Proposition 1.For the preferences given in (13)–(18), there exists an
equilibrium in which the gross risk-free interest rate is constant at(23)and the stock’s price-dividend ratio f(⋅), as a function of the state
variable zt, satisfies for all zt:(24)where for zt≤1,(25)and for zt>1,(26)We prove this formally in the Appendix. At a less formal level, our results
follow directly from the agent’s Euler equations for optimality at equilib-
rium, derived using standard perturbation arguments:
(27)(28)Readers may find it helpful to compare these equations with those de-
rived from standard asset pricing models with power utility over consump-
tion. The Euler equation for the risk-free rate is the usual one: consuming a
little less today and investing the savings in the risk-free rate does not
change the investor’s exposure to losses on the risky asset. The first term in
the Euler equation for the risky asset is also the familiar one first obtained
by Mehra and Prescott (1985). However, there is now an additional term.
Consuming less today and investing the proceeds in the risky asset exposes
the investor to the risk of greater losses. Just how dangerous this is, is de-
termined by the state variable zt.
In constructing the equilibrium in proposition 1, we follow the assump-
tion laid out in subsection 2.D, namely that buying or selling on the part of
1 =+++^1101−
ρΕ ρΕ +
γ
tt[(/)]RCCt t b vR zt[ˆ(,t t)].1 =ρΕRCCft t[( + 1 / ) ],t−γˆ(,)
()( )for
.,
,,
,vR zRR
zR RRR
tt RRtft
tt fttft
tft++
++
+=−
−≥
<
11
11
λ 1ˆ(,)
()( )for
,,
,, ,,
,vR zRR
zR R R zRRzR
tt RzRtft
tft ft t tftttft
ttft++
++
+=−
−+ −≥
<
11
11
λ 11
111 10
111= +
+ +
+ −++ +++ρρE ()γ
()E ˆ()
(),
,()( )
tt
tgt
t
tg
tfz
fzebvfz
fzezCCtCCtσσRef
gC C
= −
−
ρ
1 γγ^22 σ/^2
,242 BARBERIS, HUANG, SANTOS