CHAPTER 30
EXCHANGE
Partial equilibrium analysis: The equilibrium
conditions of ONE particular market, leaving
other markets untreated.
 General equilibrium analysis: The
equilibrium conditions of ALL markets,
allowing interactions between different
markets.

30.1 The Edgeworth Box
Two consumers: A and B.
 Two goods: 1 and 2.
1
2
 Initial endowment:  A ,  A  ,


1
B
,
2
B

Allocation:  x , x  ,  x , x 
 Feasible allocation: total consumption does not
exceed total endowment for both goods.
x1A  x1B  1A  1B

1
A
2
A
1
B
2
B
x  x   
2
A
2
B
2
A
2
B
30.1 The Edgeworth Box
30.1 The Edgeworth Box
Each point in the Edgeworth box represents a
feasible allocation.
 From W to M:

1
1
2
2
w

x
x

w
 Person A trades
A
A units of good 1 for
A
A
units of good 2;
2
2
1
1
w

x
x

w
 Person B trades
B
B units of good 2 for
B
B
units of good 1.
30.2 Trade
30.2 Trade
Trade happens whenever both consumers are
better off.
 Starting from W, M is a possible outcome of
the exchange economy because:

 Person A is
strictly better off with
1
2
with  A ,  A ;
 Person B is strictly better off with
1
2
with  B ,  B .




x
1
A
x
1
B
,x
2
A
 than
,x
2
B
 than
30.3 Pareto Efficient Allocations

An allocation is Pareto efficient whenever:
 There
is no way to make everyone strictly better
off;
 There is no way to make some strictly better off
without making someone else worse off;
 All of the gains from trade have been exhausted;
 There are no (further) mutually advantageous
trades to be made.
30.3 Pareto Efficient Allocations
30.3 Pareto Efficient Allocations
30.3 Pareto Efficient Allocations
Pareto efficiency is given by the tangency of
the indifference curves.
 Contract curve: the locus of all Pareto
efficient allocations.
 Any allocation off the contract curve is Pareto
inefficient.

30.4 Market Trade
Gross demand: Quantity demanded for a good
by a particular consumer at the market price.
 Excess demand: The difference between the
gross demand and the initial endowment of a
good by a particular consumer.
 Disequilibrium: Excess demands by both
consumers do not sum up to zero.

30.4 Market Trade
30.4 Market Trade

Competitive equilibrium: A relative price p1 p2
1
2
1
2
x
,
x
,
x
,
x
and an allocation  A A   B B  , such that:

 The

allocation matches the gross demands by both
consumers, given the relative price and initial
endowments;
 The allocation is feasible.
30.4 Market Trade
30.5 The Algebra of Equilibrium
1
*
*
2
*
*
x
p
,
p
,
x
p
,
p
 Consumer A’s demands: A  1 2 
A 1
2
1
*
*
2
*
*
x
p
,
p
,
x
p
,
p
 Consumer B’s demands: B  1 2 
B 1
2

The equilibrium condition:
x1A ( p1* , p2* )  x1B ( p1* , p2* )  1A  1B
xA2 ( p1* , p2* )  xB2 ( p1* , p2* )   A2  B2

Re-arrangement:
[ x1A ( p1* , p2* )  1A ]  [ x1B ( p1* , p2* )  B1 ]  0
[ xA2 ( p1* , p2* )   A2 ]  [ xB2 ( p1* , p2* )  B2 ]  0
30.5 The Algebra of Equilibrium

Net demand:
e1A ( p1 , p2 )  x1A ( p1 , p2 )  1A
e1B ( p1 , p2 )  x1B ( p1 , p2 )  1B

Aggregate excess demand:
z1 ( p1 , p2 )  e ( p1 , p2 )  e ( p1 , p2 )
1
A
1
B
z2 ( p1 , p2 )  e ( p1 , p2 )  e ( p1 , p2 )
2
A

2
B
Another expression:
z1  p , p   0
*
1
*
2
*
1
*
2
z2  p , p
0
30.6 Walras’ Law

Budget constraints:
p x ( p1 , p2 )  p x ( p1 , p2 )  p1  p2
1
1 A
2
2 A
1
A
2
A
p1 x1B ( p1 , p2 )  p2 xB2 ( p1 , p2 )  p1B1  p2B2

Re-arrange the terms:
p1e1A ( p1 , p2 )  p2eA2 ( p1 , p2 )  0
p e ( p1 , p2 )  p e ( p1 , p2 )  0
1
1 B

2
2 B
Adding up:
p1 z1 ( p1 , p2 )  p2 z2 ( p1 , p2 )  0
30.6 Walras’ Law
Walras’ Law: The value of aggregate excess
demand is always zero.
 Applications of the Walras’ law:


z1 ( p , p )  0 implies z2 ( p , p )  0 ;
*
1
 Market
*
2
*
1
*
2
clearing for one good implies that of the
other good;
 With k goods, we only need to find a set of prices
where k-1 of the markets are cleared.
30.7 Relative Prices
Walras’ law implies k-1 independent equations
for k unknown prices.
 Only k-1 independent prices.
 Numeraire prices: the price which can be
used to measure all other prices.
 If we choose p1 as the numeraire price, then it
is just like multiplying all prices by the
constant t=1/p1.

EXAMPLE: An Algebraic Example of
Equilibrium

The Cobb-Douglas utility function:
u A ( x1A , x A2 )  ( x1A ) a ( x A2 )1 a

The demand functions:
mA
x ( p1 , p2 , mA )  a
p1
mA
x ( p1 , p2 , mA )  (1  a)
p2
mB
x ( p1 , p2 , mB )  b
p1
mB
x ( p1 , p2 , mB )  (1  b)
p2
1
A
1
B
2
A
2
B
EXAMPLE: An Algebraic Example of
Equilibrium

Income from endowments:
mA  p11A  p2 A2 mB  p1B1  p2B2

Aggregate excess demand for good 1:
mA
mB
z1 ( p1 ,1)  a
b
 1A  1B
p1
p1
p11A   A2
p11B  B2
a
b
 1A  B1
p1
p1
EXAMPLE: An Algebraic Example of
Equilibrium

Equilibrium condition:
z1 ( p1* ,1)  0

Equilibrium price:
2
2
a


b

*
A
B
p1 
1
1
(1  a) A  (1  b)B
30.8 The Existence of Equilibrium
The existence of a competitive equilibrium can
be proved rigorously.
 A formal proof is quite complicated and far
beyond the scope of this course.

30.9 Equilibrium and Efficiency
Both indifference curves are tangent to the
budget line at the equilibrium allocation.
 The equilibrium allocation lies upon the
contract curve.
 The First Theorem of Welfare Economics:
Any competitive equilibrium is Pareto efficient.

EXAMPLE: Monopoly in the
Edgeworth Box

A regular monopolist
EXAMPLE: Monopoly in the
Edgeworth Box

First degree price discrimination
30.11 Efficiency and Equilibrium

Reverse engineering:
 Starting from
any Pareto efficient allocation;
 Use the common tangent line as the budget line;
 Use any allocation on the budget line as the initial
endowment.

The Second Theorem of Welfare Economics: For
convex preferences, any Pareto efficient allocation is
a competitive equilibrium for some set of prices and
some initial endowments.
30.11 Efficiency and Equilibrium
30.11 Efficiency and Equilibrium

A Pareto efficient allocation that is not a competitive
equilibrium.