18
relative risk aversion (CRRA) utility function c
1-(
/(1-(), where ( > 0 measures the
individual’s degree of risk aversion. In contrast to quadratic utility, which exhibits risk
aversion but not a precautionary savings motive, the CRRA function is consistent with
precautionary savings in the face of future uncertainty (technically, it satisfies the necessary
condition of a positive third derivative; see Huggett 2002). Therefore, I seek values for the
quadratic parameter that minimizes with respect to c
*
the function
Observe that the square root of L defines the
2
norm for metric spaces, which generalizes to
functions the idea of Euclidean distance (see, e.g., Adams and Guillemin 1986, 124-27).
Obviously, the parameter value minimizing L also minimizes distance in the
2
norm.
By Leibniz’s Rule, the necessary condition for minimizing (17) with respect to c
*
is
The second-order condition for a minimum is that
Since 0 # c # c
*
, each of the terms is positive or weakly positive for all c and therefore the
second-order condition is satisfied.
Equation (18) becomes
Convention