Chapter Thirty
Exchange
交换
Contents
The Edgeworth (艾奇沃斯 ) Box
Pareto improving trade
Pareto Efficient Allocation
The Core (核心 )
Trade in a competitive market
Walras’ Law of general equilibrium (一般均衡 )
First Fundamental Theorem of Welfare
Economics (福利经济学第一定理 )
Second Fundamental Theorem of Welfare
Economics (福利经济学第二定理 )
Exchange
Two consumers,A and B.
Their endowments of goods 1 and 2
are
E.g,
The total quantities available
A A A? (,)1 2B B B? (,).1 2and
A? (,)6 4? B? (,).2 2and
1 1 6 2 8A B
2 2 4 2 6A B
units of good 1
units of good 2.and
are
Exchange
Edgeworth and Bowley devised a
diagram,called an Edgeworth
box,to show all possible
allocations of the available
quantities of goods 1 and 2
between the two consumers.
Starting an Edgeworth Box
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
The dimensions of
the box are the
quantities available
of the goods.
Feasible Allocations
What allocations of the 8 units of
good 1 and the 6 units of good 2 are
feasible?
How can all of the feasible
allocations be depicted by the
Edgeworth box diagram?
One feasible allocation is the before-
trade allocation; i.e,the endowment
allocation.
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
The endowment
allocation is
A? (,)6 4
B? (,).2 2
and
The Endowment Allocation
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
A? (,)6 4
B? (,)2 2
The Endowment Allocation
A? (,)6 4OA
OB
6
8?
B? (,)2 2
The Endowment Allocation
A? (,)6 4OA
OB
6
8
4
6
The Endowment Allocation
B? (,)2 2
OA
OB
6
8
4
6
2
2
The Endowment Allocation
A? (,)6 4
B? (,)2 2
OA
OB
6
8
4
6
2
2
The
endowment
allocation
The Endowment Allocation
More generally,…
The Endowment Allocation
The Endowment Allocation
OA
OB
The
endowment
allocation
1 1A B?
2A
2
2
A
B
1A
1B
2B
Other Feasible Allocations
denotes an allocation to
consumer A.
denotes an allocation to
consumer B.
An allocation is feasible if and only if
(,)x xA A1 2
(,)x xB B1 2
x xA B A B1 1 1 1
x xA B A B2 2 2 2,and
Feasible Reallocations
OA
OB
1 1A B?
xA2
2
2
A
B
xA1
xB1
xB2
Feasible Reallocations
OA
OB
1 1A B?
xA2
2
2
A
B
xA1
xB1
xB2
Feasible Reallocations
All points in the box,including the
boundary,represent feasible
allocations of the combined
endowments.
Which allocations will be blocked by
one or both consumers?
Which allocations make both
consumers better off?
Adding Preferences to the Box
2A
1A
xA2
xA1OA
For consumer A.
Adding Preferences to the Box
2A
1A
xA2
xA1
For consumer A.
OA
Adding Preferences to the Box
2B
1B
xB2
xB1
For consumer B.
OB
Adding Preferences to the Box
xB2
xB1
For consumer B.
OB
2B
1B
Adding Preferences to the Box
2B
1BxB1
xB2
For consumer B,O
B
Adding Preferences to the Box
2A
1A
xA2
xA1OA
For consumer A.
Adding Preferences to the Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Edgeworth’s Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Improvement
An allocation of the endowment that
improves the welfare of a consumer
without reducing the welfare of
another is a Pareto-improving
allocation.
Where are the Pareto-improving
allocations?
Edgeworth’s Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Improvements
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving allocations
Pareto-Improvements
Since each consumer can refuse to
trade,the only possible outcomes
from exchange are Pareto-improving
allocations.
But which particular Pareto-
improving allocation will be the
outcome of trade?
Pareto-Improvements
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving reallocations
Pareto-Improvements
Pareto-Improvements
Pareto-Improvements
Trade
improves both
A’s and B’s welfares.
This is a Pareto-improvement
over the endowment allocation.
Pareto-Improvements
New mutual gains-to-trade region
is the set of all further Pareto-
improving
reallocations.
Trade
improves both
A’s and B’s welfares.
This is a Pareto-improvement
(帕累托改进 )over the endowment allocation.
Pareto-Improvements
Further trade cannot improve
both A and B’s
welfares.
