What is an open set?
Does not have own boundary
<
Every point is an interior point
What is a closed set?
Has its own boundary
<=
Every point is a limit point
What is a bounded set?
What a compact set?
Bounded if it is completely contained by an open ball
Compact is if bounded and closed
What is convexity?
Convexity if average of two points is contained
Strict Convexity is interior
Define increasing?
Wherever X > Y U(X) > U(Y)
Define Strictly Quasi Concave
An average of two points provides more utility
Define Continous?
No sudden Jumps in utility?
Prove Prop 2: If P»0, the budget set is compact
Compact = Closed and Bounded
Closed: Define X and Y as an interior and limit point] and prove
Define Z as a non interior point
Bounded: An open bowl that contains 0,0 and 3=W/P1+W/P2
This contains the budget set and is thus bounded
Conclude
What is proposition 3?
If utility is increasing, strictly quasi concave and continuous then:
Their is a solution
Which satisfies the budget constraint
And is unique
What is the weierstrass extreme value theorem?
If A is a non-empty compact set, any continous function achieves a maximum and minimum on A
Prove a solution exists for proposition 3
Solution - Continuity
Weierstrass Extreme Value Theorem-
As A is an increasing, strictly quasi concave and continuous function with P»0, which contains 0,0 therefore non empty
And via proposition 2 we know it is compact
Their exists a solution to A via the Weierstrass extreme value theorem
Prove the solution satisfies the budget constraint under proposition 3
Satisfies Constraint - Increasing
If allocation < wealth, point is internal to set
As increasing, we can find a point Y that provides more utility
This continues until equality
Thus satisfies the constraint
Diagram
Prove uniqueness under proposition 3
Unique - Strictly quasi concave
To maximise utility, must occur at tangency point,
If not tangent, an average of a point on the line will provide more utility
Graph
What is propositon 4?
If function is increasing, continuous, strictly quasi concave and P»0, The walrasian demand is a function homogenous of degree 0
Prove Walrasian demands are homogenous of degree -
Budget Line = P1W1+P2W2/P2 -P1/P2X1
Degree 0 Therefore aP=P
Place a infront of all P
Cancel back to original
As Prop 3 implies a unique solution exists on the budget line, the optimal bundle is identical for ap and p
Prove optimum occurs where MRS=P1/P2
Proceed by contradiction
P=(2,1) and MRS=1/1
If trader gives up one unit of X1 to buy X2 he will remain on same IC
However, this will give him one extra unit of currency
Thus utility can be increased = contradiciton
Define a Walrasian Equilbrium
A price vector P»_space;0 and an allocation such that all bundles are optimal and total supply = total demnd
Define excess demand
Total demand - initial endownment
Define Propostion7
Walras Law
Excess demand x price =0
Prove Walras’ law
Assume P»_space; 0
Via Proposition 3ii
P1x1+p2x2 = P1w1+p2W2
Summing for each trader
Move RHS to LHS
Collect like commodities and factor out P1/2
Thus
P1z1(P)+P2Z2(P)=0
What is propositon 8?
Market Clearing:
If P»0, the total demands = endownmetns for commodity I the market for commodity 2 is also in equilibrium
Prove market clearing
WLOG j=1 and P»0
Suppose X1=W1 (Summing)
Proceeding by contradiction, X2=!W2
Multiply equation 1 by P1 and 2 by P2
Summing these equations gives
P1Z1+P2Z2=!0, which contradicts Walras Law
Thus X2=W2
How do you find a Walrasian Equilibrium?
Find Walrasian Demands
X11+X12=W1
P2=1
Solve for other P1
What is theorem 1?
When replicating traders, if same endownments and utiltiy functions, allocaiton is identical
When defining a Walrasian Equilibrium, what do we do?
We normalise prices to unit simplex, allowing one price to equal zero
Define Proposition 9?
Existence of a Walrasian Equilbrium
Let Z:Tri->R2 such that
Z is a continous function
P1z1+p2z2=0
If p=0, max(z1(p),z2(p) > 0
The optimal bundle exists on the line
Prove the first part of propotiion 9
Define g(p) = P1+Max(0,z1) / 1+Max(0,z1)+Max(0,z2)
5(iii) shows that the max part is continous
5(II) shows the numerator and denominator is thus continous
5(i) shows the numerator and denominator is thus continous
5(IV) shows that as the bottom is not equal to zero, g(P) is continous
Define Theorem 5
Properties of a continous function:
If f,g are continous and k is part of R then
k+f,g is continous
f+g is continous
max(f,k) is continous
f/g is continous if g=!0
Continue to prove prop 9 (2)
As g(1)+g(2)=1, G is on the simplex Is also continous Via the Brouwer Fixed Point Theorem, the optimal P exists on the simplex
Reright the intial function with P*
What is the Brouwer Fixed Point Theorem
If a simplex funciton is continous, their exists a P* on simplex that is optimal
Continue to prove prop 9 (3)
Move the bottom to the top Multiply by Z(P) Sum the two equations Condense to Walras law Walras law is thus satisfied
Continue to prove prop 9 (4)
0=z1p1+z2P2
Both terms are thus 0 or zjp^2
If one term >0, LHS cannot be 0
Thus both terms =0
This implies zP*<=0
By the contrapositive of condition (iii), we conclude P»0
Prove Prop 9 (5)
Finally we prove Z(P)=0
Suppose z1(P)<0,
As P*»0 abd z2(P)<0 this fails to satisfy walras law (explain)
Same with z2(P)
We thus conclude z(P)=0
What are some key equations in the Cournot-Walras equilbrium
For price:
x1=e1+e2
For Oligopolists
x1(e)=w1-e
x2(e)=p1e/p2
What is Proposition 11?
Is a Cournot Walras Allocation Feasible
Prove Prop 11
We know X1 is as X1=e1+e2
Thus for x2 p1x1+p2x2<=p2wt Sum for t=3...n Normalise p2 and factor p1 out Redefine x1 as above Redefine X2 as P1(e)et
Thus x21+x22+sumx2
How do you solve a Cournot Walras Equilbrium?
Walras demands for small traders
Normalise P2
X1=e1+e2
Solve for P1
Solve Oligopolists problem WLOG we only consider t=1 x1=w-et x2=p1et Max Symetry e1=e2
In a repeated exchange economy, what happens if we increase the number of replications
Using de l’Hopital Rule Lim(infin)(f/g)=f’/g’
What are the assumptions of existance for Cournot Walras
Positve ammount of each commodity
Increasing, Continuous, Strictly Quasi Concave
Small traders are cobb douglas and hold both commodities
Guarantee but are not necessary