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ExchangeExchange
© 2010 W. W. Norton & Company, Inc.
ExchangeExchange
�Two consumers, A and B.�Two consumers, A and B.
�Their endowments of goods 1 and 2 �Their endowments of goods 1 and 2
are ωωωω ωωωω ωωωωA A A==== ( , )1 2 ωωωω ωωωω ωωωωB B B==== ( , ) .1 2and
�E.g.
ωωωω ωωωω ωωωω==== ( , )1 2 ωωωω ωωωω ωωωω==== ( , ) .1 2and
ωωωω A ==== ( , )6 4 ωωωω B ==== ( , ) .2 2and�E.g.
�The total quantities available
ωωωω ==== ( , )6 4 ωωωω ==== ( , ) .2 2and
A Bωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====A B
units of good 1are
ωωωω ωωωω2 2 4 2 6A B++++ ==== ++++ ==== units of good 2.and
© 2010 W. W. Norton & Company, Inc. 2
ExchangeExchange
�Edgeworth and Bowley devised a
diagram, called an Edgeworth box, to diagram, called an Edgeworth box, to
show all possible allocations of the show all possible allocations of the
available quantities of goods 1 and 2
between the two consumers.between the two consumers.
© 2010 W. W. Norton & Company, Inc. 3
Starting an Edgeworth BoxStarting an Edgeworth Box
© 2010 W. W. Norton & Company, Inc. 4
Starting an Edgeworth BoxStarting an Edgeworth Box
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
© 2010 W. W. Norton & Company, Inc. 5
Width = ωωωω ωωωω1 1 6 2 8++++ ==== ++++ ====
Starting an Edgeworth BoxStarting an Edgeworth Box
Height =Height =
ωωωω ωωωω2 2A B++++ωωωω ωωωω2 2
4 2
++++==== ++++6====
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
© 2010 W. W. Norton & Company, Inc. 6
Width = ωωωω ωωωω1 1 6 2 8++++ ==== ++++ ====
Starting an Edgeworth BoxStarting an Edgeworth Box
Height =Height =
ωωωω ωωωω2 2A B++++
The dimensions of
the box are theωωωω ωωωω2 2
4 2
++++==== ++++
the box are the
quantities available
6====quantities available
of the goods.
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
© 2010 W. W. Norton & Company, Inc. 7
Width = ωωωω ωωωω1 1 6 2 8++++ ==== ++++ ====
Feasible AllocationsFeasible Allocations
What allocations of the 8 units of �What allocations of the 8 units of
good 1 and the 6 units of good 2 are good 1 and the 6 units of good 2 are
feasible?
How can all of the feasible �How can all of the feasible
allocations be depicted by the allocations be depicted by the
Edgeworth box diagram?
© 2010 W. W. Norton & Company, Inc. 8
Feasible AllocationsFeasible Allocations
What allocations of the 8 units of �What allocations of the 8 units of
good 1 and the 6 units of good 2 are good 1 and the 6 units of good 2 are
feasible?
How can all of the feasible �How can all of the feasible
allocations be depicted by the allocations be depicted by the
Edgeworth box diagram?
One feasible allocation is the before-�One feasible allocation is the before-
trade allocation; i.e. the endowment trade allocation; i.e. the endowment
allocation.
