程序代写代做代考 1. Sets and Sequences

1. Sets and Sequences

Microeconomic Theory -1- Equilibrium and efficiency

Equilibrium and Pareto Efficiency in an exchange economy

1. Efficient economies 2

2. Gains from exchange 6

3. Edgeworth-Box analysis 15

4. Properties of a consumer’s choice 20

5. Walrasian equilibria are Pareto Efficient 24

Microeconomic Theory -2- Equilibrium and efficiency

Efficient economies

Definition: Pareto preferred allocation

The allocation ˆ{ }
h

hx H is Pareto preferred to { }
h

hx H if all consumers weakly prefer ˆ{ }
h

hx H over

{ }h hx H and at least one consumer strictly prefers ˆ{ }
h

hx H .

Microeconomic Theory -3- Equilibrium and efficiency

Definition: Pareto preferred allocation

The allocation ˆ{ }
h

hx H is Pareto preferred to { }
h

hx H if all consumers weakly prefer ˆ{ }
h

hx H over

{ }h hx H and at least one consumer strictly prefers ˆ{ }
h

hx H .

Definition: Pareto efficient allocation

ˆ{ }h hx H is Pareto efficient if there is no feasible Pareto preferred allocation.

Microeconomic Theory -4- Equilibrium and efficiency

Definition: Pareto preferred allocation

The allocation ˆ{ }
h

hx H is Pareto preferred to { }
h

hx H if all consumers weakly prefer ˆ{ }
h

hx H over

{ }h hx H and at least one consumer strictly prefers ˆ{ }
h

hx H .

Definition: Pareto efficient allocation

ˆ{ }h hx H is Pareto efficient if there is no feasible Pareto preferred allocation.

First welfare theorem for an exchange economy

If ( ), {1,…, }
h hU x h HH = satisfies the non-satiation property and { }h hx H is a Walrasian

Equilibrium allocation, then { }
h

hx H is Pareto Efficient.

Microeconomic Theory -5- Equilibrium and efficiency

2. Gains from exchange

Preliminary observation

Consider the standard utility maximization problem

with two commodities.

If the solution 0x  then the marginal utility per

dollar must be the same for each commodity

1 1 2 2

1 1
( ) ( )

U U
x x

p x p x

 


 
.

slope =

Microeconomic Theory -6- Equilibrium and efficiency

2. Gains from exchange

Preliminary observation

Consider the standard utility maximization problem

with two commodities.

If the solution 0x  then the marginal utility per

dollar must be the same for each commodity

1 1 2 2

1 1
( ) ( )

U U
x x

p x p x

 


 
.

Equivalently the marginal rate of substitution satisfies

1 1
1 2

( , )

x p
MRS x x

U p
x




 




slope =

Microeconomic Theory -7- Equilibrium and efficiency

2. Gains from exchange

Preliminary observation

Consider the standard utility maximization problem

with two commodities.

If the solution 0x  then the marginal utility per

dollar must be the same for each commodity

1 1 2 2

1 1
( ) ( )

U U
x x

p x p x

 


 
.

Equivalently the marginal rate of substitution satisfies

1 1
1 2

( , )

x p
MRS x x

U p
x




 




In the figure the slope of the budget line is 1

p
 .

At the maximum this slope is the same as the slope of the indifference curve.

Therefore 1 2( , )MRS x x is the slope of the indifference curve.

slope =

Microeconomic Theory -8- Equilibrium and efficiency

Pareto Efficient allocation in a 2 person 2 commodity economy

An allocation ˆAx and ˆBx is not a PE allocation if there

is an exchange of commodities 1 2( , )e e e such that

ˆ ˆ( ) ( )A AA AU x e U x  and ˆ ˆ( ) ( )
B B

A AU x e U x 

Microeconomic Theory -9- Equilibrium and efficiency

Pareto Efficient allocation in a 2 person 2 commodity economy

An allocation ˆAx and ˆBx is not a PE allocation if there

Is an exchange of commodities 1 2( , )e e e such that

ˆ ˆ( ) ( )A AA AU x e U x  and ˆ ˆ( ) ( )
B B

A AU x e U x 

Proposition: If ˆ 0Ax  and ˆ 0Bx  then a necessary

condition for an allocation to be a PE allocation is that

marginal rates of substitution are equal.

