L13
General Equilibrium
(cont)
Edgeworth Box
  ( 6 ,1 )   ( 4 , 4 )    (1 0 , 5 )
A
B


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
All Pareto efficient allocations=contract
curve
Pareto efficiency
  ( 6 ,1 )   ( 4 , 4 )    (1 0 , 5 )
B
A


OA
A
OB
Competitive equilibrium
Definition: Competitive equilibrium x * A , x * B , p *
*A
*B
*
x 1 , x 1 optimal given p
1)
*
2)
p such that markets clear
Two tricks
1) Only relative price p 1 / p 2 determined
2) Walras Law: second market will clear
auctomatically
Cobb-Douglass example
  (6 ,1),   (4 , 4 )
A
B
U ( x 1 , x 2 )  ln x 1  ln x 1 i  A , B
i
Geometry
  (6 ,1),   (4 , 4 )
A
B
U ( x 1 , x 2 )  ln x 1  ln x 1 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
Perfect substitutes: Efficiency

A
 ( 6 ,1 ), 
B
 ( 4 , 4 ), u ( x 1 , x 2 )  x 1  x 2


OA
A
OB
Perfect substitutes: Equilibrium
  ( 6 ,1),   ( 4 , 4 ), u ( x1 , x 2 )  x1  x 2
A
B
 Competitive
equilibrium:
i  A, B
Perfect substitutes: Equilibrium

A
 ( 6 ,1 ), 
B
 ( 4 , 4 ), u ( x 1 , x 2 )  x 1  x 2
OB

OA
A
Other Preferences
 Quasilinear
 A  ( 6 ,1 ),  B  ( 4 , 4 ), u ( x 1 , x 2 )  x 1  ln x 2
 Perfect
complements
  ( 6 ,1 ),   ( 4 , 4 ), u ( x 1 , x 2 )  min( x 1 , x 2 )
A
B
Application: Irving Fisher r1
 A  (10,0),  B  (0,10), u ( x1 , x2 )  ln x1 
ln x2
1 
 Determination of competitive interest rate
Application: Irving Fisher
r
1
  (10,0 ),   ( 0,10 ), u ( x1 , x 2 )  ln x1 
A
B
 Competitive
equilibrium
  3  u ( x1 , x 2 )  ln x1  0 .25 ln x 2
1 
ln x 2
Geometry
  (10,0 ),   (0,10 ), u ( x1 , x 2 )  ln x1  0 .25 ln x 2

A
B
OB
OA

A
Application:
Uncertainty
B
  (10,0 ),   (0,10 ), u ()  0 .5 ln x1  0 .5 ln x 2
A
Arrow Securities:
No idiosyncratic risk in equilibrium
Geometry
  (10,0 ),   ( 0,10 ), u ( x1 , x 2 )  0 .5 ln x1  0 .5 ln x 2

A
B
OB
OA

A