series: EI 2005-02
On the universal method to solve extremal problems
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Some applications of the theory of extremal problems to mathematics and economics are made more accessible to non-experts. 1.The following fundamental results are known to all users of mathematical techniques, such as economist, econometricians, engineers and ecologists: the fundamental theorem of algebra, the Lagrange multiplier rule, the implicit function theorem, separation theorems for convex sets, orthogonal diagonalization of symmetric matrices. However, full explanations, including rigorous proofs, are only given in relatively advanced courses for mathematicians. Here, we offer short ans easy proofs. We show that akk these results can be reduced to the task os solving a suitable extremal problem. Then we solve each of the resulting problems by a universal strategy. 2. The following three practical results, each earning their discoverers the Nobel prize for Economics, are known to all economists and aonometricians: Nash bargaining, the formula of Black and Scholes for the price of options and the models of Prescott and Kydland on the value of commitment. However, the great value of such applications of the theory of extremal problems deserves to be more generally appreciated. The great impact of these results on real life examples is explained. This, rather than mathematical depth, is the correct criterion for assessing their value.
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