Lecture 12 Exchange.ppt

Lecture 12: Exchange General Equilibrium theory I
Content
Pure exchange system
The existence of Walrasian equilibrium
The first theorem of welfare economics
The second theorem of welfare economics
Exchange with production.
Pure exchange system
An allocation A= (x1,…xI) is feasible if
is the endowments modity l
Edgeworth box: 2×2
Pure exchange system
Walrasian equilibrium:
An allocation and price are a Walrasian equilibrium, if:
Utility maximization
Feasible see the fig.
The existence of Walrasian equilibrium
Given an endowment w, is there any walrasian equilibria?
Assumption1:The demand function is HD0 of price.
The aggregated extra-demands

Assumption2: Walras’ law:
The existence of Walrasian equilibrium
Assumption3: market clearing
Desirability:
lemma1: free goods: if p* is a walrasian equilibrium price, and ,then
lemma2: Equivalent of demand and supply: all modities are desirable and p* is a walrasian equilibrium price, then
The existence of Walrasian equilibrium
Brouwer’s fixed-point theorem: continuous function , ( is a unit simplex) and there exist an x, that:
Scarf,1973.
when n=1;let g(x)=f(x)-x,
The existence of Walrasian equilibrium
If is a continuous function witch satisfied Walras’ law, , then there exist some in such that .
Construct a unit simplex:
Defined
The existence of Walrasian equilibrium
By brouwer’s fixed-point theorem, there is a p* , such that
So
The existence of Walrasian equilibrium
And
Sum up with i
So, we have:
Example: C-D economy