Investors choose consumption Ctand an allocation Stto the risky asset to
maximize
(6)subject to the standard budget constraint.^16 Using the Euler equation of
optimality,
(7)it is straightforward to derive expressions for stock returns and prices. The
details are in the Appendix.
We can now examine the model’s quantitative predictions for the param-
eter values in table 1.2. The endowment process parameters are taken from
U.S. data spanning the twentieth century, and are standard in the literature.
It is also standard to start out by considering lowvalues of γ. The reason is
that when one computes, for various values of γ, how much wealth an indi-
vidual would be prepared to give up to avoid a large-scale timeless wealth
gamble, low values of γmatch best with introspection as to what the an-
swers should be (Mankiw and Zeldes 1991). We take γ=1, which corre-
sponds to log utility.
In an economy with these parameter values, the average log return on the
stock market would be just 0.1 percent higher than the risk-free rate, not
the 3.9 percent observed historically. The standard deviation of log stock
returns would be only 12 percent, not 18 percent, and the price/dividend
ratio would be constant (implying, of course, that the dividend/price ratio
has no forecast power for future returns).
It is useful to recall the intuition for these results. In an economy with
power utility preferences, the equity premium is determined by risk aver-
sion γand by risk, measured as the covariance of stock returns and con-
sumption growth. Since consumption growth is very smooth in the data,
this covariance is very low, thus predicting a very low equity premium.
1 =^11
+−
ρ +γ
EC
C
t t R
tt ,E
tCt
t010 1ρ
γ−γ=∞∑ −
,24 BARBERIS AND THALER
(^16) For γ=1, we replace Ct 1 −γ/1−γwith log (Ct).
Table 1.2
Parameter Values for a Simple Consumption-based Model
Parameter gC σC gD σD ωγ ρ
Value 1.84% 3.79% 1.5% 12.0% 0.15 1.0 0.98