# Exchange Two consumers A and B Their endowments of by juanagui

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```									Exchange
Two consumers, A and B. ! Their endowments of goods 1 and 2 are A A A B B ! = (! 1 , ! 2 ) and ! B = (! 1 , ! 2 ).
!

Chapter Thirty-One
Exchange

E.g. ! A = ( 6,4 ) and ! B = ( 2, 2). ! The total quantities available A B are ! 1 + ! 1 = 6 + 2 = 8 units of good 1
!

A B and ! 2 + ! 2 = 4 + 2 = 6 units of good 2.
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Exchange
!

Starting an Edgeworth Box

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.

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4

Starting an Edgeworth Box

Starting an Edgeworth Box
Height = A B !2 + !2
= 4+ 2 =6

A B Width = ! 1 + ! 1 = 6 + 2 = 8
5

A B Width = ! 1 + ! 1 = 6 + 2 = 8
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Starting an Edgeworth Box
!

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?

Height = A B !2 + !2
= 4+ 2 =6

The dimensions of the box are the quantities available of the goods.

A B Width = ! 1 + ! 1 = 6 + 2 = 8
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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 beforetrade allocation; i.e. the endowment allocation.
!
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The Endowment Allocation
Height = A B !2 + !2
= 4+ 2 =6

The endowment allocation is ! A = ( 6,4 ) and
! B = ( 2, 2).

A B Width = ! 1 + ! 1 = 6 + 2 = 8

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The Endowment Allocation
Height = A B !2 + !2
= 4+ 2 =6

The Endowment Allocation
OB

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A B Width = ! 1 + ! 1 = 6 + 2 = 8

! A = ( 6,4 ) ! B = ( 2, 2)
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OA 8

! A = ( 6,4 ) ! B = ( 2, 2)
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The Endowment Allocation
OB

The Endowment Allocation
2 OB 2

6 4 OA 6 8
! A = ( 6,4 )
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6 4 OA 6 8
! B = ( 2, 2)
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The Endowment Allocation
2 OB 2 6 4 OA 6 8 The endowment allocation
! A = ( 6,4 ) ! B = ( 2, 2)
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The Endowment Allocation

More generally, …

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The Endowment Allocation
B !1

Other Feasible Allocations
A ( x1 , x A ) denotes an allocation to 2 consumer A. B B ! ( x1 , x 2 ) denotes an allocation to consumer B. ! An allocation is feasible if and only if
!

OB
B !2

A !2 +
A B !2 !2

The endowment allocation OA
A !1 A B !1 + !1

A A B x1 + xB ! " 1 + " 1 1
A B and x A + xB ! " 2 + " 2 . 2 2

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Feasible Reallocations
xB 1
OB
A !2 + B !2

Feasible Reallocations
xB 1
OB

xB 2

A !2 + B !2

xB 2

xA 2
OA
A x1

xA 2
OA
A x1

A B !1 + !1

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A B !1 + !1

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Feasible Reallocations
!

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?
!
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All points in the box, including the boundary, represent feasible allocations of the combined endowments.

xA 2

xA 2

For consumer A.

M

For consumer A. or e pr ef er re d

A !2

A !2 A !1
A x1
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OA

OA

A !1

A x1
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xB 2

xB 2

For consumer B.

M

B !2

B !2

For consumer B. or e pr ef er re d

OB

B !1

xB 1
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OB

B !1

xB 1
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xB 1
B For consumer B. ! 1

xA 2

OB

For consumer A.

M or

e

pr ef

B !2

er

re

d
xB 2
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A !2

OA

A !1

A x1
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xB 1

xA 2

xA 2
xB 1

Edgeworth’s Box
B !1

B !1

OB

OB

A !2

B !2

A !2

B !2

OA

A !1

A x1 xB 29 2

OA

A !1

A x1

xB 2

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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?
!

xA 2
xB 1

Edgeworth’s Box
B !1

OB

A !2

B !2

OA
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A !1

A x1

xB 2

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xA 2
xB 1

Pareto-Improvements
B !1

Pareto-Improvements
OB Since each consumer can refuse to trade, the only possible outcomes from exchange are Pareto-improving allocations. ! But which particular Paretoimproving allocation will be the outcome of trade?
!

A !2

B !2

OA The set of Paretoimproving allocations

A !1

A x1

xB 2

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xA 2
xB 1

Pareto-Improvements
B !1

Pareto-Improvements
OB

A !2

B !2

OA The set of Paretoimproving reallocations

A !1

A x1

xB 2

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Pareto-Improvements

Pareto-Improvements

Trade improves both A’s and B’s welfares. This is a Pareto-improvement over the endowment allocation.
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Pareto-Improvements

New mutual gains-to-trade region is the set of all further Paretoimproving reallocations.

Pareto-Improvements
Further trade cannot improve both A and B’s welfares.

