Chapter Thirty Exchange Tausch 交换. Hu Jingbei’s declaration to copyright May 17, 2007 u This ppt-file may come from an American source which is not known

Chapter Thirty

Exchange

Tausch

交换

Hu Jingbei’s declaration to copyrightMay 17, 2007

This ppt-file may come from an American source which is not known to me at the moment. It based on Varian’s Intermediate Microeconomics, 5th ed. In many places of this file, I made changes for my lectures with Varian’s 6th ed. I will be grateful to persons who may show me the original sources of this file.

Contents

The Edgeworth Box Pareto improving trade Pareto Efficient Allocation The Core (核 ) Trade in a competitive market Walras’ Law 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 2 B 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

The English economist Edgeworth devised a diagram, called an Edgeworth box (Edgeworth Diagramm) , to show all possible allocations of the available quantities of goods 1 and 2 between the two consumers.

Starting an Edgeworth Box

Width = 1 1 6 2 8A B

Height = 2 2

4 2

6

A B

The dimensions ofthe box are thequantities availableof 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 (urspaengliche Allokation).

Width = 1 1 6 2 8A B

Height =

2 2

4 2

6

A B

The endowmentallocation 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 4B ( , )2 2


A ( , )6 4OA

OB

6

8B ( , )2 2


A ( , )6 4OA

OB

6

8

4

6


B ( , )2 2

OA

OB

6

8

4

6

2

2


A ( , )6 4B ( , )2 2

OA

OB

6

8

4

6

2

2

Theendowmentallocation


More generally, …



OA

OB

Theendowmentallocation

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


OA

OB

1 1A B

xA2

2

2

A

B

xA1

xB1

xB2


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

xA1

OA

For consumer A.


2A

1A

xA2

xA1

More preferred

For consumer A.

OA


2B

1B

xB2

xB1

For consumer B.

OB

Adding Preferences to the BoxxB2

xB1

More preferred

For consumer B.

OB

2B

1B


2B

1B

xB1

xB2

More preferred

For consumer B. OB


2A

1A

xA2

xA1

OA

For consumer A.


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Edgeworth’s Box

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-Improvement

An allocation of the endowment that improves the welfare of at least a consumer without reducing the welfare of another is a Pareto-improving allocation (Allokation der Pareto-Verbesserung)

Where are the Pareto-improving allocations?

Edgeworth’s Box

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-Improvements

2A

1A

xA2

xA1

OA

2B

1B

xB1

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 (Allokationen der Pareto-Verbessrung).

But which particular Pareto-improving allocation will be the outcome of trade?

Pareto-Improvements

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

The set of Pareto-improving reallocations

Pareto-Improvements

Pareto-Improvements

Pareto-Improvements

Tradeimproves bothA’s and B’s welfares.This is a Pareto-improvementover the endowment allocation.

Pareto-ImprovementsNew mutual gains-to-trade region is the set of all further Pareto- improving reallocations.

Tradeimproves bothA’s and B’s welfares.This is a Pareto-improvement(帕累托改进 )over the endowment allocation.

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

Pareto-Optimality (Pareto-Optimum,帕累托最优 )

Better forconsumer B

Better forconsumer A

Pareto-OptimalityA is strictly better off but B is strictly worse off

B is strictly betteroff but A is strictlyworse off

Pareto-OptimalityA 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

Both A andB are worseoff

Pareto-Optimality

The allocation isPareto-optimal since theonly way one consumer’swelfare can be increased is todecrease the welfare of the otherconsumer.

Pareto-Optimality

The allocation isPareto-optimal since theonly way one consumer’swelfare can be increased is todecrease the welfare of the otherconsumer.

At the Pareto-optimality both of the Marginal Rates of Substitution must be equal.

Pareto-Optimality

Where are all of the Pareto-optimal allocations of the endowment?

Pareto-Optimality

Condition of the Pareto-Optimality: MRSA = MRSB

To explain the condition, we can assume one consumer’s utility constant and see how to maximize another’s utility. With both the budget constraints, a Lagrange method can be used to show the condition.

