Optimal Grid Selection for the Numerical Solution of Dynamic Stochastic Optimization Problems

This paper unites numerical literature concerning the optimal linear approximation of convex functions with theory on the consumption savings problems of households in macro economies with idiosyncratic risk and incomplete markets. Construction of a grid for the linear approximation of household savings behavior which is optimal in the sense of minimizing the largest absolute error is characterized in a standard environment with income fluctuations and a single savings asset. For wealthy households, the grid is characterized asymptotically as having a density which decreases in household wealth. For domains which include resource poor households, the optimal grid is seen to have non-monotonic grid point density for standard parameters. This feature contradicts conventional rules for constructing grids, and is related to non-monotonic curvature in the savings function for low resource holdings. Approximate optimal grids are seen to outperform standard grid constructs according to a variety of accuracy measures at the cost of significantly increased computational time, and efficiency-improving alternatives are given.