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