A Note on the Time Elimination Method for Solving Recursive Growth Models.

Abstract

The Time-Elimination Method for solving recursive dynamic economic models is described. By defining control-like and state-like variables, one can transform the equations of motion describing the economy's evolution through time into a system of differential equations that are independent of time. Unlike the transversality conditions, the boundary conditions for the system are not asymptotic boundary conditions. In theory, this reformulation of the problem greatly facilitates numerical analysis. In practice, problems which were impossible to solve with a popular algorithm - shooting - can be solved in short order.

The reader of this paper need not have any knowledge of numerical mathematics or dynamic programming or be able to draw high dimensional phase diagrams. Only a familiarity with the first order conditions of the "Hamiltonian" method for solving dynamic optimization problems is required.

The most natural application of Time-Elimination is to growth models. The method is applied here to three growth models: the Ramsey/Cass/Koopmans one sector model, Jones and Manuelli's (1990) variant of the Ramsey model, and a two sector growth model in the spirit of Lucas (1988). A very simple - but complete - computer program for numerically solving the Ramsey model is provided.

A MATLAB program is included as an Appendix of this paper. Click to download an ascii version of the program.

RAMSEY.M finds the policy function for the Ramsey Model.

Unfortunately, I cannot send you an electronic copy (the paper was written with a barbarian technology!)

Hard copies of this paper were circulated by the NBER and either available from them or in most university libraries around the world:

NBER Technical Working Paper No. 116, November 1991.