Avinash K. Dixit's Optimization in Economic Theory PDF

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Sample text

In case (a) there, gis not convex and X: does not minimize Bx over it. In case (b), &is not convex and X does not maximize 0x over it. The latter is the more co-“” case, arising from increasing returns in production. In such a case, considerations of marginal benefits and costs have to be supplemented by an examination of the discrete choice of whether to produce at all. This leads us to look at consumers’ surplus or related concepts. We must next find conditions on the underlying functions F and G which correspond to convexity of the sets ganda’.

B is not unique. None of these lines can be called a (b) L FIG. 4 mnmon tangent in the usual sense, but that is not essential for the economics of the problem. 9x = d. Thus separation is a generalization of the notion of a common tangent, and that is how we dispense with the requirements of differentiability ofFand G. In case (b), the two sets have a flat portion in common. This need not worry us unduly, for all candidates for the optimum choice along this common segment must have the same value of F(x), and that, after all, is the magnitude that interests us.

Even when such discontinuities exist, a very natural generalization of the concept of diminishing returns holds. The leftward partial is never less than the rightward, which is like saying that the marginal product of the kth dose of a resource cannot exceed that of the (k l)‘h. T h i s is a simple consequence of the concavity of V, which is really the economically important property. The asterisks having served their purpose of distinguishing a particular point in the (c, v) space for separation, let us drop them, and consider a point (c, V(r)) with its associated multipliers r, and compare it with a neighbouring point (c + hei, V(c + hei)), where h is a number and ei a vector with its ilh component equal to 1 and allathers zero.

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