Pareto-Optimality

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-Optimality

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

All the allocations marked bya are Pareto-optimal.

Pareto-Optimality

The contract curve (契约线 ) is the set of all Pareto-optimal allocations.

Pareto-Optimality

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

All the allocations marked bya 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 (Kore, 核 )

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

The set of Pareto-improving reallocations

The Core

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

The Core

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Pareto-optimal trades blocked by B

Pareto-optimal trades blocked by A

The Core

2A

1A

xA2

xA1

OA

2B

1B

xB1

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, ...


2A

1A

xA2

xA1

OA

consumer A’s budget line:

p x p x p pA A A A1 1 2 2 1 1 2 2

x A2*

x A1*


So given p1 and p2, consumer A’s net demands for commodities 1 and 2 are

x A A1 1* x A A

2 2* . and


And, similarly, for consumer B …


2B

1B

xB2

xB1

consumer B’s budget line:

OB x B1*

x B2*

p x p x p pB B B B1 1 2 2 1 1 2 2


So given p1 and p2, consumer B’s net demands for commodities 1 and 2 are

x B B1 1* x B B

2 2* . and

Trade in Competitive Markets A general equilibrium (Allgemeines

Gleichgewicht, 一般均衡 ) 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


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Can this PO allocation beachieved?


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Budget constraint for consumer A


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


x A2*

x A1*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Budget constraint for consumer B

x A2*

x A1*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


x A2*

x A1*

x B1*

x B2*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

x A2*

x A1*

x B1*

x B2*

But x xA B A B1 1 1 1* *


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

x A2*

x A1*

x B1*

x B2*

and x xA B A B2 2 2 2* *


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.


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

So this PO allocation cannot beachieved by the given prices.


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

How can PO allocations beachieved by competitive trading?


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.


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Which PO allocations can beachieved by competitive trading?


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB



2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB



2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB



2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


x A2*

x A1*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


x A2*

x A1*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB


x A2*

x A1*

x B1*

x B2*


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

x A2*

x A1*

x B1*

x B2*

So x xA B A B1 1 1 1* *


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

x A2*

x A1*

x B1*

x B2*

and x xA B A B2 2 2 2* *


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 Walras’sches Gesetz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.

Terms of demand

w: goods in initial endowment allocation x: goods in the final allocation x: gross demand (Bruttonachfrage) for the good x (x – w): net demand (Nettonachfrage) for the good x, excess demand (Ueberschussnachfrage) (xA + xB): aggregate demand (aggregierte Nachfrage)

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 (budget constraint):

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 B1 1 1 2 2 2

1 1 1 2 2 2

( ) ( )

( ) ( ).

* * * *

Summing gives

Walras’ Law

Rearranged,

p x x

p x x

A B A B

A B A B1 1 1 1 1

2 2 2 2 2 0

( )

( ) .

* *

* *

The first parenthesis can be seen as the excess demand for good1 and the second one as the excess demand for good2.

Walras’ Law

.0

)xx(p

)xx(pB2

A2

B*2

A*22

B1

A1

B*1

A*11

This says that the summed marketvalue of excess demands is zero forany positive prices p1 and p2 -- this is Walras’ Law.

Walras’ Law We often name z the excess demand

function and, because x is a demand function with p, we get

It stands for any p(p1, p2) > 0. It is the Walras’ law.

* *1 1 2 1 1 2 1 1 2 1 1

* *2 1 2 2 1 2 2 1 2 2 2

1 1 1 2 2 2 1 2

( , ) ( ( , ) ( , ) )

( , ) ( ( , ) ( , ) )

( , ) ( , ) 0.

A B A B

A B A B

z p p x p p x p p

z p p x p p x p p

thenp z p p p z p p

Implications of Walras’ Law

0)xx(p

)xx(pB2

A2

B*2

A*22

B1

A1

B*1

A*11

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

.0xx B1

A1

B*1

A*1

Then

implies

.0xx B2

A2

B*2

A*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.