Pareto-Optimality (帕累托最优 )
Better for
consumer B
Better for
consumer A
Pareto-Optimality
A is strictly better off
but B is strictly worse
off
B is strictly better
off but A is strictly
worse off
Pareto-Optimality
A is strictly better off
but B is strictly worse
off
B is strictly better
off but A is strictly
worse off
Both A
and B are
worse
off
Both A and
B are worse
off
Pareto-Optimality
The allocation is
Pareto-optimal since the
only way one consumer’s
welfare can be increased is to
decrease the welfare of the other
consumer.
Pareto-Optimality
The allocation is
Pareto-optimal since the
only way one consumer’s
welfare can be increased is to
decrease the welfare of the other
consumer.
The Marginal Rates of
Substitution Are Equal.
Pareto-Optimality
Where are all of the Pareto-optimal
allocations of the endowment?
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
All the allocations marked by
a are Pareto-optimal.
Pareto-Optimality
The contract curve (契约线 ) is the set
of all Pareto-optimal allocations.
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
All the allocations marked by
a are Pareto-optimal.
The contract curve
Pareto-Optimality
But to which of the many allocations
on the contract curve will consumers
trade?
That depends upon how trade is
conducted.
In perfectly competitive markets? By
one-on-one bargaining?
The Core (核心 )
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving reallocations
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-optimal trades blocked
by B
Pareto-optimal trades blocked
by A
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-optimal trades not blocked
by A or B are the core.
The Core
The core is the set of all Pareto-
optimal allocations that are welfare-
improving for both consumers
relative to their own endowments.
Rational trade should achieve a core
allocation.
The Core
But which core allocation?
Again,that depends upon the
manner in which trade is conducted.
Trade in Competitive Markets
Consider trade in perfectly
competitive markets.
Each consumer is a price-taker trying
to maximize her own utility given p1,
p2 and her own endowment,That is,
...
Trade in Competitive Markets
2A
1A
xA2
xA1OA
For consumer A.
p x p x p pA A A A1 1 2 2 1 1 2 2
x A2*
x A1*
Trade in Competitive Markets
So given p1 and p2,consumer A’s net
demands for commodities 1 and 2
are x A A
1 1* x A A2 2*,and
Trade in Competitive Markets
And,similarly,for consumer B …
Trade in Competitive Markets
2B
1B
xB2
xB1
For consumer B.
OB x B1*
x B2*
p x p x p pB B B B1 1 2 2 1 1 2 2
Trade in Competitive Markets
So given p1 and p2,consumer B’s net
demands for commodities 1 and 2
are x B B
1 1* x B B2 2*,and
Trade in Competitive Markets
A general equilibrium occurs when
prices p1 and p2 cause both the
markets for commodities 1 and 2 to
clear; i.e.
x xA B A B1 1 1 1* *
x xA B A B2 2 2 2* *,and
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this PO allocation be
achieved?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
x B1*
x B2*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
But x xA B A B1 1 1 1* *
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
and x xA B A B2 2 2 2* *
Trade in Competitive Markets
So at the given prices p1 and p2 there
is an
– excess supply of commodity 1
– excess demand for commodity 2.
Neither market clears so the prices
p1 and p2 do not cause a general
equilibrium.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
So this PO allocation cannot be
achieved by competitive trading.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
Since there is an excess demand for
commodity 2,p2 will rise.
Since there is an excess supply of
commodity 1,p1 will fall.
The slope of the budget constraints
is - p1/p2 so the budget constraints
will pivot about the endowment point
and become less steep.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
x B1*
x B2*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
So x xA B A B1 1 1 1* *
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
and x xA B A B2 2 2 2* *
Trade in Competitive Markets
At the new prices p1 and p2 both
markets clear; there is a general
equilibrium.
Question 1,Are prices p1 and p2
determined independently at
equilibrium?
Question 2,Does the equilibrium
exist?
Walras’ Law (瓦尔拉斯法则 )
Walras’ Law is an identity; i.e,a
statement that is true for any
positive prices (p1,p2),whether these
are equilibrium prices or not.
Walras’ Law
Every consumer’s preferences are
well-behaved so,for any positive
prices (p1,p2),each consumer spends
all of his budget.
