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 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 ∑x=∑。、"andx!xx, for all i∈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 If x is in the core.x must be pareto efficient See the fig
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 X to x,and∑x=∑ But Walrasian equilibrium implies ∑x>p∑。 w, for px>pn
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:not:Letx=∑1x1,x=1∑ j So∑。x+∑ /=1MB= Ir 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 fig. 8=CWA+(1-aya for some 8>0 Let &=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: V8 +(V-1)yB =w4+(1-)y+(-)yB TwA+(V-TLyA+yBI =74+(V-m)v4+g =4+(V-)wg So the coalition with v agents of type A and (v-t) 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 and j are gross substitutes at price p, if (p) ≥0fori≠ 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
