Title: Chapter Thirty-One
1Chapter Thirty-One
2Exchange
- Two consumers, A and B.
- Their endowments of goods 1 and 2 are
- E.g.
- The total quantities available
and
and
units of good 1
are
units of good 2.
and
3Exchange
- 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.
4Starting an Edgeworth Box
5Starting an Edgeworth Box
Width
6Starting an Edgeworth Box
Height
Width
7Starting an Edgeworth Box
Height
The dimensions ofthe box are thequantities
availableof the goods.
Width
8Feasible 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?
9Feasible 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.
10The Endowment Allocation
The endowmentallocation is
Height
and
Width
11The Endowment Allocation
Height
Width
12The Endowment Allocation
OB
6
OA
8
13The Endowment Allocation
OB
6
4
OA
6
8
14The Endowment Allocation
2
OB
2
6
4
OA
6
8
15The Endowment Allocation
2
OB
2
6
Theendowmentallocation
4
OA
6
8
16The Endowment Allocation
More generally,
17The Endowment Allocation
OB
Theendowmentallocation
OA
18Other Feasible Allocations
- denotes an allocation to consumer
A. - denotes an allocation to consumer
B. - An allocation is feasible if and only if
and
19Feasible Reallocations
OB
OA
20Feasible Reallocations
OB
OA
21Feasible Reallocations
- All points in the box, including the boundary,
represent feasible allocations of the combined
endowments.
22Feasible 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?
23Adding Preferences to the Box
For consumer A.
OA
24Adding Preferences to the Box
For consumer A.
More preferred
OA
25Adding Preferences to the Box
For consumer B.
OB
26Adding Preferences to the Box
For consumer B.
More preferred
OB
27Adding Preferences to the Box
For consumer B.
OB
More preferred
28Adding Preferences to the Box
For consumer A.
OA
29Adding Preferences to the Box
OB
OA
30Edgeworths Box
OB
OA
31Pareto-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?
32Edgeworths Box
OB
OA
33Pareto-Improvements
OB
OA
The set of Pareto-improving allocations
34Pareto-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?
35Pareto-Improvements
OB
OA
The set of Pareto-improving reallocations
36Pareto-Improvements
37Pareto-Improvements
38Pareto-Improvements
Tradeimproves bothAs and Bs welfares.This is
a Pareto-improvementover the endowment
allocation.
39Pareto-Improvements
New mutual gains-to-trade region is the
set of all further Pareto-
improving
reallocations.
Tradeimproves bothAs and Bs welfares.This is
a Pareto-improvementover the endowment
allocation.
40Pareto-Improvements
Further trade cannot improve
both A and Bs
welfares.
41Pareto-Optimality
Better forconsumer A
Better forconsumer B
42Pareto-Optimality
A is strictly better off but B is strictly
worse off
43Pareto-Optimality
A is strictly better off but B is strictly
worse off
B is strictly betteroff but A is strictlyworse
off
44Pareto-Optimality
Both A andB are worseoff
A is strictly better off but B is strictly
worse off
B is strictly betteroff but A is strictlyworse
off
45Pareto-Optimality
Both A andB are worseoff
A is strictly better off but B is strictly
worse off
B is strictly betteroff but A is strictlyworse
off
Both Aand B are worse off
46Pareto-Optimality
The allocation isPareto-optimal since theonly
way one consumers welfare can be increased is
todecrease the welfare of the otherconsumer.
47Pareto-Optimality
An allocation where convexindifference curves
are only just back-to-back is
Pareto-optimal.
The allocation isPareto-optimal since theonly
way one consumers welfare can be increased is
todecrease the welfare of the otherconsumer.
48Pareto-Optimality
- Where are all of the Pareto-optimal allocations
of the endowment?
49Pareto-Optimality
OB
OA
50Pareto-Optimality
All the allocations marked bya are
Pareto-optimal.
OB
OA
51Pareto-Optimality
- The contract curve is the set of all
Pareto-optimal allocations.
52Pareto-Optimality
All the allocations marked bya are
Pareto-optimal.
OB
OA
The contract curve
53Pareto-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?