For consumer A:
For consumer B:
p x p x p pA A A A1 1 2 2 1 1 2 2* *
p x p x p pB B B B1 1 2 2 1 1 2 2* *
Walras’ Law
p x p x p pA A A A1 1 2 2 1 1 2 2* *
p x p x p pB B B B1 1 2 2 1 1 2 2* *
p x x p x x
p p
A B A B
A B B B
1 1 1 2 2 2
1 1 1 2 2 2
( ) ( )
( ) ( ).
* * * *
Summing gives
Walras’ Lawp x x p x x
p p
A B A B
A B B B
1 1 1 2 2 2
1 1 1 2 2 2
( ) ( )
( ) ( ).
* * * *
Rearranged,p x x
p x x
A B A B
A B A B
1 1 1 1 1
2 2 2 2 2 0
( )
( ),
* *
* *
That is,...
Walras’ Law
.0
)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
This says that the summed market
value of excess demands is zero for
any positive prices p1 and p2 --
this is Walras’ Law.
Implications of Walras’ Law
0)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
Suppose the market for commodity A
is in equilibrium; that is,
.0xx B1A1B*1A*1
Then
implies
.0xx B2A2B*2A*2
So one implication of Walras’ Law for
a two-commodity exchange economy
is that if one market is in equilibrium
then the other market must also be in
equilibrium.
In general,if there are k markets,
then only k-1 prices are determined
at equilibrium.
I.e.,the general equilibrium
determines only relative prices.
Implications of Walras’ Law
Implications of Walras’ Law
What if,for some positive prices p1 and
p2,there is an excess quantity supplied
of commodity 1? That is,
.0xx B1A1B*1A*1
0)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
Then
implies
.0xx B2A2B*2A*2
Implications of Walras’ Law
So a second implication of Walras’ Law
for a two-commodity exchange economy
is that an excess supply in one market
implies an excess demand in the other
market.
Existence of Equilibrium
Require that the aggregate excess
demand functions be a continuous
function,which in turn require that
–Each individual’s demand function
be continuous,or
–Each consumer is small relative to
the market.
Trade in Competitive Markets
Trading in competitive markets
achieves a particular Pareto-optimal
allocation of the endowments.
This is an example of the First
Fundamental Theorem of Welfare
Economics.
First Fundamental Theorem of
Welfare Economics
Given that consumers’ preferences
are well-behaved,trading in perfectly
competitive markets implements a
Pareto-optimal allocation of the
economy’s endowment.
The Equilibrium with Monopoly
See Fig,30.5 in textbook.
The Equilibrium with Monopoly
Figure 30.6 in book.
Implicit Assumptions
No consumption externality (外部效应 )
Agents behave competitively
Equilibrium exists
Market Socialism,The Debate
Lange vs,Hayek
Socialism with economic planning
–State-ownership
–Government chooses prices to
clear markets
Advantage of competitive markets
–Information
Second Fundamental Theorem of
Welfare Economics
The First Theorem is followed by a
second that states that any Pareto-
optimal allocation (i.e,any point on
the contract curve) can be achieved
by trading in competitive markets
provided that endowments are first
appropriately rearranged amongst
the consumers.
Given that consumers’ preferences
are well-behaved,for any Pareto-
optimal allocation there are prices
and an allocation of the total
endowment that makes the Pareto-
optimal allocation implementable by
trading in competitive markets.
Second Fundamental Theorem of
Welfare Economics
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The contract curve
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
A*2x B*2x
A*1x
B*1x
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
A*2x B*2x
A*1x
B*1x
Implemented by competitive
trading from the endowment?.
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this allocation be implemented
by competitive trading from
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this allocation be implemented
by competitive trading from No.
Second Fundamental Theorem
xA2
xA1OA
xB1
xB2
OB
But this allocation is implemented
by competitive trading from q.
A1q
B2q
B1q
A2q
The Assumption about Convexity
Figure 30.8 in book.
Implications of the 2nd Theorem
Market equilibrium is distributionally
neutral.
Redistribution can be carried out for
endowments.
The market plays only the allocation
role.
Exchange
交换
Contents
The Edgeworth (艾奇沃斯 ) Box
Pareto improving trade
Pareto Efficient Allocation
The Core (核心 )
Trade in a competitive market
Walras’ Law of general equilibrium (一般均衡 )
First Fundamental Theorem of Welfare
Economics (福利经济学第一定理 )
Second Fundamental Theorem of Welfare
Economics (福利经济学第二定理 )
Exchange
Two consumers,A and B.
