Lecture 12: Exchange General Equilibrium theory
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
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=(x, x))is feasible if , for any I=l,…L @, is the endowments of commodity l Edgeworth box:2×2
Pure exchange system • An allocation A= (x1 ,…xI ) is feasible if • is the endowments of commodity l • Edgeworth box: 2×2 1 , for any 1, I li l i x l L = = l
Pure exchange system Walrasian equilibrium An allocation(x2…x) and price p'∈are a Walrasian equilibrium if Utility maximization x∈maxa1(x) x∈X St.px≤pOl Feasible x;≤O see the fig
Pure exchange system • Walrasian equilibrium: – An allocation and price are a Walrasian equilibrium, if: • Utility maximization • Feasible see the fig. 1 ( , )I x x L p max ( ) . . i i i i x X i i x u x s t x p p 1 I li l i x =
The existence of walrasian equilibrium Given an endowment w, is there any Walrasian equilibria? Assumption 1: The demand function is Hdo of price The aggregated extra-demands (p)=∑x(p,p)- Assumption2: Walras law: p=(P)=0
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: 1 ( ) [ ( , ) ] I i i i i z p x p pw w = = − pz p( ) 0 =
The existence of walrasian equilibrium Assumption3: market clearing Desirability: z(p)>0 for p,=0 lemma: free goods: if p" is a walrasian equilibrium price, andz, (p)<o, then p*=0 lemma: Equivalent of demand and supply: all the commodities are desirable and p is a walrasian equilibrium prIce, then, ( p)=0
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 the commodities are desirable and p * is a walrasian equilibrium price, then ( ) 0 for 0 i z p p = ( ) 0 j z p 0 j p = ( ) 0 j z p =
The existence of walrasian equilibrium Brouwer's fixed-point theorem: continuous function f: s">",(s is a unit simplex) and there exist an x, that x=f(x) Scarf, 1973 when n=1; let g(x)=f(x)-J g(0)=f(0)-0≥0;g(1)=f(1)-1≤0 彐x,g(x)=0=f(x)-x
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, : n n f s s → x x = f ( ) g f g f (0) (0) 0 0; (1) (1) 1 0 = − = − = = − x g x f x x , ( ) 0 ( ) n s
The existence of walrasian equilibrium fz:s4-1→>鸦 is a continuous function witch satisfied Walras' law, pz(p)=0, then there exist some p in s-l such that z(p) 8, (p) p,+max(0, z, (p)) 1+ j-max(0, =, (p)
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 1 : k k z s − → p p z( ) 0 = p k 1 s − z( ) 0 p 1 i i K j j p p p = = 1 1 { in : 1 k k k i i s p − = = = p 1 1 : k k g s s − − → 1 max(0, ( )) ( ) 1 max(0, ( ) i i i K j j p z g z = + = + p p p
The existence of walrasian equilibrium By brouwer's fixed-point theorem there is a p, such that p=gp) p:=g (p) Pi+max(0,z(p)) K 1+ max 0,=/(p) So pt>i-max(0, 2 (p")]=max(0, = (p) for i=1.k
The existence of Walrasian equilibrium – By brouwer’s fixed-point theorem, there is a p * , such that – So g( ) p p = 1 max(0, ( )) ( ) 1 max(0, ( ) i i i i K j j p z p g p z = + = = + p p 1 [ max(0, ( )] max(0, ( )) for 1 K i j i j p z z i k = p p = =
The existence of walrasian equilibrium And=(p)∑ max(0.二 (P)=(p)max(0,=(p) Sum up with i ∑max(2(p)∑p)=∑=p)max0.(p ∑(p)max(0,(p)=0fr∑n=(p)=0 so, we have:z1(p)≤0 Example: C-D economy
The existence of Walrasian equilibrium – And – Sum up with i – So, we have: • Example: C-D economy 1 ( ) [ max(0, ( )] ( )max(0, ( )) K i i j i i j z p z z z = p p p p = 1 1 1 [ max(0, ( )] ( ) ( ) max(0, ( )) k k K j i i i i j i i z p z z z = = = p p p p = 1 1 ( ) max(0, ( )) =0 for ( ) 0 k k i i i i i i z z p z = = p p p = ( ) 0 i z p