00Thaler_FM i-xxvi.qxd

Appendix A

We show that for the economy laid out in Equations (3–6), there is an equi-
librium in which the risk-free rate is constant and given by

(18)

and in which the price/dividend ratio is a constant f, and satisfies

(19)

In this equilibrium, returns are therefore given by

(20)

To see this, start from the Euler equations of optimality, obtained
through the usual perturbation arguments,

(21)

(22)

Computing the expectation in Eq. (21) gives Eq. (18). We conjecture that
in this economy, there is an equilibrium in which the price/dividend ratio
is a constant f, so that returns are given by Eq. (20). Substituting this into
Eq. (22) and computing the expectation gives Eq. (19), as required. For given
parameter values, the quantitative implications for P/Dratios and returns
are now easily computed.

1 = 1 1

































+

+

−

ρ

γ

Ett t

t

R

C

C

.

1 =^1

































+

−

ρ

γ

R

C

ft C

t

t

E ,

R

DP

P

PD

PD

D

D

f

f

t tt g

t

tt

tt

t

t

+ DDt

++ ++ +

= +

+

=

+

⋅=

+

1 11 11 1 + 1

11 /

/

exp[ σε].

1

1

= 21 2222

+

ργσγσγσσω−+ + −

f

f

exp[ggDC D C CD( )].

RgfCC=−

(^11)
2
22
ρ
exp[γγσ],
66 BARBERIS AND THALER