Their endowments of goods 1 and 2
are
E.g,
The total quantities available
A A A? (,)1 2B B B? (,).1 2and
A? (,)6 4? B? (,).2 2and
1 1 6 2 8A B
2 2 4 2 6A B
units of good 1
units of good 2.and
are
Exchange
Edgeworth and Bowley devised a
diagram,called an Edgeworth
box,to show all possible
allocations of the available
quantities of goods 1 and 2
between the two consumers.
Starting an Edgeworth Box
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
Starting an Edgeworth Box
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
The dimensions of
the box are the
quantities available
of the goods.
Feasible Allocations
What allocations of the 8 units of
good 1 and the 6 units of good 2 are
feasible?
How can all of the feasible
allocations be depicted by the
Edgeworth box diagram?
One feasible allocation is the before-
trade allocation; i.e,the endowment
allocation.
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
The endowment
allocation is
A? (,)6 4
B? (,).2 2
and
The Endowment Allocation
Width =1 1 6 2 8A B
Height =
2 2
4 2
6
A B?
A? (,)6 4
B? (,)2 2
The Endowment Allocation
A? (,)6 4OA
OB
6
8?
B? (,)2 2
The Endowment Allocation
A? (,)6 4OA
OB
6
8
4
6
The Endowment Allocation
B? (,)2 2
OA
OB
6
8
4
6
2
2
The Endowment Allocation
A? (,)6 4
B? (,)2 2
OA
OB
6
8
4
6
2
2
The
endowment
allocation
The Endowment Allocation
More generally,…
The Endowment Allocation
The Endowment Allocation
OA
OB
The
endowment
allocation
1 1A B?
2A
2
2
A
B
1A
1B
2B
Other Feasible Allocations
denotes an allocation to
consumer A.
denotes an allocation to
consumer B.
An allocation is feasible if and only if
(,)x xA A1 2
(,)x xB B1 2
x xA B A B1 1 1 1
x xA B A B2 2 2 2,and
Feasible Reallocations
OA
OB
1 1A B?
xA2
2
2
A
B
xA1
xB1
xB2
Feasible Reallocations
OA
OB
1 1A B?
xA2
2
2
A
B
xA1
xB1
xB2
Feasible Reallocations
All points in the box,including the
boundary,represent feasible
allocations of the combined
endowments.
Which allocations will be blocked by
one or both consumers?
Which allocations make both
consumers better off?
Adding Preferences to the Box
2A
1A
xA2
xA1OA
For consumer A.
Adding Preferences to the Box
2A
1A
xA2
xA1
For consumer A.
OA
Adding Preferences to the Box
2B
1B
xB2
xB1
For consumer B.
OB
Adding Preferences to the Box
xB2
xB1
For consumer B.
OB
2B
1B
Adding Preferences to the Box
2B
1BxB1
xB2
For consumer B,O
B
Adding Preferences to the Box
2A
1A
xA2
xA1OA
For consumer A.
Adding Preferences to the Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Edgeworth’s Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Improvement
An allocation of the endowment that
improves the welfare of a consumer
without reducing the welfare of
another is a Pareto-improving
allocation.
Where are the Pareto-improving
allocations?
Edgeworth’s Box
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Improvements
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving allocations
Pareto-Improvements
Since each consumer can refuse to
trade,the only possible outcomes
from exchange are Pareto-improving
allocations.
But which particular Pareto-
improving allocation will be the
outcome of trade?
Pareto-Improvements
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving reallocations
Pareto-Improvements
Pareto-Improvements
Pareto-Improvements
Trade
improves both
A’s and B’s welfares.
This is a Pareto-improvement
over the endowment allocation.
Pareto-Improvements
New mutual gains-to-trade region
is the set of all further Pareto-
improving
reallocations.
Trade
improves both
A’s and B’s welfares.
This is a Pareto-improvement
(帕累托改进 )over the endowment allocation.
Pareto-Improvements
Further trade cannot improve
both A and B’s
welfares.