© 2010 W. W. Norton & Company, Inc. 9
The Endowment AllocationThe Endowment Allocation
Height = The endowmentHeight =
ωωωω ωωωω2 2A B++++
The endowment
allocation isAωωωω ωωωω2 2
4 2
A B++++==== ++++
ωωωω A ==== ( , )6 4and4 2
6
==== ++++==== ωωωω B ==== ( , ) .2 2
and
ωωωω ==== ( , ) .2 2
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
© 2010 W. W. Norton & Company, Inc. 10
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
The Endowment AllocationThe Endowment Allocation
Height =Height =
ωωωω ωωωω2 2A B++++ωωωω ωωωω2 2
4 2
A B++++==== ++++4 2
6
==== ++++====
ωωωω A ==== ( , )6 4
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ====
ωωωω A ==== ( , )6 4
ωωωω B ==== ( , )2 2
© 2010 W. W. Norton & Company, Inc. 11
Width = ωωωω ωωωω1 1 6 2 8A B++++ ==== ++++ ==== ωωωω ==== ( , )2 2
The Endowment Allocation
O
The Endowment Allocation
OB
6
ωωωω A ==== ( , )6 4OA ωωωω A ==== ( , )6 4OA
8ωωωω B ==== ( , )2 2
© 2010 W. W. Norton & Company, Inc. 12
8ωωωω ==== ( , )2 2
The Endowment Allocation
O
The Endowment Allocation
OB
6
44
ωωωω A ==== ( , )6 4OA ωωωω A ==== ( , )6 4OA
8
6
© 2010 W. W. Norton & Company, Inc. 13
8
The Endowment Allocation
O2
The Endowment Allocation
OB
2
2
2
6
44
OA
ωωωω B ==== ( , )2 2
OA
8
6
© 2010 W. W. Norton & Company, Inc. 14
ωωωω ==== ( , )2 28
The Endowment Allocation
O2
The Endowment Allocation
OB
2
2
2
6
4The
4 endowment
allocation
ωωωω A ==== ( , )6 4OA
allocation
ωωωω A ==== ( , )6 4
ωωωω B ==== ( , )2 2
OA
8
6
© 2010 W. W. Norton & Company, Inc. 15
ωωωω ==== ( , )2 28
The Endowment AllocationThe Endowment Allocation
More generally, …More generally, …
© 2010 W. W. Norton & Company, Inc. 16
The Endowment AllocationThe Endowment Allocation
Oωωωω 1
B
OB
ωωωω A ωωωω 2B
ωωωω 2A
++++
ωωωω 2B
Theωωωω 2A
ωωωω B
++++
endowment
allocation
ωωωω 2A
ωωωω 2B
OA
allocation
AOA
ωωωω ωωωωA B++++ωωωω 1
A
© 2010 W. W. Norton & Company, Inc. 17
ωωωω ωωωω1 1A B++++
Other Feasible AllocationsOther Feasible Allocations
denotes an allocation to ( , )x xA A� denotes an allocation to
consumer A.
( , )x xA A1 2
consumer A.
� denotes an allocation to
consumer B.
( , )x xB B1 2
consumer B.
�An allocation is feasible if and only if�An allocation is feasible if and only if
x xA B A B1 1 1 1++++ ≤≤≤≤ ++++ωωωω ωωωωx x1 1 1 1++++ ≤≤≤≤ ++++ωωωω ωωωω
x xA B A B2 2 2 2++++ ≤≤≤≤ ++++ωωωω ωωωω .and x x2 2 2 2++++ ≤≤≤≤ ++++ωωωω ωωωω .and
© 2010 W. W. Norton & Company, Inc. 18
Feasible ReallocationsFeasible Reallocations
Ox B1
OB
ωωωω A Bωωωω 2A
++++x B2
ωωωω B
++++
x A2
ωωωω 2B
OA
x A2
AOA
ωωωω ωωωωA B++++x A1
© 2010 W. W. Norton & Company, Inc. 19
ωωωω ωωωω1 1A B++++
Feasible ReallocationsFeasible Reallocations
Ox B1
OB
ωωωω A x Bωωωω 2A
++++
x B2
ωωωω B
++++
x A2
ωωωω 2B
OAA
OA
ωωωω ωωωωA B++++x A1
© 2010 W. W. Norton & Company, Inc. 20
ωωωω ωωωω1 1A B++++
Feasible ReallocationsFeasible Reallocations
�All points in the box, including the
boundary, represent feasible boundary, represent feasible
allocations of the combined allocations of the combined
endowments.
© 2010 W. W. Norton & Company, Inc. 21
Feasible ReallocationsFeasible Reallocations
�All points in the box, including the
boundary, represent feasible boundary, represent feasible
allocations of the combined allocations of the combined
endowments.
�Which allocations will be blocked by
one or both consumers?one or both consumers?
�Which allocations make both �Which allocations make both
consumers better off?
