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Polar production and cost functions
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analysis Appendix A.3 approximation Assumptions F.11 Chapter Cholesky factorizable closed Cobb-Douglas commodities concave constant constraint continuously differentiable convex cone convex function convex set corresponding cost function cost minimizing DANIEL McFADDEN defined demand functions denote Diewert distance function dual duality econometric economic efficiency equation estimation ex post example exists finite firm free disposal functional forms gauge function given Hanoch Hence Hessian matrix homogeneous function homogeneous of degree homothetic hypothesis implies input bundle input prices input requirement sets input-conventional isoquant Legendre transformation Lemma mapping McFadden monotonicity netput non-decreasing non-empty non-negative normalized profit function optimal output bundle parameters partial derivatives polar positive definite positive semidefinite price vector production function production possibility set profit maximization Proof properties quasi-convex real symmetric relative interior restricted profit function result Rockafellar 1970 satisfies scalar semi-bounded separable structure Suppose symmetric matrix Theorem transformation variable yields zero