程序代写代做代考 1. Sets and Sequences
1. Sets and Sequences
Microeconomic Theory -1- Equilibrium and efficiency
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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 HH = 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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
2
2
( , )
U
x p
MRS x x
U p
x
*
slope =
Microeconomic Theory -7- Equilibrium and efficiency
© John Riley October 15, 2018
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
2
2
( , )
U
x p
MRS x x
U p
x
In the figure the slope of the budget line is 1
2
p
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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 =
slope =
Microeconomic Theory -11- Equilibrium and efficiency
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
Pareto efficient
allocation
Microeconomic Theory -15- Equilibrium and efficiency
© John Riley October 15, 2018
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
© John Riley October 15, 2018
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
© John Riley October 15, 2018
Box rotated 180
Microeconomic Theory -18- Equilibrium and efficiency
© John Riley October 15, 2018
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 -19- Equilibrium and efficiency
© John Riley October 15, 2018
The allocation ˆAx and ˆ ˆB Ax x
is Pareto- efficient since the
marginal rates of substitution are equal.
Microeconomic Theory -20- Equilibrium and efficiency
© John Riley October 15, 2018
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 -21- Equilibrium and efficiency
© John Riley October 15, 2018
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 -22- Equilibrium and efficiency
© John Riley October 15, 2018
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 .
(i) If ˆ( ) ( )
h h h hU x U x then ˆ
h hp x p x (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 -23- Equilibrium and efficiency
© John Riley October 15, 2018
(ii) If ˆ( ) ( )
h h h hU x U x then ˆ
h hp x p x
Proof: If ˆ( ) ( )
h h h hU x U x this follows from (i)
Suppose instead that ˆ( ) ( )
h h h hU x U x and ˆ
h hp x p x .
Define
1 1 1
ˆ̂ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, )h j j j nx x x x x x .
Then for some j and all small 0
1 1 1 1 1 1
ˆ̂ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ,…. , , ,…, ) ( ,…. , , ,…, )h h h hj j j n j j j nU x U x x x x x U x x x x x .
*
Microeconomic Theory -24- Equilibrium and efficiency
© John Riley October 15, 2018
(ii) If ˆ( ) ( )
h h h hU x U x then ˆ
h hp x p x
Proof: If ˆ( ) ( )
h h h hU x U x this follows from (i)
Suppose instead that ˆ( ) ( )
h h h hU x U x and ˆ
h hp x p x .
Define
1 1 1
ˆ̂ ˆ ˆ ˆ ˆ ˆ( ,…. , , ,…, )h j j j nx x x x x x .
Then for some j and all small 0
1 1 1 1 1 1
ˆ̂ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ( ) ( ,…. , , ,…, ) ( ,…. , , ,…, )h h h hj j j n j j j nU x U x x x x x U 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 -25- Equilibrium and efficiency
© John Riley October 15, 2018
3. Walrasian equilibria are Pareto Efficient
First welfare theorem for an exchange economy
If ( ), {1,…, }
h hU x h HH = 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 -26- Equilibrium and efficiency
© John Riley October 15, 2018
First welfare theorem for an exchange economy
If ( ), {1,…, }
h hU x h HH = 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 -27- Equilibrium and efficiency
© John Riley October 15, 2018
Summarizing,
(a) ˆ
h hp x p for some hH
(b) ˆ
h hp x p for all hH
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
h
h
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