© 2010 W. W. Norton & Company, Inc. 22
Adding Preferences to the BoxAdding Preferences to the Boxx A2
For consumer A.x 2
ωωωω Aωωωω 2A
Oωωωω 1
A x A1
OA
© 2010 W. W. Norton & Company, Inc. 23
Adding Preferences to the BoxAdding Preferences to the Boxx A2
For consumer A.x 2
ωωωω Aωωωω 2A
Oωωωω 1
A x A1
OA
© 2010 W. W. Norton & Company, Inc. 24
Adding Preferences to the BoxAdding Preferences to the Boxx B2
For consumer B.x 2
ωωωω Bωωωω 2B
Oωωωω 1
Bx B1
OB
© 2010 W. W. Norton & Company, Inc. 25
Adding Preferences to the BoxAdding Preferences to the Boxx B2
For consumer B.x 2
ωωωω Bωωωω 2B
Ox B1
OB ωωωω 1B
© 2010 W. W. Norton & Company, Inc. 26
Adding Preferences to the BoxAdding Preferences to the Boxωωωω 1
Bx B1
For consumer B. OBx 1 OB
ωωωω 2B
x B2
© 2010 W. W. Norton & Company, Inc. 27
Adding Preferences to the BoxAdding Preferences to the Boxx A2
For consumer A.x 2
ωωωω Aωωωω 2A
Oωωωω 1
A x A1
OA
© 2010 W. W. Norton & Company, Inc. 28
Adding Preferences to the BoxAdding Preferences to the Boxx A2
Bx 2
ωωωω 1B
x B1 OB
ωωωω A
ωωωω 2B
ωωωω 2A
Oωωωω 1
A x A1
OA
© 2010 W. W. Norton & Company, Inc. 29x B2
Edgeworth’s BoxEdgeworth’s Boxx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 30x B2
Pareto-ImprovementPareto-Improvement
�An allocation of the endowment that
improves the welfare of a consumer improves the welfare of a consumer
without reducing the welfare of without reducing the welfare of
another is a Pareto-improving
allocation.allocation.
�Where are the Pareto-improving �Where are the Pareto-improving
allocations?
© 2010 W. W. Norton & Company, Inc. 31
Edgeworth’s BoxEdgeworth’s Boxx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 32x B2
Pareto-ImprovementsPareto-Improvementsx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
The set of Pareto-
© 2010 W. W. Norton & Company, Inc. 33x B2
The set of Pareto-
improving allocations
Pareto-ImprovementsPareto-Improvements
�Since each consumer can refuse to
trade, the only possible outcomes trade, the only possible outcomes
from exchange are Pareto-improving from exchange are Pareto-improving
allocations.
�But which particular Pareto-
improving allocation will be the improving allocation will be the
outcome of trade?
© 2010 W. W. Norton & Company, Inc. 34
Pareto-ImprovementsPareto-Improvementsx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
The set of Pareto-
© 2010 W. W. Norton & Company, Inc. 35x B2
The set of Pareto-
improving reallocations
Pareto-ImprovementsPareto-Improvements
© 2010 W. W. Norton & Company, Inc. 36
Pareto-ImprovementsPareto-Improvements
© 2010 W. W. Norton & Company, Inc. 37
Pareto-ImprovementsPareto-Improvements
Trade
improves bothimproves both
A’s and B’s welfares.
This is a Pareto-improvement
over the endowment allocation.© 2010 W. W. Norton & Company, Inc. 38
over the endowment allocation.
Pareto-ImprovementsPareto-ImprovementsNew mutual gains-to-trade region
is the set of all further Pareto-is the set of all further Pareto-
improvingimproving
reallocations.
Trade
improves bothimproves both
A’s and B’s welfares.
This is a Pareto-improvement
over the endowment allocation.© 2010 W. W. Norton & Company, Inc. 39
over the endowment allocation.
Pareto-ImprovementsPareto-ImprovementsFurther trade cannot improveFurther trade cannot improve
both A and B’s
welfares.welfares.
