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Polar production and cost functions
The polar profit and production functions
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analysis Appendix A.3 approximation Assumptions F.1 Axioms Chapter 1.1 Cholesky factorizable closed Cobb-Douglas commodity concave convex cone convex function convex hull convex set Corollary corresponding cost function cost minimizing defined demand functions denote Diewert distance function dual econometric economic equation estimation ex post exists finite firm free disposal functional forms G T(z gauge function given Hanoch Hence Hessian matrix holds homogeneous function homogeneous of degree homothetic hypothesis implies input bundle input prices input requirement sets input-conventional isoquant Jorgenson Legendre transformation Lemma lower hemicontinuous McFadden netput non-decreasing non-empty non-negative normalized profit function optimal output bundle parameters partial derivatives polar positive definite positive semidefinite positively linear homogeneous 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 transformation variable yields zero