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Extra info for Maxima and Minima: Theory and Economic Applications

Sample text

Thus we have found at least a necessary criterion, which our point x must satisfy: it must besuchthat there does not exist another point x' having the property stated above. A point x satisfying this requirement is said to be a Paretooptimal point (after the Italian economist and mathematician Vilfredo Pareto). In exact terms, we can state this definition in the following form. 4). A point x=(x 1 , x 2 , ••• , xn) is said to be a Paretooptimal point when- within the Iimits ofthe domain of variation considered- there exists no point x' = (x~, x;, ...

1') (p. 120), up to and excluding "The matrix of the second-order derivatives ... 1) (T marking a summit and not a depression of the "terrain"). If we wish to find the maximum of the function without imposing any constraint on the variables x 1 , x 2 , we reach the point Tin the way indicated in Chapter 5. We now suppose that the variables are subjected to a constraint in the form of an equation. 2). The imposed constraint is now that we cannot move freely in the (x 1 , x 2 ) plane, but only along this curve.

That is the clearest way to carry out the analysis of extremal problems, and to train the reader how to handle the mathematical tools. Should the occasion arise that one wishes to take the endpoints into consideration, one could, for example, simply calculate the values of the function at the end-points, and compare them directly with the values it takes elsewhere in the interval. As for concrete problems, it is in general very important to take the end-points - which may have a specific significance - into account, and it is advisable to carry out the above mentioned comparison of values.

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