sluggishly, allowing past gains and losses to linger and affect the investor
for a long time; in other words, the investor has a long memory.^11
E. The Scaling Term btWe scale the prospect theory term in the utility function to ensure that
quantities like the price/dividend ratio and risky asset risk premium remain
stationary even as aggregate wealth increases over time. Without a scaling
factor, this will not be the case because the second term of the objective
function will come to dominate the first as aggregate wealth grows. One
reasonable specification of the scaling term is
(11)where is the aggregateper capita consumption at time t, and hence ex-
ogenous to the investor. By using an exogenous variable, we ensure that bt
simply acts as a neutral scaling factor, without affecting the economic intu-
ition of the previous paragraphs.
The parameter b 0 is a nonnegative constant that allows us to control the
overall importance of utility from gains and losses in financial wealth rela-
tive to utility from consumption. Setting b 0 =0 reduces our framework to
the much studied consumption-based model with power utility.
3 .Evidence from PsychologyThe design of our model is influenced by some long-standing ideas from
psychology. The idea that people care about changes in financial wealth
and that they are loss averse over these changes is a central feature of the
prospect theory of Kahneman and Tversky (1979). Prospect theory is a de-
scriptive model of decision making under risk that was originally developed
to help explain the numerous violations of the expected utility paradigm
documented over the years.
While our model is influenced by the work of Kahneman and Tversky
(1979), we do not attempt an exhaustive implementation of all aspects of
prospect theory. Figure 7.2 shows Kahneman and Tversky’s utility function
over gains and losses:^12
wX (12)X
XX
X()
.( )for
..
= −−.≥
<
088225 0880
0CtbbCtt=−
0γ,236 BARBERIS, HUANG, SANTOS
(^11) A simple mathematical argument can be used to show that the “half-life” of the investor’s
memory is equal to −0.693/log η. In other words, after this amount of time, the investor has lost
half of his memory. When η=0.9, this quantity is 6.6 years, and when η=0.8, it equals 3.1 years.
(^12) This functional form is proposed in Tversky and Kahneman (1992) and is based on ex-
perimental findings.