© 2010 W. W. Norton & Company, Inc. 40
Pareto-OptimalityPareto-Optimality
Better forBetter for
consumer Aconsumer A
Better forBetter for
consumer B
© 2010 W. W. Norton & Company, Inc. 41
consumer B
Pareto-OptimalityPareto-Optimality
A is strictly better offA is strictly better off
but B is strictly worse
offoff
© 2010 W. W. Norton & Company, Inc. 42
Pareto-OptimalityPareto-Optimality
A is strictly better offA is strictly better off
but B is strictly worse
offoff
B is strictly better
off but A is strictlyoff but A is strictly
worse off
© 2010 W. W. Norton & Company, Inc. 43
worse off
Pareto-OptimalityPareto-Optimality
A is strictly better offBoth A and
B are worse A is strictly better off
but B is strictly worse
off
B are worse
offoff
B is strictly better
off but A is strictlyoff but A is strictly
worse off
© 2010 W. W. Norton & Company, Inc. 44
worse off
Pareto-OptimalityPareto-Optimality
A is strictly better offBoth A and
B are worse A is strictly better off
but B is strictly worse
off
B are worse
offoff
B is strictly better
off but A is strictlyBoth A
off but A is strictly
worse off
Both A
and B are
worse© 2010 W. W. Norton & Company, Inc. 45
worse off worse
off
Pareto-OptimalityPareto-Optimality
The allocation isThe allocation is
Pareto-optimal since the
only way one consumer’sonly way one consumer’s
welfare can be increased is towelfare can be increased is to
decrease the welfare of the other
consumer.© 2010 W. W. Norton & Company, Inc. 46
consumer.
Pareto-OptimalityPareto-OptimalityAn allocation where convexAn allocation where convex
indifference curves are “only
just back-to-back” isjust back-to-back” is
Pareto-optimal.
The allocation is
Pareto-optimal.
The allocation is
Pareto-optimal since the
only way one consumer’sonly way one consumer’s
welfare can be increased is towelfare can be increased is to
decrease the welfare of the other
consumer.© 2010 W. W. Norton & Company, Inc. 47
consumer.
Pareto-OptimalityPareto-Optimality
�Where are all of the Pareto-optimal
allocations of the endowment?allocations of the endowment?
© 2010 W. W. Norton & Company, Inc. 48
Pareto-OptimalityPareto-Optimalityx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 49x B2
Pareto-OptimalityPareto-Optimalityx A2 All the allocations marked by
a are Pareto-optimal.
ωωωω 1B
x B O
a are Pareto-optimal.
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 50x B2
Pareto-OptimalityPareto-Optimality
�The contract curve is the set of all
Pareto-optimal allocations.Pareto-optimal allocations.
© 2010 W. W. Norton & Company, Inc. 51
Pareto-OptimalityPareto-Optimalityx A2 All the allocations marked by
a are Pareto-optimal.
ωωωω 1B
x B O
a are Pareto-optimal.
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
The contract curve© 2010 W. W. Norton & Company, Inc. 52x B
2
The contract curve
Pareto-OptimalityPareto-Optimality
�But to which of the many allocations
on the contract curve will consumers on the contract curve will consumers
trade?trade?
�That depends upon how trade is
conducted.
� In perfectly competitive markets? By � In perfectly competitive markets? By
one-on-one bargaining?one-on-one bargaining?
© 2010 W. W. Norton & Company, Inc. 53
The CoreThe Corex A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
The set of Pareto-
© 2010 W. W. Norton & Company, Inc. 54x B2
The set of Pareto-
improving reallocations
The CoreThe Corex A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 55x B2
The CoreThe Corex A2 Pareto-optimal trades blocked
ωωωω 1B
x B O
Pareto-optimal trades blocked
by Bωωωω 1x B
1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
Pareto-optimal trades blocked
© 2010 W. W. Norton & Company, Inc. 56x B2
Pareto-optimal trades blocked
by A
The CoreThe Corex A2 Pareto-optimal trades not blocked
by A or B
ωωωω 1B
x B O
by A or B
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 57x B2
The CoreThe Corex A2 Pareto-optimal trades not blocked
by A or B are the core.
ωωωω 1B
x B O
by A or B are the core.
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 58x B2
The CoreThe Core�The core is the set of all Pareto-�The core is the set of all Pareto-
optimal allocations that are welfare-
improving for both consumers improving for both consumers
relative to their own endowments.relative to their own endowments.
�Rational trade should achieve a core
allocation.allocation.
© 2010 W. W. Norton & Company, Inc. 59
The CoreThe Core
�But which core allocation?
�Again, that depends upon the
manner in which trade is conducted.manner in which trade is conducted.
© 2010 W. W. Norton & Company, Inc. 60
Trade in Competitive MarketsTrade in Competitive Markets
�Consider trade in perfectly
competitive markets.competitive markets.
�Each consumer is a price-taker trying �Each consumer is a price-taker trying
to maximize her own utility given p1, 1p2 and her own endowment. That is,
......