Trade improves both A’s and B’s welfares. This is a Pareto-improvement over the endowment allocation.
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Pareto-Optimality
Better for consumer A

Pareto-Optimality
A is strictly better off but B is strictly worse off

Better for consumer B
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Pareto-Optimality
A is strictly better off but B is strictly worse off

Both A and B are 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
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B is strictly better off but A is strictly worse off
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Both A and B are worse off

Pareto-Optimality
A is strictly better off but B is strictly worse off

Pareto-Optimality

B is strictly better off but A is strictly worse off

Both A and B are worse 45 off

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.

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Pareto-Optimality
An allocation where convex indifference curves are “only just back-to-back” is Pareto-optimal. 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
Where are all of the Pareto-optimal allocations of the endowment?

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xA 2
xB 1

Pareto-Optimality
B !1

xA 2

Pareto-Optimality
All the allocations marked by a are Pareto-optimal. B !1 O
B

OB

xB 1

A !2

B !2

A !2
A x1

B !2

OA

A !1

OA

A !1

A x1

xB 2

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xB 2

50

Pareto-Optimality
!

xA 2
xB 1

Pareto-Optimality
All the allocations marked by a are Pareto-optimal. B !1 O
B

The contract curve is the set of all Pareto-optimal allocations.

A !2

B !2

OA The contract curve
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A !1

A x1

xB 2

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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?
!

xA 2
xB 1

The Core
B !1

OB

A !2

B !2

OA The set of Paretoimproving reallocations

A !1

A x1

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xB 2

54

xA 2
xB 1

The Core
B !1

OB

xB 1

x A Pareto-optimal trades blocked 2 by B B !1 OB

The Core

A !2

B !2

A !2
A x1

B !2 A x1

OA

A !1

xB 2

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A !1 Pareto-optimal trades blocked by A

OA

xB 2

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xB 1

x A Pareto-optimal trades not blocked 2 by A or B B !1 O
B

The Core

xB 1

x A Pareto-optimal trades not blocked 2 by A or B are the core. B !1 O
B

The Core

A !2

B !2

A !2
A x1

B !2

OA

A !1

OA

A !1

A x1

xB 2

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xB 2

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The Core
The core is the set of all Paretooptimal allocations that are welfareimproving 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.

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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, ...
!

xA 2

For consumer A.
A A A p1x1 + p 2x A = p1! 1 + p 2! 2 2

x*A 2
A !2

OA
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x*A 1

A !1

A x1
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!

!

So given p1 and p2, consumer A’s net demands for commodities 1 and 2 are *A A A x*A ! " 1 and x 2 ! " 2 . 1

And, similarly, for consumer B …

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xB 2

!

For consumer B.
B B p1xB + p 2xB = p1! 1 + p 2! 2 1 2

x*B 2

B !2

So given p1 and p2, consumer B’s net demands for commodities 1 and 2 are *B B B x*B ! " 1 and x 2 ! " 2 . 1

OB

B !1

x*B 1

xB 1
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!

x2
xB 1
B !1

A general equilibrium occurs when prices p1 and p2 cause both the markets for commodities 1 and 2 to clear; i.e.
A B x*A + x*B = ! 1 + ! 1 1 1
A B and x*A + x*B = ! 2 + ! 2 . 2 2

OB

A !2

B !2

OA
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A !1

A x1

xB 2

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x2
xB 1

Can this PO allocation be achieved? B !1 O

xB 1
B !1

x 2 Budget constraint for consumer A

B

OB

A !2

B !2

A !2
A x1

B !2

OA

A !1

OA

A !1

A x1

xB 2

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xB 2

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xB 1
B !1

x 2 Budget constraint for consumer A

x2
xB 1
B !1

OB

OB

x*A 2

A !2

B !2

x*A 2
A x1

A !2

B !2

OA

x*A 1

A !1

OA

x*A 1

A !1

A x1
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xB 2

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Budget constraint for consumer B xB 2

x2
x*B 1
B !1

x2
x*B 1
B !1

xB 1

OB
x*B 2

xB 1

OB
x*B 2

x*A 2

A !2

B !2

x*A 2
A x1
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A !2

B !2

OA

x*A 1

A !1

OA But

x*A 1

A !1

A x1

Budget constraint for consumer B xB 2

A B x*A + x*B < ! 1 + ! 1 1 1

xB 2

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x2
x*B 1
B !1

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.
!
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xB 1

OB
x*B 2

x*A 2

A !2

B !2

OA and

x*A 1

A !1

A x1

A B x*A + x*B > ! 2 + ! 2 2 2

xB 2

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x2
xB 1
B

So this PO allocation cannot be achieved by competitive trading. B !1 O

x2
xB 1
B

Which PO allocations can be achieved by competitive trading? B !1 O

A !2

B !2

A !2
A x1

B !2

OA

A !1

OA

A !1

A x1

xB 2

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xB 2

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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.
!
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x2
xB 1
B

