L14
General Equilibrium
(cont)
Big ideas:
Tuesday:
 Edgeworth box
 Pareto efficiency (normative theory)
Today:
 Competitive equilibrium (positive theory)
 First welfare theorem
Edgeworth Box

A
 ( 6 ,1 ) 
B
 ( 4 , 4 )    (1 0 , 5 )


OA
A
OB
Desirable Allocation: Pareto Efficient
 Allocation
x Pareto efficient, if there
does not exist allocation y that is
A) at least as good as x for all
B) is strictly better for at least one
 Pareto
 All
efficiency = equality of MRS
Pareto efficient allocations=contract
curve
Pareto efficiency

A
 ( 6 ,1 )
  ( 4 , 4 )    (1 0 , 5 )
B


OA
A
OB
Competitive (Walrasian) Equilibrium

Competitive Equilibrium
A positive model of free market economy
 Walras, then Arrow and Debreu


Extensively used by ``practitioners’’
Competitive (Walrasian) Equilibrium
*A
Consider

*B
x ,x , p
Individuals respond optimally to prices
x

*
*A
,x
*
p
*B
Prices are such that markets clear
x
*A
1
x
*B
1
 1
x 2* A  x 2* B   2
*A
*B
*
x
,
x
,
p
We call
a competitive equilibrium
Excess supply, Demand
p1  10, p 2  1
OB

OA
A
Excess Demand, Supply, Equilibrium
p1  1, p 2  10
OB

OA
A
Excess Demand, Supply, Equilibrium
p1  5, p 2  5
OB

OA
p1  20, p 2  20 ?
A
Cobb-Douglass Calculation

Equilibrium = 6 numbers

3 tricks that simplify calculation
– Market clearing for one market
(Walras Law)
– Use Magic Formulas
– Solve for relative price (only)
Cobb-Douglass general
i  A,B
Example
Geometry
  (6 ,1),   ( 4 , 4 )
A
B
U ( x1 , x 2 )  ln x1  ln x1 i  A , B
i
OB

OA
A
Invisible Hand (Adam Smith)
Are markets (Pareto) efficient?
 First Welfare Theorem: allocation in
Competitive equilibrium is Pareto optimal
 Proof

OB

OA
A
Other Preferences
 Quasilinear

A
 ( 6 ,1 ) , 
 Perfect

A
B
 ( 4 , 4 ) , u ( x 1 , x 2 )  x 1  ln x 2
complements
 ( 6 ,1 ) , 
B
 ( 4 , 4 ) , u ( x 1 , x 2 )  m in ( x 1 , x 2 )