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Discrete-Time Finite Horizon Approximation of Infinite Horizon Optimization Problems with Steady-State Invariance
Jean Mercenier
Philippe Michel
Abstract
The computation of large-scale nonlinear intertemporal optimization problems requires time-aggregation assumptions. This consists in choosing a specific gridding of the time horizon, and considering the optimal solution computed using this restricted set of dates as an approximation to the underlying continuous- or more disaggregated discrete-time formulation. A procedure generally adopted for time aggregation in applied intertemporal optimization economic models using mathematical programming techniques is shown to be inappropriate, as it introduces a dependency of the solution steady state to a specific choice of sequence of time intervals. We establish necessary and sufficient conditions to avoid this dependency, i.e., to ensure that the discretization satisfies steady-state invariance. The result is a considerable improvement in the numerical accuracy of the time-aggregated approximation. These easy-to-handle conditions are shown to apply to a broad class of models such as multidimensional intertemporal problems and endogenous growth models and may therefore prove extremely powerful in applied works such as large-scale applied general-equilibrium modeling. This conclusion is further highlighted by a comparison between this approach and a spectral-projection method using optimal orthogonal collocation as recently introduced by Judd (1992).
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