and Atsushi Kajii. 1999. "The Structure of Sunspot Equilibria: The
Role of Multiplicity." Rev. Econ. Studies 66 (July): 713-732.
[10] Hens, Thorsten. 2000. "Do Sunspots Matter when Spot Market Equilibria are Unique?"
Econometrica 68 (March): 435-441.
[11] Johnsen, Thore H. and John B. Donaldson. 1985. "The Structure of Intertemporal Preferences under Uncertainty and Time Consistent Plans." Econometrica 53 (November):
1451-1458.
[12] Kubler, Felix and Karl Schmedders. 2010. "Tackling Multiplicity of Equilibria with
Gröbner Bases." Operations Research 58 (July-August): 1037-1050.
16
[13] Skiadas, Costis. 1998. "Recursive Utility and Preferences for Information." Econ Theory
12 (August): 293-312.
17

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