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 lecture 4 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 lecture 4 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: &=xE R": px 2c) Normal vecto:p∈界 Hyperplane:H={x∈界":px=c} K is convex closure VX eK and x∈K,p∈界"andc∈界,3px<C≤pX K is concave, k is the closed convex hull K=∩(=K) lecture 4 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)=nfpx:x∈K} ux(p) given an alternative description for K K={X∈9”:px≥/k(p) for every p} Proposition8: Pk(p)is HD1 and concave See thefig lecture 4 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
1. duality in mathematics Duality theorem: K is nonempty closure, and its support function ux() is differentiable at p, then there is only onexe K, that px=HK(p)and Vuk(p)=x lecture 4 for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 1.Duality in mathematics ▪ Duality theorem: K is nonempty closure, and its support function is differentiable at , then there is only one , that See the fig. K ( ) p p x K and K K p x = p p x ( ) ( ) =
2S.C. for cost function Differentiable function o(w, g) satisfied HD1 of w Concave of w ° Non-decreasing of w Nonnegative for w≥0,q≥0 Then o(w, q)is the cost function definite by the tech.ofv(q)={x∈9w·x≥p(w,q)} lecture 4 for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 2.S.C. for cost function ▪ Differentiable function satisfied: • HD1 of w • Concave of w • Non-decreasing of w • Nonnegative for . ▪ Then is the cost function definite by the tech. of ( , ) w q w 0, 0 q ( , ) w q 1 ( ) { : ( , )} n V q q − = + x w x w
3 what about demand function Duality indicate that HD 1 and concave is what the convex tech need for the cost What about the other function such as factor demand function g(w, q )s HDO and ag, (w, q) is symmetric negative semidefinite. Then it is the conditional factor demand function of a certain tech lecture 4 for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 3.what about demand function ▪ Duality indicate that HD1 and concave is what the convex tech need for the cost. ▪ What about the other function, such as factor demand function. ▪ is HD0 and is symmetric negative semidefinite. Then it is the conditional factor demand function of a certain tech. g w( , ) q ( , ) i j g q w w
3 what about demand function Example: given a cost function c(w, q)=qwiw2 then what's the corresponded tech and factor demand function? x, (w, q=aqui w2 x,w, g)=(1-aqwi w? g=a aXx Proposition; elasticity of scale e(x, q AC(q MC(g lecture 4 for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 3.what about demand function ▪ Example: given a cost function , then what’s the corresponded tech. and factor demand function? ▪ Proposition9: elasticity of scale 1 1 2 c q qw w ( , ) − w = ( ) ( , ) ( ) AC q e x q MC q = 1 1 1 1 2 x q qw w ( , ) − − w = 2 1 2 x q qw w ( , ) (1 ) − w = − 1 1 1 2 q x x (1 ) − − − = −
4.Geometry of duality 1W2 isoquant isocost factor demand x lecture 4 for Chu Kechen Honors College
lecture4 for Chu Kechen Honors College 4.Geometry of duality x2 x1 isoquant w1 w2 isocost x1 w1 /w2 factor demand