Microeconomic Theory -10- Equilibrium and efficiency

Pareto Efficient allocation in a 2 person 2 commodity economy

An allocation ˆAx and ˆBx is not a PE allocation if there

Is an exchange of commodities 1 2( , )e e e such that

ˆ ˆ( ) ( )A AA AU x e U x  and ˆ ˆ( ) ( )
B B

A AU x e U x 

Proposition: If ˆ 0Ax  and ˆ 0Bx  then a necessary

condition for an allocation to be a PE allocation is that

marginal rates of substitution are equal.

Suppose instead that, as depicted,

ˆ ˆ( ) ( )
A B

A BMRS x MRS x

Consider a proposal by Alex of 1 2( , )e e e where

1 20e e 

and the exchange rate lies between the

two marginal rates of substitution

slope =

Microeconomic Theory -11- Equilibrium and efficiency

Such an exchange is depicted.

On the margin, Alex is willing to give up more

of commodity 2 In exchange for commodity 1.

Therefore Alex offers Bev some of commodity 2

In exchange for commodity 1.

Microeconomic Theory -12- Equilibrium and efficiency

Such an exchange is depicted.

On the margin, Alex is willing to give up more

of commodity 2 In exchange for commodity 1.

Therefore Alex offers Bev some of commodity 2

In exchange for commodity 1.

If the proposed trade is too large it may not be better

for both consumers due to the curvature of the level sets.

But for all sufficiently small  , the proposed

trade e must raise the utility of both consumers.

So the initial allocation ˆ ˆ,A Bx x is not a Pareto efficient

allocation.

Microeconomic Theory -13- Equilibrium and efficiency

What if there are more than two commodities?

For all possible allocations we can, in principle

compute the utilities and hence the set of feasible

utilities.

For any point in the interior of this set there is

another allocation such that Bev is no worse off

and Alex is strictly better off.

Pareto preferred

allocation

Microeconomic Theory -14- Equilibrium and efficiency

What if there are more than two commodities?

For all possible allocations we can, in principle

compute the utilities and hence the set of feasible

utilities.

For any point in the interior of this set there is

another allocation such that Bev is no worse off

and Alex is strictly better off.

Consider the following maximization problem.

ˆ ˆ ˆ{ ( ) | ( ) ( )}B BA B B
e

Max U x e U x e U x  

Class Exercise

What exchange *e solves this problem if the allocation

ˆ ˆ,A Bx x is Pareto efficient?

Pareto preferred

allocation

Microeconomic Theory -15- Equilibrium and efficiency

3. Efficiency in an Edgeworth-Box diagram

Consider Alex and Bev with endowments
A and B .

( ) ( )A BMRS MRS  so there are gains from exchange.

Microeconomic Theory -16- Equilibrium and efficiency

Efficiency in an Edgeworth-Box diagram

If the endowments are
A and B ,

the set of feasible allocations for Bev is the

set of allocation in the rectangle or “box”

The set of allocations preferred by Bev

Is the dotted region in the lower box.

On the next slide we rotate the box 180 .

Microeconomic Theory -17- Equilibrium and efficiency

We also add the level set for Alex through the endowment. Because ( ) ( )
A B

A BMRS MRS 

there is a vertically lined region of Pareto preferred allocations

Microeconomic Theory -18- Equilibrium and efficiency

The allocation ˆAx and ˆ ˆB Ax x 

is Pareto- efficient since the

marginal rates of substitution are equal.

Microeconomic Theory -19- Equilibrium and efficiency

Group exercise

Suppose that
3/4 3/4

1 2( ) 2( ) 3( )
A A A

AU x x x  and
3/4 3/4

1 2( ) 2( ) 3( )
B B B

BU x x x 

The aggregate endowment is (100,200) .

(a) Show that for all allocation to be a PE allocation, both consumers are allocated twice as much of

commodity 2.

(b) What is the MRS if an allocation is Pareto Efficient?

Microeconomic Theory -20- Equilibrium and efficiency

4. Properties of a consumer’s choice

Non satiation property

For every x , there is at a commodity j such that for all sufficiently small 0  ,

1 1 1 1 1 1( ,…. , , ,…, ) ( ,…. , , ,…, )
h h

j j j n j j j nU x x x x x U x x x x x    

Consider a consumer with a utility function whose utility satisfies this very weak property.

