L13
General Equilibrium
(Social Efficiency)
Review


Model of choice of individual
We know preferences U  x1 x 2 and
p1 , p 2 ,  1 ,  2
we find demands
*
1
x ,x

*
2
With many such agents:
 Q1: How prices p1 , p 2 are formed?
 Q2: Are markets efficient?
Big ideas:
Today:
 Edgeworth box
 Pareto efficiency
Next lecture
 Competitive equilibrium
 First welfare theorem
“Economy” with apples and oranges
 Two
consumers, A and B.
  (6,4 )
A
 Total
and  B  ( 4 , 2 ).
resources available
1     
A
1
 Feasible
B
1
2     
A
2
A
1
A
2
allocation ( x , x )
x1A  x1B   1
x x
A
2
B
2
B
2
B
1
B
2
(x , x )
 2
Geometric representation
 Four
numbers and geometric representation
 Insane?
 No:
Edgeworth box
 Collection of all feasible allocations
Edgeworth Box
  ( 6 ,1)   (4, 4)    (10, 5)
A
B


OA
A
OB
Socially Desirable Allocation
Pareto Efficiency
 When
allocation is “socially” efficient?
- Maximizing sum of utilities? NO!
- Weaker notion: Pareto efficiency!
 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
Quiz

Consider a two agent economy with
Q: Is allocation
A) Yes
B) No
C) Depends
Pareto efficient
Pareto Efficiency
 Assume
Cobb Douglass preferences
 Necessity of tangency
OB
x
OA
Pareto Efficiency
 Sufficiency
of tangency
OB
x
OA
Equivalent Characterization
 Assume
Interior Allocations
 Allocation is Pareto efficient
if and only if indifference curves are
tangent (equal MRS)
 Are
initial endowments efficient?
Example:


OA
A
OB
Quiz
 Let
 Is
allocation
A) Yes
B) No
C) Depends
Pareto efficient
Contract Curve
 Contract
curve is the set of all
Pareto-optimal allocations.
OB
OA
Cobb-Douglass example
  (10 ,5 )
U ( x1 , x 2 )  a ln x1  b ln x1
i
i  A, B
Contract Curve
 Cobb
Douglass
OB
OA
Contract Curve
 Perfect
substitutes
OB
OA
Contract Curve
 Other
preferences
A) General Perfect substitutes
B) Perfect Complements
C) Quasilinear