Pareto-Optimality (帕累托最优 )
Better for
consumer B
Better for
consumer A
Pareto-Optimality
A is strictly better off
but B is strictly worse
off
B is strictly better
off but A is strictly
worse off
Pareto-Optimality
A is strictly better off
but B is strictly worse
off
B is strictly better
off but A is strictly
worse off
Both A
and B are
worse
off
Both A and
B are worse
off
Pareto-Optimality
The allocation is
Pareto-optimal since the
only way one consumer’s
welfare can be increased is to
decrease the welfare of the other
consumer.
Pareto-Optimality
The allocation is
Pareto-optimal since the
only way one consumer’s
welfare can be increased is to
decrease the welfare of the other
consumer.
The Marginal Rates of
Substitution Are Equal.
Pareto-Optimality
Where are all of the Pareto-optimal
allocations of the endowment?
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
All the allocations marked by
a are Pareto-optimal.
Pareto-Optimality
The contract curve (契约线 ) is the set
of all Pareto-optimal allocations.
Pareto-Optimality
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
All the allocations marked by
a are Pareto-optimal.
The contract curve
Pareto-Optimality
But to which of the many allocations
on the contract curve will consumers
trade?
That depends upon how trade is
conducted.
In perfectly competitive markets? By
one-on-one bargaining?
The Core (核心 )
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The set of Pareto-
improving reallocations
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-optimal trades blocked
by B
Pareto-optimal trades blocked
by A
The Core
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Pareto-optimal trades not blocked
by A or B are the core.
The Core
The core is the set of all Pareto-
optimal allocations that are welfare-
improving for both consumers
relative to their own endowments.
Rational trade should achieve a core
allocation.
The Core
But which core allocation?
Again,that depends upon the
manner in which trade is conducted.
Trade in Competitive Markets
Consider trade in perfectly
competitive markets.
Each consumer is a price-taker trying
to maximize her own utility given p1,
p2 and her own endowment,That is,
...
Trade in Competitive Markets
2A
1A
xA2
xA1OA
For consumer A.
p x p x p pA A A A1 1 2 2 1 1 2 2
x A2*
x A1*
Trade in Competitive Markets
So given p1 and p2,consumer A’s net
demands for commodities 1 and 2
are x A A
1 1* x A A2 2*,and
Trade in Competitive Markets
And,similarly,for consumer B …
Trade in Competitive Markets
2B
1B
xB2
xB1
For consumer B.
OB x B1*
x B2*
p x p x p pB B B B1 1 2 2 1 1 2 2
Trade in Competitive Markets
So given p1 and p2,consumer B’s net
demands for commodities 1 and 2
are x B B
1 1* x B B2 2*,and
Trade in Competitive Markets
A general equilibrium occurs when
prices p1 and p2 cause both the
markets for commodities 1 and 2 to
clear; i.e.
x xA B A B1 1 1 1* *
x xA B A B2 2 2 2* *,and
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this PO allocation be
achieved?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
x B1*
x B2*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
But x xA B A B1 1 1 1* *
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
and x xA B A B2 2 2 2* *
Trade in Competitive Markets
So at the given prices p1 and p2 there
is an
– excess supply of commodity 1
– excess demand for commodity 2.
Neither market clears so the prices
p1 and p2 do not cause a general
equilibrium.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
So this PO allocation cannot be
achieved by competitive trading.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
Since there is an excess demand for
commodity 2,p2 will rise.
Since there is an excess supply of
commodity 1,p1 will fall.
The slope of the budget constraints
is - p1/p2 so the budget constraints
will pivot about the endowment point
and become less steep.
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Which PO allocations can be
achieved by competitive trading?
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer A
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Budget constraint for consumer B
x A2*
x A1*
x B1*
x B2*
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
So x xA B A B1 1 1 1* *
Trade in Competitive Markets
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
x A2*
x A1*
x B1*
x B2*
and x xA B A B2 2 2 2* *
Trade in Competitive Markets
At the new prices p1 and p2 both
markets clear; there is a general
equilibrium.
Question 1,Are prices p1 and p2
determined independently at
equilibrium?
Question 2,Does the equilibrium
exist?
Walras’ Law (瓦尔拉斯法则 )
Walras’ Law is an identity; i.e,a
statement that is true for any
positive prices (p1,p2),whether these
are equilibrium prices or not.
Walras’ Law
Every consumer’s preferences are
well-behaved so,for any positive
prices (p1,p2),each consumer spends
all of his budget.