Let hx be the choice of consumer h .

(i) If ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   and (ii) if ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   .

Microeconomic Theory -21- Equilibrium and efficiency

2. Properties of a consumer’s choice

Non satiation property

For every x , there is at a commodity j such that for all sufficiently small 0  ,

1 1 1 1 1 1( ,…. , , ,…, ) ( ,…. , , ,…, )
h h

j j j n j j j nU x x x x x U x x x x x    

Consider a consumer with a utility function whose utility satisfies this very weak property.

Let hx be the choice of consumer h .

(ii) If ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   and (ii) if ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   .

Proof of (i): If ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x  

Suppose instead that ˆ( ) ( )
h h h hU x U x and ˆ

h hp x p x   .

Then hx is not the choice of the consumer since it does not maximize utility among commodity

bundles in the budget set p x I  .

Microeconomic Theory -22- Equilibrium and efficiency

Proof of (ii): If ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   .

Suppose instead that ˆ( ) ( )
h h h hU x U x and ˆ

h hp x p x   .

Then for some j and all small 0 

1 1 1 1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, ) ( ,…. , , ,…, )h hj j j n j j j nU x x x x x U x x x x x     .

Define
1 1 1

ˆ̂ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, )h j j j nx x x x x x  .

Microeconomic Theory -23- Equilibrium and efficiency

Proof of (ii): If ˆ( ) ( )
h h h hU x U x then ˆ

h hp x p x   .

Suppose instead that ˆ( ) ( )
h h h hU x U x and ˆ

h hp x p x   .

Then for some j and all small 0 

1 1 1 1 1 1
ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, ) ( ,…. , , ,…, )h hj j j n j j j nU x x x x x U x x x x x     .

Define
1 1 1

ˆ̂ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, )h j j j nx x x x x x  .

Since ˆ
h hp x p x   we can choose 0  such that ˆ̂hx is in the budget set and ˆ̂ ˆ( ) ( )h h hU x U x .

Then again hx is not the choice of the consumer since it does not maximize utility among commodity

bundles in the budget set p x I  .

Microeconomic Theory -24- Equilibrium and efficiency

3. Walrasian equilibria are Pareto Efficient

First welfare theorem for an exchange economy

If ( ), {1,…, }
h hU x h HH = satisfies the non-satiation property and { }h hx H is a Walrasian

Equilibrium allocation, then { }
h

hx H is Pareto Efficient.

Proof:

Remember that { }
h

hx H is an equilibrium allocation.

Consider any Pareto preferred allocation ˆ{ }
h

hx H

Step 1:

For some h , ˆ( ) ( )
h h hU x U x .

By the non-satiation property (i)

ˆh hp x p    .

Microeconomic Theory -25- Equilibrium and efficiency

First welfare theorem for an exchange economy

If ( ), {1,…, }
h hU x h HH = satisfies the non-satiation property and { }h hx H is a Walrasian

Equilibrium allocation, then { }
h

hx H is Pareto Efficient.

Proof:

Remember that { }
h

hx H is an equilibrium allocation.

Consider any Pareto preferred allocation ˆ{ }
h

hx H .

Step 1

For some h , ˆ( ) ( )
h h hU x U x .

By the non-satiation property (i)

ˆh hp x p    .

Step 2

For all h , ˆ( ) ( )
h h hU x U x .

By the non-satiation properties (i) and (ii)

ˆh hp x p   

Microeconomic Theory -26- Equilibrium and efficiency

Summarizing,

(a) ˆ
h hp x p    for some hH

(b) ˆ
h hp x p    for all hH

Summing over consumers,

ˆ ˆh h

h h

p x p x p p 
 

       
H H

Step 3:

For any feasible allocation { }
h

hx H the total consumption vector must satisfy

x x 


 
H

Since 0p  it follows that for any feasible allocation

p x p    .

Since ˆp x p    it follows that x̂ is not a feasible allocation.

Remark: An almost identical argument can be used to show that a Walrasian Equilibrium allocation

for an economy with production is also Pareto Efficient

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