Duality Ye Jianliang
Duality Ye Jianliang
Duality Given the technology, we can obtain the cost function are the cost function contains the same information of the technology (production function) If the answer is yes, then the cost minimization behavior will indicate the technology of the firm lectured for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College Duality ▪ Given the technology, we can obtain the cost function, are the cost function contains the same information of the technology (production function)? ▪ If the answer is “yes”, then the cost minimization behavior will indicate the technology of the firm
Duality Content Duality in mathematics Sufficient condition for cost function Factor demand function Geometry of duality lectured for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College Duality ▪ Content: • Duality in mathematics • Sufficient condition for cost function • Factor demand function • Geometry of duality
1. Duality in mathematics Some concepts and properties Half-spaces:x={x∈界":px≥c} ° Normal vector:p∈界 Hyperplane:H={X∈界":px=c} K is convex closure Vx≠ K x∈K,p∈界andc∈界3px<C≤pX K is concave,k is the closed convex hull K=0(Z=K) lectured for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 1.Duality in mathematics ▪ Some concepts and properties: • Half-spaces: • Normal vector: • Hyperplane: • K is convex closure: • K is concave, K* is the closed convex hull: { : } n H = x px c n p { : } n H c = = x px K x K c c and , and , n x p px px * ( ) p K K = H
1. Duality in mathematics Support function: (infimum) k(p)= inf(px:x∈K} Ak(p) given an alternative description for K K={x∈界”:px≥/k(p) for every p} Proposition8 uk(p)is HD 1 and concave seethefig lectured for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 1.Duality in mathematics ▪ Support function: (infimum) ▪ given an alternative description for K. ▪ Proposition8: is HD1 and concave. See the fig. inf{ : } K ( ) = p px x K K ( ) p { : for every } n K = ( ) K x px p p K ( ) p