Advanced microeconomics (LECTURE 3: production theory l) Ye Jianliang
Advanced Microeconomics (LECTURE 3:production theory III) Ye Jianliang
Cost Minimization Content Definitions Properties of cost function WACM Some forms of cost functions lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College Cost Minimization ▪ Content: • Definitions • Properties of cost function • WACM • Some forms of cost functions
1. Definitions One production, cost function is c(w, g=min WX x≥0 St.f(x)≥q The optimal solution x(w, q), is the conditional factor demand function Question1: calculate the conditional factor demand function of c-d tech and ces tech lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 1.Definitions ▪ One production, cost function is ▪ The optimal solution x(w,q), is the conditional factor demand function. ▪ Question1:calculate the conditional factor demand function of C-D tech. and CES tech. 0 ( , ) min . . ( ) x c q s t f x q = w w x
1. Definitions Recall the cost minimization condition let x>0, then set Lagrangian (1,x)=Wx-4(x)-q) We got w=n Vf(x) 母g What n is? lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 1.Definitions ▪ Recall the cost minimization condition, let x>0, then set Lagrangian: ▪ We got: see the fig. ▪ What is? L ( , ) ( ( ) ) x wx x = − − f q f ( ) w x =
2. Properties of cost function Proposition: c(w, q) is homogeneous of degree 1 in w, and non-decreasing in q Proposition 2: c(w, q) is concave function of Proposition3 x(W, g) is homogeneous of degree 0 in w Proposition4: if v(g is convex, then is x(. if v(a is strictly convex, x( is single point lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 2.Properties of cost function ▪ Proposition1: c(w,q) is homogeneous of degree 1 in w, and non-decreasing in q. ▪ Proposition2: c(w,q) is concave function of w. ▪ Proposition3: x(w,q) is homogeneous of degree 0 in w. ▪ Proposition4:if V(q) is convex, then is x(.) if V(q) is strictly convex, x(.) is single point
2. Properties of cost function Proposition: Shephards lemma)if x(w, q) is single point, then x(w,g=Vc(w,g Proposition6: D, X(w, q)=Dw c(w, q)is symmetric negative semi-definite, and D x(w, a)w=0 e the. Proposition: if f(?) is HD1, co and x(.is HD1 too, if f() is concave, co is convex in lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 2.Properties of cost function ▪ Proposition5: (Shephard’s lemma) if x(w,q) is single point, then ▪ Proposition6: is symmetric negative semi-definite, and see the fig. ▪ Proposition7: if f(.) is HD1, c(.) and x(.) is HD1 too, if f(.) is concave, c(.) is convex in q. ( , ) ( , ) w x w w q c q = 2 ( , ) ( , ) D q D c q w w x w w = ( , ) 0 D q w x w w =
3 WACM Weak Axiom of Cost Minimization( WACM) if xs, xt are in Y, and choice by firm under price ws and wt. thenwx'swx'. We can get △W△x<0 lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 3.WACM ▪ Weak Axiom of Cost Minimization (WACM): if x s , x t are in Y, and choice by firm under price ws and wt . then . we can get: t t t s w x w x w x 0
3 WACM 2 X X B xB lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 3.WACM x2 x A x1 x B x2 x A x1 x B
3 WACM X X B B 1 lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 3.WACM x2 x A x1 x B x2 x A x1 x B VI VO
4 Some forms of cost functions If x=(x,,x), corresponding price w=(w,,w) and x,=2 is the limited factor(fixed assets) while x, =X,(w, g,x,) is the variable factor. The total cost is c(w, g ,x=stc= scy+ Fc=w,x, (w, g, x )+wX faf SAC=c(w,g,/9, SAVC=Wx(w, q,x,)/ SAFC=W X / q, SMC=ac(w,, xr)/ac lecture 3 for Chu Kechen Honors College
lecture3 for Chu Kechen Honors College 4.Some forms of cost functions ▪ If , corresponding price , and is the limited factor (fixed assets), while is the variable factor. The total cost is ( , ) = v f x x x ( , ) w w w = v f f x = z ( , , ) v v f x x w x = q ( , , ) ( , , ) f v v f f f c q STC SCV FC q w x w x w x w x = = + = + ( , , ) / , ( , , ) / / , ( , , ) / f v v f f f f SAC c q q SAVC q q SAFC q SMC c q q = = = = w x w x w x w x w x