Lecture10: Competitive Market local equilibrium theory i
Lecture10:Competitive Market local equilibrium theory I
Content ◆ Competitive equilibrium ◆ Local analysis cOmplete compete market ◆ Monopoly
Content Competitive equilibrium Local analysis Complete compete market Monopoly
Competitive equilibrium ◆ An allocation a=(x1…xy1……yisa combine of consumption vector x and production vector y. A is feasible if ∑ x1≤+>y for any l=1……L ×J×L
Competitive equilibrium An allocation A= (x1 ,…xI ;y1……yJ ) is a combine of consumption vector x and production vector y. A is feasible if 1 1 , for any 1, I J li l lj i j x y l L = = + = I J L
Competitive equilibrium pAreto Optimal( pareto efficient) ◆ An allocation(x…x;y1…y) s Pareto efficient( optimal ) if there isn't any the other feasible allocation(x,…x;y2…y) made u()2u(x) for any i and u ()>u,(x) for some i See the
Competitive equilibrium Pareto Optimal ( Pareto efficient ) : An allocation is Pareto efficient ( optimal ) if there isn’t any the other feasible allocation , made for any i and for some i. See the fig. 1 1 ( , ; , ) I J x x y y 1 1 ( , ; , ) I J x x y y ( ) ( ) i i u u x x ( ) ( ) i i u u x x
Competitive equilibrium e Competitive equilibrium p'∈9 ■ An allocation(x;…x1;y,…y) and price are a competitive(Walrasian) equilibrium Profit maximization y emax py e Utility maximization x∈maxu(x)Mist. pspa+∑9,p∵y ◆ Market clearing ∑x=0+∑功 j=1
Competitive equilibrium Competitive equilibrium: ◼ An allocation and price are a competitive (Walrasian) equilibrium, if: Profit maximization Utility maximization Market clearing 1 1 ( , ; , ) I J x x y y L p max j j j y Y y p y j 1 max ( ) . . i J i i i i i ij j x X j x u x i s t p x p p y = + 1 1 I J li l lj i j x y = = = +
Local analysis ◆ Hicksian Separability Divide the consumption bundle into two sub-bundles x=(x, z), and price p=(p,p, The prices of z are change homogenously P2= ■ Choice:(x,z)∈maxl(x,z)st.px+p0z=1 Let poz= then x emax u(x, t)st. px+tu Sol(x,2)=l(x,D=(-n)+(x)=m+(x)=l(x,m)
Local analysis Hicksian Separability ◼ Divide the consumption bundle into two sub-bundles , and price ◼ The prices of z are change homogenously ◼ Choice: ◼ Let then ◼ so x z = ( , ) x = ( , ) p p pz 0 = t p p z , ( , ) max ( , ) . . x x u x s t px t w + = 0 z z z p z p z0 = wz max ( , ) . . z x x u x t s t px tw w + = ( , ) ( , ) ( ) ( ) ( ) ( , ) z u x u x t w tw x m x u x m z = = − + = + =
Local analysis OFor every i=1,I, they have the quasi linear utility function l1(x,m1)=m1+d(x) and(x)>0;(x)<0,0(0)=0 ◆( Inada condition.) e Standardization the price of m as 1, and pricing commodity I as p
Local analysis For every i=1,…I, they have the quasilinear utility function: and (Inada condition.) Standardization the price of m as 1,and pricing commodity l as p. ( , ) ( ) i i i i i i u x m m x = + ( ) 0; ( ) 0; (0) 0 i i i i i x x =
Local analysis ◆ For the firm j {(-21,91):q≥0andz≥c (q,)} ◆ Profit maximization max pg-c q ◆ First order condition p=c (,), for g>0
Local analysis For the firm j Profit maximization First order condition: {( , ) : 0 and ( )} Y z q q z c q j j j j j j j = − 0 max ( ) j j j j q p q c q − ( ), for 0 j j p c q q =
Local analysis ◆ For the consumer i y={( q,):90andz≥C (q)} ◆ Utility maximization max m,+o,(r,) mi, xi st.m+px≤On+∑Q2·(P·9-c(q,) ◆ First order condition: 6 (x'=p for x>0
Local analysis For the consumer i Utility maximization First order condition: {( , ) : 0 and ( )} Y z q q z c q j j j j j j j = − , 1 max ( ) . . ( ( )) i i i i i m x J i i mi ij j j j j m x s t m px p q c q = + + + − ( ) for 0 i i i x p x =
Local analysis ◆ Market clearing ◆ i's Demand function if p<o(o) x (p) x01 jfp≥g( x(P)=<0p<q(0) d"(x)
Local analysis Market clearing i’s Demand function 1 1 I J i j i j x q = = = 1 ( ) (0) ( ) 0 (0) i i i i i x if p x p if p − = 1 ( ) <0 (0) ( ) i i i i x p if p x =
