《高级微观经济学Advanced Microeconomics》PPT课件：Lecture 13 market as a process General Equilibrium theory II

Lecture 13: market as a process General Equilibrium theory ll

Lecture 13: market as a process General Equilibrium theory II

Content The“core” Uniqueness of equilibrium Stability of equilibrium Welfare

Content • The “core” • Uniqueness of equilibrium • Stability of equilibrium • Welfare

The“core mprove upon an allocation: a group of agents S is said to improve upon a given allocation. if there is some allocationx' such that ∑x=∑"andx;>x, for allie s If an allocation can be improved upon then there is some group of agents can do better without market!

The “core” • Improve upon an allocation: a group of agents S is said to improve upon a given allocation x, if there is some allocation x’ such that: and • If an allocation can be improved upon, then there is some group of agents can do better without market! i i i S i S w     x = for all i i i x x  i S 

The“core Core of an economy: a feasible allocation x is in the core of the economy if it cannot be improved upon by any coalition Ifx is in the core x must be pareto efficient See the

The “core” • Core of an economy: a feasible allocation x is in the core of the economy if it cannot be improved upon by any coalition. • If x is in the core, x must be Pareto efficient. See the fig

The“core Walrasian equilibrium is in core Proof: let (x, p) be the Walrasian equilibrium with initial endowment wi If not there is some coalition s and some feasible allocation x,, such that all agents i in S strictly prefer Xi to X,,and∑x=∑ But Walrasian equilibrium implies x>p∑ w, for px>

The “core” • Walrasian equilibrium is in core. – Proof: let (x,p) be the Walrasian equilibrium with initial endowment wi . – If not , there is some coalition S and some feasible allocation x’, such that all agents i in S strictly prefer to , and – But Walrasian equilibrium implies i x i  x i i i S i S w     x = for i i i i S i S w w   p x p px p      

The“core Equal treatment in the core: if X is an allocation in the r-core of a given economy, then any two agents of the same type must receive the same bundle Proof: if not LetsI\rxA1BrLj= B So∑x+∑1x=∑+∑ =16 That is x+XB=w+WB Every agent below the average will coalize to improve upon the allocation

The “core” • Equal treatment in the core: if x is an allocation in the r-core of a given economy, then any two agents of the same type must receive the same bundle. Proof: if not. Let , – So – That is – Every agent below the average will coalize to improve upon the allocation. 1 1 r A A r j x x = =  1 1 r B B r j x x = =  1 1 1 1 1 1 1 1 r r r r r r r r A B Aj Bj j j j j x x w x = = = =     + = + A B A B x x w w + = +

The“core Shrinking core: there is a unique market equilibrium x' from initial endowment w if y is not the equilibrium, there is some replication r, such that y is not in the r-core Proof: since y is not the equilibrium, there is another allocation g improve upon A(or B)at least. That means see the 8=8w+(1-a)y for some 8>0 Let a=T/V(T and V are integers)

The “core” • Shrinking core: there is a unique market equilibrium x * from initial endowment w. if y is not the equilibrium, there is some replication r, such that y is not in the r-core. Proof: since y is not the equilibrium, there is another allocation g improve upon A(or B) at least. That means see the fig. – Let (T and V are integers) (1 ) for some 0 A A g w y = + −      =T V/

The“core Replicated V times of the economy, we have:Vga+ (-7)yB =[w4+(1-)y4]+(-T)yg 7Yv4+(-T)y4+y] =hw4+(-Tw4+ So the coalition with v agents of type A and (-D)of type B can improve upon y

The “core” • Replicated V times of the economy, we have: • So the coalition with V agents of type A and (V-T) of type B can improve upon y. ( ) [ (1 ) ] ( ) ( )[ ] ( )[ ] ( ) A B A A B A A B A A B A B Vg V T y T T V w y V T y V V Tw V T y y Tw V T w w Vw V T w + − = + − + − = + − + = + − + = + −

The“core Convexity and size If agent has non-convex preference, is there still a equilibrium? See the fig Replication the economy

The “core” • Convexity and size: • If agent has non-convex preference, is there still a equilibrium? See the fig. • Replication the economy

Uniqueness of equilibrium Gross substitutes: two goods i andj are gross substitutes at price p, if 0z,(p) ≥0fori≠j Proposition: If all goods are gross substitutes at all price, then if p is an equilibrium price, then it's the unique equilibrium price

Uniqueness of equilibrium • Gross substitutes: two goods i and j are gross substitutes at price p, if : • Proposition: If all goods are gross substitutes at all price, then if p * is an equilibrium price, then it’s the unique equilibrium price. ( ) 0 for j i z i j p     p