Download Walras` Law

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:


1
A
2
A

A is strictly better off with x , x than
1
2
with  A ,  A ;
1
2
 Person B is strictly better off with xB , xB than
1
2
with  B ,  B .
 Person






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
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

The Second Theorem of Welfare Economics
30.11 Efficiency and Equilibrium

A Pareto efficient allocation that is not a competitive
equilibrium.