© 2010 W. W. Norton & Company, Inc. 61
Trade in Competitive MarketsTrade in Competitive Marketsx A2
For consumer A.x 2
p x p x p pA A A A1 1 2 2 1 1 2 2++++ ==== ++++ωωωω ωωωω
A*
ωωωω A
x A2*
ωωωω 2A
Oωωωω 1
A x A1
OA x A1*
© 2010 W. W. Norton & Company, Inc. 62
Trade in Competitive MarketsTrade in Competitive Markets
�So given p1 and p2, consumer A’s net
demands for commodities 1 and 2 1 2
demands for commodities 1 and 2
arearex A A1 1* −−−− ωωωω x A A
2 2*
.−−−− ωωωωand
© 2010 W. W. Norton & Company, Inc. 63
Trade in Competitive MarketsTrade in Competitive Markets
�And, similarly, for consumer B …
© 2010 W. W. Norton & Company, Inc. 64
Trade in Competitive MarketsTrade in Competitive Marketsx B2
For consumer B.x 2
p x p x p pB B B B1 1 2 2 1 1 2 2++++ ==== ++++ωωωω ωωωω
ωωωω Bωωωω 2B
x B2*x B2*
Oωωωω 1
Bx B1
OB x B1*
© 2010 W. W. Norton & Company, Inc. 65
Trade in Competitive MarketsTrade in Competitive Markets
�So given p1 and p2, consumer B’s net
demands for commodities 1 and 2 1 2
demands for commodities 1 and 2
arearex B B1 1* −−−− ωωωω x B B
2 2*
.−−−− ωωωωand
© 2010 W. W. Norton & Company, Inc. 66
Trade in Competitive MarketsTrade in Competitive Markets
�A general equilibrium occurs when
prices p and p cause both the prices p1 and p2 cause both the
markets for commodities 1 and 2 to markets for commodities 1 and 2 to
clear; i.e.
x xA B A B1 1 1 1* *++++ ==== ++++ωωωω ωωωωA B A B* *x xA B A B2 2 2 2* *
.++++ ==== ++++ωωωω ωωωωand
© 2010 W. W. Norton & Company, Inc. 67
Trade in Competitive MarketsTrade in Competitive Marketsx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 68x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2 Can this PO allocation be
achieved?
ωωωω 1B
x B O
achieved?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 69x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2
Budget constraint for consumer A
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 70x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2
Budget constraint for consumer A
ωωωω 1B
x B Oωωωω 1x B1 OB
A*
ωωωω 2A ωωωω 2
Bx A2*
ωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 71x B2
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
A*
ωωωω 2A ωωωω 2
Bx A2*
ωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 72x B2Budget constraint for consumer B
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x B1*
ωωωω 1x B1 OB
B*
A*
x B2*
ωωωω 2A ωωωω 2
Bx A2*
ωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 73x B2Budget constraint for consumer B
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x B1*
ωωωω 1x B1 OB
B*
A*
x B2*
ωωωω 2A ωωωω 2
Bx A2*
ωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
x A1*
But© 2010 W. W. Norton & Company, Inc. 74x B
2
1But
x xA B A B1 1 1 1* *++++ <<<< ++++ωωωω ωωωω
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x B1*
ωωωω 1x B1 OB
B*
A*
x B2*
ωωωω 2A ωωωω 2
Bx A2*
ωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
x A1*
and© 2010 W. W. Norton & Company, Inc. 75x B
2
1and
x xA B A B2 2 2 2* *++++ >>>> ++++ωωωω ωωωω
Trade in Competitive MarketsTrade in Competitive Markets
�So at the given prices p1 and p2 there
is an1 2
is an
– excess supply of commodity 1– excess supply of commodity 1
– excess demand for commodity 2.– excess demand for commodity 2.
�Neither market clears so the prices
p and p do not cause a general p1 and p2 do not cause a general
equilibrium.equilibrium.
© 2010 W. W. Norton & Company, Inc. 76
Trade in Competitive MarketsTrade in Competitive Marketsx A2 So this PO allocation cannot be
achieved by competitive trading.
ωωωω 1B
x B O
achieved by competitive trading.
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 77x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2 Which PO allocations can be
achieved by competitive trading?
ωωωω 1B
x B O
achieved by competitive trading?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 78x B2
Trade in Competitive MarketsTrade in Competitive Markets
�Since there is an excess demand for
commodity 2, p will rise.commodity 2, p2 will rise.
