Production Correspondences
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An analysis of the production correspondences of multi-commodity production structures. Properties of the distance function of a production structure are studied and are related to properties of the correspondence. Homotheticity is generalized to allow discontinuities in returns to scale and to take cognizance of several outputs. The factorization of the distance function into the product of two functions, one depending only on output and the other only on input, is then demonstrated. A cost function is shown to give rise to a production structure. Proof of a Shephard-type duality theorem implies a dual relation between the cost and distance functions; given one, a minimization problem yields the other. The duality theorem is used to deduce the necessary and sufficient conditions, in terms of the cost function, for a production correspondence to be homothetic. A few simple applications of the duality theorem are given which relate properties of the cost function to geometric properties of the correspondence.
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