54The Core
OB
OA
The set of Pareto-improving reallocations
55The Core
OB
OA
56The Core
Pareto-optimal trades blocked by B
OB
OA
Pareto-optimal trades blocked by A
57The Core
Pareto-optimal trades not blocked by A or B
OB
OA
58The Core
Pareto-optimal trades not blocked by A or B
are the core.
OB
OA
59The 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.
60The Core
- But which core allocation?
- Again, that depends upon the manner in which
trade is conducted.
61Trade 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, ...
62Trade in Competitive Markets
For consumer A.
OA
63Trade in Competitive Markets
- So given p1 and p2, consumer As net demands for
commodities 1 and 2 are
and
64Trade in Competitive Markets
- And, similarly, for consumer B
65Trade in Competitive Markets
For consumer B.
OB
66Trade in Competitive Markets
- So given p1 and p2, consumer Bs net demands for
commodities 1 and 2 are
and
67Trade 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.
and
68Trade in Competitive Markets
OB
OA
69Trade in Competitive Markets
Can this PO allocation be achieved?
OB
OA
70Trade in Competitive Markets
Budget constraint for consumer A
OB
OA
71Trade in Competitive Markets
Budget constraint for consumer A
OB
OA
72Trade in Competitive Markets
OB
OA
Budget constraint for consumer B
73Trade in Competitive Markets
OB
OA
Budget constraint for consumer B
74Trade in Competitive Markets
OB
OA
But
75Trade in Competitive Markets
OB
OA
and
76Trade 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.
77Trade in Competitive Markets
So this PO allocation cannot be achieved by
competitive trading.
OB
OA
78Trade in Competitive Markets
Which PO allocations can beachieved by
competitive trading?
OB
OA
79Trade 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.
80Trade in Competitive Markets
Which PO allocations can beachieved by
competitive trading?
OB
OA
81Trade in Competitive Markets
Which PO allocations can beachieved by
competitive trading?
OB
OA
82Trade in Competitive Markets
Which PO allocations can beachieved by
competitive trading?
OB
OA
83Trade in Competitive Markets
Budget constraint for consumer A
OB
OA
84Trade in Competitive Markets
Budget constraint for consumer A
OB
OA
85Trade in Competitive Markets
OB
OA
Budget constraint for consumer B
86Trade in Competitive Markets
OB
OA
Budget constraint for consumer B
87Trade in Competitive Markets
OB
OA
So
88Trade in Competitive Markets
OB
OA
and
89Trade in Competitive Markets
- At the new prices p1 and p2 both markets clear
there is a general equilibrium. - 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.
90First Fundamental Theorem of Welfare Economics
- Given that consumers preferences are
well-behaved, trading in perfectly competitive
markets implements a Pareto-optimal allocation of
the economys endowment.
91Second 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.
92Second Fundamental Theorem of Welfare Economics
- 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.
93Second Fundamental Theorem
OB
OA
The contract curve
94Second Fundamental Theorem
OB
OA
95Second Fundamental Theorem
Implemented by competitive trading from the
endowment w.
OB
OA
96Second Fundamental Theorem
Can this allocation be implementedby competitive
trading from w?
OB
OA
97Second Fundamental Theorem
Can this allocation be implementedby competitive
trading from w? No.
OB
OA
98Second Fundamental Theorem
But this allocation is implementedby competitive
trading from q.
OB
OA
99Walras Law
- 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.
100Walras Law
- Every consumers preferences are well-behaved so,
for any positive prices (p1,p2), each consumer
spends all of his budget. - For consumer AFor consumer B
101Walras Law
Summing gives
102Walras Law
Rearranged,
That is, ...
103Walras Law
This says that the summed marketvalue of excess
demands is zero for any positive prices p1 and p2
-- this is Walras Law.
104Implications of Walras Law
Suppose the market for commodity A is in
equilibrium that is,
Then
implies
105Implications of Walras Law
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.
106Implications of Walras Law
What if, for some positive prices p1 and p2,
there is an excess quantity supplied of commodity
1? That is,
Then
implies
107Implications 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.