Numerical Solution of Nonlinear Planning Models

Abstract

The use of numerical techniques for solving dynamic nonlinear multisectoral models for development planning is analyzed. A four sector model with nonlinear welfare, production, and investment functions is developed and solved using conjugate gradient and neighboring extremal methods. The model draws on recent developments in nonlinear theoretical growth models and linear development planning models. Sensitivity tests on the turnpike properties of the model and on changes in the elasticity of substitution parameters in the production function are discussed. The numerical techniques used have their origins in control theory applications and exploit the dynamic structure of the planning model. THIS PAPER discusses the application of numerical methods to the solution of nonlinear planning models of a type that could be used in the formulation of development programs for less developed countries. The numerical methods were originally developed by control theorists and our chief interest has been in testing their capability to solve moderately sized economic problems. In general, we have been satisfied with the results of these tests, and wish to demonstrate the usefulness of the nonlinear specification with a realistically formulated four sector model. This model is based on ideas drawn from two recent lines of thought about economic growth over time. The first is that of neoclassical theoretical models designed to analyze the characteristics of an economy in asymptotic optimal growth, viz. Samuelson and Solow [24] and Koopmans [19]. The other line is that of finite horizon linear programming planning models, viz. Bruno [4], Eckaus and Parikh [10], and Chakravarty and Lefeber [7]. We have attempted to blend the nonlinear production and welfare functions of the neoclassical models with the disaggregation and emphasis on foreign trade of the linear programs. In future applications this combination should offer growth theorists the possibility of using greater disaggregation than their present closed-form solution methods permit and at the same time give economic planners the opportunity to specify their models with nonlinear functions in both the performance index and the constraints.2