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By Avinash K. Dixit

Development on a base of easy monetary concept and trouble-free linear algebra and calculus, this large therapy of static and dynamic optimization tools discusses the significance of shadow costs and features outlined by way of options of optimization difficulties.

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A) The generalization to several constraints is straightforward. The set di of points for which G’(x) S ci will be convex if G’ is quasi-convex. _. FIG. ,.. ~-. -. ,,. -- 44 Inequality Constraints 45 Optimization in Economic Theory a strict inequality depends on the criterion function. If the optimum is on one of the three faces, then two constraints will not be binding. In the case of linear programming, where F and G are linear functions, it is possible to make more precise statements about the number of binding constraints.

The economic reason for this will soon become clear. The results generalize very easily. In a space of any dimension, given two convex sets which have only boundary points in common, we can find a hyperplane such that the sets lie one on each side of it, or in other words, the hyperplane separafes the sets. A hyperplane has a linear equation, 0x =d, where 0 is a non-zero row vector. x < d, and for the other, 0x > d. This is quite obvious from geometric intuition, and I shall leave it to the reada to convince himself by drawing a few pictures, and omit the proof.

Is the true global maximizing choice, without concavity we cannot be sure that it maximizes the Lagrangean; it may merely yield a stationary point of it. The problem is similar to that of determining the optimum output when there are economies of scale. The first-order condition of equality between price and marginal cost is still necessary, but profit need not be maximized even locally. The interpretation of Lagrange’s method as converting constrained ‘value’ maximization to unconstrained ‘profit’ nzximization must be confined to the case of concave programming.

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