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INVESTING RETIREMENT WEALTH: A LIFE-CYCLE MODEL – The household’s optimization problem

Next-period liquid and retirement wealth are then given by:
Formula 10-11
where fu is a binary variable which equals zero until the investor pays the fixed cost of entering the stock market and equals one thereafter, and — it+г is the return on the portfolio held from period t to period t + 1:
Formula 12
Here a^t is freely chosen when fu = 1, and equals zero when fu = 0.
After retirement (t > К), the problem takes the same form except that retirement wealth no longer accumulates. Instead, it is annuitized and provides riskless income A(W^). After-tax labor income (1 — 9)Yit in (9) and (10) is replaced by A(W^).
The problem the investor faces is to maximize (1) subject to the working-life and retirement versions of (2) through (12), plus the constraints that consumption must be non-negative at all dates. The control variables of the problem are {С, &u, fit}f= i. The state variables are jt, Xu,Wtf,vu,fi,t-i}. The problem is to solve for the policy rules as a function of the state variables, i.e., Си(Xu,Wtf,vu,fi,t-1), au (Xn, Wit, V, fi,t-1) and fit (Xit, Wi, vit, 0.
This problem cannot be solved analytically. We derive the policy functions numerically by discretizing the state-space and the variables over which the choices are made, and by using Gaussian quadrature to approximate the distributions of the innovations to the labor income process and risky asset returns (Tauchen and Hussey, 1991). The problem is then solved by using backward induction. In period T the investor consumes all his wealth and the value function coincides with the instantaneous utility. In every period t prior to T, and for each admissible combination of the state variables, we compute the value associated with each level of consumption, decision to pay the fixed cost of entering the stock market, and share of liquid wealth invested in stocks. This value is equal to current utility plus the expected discounted continuation value. To compute this continuation value for points which do not lie on the grid we use cubic spline interpolation. The combinations of the choice variables ruled out by the constraints of the problem are given a very large (negative) utility such that they are never optimal. We optimize over the different choices using grid search.
When the fixed cost of equity market participation F is equal to zero, we simplify the solution by exploiting the scale-independence of the maximization problem and rewriting all variables as ratios to the permanent component of labor income.