Chuong 29

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..[r]

Chapter Twenty-Nine

Exchange  Two consumers, A and B.

 Their endowments of goods and are

 E.g

 The total quantities available

 A (1A,2A ) and B (1B,2B).  A ( , )6 4 and B ( , ).2 2

1A  1B  6 8 2A  2B  4 6

units of good 1 units of good 2. and

Exchange

 Edgeworth and Bowley devised a

Starting an Edgeworth Box

Starting an Edgeworth Box

Width = 1A  1B  6 8

Height =

2 2 4 2 6

A B

Starting an Edgeworth Box

Width = 1A  1B  6 8

Height =

2 2 4 2 6

A B

   

The dimensions of the box are the

Feasible Allocations

 What allocations of the units of

good and the units of good are feasible?

 How can all of the feasible

Feasible Allocations

 What allocations of the units of

good and the units of good are feasible?

 How can all of the feasible allocations be depicted by the Edgeworth box

diagram?

Width = 1A  1B  6 8

Height =

2 2 4 2 6

A B

   

The endowment allocation is

 A ( , )6 4

B ( , ).2 2

and

Width = 1A  1B  6 8

Height =

2 2 4 2 6

A B

   

 A ( , )6 4 B ( , )2 2

 A ( , )6 4

OA

OB

6

8 

B

( , )2 2

 A ( , )6 4

OA

OB

6

8 4

6

B ( , )2 2

OA

OB

6

8 4

6

2 2

 A ( , )6 4 B ( , )2 2

OA

OB

6

8 4

6

2 2

The

endowment allocation

More generally, …

The Endowment Allocation OA OB The endowment allocation

1A  1B  2A

  2 2 A B 

1A

1B

Other Feasible Allocations  denotes an allocation to

consumer A.

 denotes an allocation to consumer B.

 An allocation is feasible if and only if

(x1A ,xA2 ) (x xB1 , B2 )

x1A  xB1 1A  1B xA2  xB2 2A  2B.

Feasible Reallocations

OA

OB

1A  1B

xA2





2

2

A

B



x1A

xB1

Feasible Reallocations

OA

OB

1A  1B

xA2





2

2

A

B



x1A

xB1

Feasible Reallocations

 All points in the box, including the boundary, represent feasible

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?

Adding Preferences to the Box

2A

1A

xA2

x1A OA

Adding Preferences to the Box

2A

1A

xA2

x1A

Mo

re p

refe

rred

For consumer A.

Adding Preferences to the Box

2B

1B

xB2

xB1

For consumer B.

Adding Preferences to the Box

xB2

xB1 Mo

re p

refe

rred

For consumer B.

OB

2B

Adding Preferences to the Box

2B 1B

xB1

xB2

Mo

re p

refe

rred

For consumer B. O

Adding Preferences to the Box

2A

1A

xA2

x1A OA

Adding Preferences to the Box

2A

1A

xA2

x1A OA

2B 1B

xB1

Edgeworth’s Box

2A

1A

xA2

x1A OA

 2B 1B

xB1

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.

Edgeworth’s Box

2A

1A

xA2

x1A OA

 2B 1B

xB1

Pareto-Improvements

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

The set of

Pareto-Improvements

 Since each consumer can refuse to trade, the only possible outcomes

from exchange are Pareto-improving allocations.

 But which particular

Pareto-Improvements

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

The set of

Pareto-Improvements

Trade

improves both

A’s and B’s welfares.

Pareto-Improvements

New mutual gains-to-trade region is the set of all further improving

reallocations.

Trade

improves both

A’s and B’s welfares.

Pareto-Improvements

Further trade cannot improve both A and B’s

Pareto-Optimality

Better for

consumer B

Better for

Pareto-Optimality

A is strictly better off

Pareto-Optimality

A is strictly better off

but B 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

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

Pareto-Optimality

The allocation is

Pareto-optimal since the only way one consumer’s

welfare can be increased is to

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.

An allocation where convex indifference curves are “only just back-to-back” is

Pareto-Optimality

Pareto-Optimality

2A

1A

xA2

x1A OA

 2B 1B

xB1

Pareto-Optimality

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Pareto-Optimality

Pareto-Optimality

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

All the allocations marked by a are Pareto-optimal.