�Since there is an excess supply of �Since there is an excess supply of
commodity 1, p1 will fall.1
�The slope of the budget constraints
is - p /p so the budget constraints is - p1/p2 so the budget constraints
will pivot about the endowment point will pivot about the endowment point
and become less steep.
© 2010 W. W. Norton & Company, Inc. 79
Trade in Competitive MarketsTrade in Competitive Marketsx A2 Which PO allocations can be
achieved by competitive trading?
ωωωω 1B
x B O
achieved by competitive trading?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 80x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2 Which PO allocations can be
achieved by competitive trading?
ωωωω 1B
x B O
achieved by competitive trading?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 81x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2 Which PO allocations can be
achieved by competitive trading?
ωωωω 1B
x B O
achieved by competitive trading?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 82x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2
Budget constraint for consumer A
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 83x B2
Trade in Competitive MarketsTrade in Competitive Marketsx A2
Budget constraint for consumer A
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bx A* ωωωω 2
A
ωωωω A AOA
ωωωω 2B
x A2*
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 84x B2
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bx A* ωωωω 2
A
ωωωω A AOA
ωωωω 2B
x A2*
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 85x B2Budget constraint for consumer B
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x 1
ωωωω 1x B1 OB
x B*x B2*
ωωωω 2A ωωωω 2
Bx A* ωωωω 2
A
ωωωω A AOA
ωωωω 2B
x A2*
ωωωω 1A x A
1
OA
x A1*
© 2010 W. W. Norton & Company, Inc. 86x B2Budget constraint for consumer B
1
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x 1
ωωωω 1x B1 OB
x B*x B2*
ωωωω 2A ωωωω 2
Bx A* ωωωω 2
A
ωωωω A AOA
ωωωω 2B
x A2*
ωωωω 1A x A
1
OA
x A1*
So© 2010 W. W. Norton & Company, Inc. 87x B
2
1So
x xA B A B1 1 1 1* *++++ ==== ++++ωωωω ωωωω
Trade in Competitive MarketsTrade in Competitive Marketsx A2
x B1*
ωωωω 1B
x B O
x 1
ωωωω 1x B1 OB
x B*x B2*
ωωωω 2A ωωωω 2
Bx A* ωωωω 2
A
ωωωω A AOA
ωωωω 2B
x A2*
ωωωω 1A x A
1
OA
x A1*
and© 2010 W. W. Norton & Company, Inc. 88x B
2
1and
x xA B A B2 2 2 2* *++++ ==== ++++ωωωω ωωωω
Trade in Competitive MarketsTrade in Competitive Markets
�At the new prices p and p both �At the new prices p1 and p2 both
markets clear; there is a general
equilibrium.
�Trading in competitive markets �Trading in competitive markets
achieves a particular Pareto-optimal achieves a particular Pareto-optimal
allocation of the endowments.
�This is an example of the First �This is an example of the First
Fundamental Theorem of Welfare Fundamental Theorem of Welfare
Economics.
© 2010 W. W. Norton & Company, Inc. 89
First Fundamental Theorem of
Welfare EconomicsWelfare Economics
�Given that consumers’ preferences
are well-behaved, trading in perfectly are well-behaved, trading in perfectly
competitive markets implements a competitive markets implements a
Pareto-optimal allocation of the
economy’s endowment.economy’s endowment.
© 2010 W. W. Norton & Company, Inc. 90
Second Fundamental Theorem of
Welfare EconomicsWelfare Economics
�The First Theorem is followed by a
second that states that any Pareto-second that states that any Pareto-
optimal allocation (i.e. any point on optimal allocation (i.e. any point on
the contract curve) can be achieved
by trading in competitive markets by trading in competitive markets
provided that endowments are first provided that endowments are first
appropriately rearranged amongst
the consumers.the consumers.
© 2010 W. W. Norton & Company, Inc. 91
Second Fundamental Theorem of
Welfare EconomicsWelfare Economics
�Given that consumers’ preferences
are well-behaved, for any Pareto-are well-behaved, for any Pareto-
optimal allocation there are prices optimal allocation there are prices
and an allocation of the total
endowment that makes the Pareto-endowment that makes the Pareto-
optimal allocation implementable by optimal allocation implementable by
trading in competitive markets.
