Lecture10: Competitive Market local equilibrium theory I
Lecture10:Competitive Market local equilibrium theory I
Content Competitive equilibrium ◆ Local analysis o Complete compete market ◆ Monopoly
Content Competitive equilibrium Local analysis Complete compete market Monopoly
Competitive equilibrium ◆ An allocation a=(x1…xy1……y)isa combine of consumption vector x and production vector y. A is feasible if ∑ x1≤O,+ ∑y, for any l=1,…L I×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 o Pareto Optimal( pareto efficient ◆ An allocation(x;…x:…y) is Pareto efficient( optimal if there isn't any the other feasible allocation (xi,. yi, yn) made u (x,)2u,(x) for any i and u, (x)>(x) for some i See the fig
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 o Competitive equilibrium ∈汎 An allocation(x,…xy,…y) and price are a competitive(Walrasian) equilibrium Profit maximization y E max p y Vj Utility maximization x∈maxu1(x)ⅵist.p'x≤p+ x.∈X ∑Qp·y Market clearing >x=O,+2y
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, The prices of z are change homogenously p,=tp Choice:(x", zEmax u(x, z)st. px+tpz=w x-Z Let poz=w then x emax u(x, t)st. px+tw=w so u(x, z=u(x, t)=(w-t)+o(x)=m+o(x=u(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 o For every i=1,I, they have the quasi linear utility function: l(x,m2)=m2+(x) and(x)>0,g(x)<0;的(0)=0 ◆( Inada condition) 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 y=(-=1,9):9,≥0and=1≥c(q) ◆ Profit maximization max pq-c, q) First order condition p=c g ), 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={(-x,9):120and=2c(q)} ◆ Utility maximization max m,+o,(x) s!.m+px≤Om+∑(Pq-c(q1) ◆Fi irst order condition B (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 ◆Mar ket clearing ∑x=∑ ◆ i's Demand function n、-(x)∥P<(0) x (p) 0jfp≥g(0) x(p)=<0Jp<(0) (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 =
