Working on Fisher's Problem With Computer Algebra
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Application of optimal control theory to applied resource problems has been limited by the difficulty of numerical solutions. Typically, terminal values for the length of the production period, price, or production have been assumed rather than optimized. The use of an objective functional with explicit discounting gives direct solution values for n, y(t), p(t), and monopoly profit (or consumer surplus) for continuous or discrete problems. The method can be used for numerical solutions to problems with linear demand, cost trend, or risk of expropriation. Computer mathematics is a useful tool in exploring solution values for specific parameters. The techniques are illustrated with Fisher's wildly used discrete problem, and with application to parameters representing remaining world oil resources for competitive and monopolistic assumptions.