Which PO allocations can be achieved by competitive trading? B !1 O

A !2

B !2

OA

A !1

A x1

xB 2

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x2
xB 1
B

Which PO allocations can be achieved by competitive trading? B !1 O

x2
xB 1
B

Which PO allocations can be achieved by competitive trading? B !1 O

A !2

B !2

A !2
A x1

B !2

OA

A !1

OA

A !1

A x1

xB 2

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xB 2

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xB 1
B !1

x 2 Budget constraint for consumer A

xB 1
B !1

x 2 Budget constraint for consumer A

OB

OB

A !2

B !2

OA

A !1

A x1

A x*A ! 2 2 OA

B !2

x*A 1

A !1

A x1

xB 2

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xB 2

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x2
xB 1
B !1

x2
x*B 1

OB

xB 1

B !1

OB
x*B 2

A x*A ! 2 2 OA

B !2

x*A 1

A !1

A x1

A x*A ! 2 2 OA

B !2

x*A 1

A !1

A x1

Budget constraint for consumer B

xB 2

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Budget constraint for consumer B

xB 2

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x2
x*B 1

x2
x*B 1

xB 1

B !1

OB
x*B 2

xB 1

B !1

OB
x*B 2

A x*A ! 2 2 OA

B !2

So

x*A 1

A !1

A x1

A x*A ! 2 2 OA

B !2

A B x*A + x*B = ! 1 + ! 1 1 1

xB 2

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and

x*A 1

A !1

A x1

A B x*A + x*B = ! 2 + ! 2 2 2

xB 2

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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.
!
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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.

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Second Fundamental Theorem of Welfare Economics
!

Second Fundamental Theorem of Welfare Economics
!

The First Theorem is followed by a second that states that any Paretooptimal 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.
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Given that consumers’ preferences are well-behaved, for any Paretooptimal allocation there are prices and an allocation of the total endowment that makes the Paretooptimal allocation implementable by trading in competitive markets.

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Second Fundamental Theorem
xA 2
xB 1
B !1

Second Fundamental Theorem
xA 2
xB 1
* x 2A

OB

* x1B

B !1

OB
* x 2B

A !2

B !2

A !2

B !2

OA The contract curve

A !1

A x1

OA

* x1A

A !1

A x1

xB 2

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xB 2

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Second Fundamental Theorem
xA 2

Second Fundamental Theorem
xB 1
B !1

Implemented by competitive trading from the endowment !.
* x1B
B !1

x A Can this allocation be implemented 2 by competitive trading from !?

xB 1
* x 2A

OB
* x 2B

OB

A !2

B !2

A !2

B !2

OA

* x1A

A !1

A x1

OA

A !1

A x1

xB 2

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xB 2

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Second Fundamental Theorem
xA 2
Can this allocation be implemented by competitive trading from !? No.
B !1

Second Fundamental Theorem
xB 1
A !2

x A But this allocation is implemented 2 by competitive trading from ".

xB 1

OB

!B 1

OB

A !2

B !2

!B 2
A !1
A x1

OA

A !1

A x1

OA

xB 2

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xB 2

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Walras’ Law
!

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: A A p1x*A + p 2x*A = p1! 1 + p 2! 2 1 2 For consumer B: B B p1x*B + p 2x*B = p1! 1 + p 2! 2 1 2
!

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.

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Walras’ Law
A A p1x*A + p 2x*A = p1! 1 + p 2! 2 1 2

Walras’ Law
p1 ( x*A + x*B ) + p 2 ( x*A + x*B ) 1 1 2 2
A B B B = p1 (! 1 + ! 1 ) + p 2 (! 2 + ! 2 ).

B B p1x*B + p 2x*B = p1! 1 + p 2! 2 1 2

Summing gives
p1 ( x*A + x*B ) + p 2 ( x*A + x*B ) 1 1 2 2 A B B B = p1 (! 1 + ! 1 ) + p 2 (! 2 + ! 2 ).

Rearranged,
A B p1 ( x*A + x*B ! " 1 ! " 1 ) + 1 1 A B p 2 ( x*A + x*B ! " 2 ! " 2 ) = 0. 2 2

That is, ...
101 102

Walras’ Law
A B p1 ( x*A + x*B " !1 " !1 ) + 1 1

Implications of Walras’ Law
Suppose the market for commodity A is in equilibrium; that is, Then
A B x*A + x*B " !1 " !1 = 0. 1 1

p 2 ( x*A 2 = 0.

+ x*B 2

A " !2

" !B ) 2

This says that the summed market value of excess demands is zero for any positive prices p1 and p2 -this is Walras’ Law.
103

A B p1 ( x*A + x*B " !1 " !1 ) + 1 1 A p 2 ( x*A + x*B " !2 " !B ) = 0 2 2 2

implies
A x*A + x*B " !2 " !B = 0. 2 2 2
104

Implications 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.

Implications of Walras’ Law
What if, for some positive prices p1 and p2, there is an excess quantity supplied of commodity 1? That is, Then
A B x*A + x*B " !1 " !1 < 0. 1 1

A B p1 ( x*A + x*B " !1 " !1 ) + 1 1 A p 2 ( x*A + x*B " !2 " !B ) = 0 2 2 2

implies
A x*A + x*B " !2 " !B > 0. 2 2 2
105 106

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.

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