AppendixProof of Proposition 1. Proposition 1 is a special case of Proposition 2
with Yt=0 and Ct=Dtfor all t.
Proof of Proposition 2. We first show that if the risk-free interest rate is
given by (33) and stock returns are determined by (18), (19), (32), and (34),
then the strategy of consuming Ct=Ct=Dt+Ytand holding the market
supply of financial securities indeed satisfies the Euler equations of optimal-
ity (27) and (28). We then show that these Euler equations are necessary
and sufficient conditions for optimality.
Given (29), the interest rate Rfin (33) satisfies the Euler equation (27).
For Euler equation (28), note that zt+ 1 is determined only by ztand t+ 1
through (18) and (32), so we have
Applying this, we find that the strategy of consuming and holding the
market securities indeed satisfies Euler equation (28).
Euler equations are necessary conditions for optimality. To prove that
they are sufficient conditions as well, we apply a method used by Duffie
and Skiadas [1994] and Constantinides and Duffie [1996].
To simplify notation, let and Assume
that the strategy (C,S) satisfies the Euler equations,
(47)(48)Consider any alternative strategy (C+δC, S+δS) that satisfies the bud-
get constraint
δδδ δWWCRSRRtttfttf++^11 =− +()().− (49)uCtt′=( *)E[tR u Ct t++ +11 1′(t* )]+bttE [υˆ(,R zt+ 1 t)].uCtt′=( *)E[(Rftu Ct t′++^11 *)],uCtt()=−ργtC^1 t−γ/( ) 1 bbt=ρt+^1 t.CtEE
()
()EE[ ](()
ttt
ttt
tgggg
tt tR
C
Cfz
fzeeeeefDDt CCtDC Ct Dt++−
+ +−+−−
+
= +
=+++++1111111111γ
σγσηγγσση zz
fze
fz
fzet
t
gg t
tDC c D Ct+−+ − + −
= +
+1(^22122111)
)
()
E
()
()
γγσ ω()/t (σγωσ ).
266 BARBERIS, HUANG, SANTOS