General Equilibrium Theory
A General Economy
• m consumers
• n producers
(n goods)
• Resources
• m X n demand
equations
• n supply equations
Prices
A Pure Exchange Economy
An economy in which there is no production. A special case of a general
economy in which economic activities consist only of trading and consuming.
The simplest form of a pure exchange economy is the two-agent, two-good
exchange economy, which may be illustrated graphically using the Edgeworth –
Bowley Box.
The “Edgeworth Box”: a pure exchange economy
x1
x2
 A2
1B
The “Contract Curve”: The set of all
Pareto efficient points.
B

·
 B2
x
·
U A0
U B0
x2
A
1A
x
1
The “Core”: A set of feasible
allocations that cannot be improved upon.
The “Edgeworth Box”: a pure exchange economy
x1
x2
 A2
1B
B

·
 B2
·
x
-p1/p2
U A0
U B0
-p1/p2
-p1*/p2*
x2
A
1A
x1
The Algebra of Equilibrium
Assumptions:
Pure exchange economy
Two goods: x1 and x2
Two agents, A and B, with …
Identical preferences:
U A x1 , x2   x1 x12 

0    1
 1  
U B x1 , x2   x1 x2 

Arbitrarily determined, but different, endowments:
   A ,  B 



 A  1A ,  A2   B  1B ,  B2



2
Equilibrium is defined as a consumption bundle x  x1A , x A
; x1B , xB2
Where aggregate excess demands are zero in both markets:
z1  p1 , p2   0  Hence, we are seeking a set
 of prices,  p , p2  , that
z 2  p1 , p2   0 satisfies these1 equilibrium
conditions.

Pure Exchange and Equilibrium
x1
1B  6
x1B  4
B
x2
21   A2

·
 B2  6
x
15  x A2
·
xB2  12
U 1A
U A0
U B1 U B0
 P1 P = - 31
2
A
1A  3
x1A  5
x2
x1
First and Second Theorems of Welfare Economics
1. All market equilibria are Pareto efficient.
“With such a definition it is almost self-evident that this so-called
maximum [Pareto-optimality] obtains under free competition …
But this is not to say that the result of production and exchange will
be satisfactory from a social point of view or will, even
approximately, produce the greatest possible social advantage.”
Knut Wicksell, “On the Problem of Distribution” (1902)
What are the implicit assumptions underlying this result?
1.
No consumption externalities
2.
Agents behave “competitively” (large # of agents)
3.
A competitive equilibrium actually exists
Pure Exchange and Redistribution

1B  7
x1
1B  6

x1B  4.5
x1B  4
B
x2
21   A2
’ 
· ·
15  x A2

13.5  x A2
 B2  6
x
·
x’
·
xB2  12

x B2  13 .5
U 1A
U A0
U B1
U B0
x2
A
1
1A  3
A  2
x1A  5

x1A  4.5
x1
Pure Exchange and Redistribution
x1
1B  6

x1B  4.5
x1B  4
B
x2
21   A2
′
18 =  A2

·’’
 B2  6
′
B2 = 9
·
15  x A2

13.5  x A2
x’’
·
x
·
xB2  12

x B2  13 .5
U 1A
U A0
U B1
U B0
x2
A
1A  3
x1A  5

x1A  4.5
x1
First and Second Theorems of Welfare Economics
2. Every Pareto efficient allocation can be achieved
as a competitive equilibrium (given an appropriate
initial endowment and convexity of preferences).
“[Pareto optimality] does not define, uniquely, a best situation in
any sense of the word … Other criteria – roughly speaking,
those we associate with the term ‘distributive justice’ – have to
be called into play.”
Kenneth Arrow, The Limits of Organization (1974)
What is the important implication of this result?
Don’t manipulate prices to achieve equity-related
goals – use lump sum transfers instead.
“However, there are practical matters involved …”