For consumer A:
For consumer B:
p x p x p pA A A A1 1 2 2 1 1 2 2* *
p x p x p pB B B B1 1 2 2 1 1 2 2* *
Walras’ Law
p x p x p pA A A A1 1 2 2 1 1 2 2* *
p x p x p pB B B B1 1 2 2 1 1 2 2* *
p x x p x x
p p
A B A B
A B B B
1 1 1 2 2 2
1 1 1 2 2 2
( ) ( )
( ) ( ).
* * * *
Summing gives
Walras’ Lawp x x p x x
p p
A B A B
A B B B
1 1 1 2 2 2
1 1 1 2 2 2
( ) ( )
( ) ( ).
* * * *
Rearranged,p x x
p x x
A B A B
A B A B
1 1 1 1 1
2 2 2 2 2 0
( )
( ),
* *
* *
That is,...
Walras’ Law
.0
)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
This says that the summed market
value of excess demands is zero for
any positive prices p1 and p2 --
this is Walras’ Law.
Implications of Walras’ Law
0)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
Suppose the market for commodity A
is in equilibrium; that is,
.0xx B1A1B*1A*1
Then
implies
.0xx B2A2B*2A*2
So one implication of Walras’ Law for
a two-commodity exchange economy
is that if one market is in equilibrium
then the other market must also be in
equilibrium.
In general,if there are k markets,
then only k-1 prices are determined
at equilibrium.
I.e.,the general equilibrium
determines only relative prices.
Implications of Walras’ Law
Implications of Walras’ Law
What if,for some positive prices p1 and
p2,there is an excess quantity supplied
of commodity 1? That is,
.0xx B1A1B*1A*1
0)xx(p
)xx(p
B
2
A
2
B*
2
A*
22
B
1
A
1
B*
1
A*
11
Then
implies
.0xx B2A2B*2A*2
Implications of Walras’ Law
So a second implication of Walras’ Law
for a two-commodity exchange economy
is that an excess supply in one market
implies an excess demand in the other
market.
Existence of Equilibrium
Require that the aggregate excess
demand functions be a continuous
function,which in turn require that
–Each individual’s demand function
be continuous,or
–Each consumer is small relative to
the market.
Trade in Competitive Markets
Trading in competitive markets
achieves a particular Pareto-optimal
allocation of the endowments.
This is an example of the First
Fundamental Theorem of Welfare
Economics.
First Fundamental Theorem of
Welfare Economics
Given that consumers’ preferences
are well-behaved,trading in perfectly
competitive markets implements a
Pareto-optimal allocation of the
economy’s endowment.
The Equilibrium with Monopoly
See Fig,30.5 in textbook.
The Equilibrium with Monopoly
Figure 30.6 in book.
Implicit Assumptions
No consumption externality (外部效应 )
Agents behave competitively
Equilibrium exists
Market Socialism,The Debate
Lange vs,Hayek
Socialism with economic planning
–State-ownership
–Government chooses prices to
clear markets
Advantage of competitive markets
–Information
Second Fundamental Theorem of
Welfare Economics
The First Theorem is followed by a
second that states that any Pareto-
optimal allocation (i.e,any point on
the contract curve) can be achieved
by trading in competitive markets
provided that endowments are first
appropriately rearranged amongst
the consumers.
Given that consumers’ preferences
are well-behaved,for any Pareto-
optimal allocation there are prices
and an allocation of the total
endowment that makes the Pareto-
optimal allocation implementable by
trading in competitive markets.
Second Fundamental Theorem of
Welfare Economics
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
The contract curve
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
A*2x B*2x
A*1x
B*1x
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
A*2x B*2x
A*1x
B*1x
Implemented by competitive
trading from the endowment?.
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this allocation be implemented
by competitive trading from
Second Fundamental Theorem
2A
1A
xA2
xA1OA
2B
1BxB1
xB2
OB
Can this allocation be implemented
by competitive trading from No.
Second Fundamental Theorem
xA2
xA1OA
xB1
xB2
OB
But this allocation is implemented
by competitive trading from q.
A1q
B2q
B1q
A2q
The Assumption about Convexity
Figure 30.8 in book.
Implications of the 2nd Theorem
Market equilibrium is distributionally
neutral.
Redistribution can be carried out for
endowments.
The market plays only the allocation
role.
