浙江大学：《高级微观经济学》课程PPT教学课件（英文版）Lecture 4 for Chu Kechen Honors College

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

1. Duality in mathematics Duality theorem: K is nonempty closure, and its support function Ak(p)is differentiable at p, then there is only onexe K, that pX=uK(p)and VuK(p)=X lectured 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 (w, g satisfied HD1 of w Concave of w Non-decreasing of w Nonnegative for w≥0.q≥0 Then p(w, q)is the cost function definite by the tech of V(=xEm:wx>p(w, g) lectured 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 HD1 and concave is what the convex tech need for the cost What about the other function such as factor demand function g(w, q is HDO and ag, (w, q) is symmetric C negative semidefinite. Then it is the conditional factor demand function of a certain tech lectured 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, g=agwi w2 x2 (w,9=(1-a)qwi w2 q=a-a(1-a) Propositions: elasticity of scale e(x, q AC(g MC(g lectured 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 lectured 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