Lecture 13 market as a process.ppt

Lecture 13: market as a process General Equilibrium theory II
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:
and
If an allocation can be improved upon, then there is some group of agents can do better without market!
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 to , and
But Walrasian equilibrium implies
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.
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)
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.
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 :
Proposition: If all goods are gross substitutes at all price, then if p* is an equilibrium price, then it’s the unique equilibrium pr