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1 Scale Economies The description of scale economies begins with some expositIOns on the homogeneity of correspondences underlying the criterion of the homogeneity of functions. The exact definitions of the used terms of convex analysis are introduced 2 Theory of the Firm 37 in Appendix B. 2 contain explanations of the concepts of a function as single-valued mapping and of a correspondence as multivalued mapping. "* 0 we get the equivalence relation x E >"C x/>" <=} E C. The scalar multiplication suggests examining sets of a certain structure.

Doubling the quantity of steel in producing oil pipes by increasing the diameter of the pipe results in an overproportionate enlargement of the volume of a pipe. Thus. we speak of scale economies of the third dimension. 3 reflects the problem at hand more accurately. (3 . 4) by A 2/3 leads to A(V. x) (2,8/3), with Multiplication of the activity (v. x) the completed fraction of the last unit of the indivisible good given by X4 = X - LxJ =2/3 . 72KooPMANS = = = 2 Theory of the Firm 43 Although each possible activity can theoretically be repeated exactly, it is useful with respect to scarce resources to also permit decreasing returns to scale.

23, the convex hull of the graph 4hLuJJ} GR= {(U,X)IXEZ+, x~ L is an (integer) convex cone with the origin (u, x) = (0,0). The bold dots are associated with quantities of input corresponding to multiples of 3, whereas the quantities of output are multiples of 4. Since no possible activity lies above the ray through the origin, the activities marked by bold dots obey constant returns to scale. Similarly, we can identify increasing returns to scale. Starting with the activities marked by 0, each possible variation of the production level generates a possible activity lying in the adjoined ray through the origin and the points 0.

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