© 2010 W. W. Norton & Company, Inc. 92
Second Fundamental TheoremSecond Fundamental Theoremx A2
ωωωω 1B
x B Oωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
The contract curve© 2010 W. W. Norton & Company, Inc. 93x B
2
The contract curve
Second Fundamental TheoremSecond Fundamental Theoremx A2
ωωωω 1B
x B OB*
1x ωωωω 1x B1 OB
1x
A*x B*x
ωωωω 2A ωωωω 2
B
A*2x B*
2x
ωωωω 2A
ωωωω A AOA
ωωωω 2B
A*x ωωωω 1A x A
1
OA A*1x
© 2010 W. W. Norton & Company, Inc. 94x B2
Second Fundamental TheoremSecond Fundamental Theoremx A2
Implemented by competitive
trading from the endowment ωωωω.ωωωω 1
Bx B O
B*1x
trading from the endowment ωωωω.ωωωω 1x B
1 OB1x
A*x B*x
ωωωω 2A ωωωω 2
B
A*2x B*
2x
ωωωω 2A
ωωωω A AOA
ωωωω 2B
A*x ωωωω 1A x A
1
OA A*1x
© 2010 W. W. Norton & Company, Inc. 95x B2
Second Fundamental TheoremSecond Fundamental Theoremx A2
Can this allocation be implemented
by competitive trading from ωωωω?
ωωωω 1B
x B O
by competitive trading from ωωωω?
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 96x B2
Second Fundamental TheoremSecond Fundamental Theoremx A2
Can this allocation be implemented
by competitive trading from ωωωω? No.
ωωωω 1B
x B O
by competitive trading from ωωωω? No.
ωωωω 1x B1 OB
ωωωω 2A ωωωω 2
Bωωωω 2A
ωωωω A AOA
ωωωω 2B
ωωωω 1A x A
1
OA
© 2010 W. W. Norton & Company, Inc. 97x B2
Second Fundamental TheoremSecond Fundamental Theoremx A2
But this allocation is implemented
by competitive trading from θθθθ.
x B O
by competitive trading from θθθθ.B1θθθθx B
1 OB1θθθθ
BθθθθAθθθθ B2θθθθA
2θθθθ
AOA Aθθθθ x A1
OA A1θθθθ
© 2010 W. W. Norton & Company, Inc. 98x B2
Walras’ LawWalras’ Law
�Walras’ Law is an identity; i.e. a
statement that is true for any statement that is true for any
positive prices (p1,p2), whether these
are equilibrium prices or not.are equilibrium prices or not.
© 2010 W. W. Norton & Company, Inc. 99
Walras’ LawWalras’ Law
�Every consumer’s preferences are �Every consumer’s preferences are
well-behaved so, for any positive
prices (p ,p ), each consumer spends prices (p1,p2), each consumer spends
all of his budget.all of his budget.
�For consumer A:
p x p x p pA A A A* *++++ ==== ++++ωωωω ωωωωFor consumer B:
p x p x p pA A A A1 1 2 2 1 1 2 2
* *++++ ==== ++++ωωωω ωωωωFor consumer B:
p x p x p pB B B B1 1 2 2 1 1 2 2
* *++++ ==== ++++ωωωω ωωωω
© 2010 W. W. Norton & Company, Inc. 100
Walras’ LawWalras’ Law
A A A A* *++++ ==== ++++ωωωω ωωωωp x p x p pA A A A1 1 2 2 1 1 2 2
* *++++ ==== ++++ωωωω ωωωω
p x p x p pB B B B* *++++ ==== ++++ωωωω ωωωωp x p x p pB B B B1 1 2 2 1 1 2 2
* *++++ ==== ++++ωωωω ωωωω
Summing gives
p x x p x xA B A B1 1 1 2 2 2( ) ( )
* * * *++++ ++++ ++++
Summing gives
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
( ) ( )
( ) ( ) .
* * * *++++ ++++ ++++
==== ++++ ++++ ++++ωωωω ωωωω ωωωω ωωωωp pA B B B1 1 1 2 2 2( ) ( ) .==== ++++ ++++ ++++ωωωω ωωωω ωωωω ωωωω
© 2010 W. W. Norton & Company, Inc. 101
Walras’ LawWalras’ Law
p x x p x xA B A B( ) ( )
* * * *++++ ++++ ++++p x x p x x
p p
A B A B
A B B B
1 1 1 2 2 2( ) ( )
( ) ( ) .
