Appendix A: Solving Technique for the Normative Problem
The optimization problem (3)-(6) is solved backward via stochastic dynamic
programming. The Bellman equation for this problem depends on three state
variables: time t, cash on hand Wt, and the expected labor income path, represented
by Lt. The Bellman equation (with V denoting the value function) is given by t = 0, 1
, ∙∙∙, T- x- 1,
Vt(Wt,Lt)=mαta,Cxt{Ut(Ct)+ptδEt(Vt+1(Wt+1,Lt+1))}, (A1)
subject to constraints (4) through (6). In the last period, the remaining wealth is
consumed, and the value function is simply given by
UT-x(WT-x). In general, for CRRA utility, the Lt-state can be reduced by dividing Wt
through Lt (see Carroll, 2004). But, since our econometric results show that, in
reality, consumers do not behave exactly according to CRRA, and thus in order to
integrate empirical asset allocations into the model (which depends on both state
variables, see section 5.3), the Lt state should not be dropped. Nevertheless, problem
(A1) is solved by referring explicitly only to the Wt state. The Lt state is considered
implicitly, because equation (A1) is calculated for each individual separately, thus
referring to each individual’s expected labor income path.
The Bellman equation (A1) cannot be solved analytically, and hence a numerical
technique is used. First, at each point in time t, the Wt-state space is discretized into
I ∈ N points Wt, with i = 1, 2, ..., I. The upper and lower bounds of this Wt1 grid
were chosen to be nonbinding in all periods. The distributions of the risky return Rt
and the labor income Lt were discretized using Gaussian quadrature methods. Since
in the last period (i.e., at t = T - x), the value function VT - x(WT - x) is given by UT-
x(WT-x), the numerical solution algorithm starts at the penultimate period (i.e., at
t = T - x - 1). For each Wti, equation (A1) is solved with the MATHEMATICA® 6.0
implemented nonlinear optimizer NMaximize, yielding the optimal decisions
αti(Wti), Cti (Wti), and the function value of Vt(Wti). Next, a continuous function is
fitted to the points Vt(Wti), which delivers a continuous approximation of the value