What if, for some positive prices p1 andp2, there is an excess quantity suppliedof commodity 1? That is,

.0xx B1

A1

B*1

A*1

0)xx(p

)xx(pB2

A2

B*2

A*22

B1

A1

B*1

A*11

Then

implies

.0xx B2

A2

B*2

A*2


So a second implication of Walras’ Lawfor a two-commodity exchange economyis that an excess supply in one marketimplies an excess demand in the othermarket.It also means that not any p(p1, p2) > 0 can ensure z1 = 0 and z2 = 0 at the same time, even if any p(p1, p2) > 0 can ensure p1z1+p2z2 =0


Because the equilibrium prices are prices which lead demand for and supply of every good to be equal at the same time, the Walras’ law does not mean there must be at least a set of prices pE(pE

1, pE2) > 0

which could result in z1 = 0 and z2 = 0 simultaneously.

Therefore: We must find the equilibrium prices pE(pE

1, pE2) > 0 if it/they would exist.

Relative Prices Walras’ law means there are only k-1

prices that can be determined independently in an economy with k goods. That means also, we can only get pi for good i that

pi = pi(pj) i=1,2,…,k, i≠j Only after pj is determined, can we get all

pi. pj is called a numeraire price. Good j is the

numeraire.

Relative prices All prices except the numeraire price are prices

relative to the numeraire price. Making it as 1, all prices are relative to the numeraire.

That is: All prices are relative prices. There are no absolute prices.

Changing pj from being 1 to being 2, all other prices will be double. But the exchange relations between each pair of all kinds of goods remain the same.

In the economy with paper money, governments can multiply the numeraire price. We refer it to change in price level or in absolute price level.

Solve for General Equilibrium

Individual optimization problem Xi

j=Xij (P1, P2, Mj), i=1, 2; j=A, B

Mj=Mj (P1, P2, j1, j

2)

Xij=Xi

j (P1, P2 ) Market clearing for market 1:

X1A (P1,P2)+ X1

B (P1,P2) = 1A + 1

B

Can solve for equilibrium P1/P2? Can solve for demand at equilibrium?

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.The second requirement is just one of

conditions for perfectly competitive makerts. Therefore: Competitive markets will necessarily reach equilibriums.

Existence of Equilibrium

Example: The endowment allocation Apple Orange (good 1) (good 2) A wa

1 wa2

B wb1 wb

2 (The whole example will come…)

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.

It means that competitive equilibrium will necessarily be Pareto efficient.

The Equilibrium with Monopoly

Figure 30.5 in textbook.

It was taken off to reduce the size of this ppt.

The Equilibrium with Monopoly

Figure 30.6 in textbook.

It was taken off to reduce the size of this ppt.

Implicit Assumptions

No consumption externality (外部效应 ) Agents behave competitively Equilibrium exists

This theorem gives a general mechanism—the competitive market—to ensure Pareto efficiency or Pareto optimality.

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.

That means any Pareto efficient or optimal allocation is the competitive equilibrium allocation.

Second Fundamental Theorem of Welfare Economics

Second Fundamental Theorem

2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

The contract curve


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

A*2x B*

2x

A*1x

B*1x


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

A*2x B*

2x

A*1x

B*1x

Implemented by competitivetrading from the endowment .


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Can this allocation be implementedby competitive trading from ?


2A

1A

xA2

xA1

OA

2B

1B

xB1

xB2

OB

Can this allocation be implementedby competitive trading from ? No.

Second Fundamental TheoremxA2

xA1

OA

xB1

xB2

OB

But this allocation is implementedby competitive trading from .

A1

B2

B1

A2

The Assumption about Convexity

Implications of the 2nd Theorem

Market equilibrium is distributionally neutral.

Redistribution can be carried out for endowments.

The market plays only the allocation role.

Documents

Chapter Thirty Exchange Tausch 交换. Hu Jingbei’s declaration to copyright May 17, 2007 u This ppt-file may come from an American source which is not known