* * * *++++ ++++ ++++
==== ++++ ++++ ++++ωωωω ωωωω ωωωω ωωωωp pA B B B1 1 1 2 2 2( ) ( ) .==== ++++ ++++ ++++ωωωω ωωωω ωωωω ωωωω
Rearranged,Rearranged,
p x xA B A B1 1 1 1 1( )
* *++++ −−−− −−−− ++++ωωωω ωωωωp x x
p x x
A B A B
A B A B
1 1 1 1 1
2 2 2 2 2 0
( )
( ) .
* *
* *
++++ −−−− −−−− ++++
++++ −−−− −−−− ====
ωωωω ωωωω
ωωωω ωωωωp x xA B A B2 2 2 2 2 0( ) .
* *++++ −−−− −−−− ====ωωωω ωωωω
That is, ...That is, ...
© 2010 W. W. Norton & Company, Inc. 102
Walras’ LawWalras’ Law
)xx(pBAB*A* ++++ωωωω−−−−ωωωω−−−−++++
)xx(p
)xx(p
BAB*A*
B1
A1
B*1
A*11
ωωωω−−−−ωωωω−−−−++++
++++ωωωω−−−−ωωωω−−−−++++
.0
)xx(pB2
A2
B*2
A*22
====ωωωω−−−−ωωωω−−−−++++
.0====This says that the summed marketThis says that the summed market
value of excess demands is zero for
any positive prices p and p --any positive prices p1 and p2 --
this is Walras’ Law.this is Walras’ Law.
© 2010 W. W. Norton & Company, Inc. 103
Implications of Walras’ LawImplications of Walras’ Law
Suppose the market for commodity A
is in equilibrium; that is,is in equilibrium; that is,
.0xxB1
A1
B*1
A*1 ====ωωωω−−−−ωωωω−−−−++++
)xx(pBAB*A* ++++ωωωω−−−−ωωωω−−−−++++
.0xx 1111 ====ωωωω−−−−ωωωω−−−−++++Then
0)xx(p
)xx(p
BAB*A*
B1
A1
B*1
A*11
====ωωωω−−−−ωωωω−−−−++++
++++ωωωω−−−−ωωωω−−−−++++
0)xx(pB2
A2
B*2
A*22 ====ωωωω−−−−ωωωω−−−−++++
impliesimplies
.0xxB2
A2
B*2
A*2 ====ωωωω−−−−ωωωω−−−−++++
© 2010 W. W. Norton & Company, Inc. 104
.0xx 2222 ====ωωωω−−−−ωωωω−−−−++++
Implications of Walras’ LawImplications of Walras’ Law
So one implication of Walras’ Law forSo 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 inthen the other market must also be in
equilibrium.
© 2010 W. W. Norton & Company, Inc. 105
Implications of Walras’ LawImplications of Walras’ Law
What if, for some positive prices p1 andWhat if, for some positive prices p1 and
p2, there is an excess quantity supplied
of commodity 1? That is,of commodity 1? That is,
.0xxB1
A1
B*1
A*1 <<<<ωωωω−−−−ωωωω−−−−++++ .0xx
B1
A1
B*1
A*1 <<<<ωωωω−−−−ωωωω−−−−++++
)xx(pBAB*A* ++++ωωωω−−−−ωωωω−−−−++++
Then
0)xx(p
)xx(p
BAB*A*
B1
A1
B*1
A*11
====ωωωω−−−−ωωωω−−−−++++
++++ωωωω−−−−ωωωω−−−−++++
0)xx(pB2
A2
B*2
A*22 ====ωωωω−−−−ωωωω−−−−++++
impliesimplies
.0xxB2
A2
B*2
A*2 >>>>ωωωω−−−−ωωωω−−−−++++
© 2010 W. W. Norton & Company, Inc. 106
.0xx 2222 >>>>ωωωω−−−−ωωωω−−−−++++
Implications of Walras’ LawImplications of Walras’ Law
So a second implication of Walras’ LawSo a second implication of Walras’ Law
for a two-commodity exchange economy
is that an excess supply in one marketis that an excess supply in one market
implies an excess demand in the otherimplies an excess demand in the other
market.
© 2010 W. W. Norton & Company, Inc. 107