An optimum seeking method and examples of its use in economics



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Mathematical optimization methods have enjoyed a bonanza since the advent of high speed computation equipment. Continually new methods are being developed that are more efficient or faster than older methods. Because some of the problems are quite difficult, algorithms are usually designed to solve specific problems. An interpolation method is devised that has several excellent features. This method is based on the use of special interpolating quadratic functions that adapt themselves to the function to be optimized. The method can be used to optimized unrestricted objective functions. This method can be utilized even if the analytical form of the function is not known. Because of the way the method iterates, the procedure always generates improved values until a local or global optimum is found. The method was utilized to solve several problems. The first one was the optimization of a production function and average product function is one independent variable. Another problem solved was the finding of the optimum replacement life of capital equipment. To better evaluate the interpolation, the same problems were solved using the Fibonacci method. This method is recognized as very efficient in the calculation of the optimum of unimodal function. Multivariable functions were also utilized. Two problems were solved. One was to find the optimum of a three variable production function devised by Ragnar Frisch. Another example was the finding of a triplet of weighting parameters used in a forecasting method derived by Peter R. Winters. These last two problems were also solved by the Fletcher-Powell Deflected Gradient Method for comparison purposes. Using as a basis the cases solved and the methods used the interpolation method shows to be as good as any of the methods available in the literature and in some cases better. The values obtained and the number of iterations required to obtain these values were always as reduced as the best methods available and most of the times better. All the methods were programmed for computer use. The programming languages used were Fortran II and IV for the IBM 1620 and SDS Sigma 7 computers. To improve the computational efficiency of the method several improvements are spelled out for future work. It is hoped that these methods can be used more in the area of normative economics, at the microeconomic or macroeconomic level.