Pareto-Optimality

 But to which of the many allocations on the contract curve will consumers trade?

 That depends upon how trade is conducted.

The Core

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

The set of

The Core

2A

1A

xA2

x1A OA

 2B 1B

xB1

The Core

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Pareto-optimal trades blocked by B

The Core

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

The Core

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

The Core

 The core is the set of all

Pareto-optimal allocations that are welfare-improving for both consumers

relative to their own endowments.

The Core

 But which core allocation?

 Again, that depends upon the

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

Trade in Competitive Markets

2A

1A

xA2

x1A OA

For consumer A.

p x1 1A  p x2 2A p1 1 A  p2 2 A

x*2A

Trade in Competitive Markets

 So given p1 and p2, consumer A’s net demands for commodities and

are

Trade in Competitive Markets

Trade in Competitive Markets

2B

1B

xB2

xB1

For consumer B.

OB

x*1B x*2B

Trade in Competitive Markets

 So given p1 and p2, consumer B’s net demands for commodities and

are

Trade in Competitive Markets

 A general equilibrium occurs when prices p1 and p2 cause both the

markets for commodities and to clear; i.e.

x*1A  x*1B 1A  1B

x*2A  x*2B 2A  2B.

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer A

x*2A

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer B x*2A

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer B x*2A

x*1A

x*1B

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

x*2A

x*1A

x*1B

x*2B

But x A x B A B

1 1 1 1

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

x*2A

x*1A

x*1B

x*2B

and x A x B A B

2 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

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB Which PO allocations can be

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

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB Which PO allocations can be

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB Which PO allocations can be

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB Which PO allocations can be

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer A

x*2A

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer B x*2A

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Budget constraint for consumer B x*2A

x*1A

x*1B

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

x*2A

x*1A

x*1B

x*2B

So x A x B A B

1 1 1 1

* *

Trade in Competitive Markets

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

x*2A

x*1A

x*1B

x*2B

and x A x B A B

2 2 2 2

* *

Trade 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

First Fundamental Theorem of Welfare Economics

 Given that consumers’ preferences are well-behaved, trading in perfectly competitive markets implements a

Second 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.

Second Fundamental Theorem

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Second Fundamental Theorem

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

A * 2

x x2*B

Second Fundamental Theorem

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

A * 2

x x2*B

A * 1 x B * 1 x

Implemented by competitive

Second Fundamental Theorem

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Second Fundamental Theorem

2A

1A

xA2

x1A OA

 2B 1B

xB1

xB2 OB

Second Fundamental Theorem xA2

x1A OA

xB1

xB2 OB

But this allocation is implemented by competitive trading from .

Walras’ Law

 Walras’ Law is an identity; i.e a statement that is true for any

positive prices (p1,p2), whether these

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 x1 1*A  p x2 2*A p1 1 A  p2 2 A

Walras’ Law

p x1 1*A  p x2 2*A p1 1 A  p2 2 A

p x1 1*B  p x2 2*B p1 1B  p2 2B

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

( ) ( )

( ) ( ).

*  *  *  *

       

Walras’ Law

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

( ) ( ) ( ) ( ). * * * *            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

( ) ( ) . * * * *            

Walras’ Law . 0 ) x x ( p ) x x ( p B 2 A 2 B * 2 A * 2 2 B 1 A 1 B * 1 A * 1 1            

This says that the summed market value of excess demands is zero for any positive prices p1 and p2

Implications of Walras’ Law 0 ) x x ( p ) x x ( p B 2 A 2 B * 2 A * 2 2 B 1 A 1 B * 1 A * 1 1            

Suppose the market for commodity A is in equilibrium; that is,

. 0 x

x*1A  *1B  1A  1B  Then

implies

. 0 x

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,

. 0 x

x*1A  *1B  1A  1B 

0 ) x x ( p ) x x ( p B 2 A 2 B * 2 A * 2 2 B 1 A 1 B * 1 A * 1 1             Then implies . 0 x

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