1. Sets and Sequences
Microeconomic Theory -1- Equilibrium and efficiency
John RileyOctober15, 2018
Equilibrium and Pareto Efficiency in an exchange economy
1. Efficient economies2
2. Gains from exchange6
3. Edgeworth-Box analysis 15
4. Properties of a consumers choice 20
5. Walrasian equilibria are Pareto Efficient24
Microeconomic Theory -2- Equilibrium and efficiency
John RileyOctober15, 2018
Efficient economies
Definition: Pareto preferred allocation
The allocation { }
h
hx His Pareto preferred to { }
h
hx Hif all consumers weakly prefer{ }
h
hx Hover
{ }h hx H and at least one consumer strictly prefers { }
h
hx H .
Microeconomic Theory -3- Equilibrium and efficiency
John RileyOctober15, 2018
Definition: Pareto preferred allocation
The allocation { }
h
hx His Pareto preferred to { }
h
hx Hif all consumers weakly prefer{ }
h
hx Hover
{ }h hx H and at least one consumer strictly prefers { }
h
hx H .
Definition: Pareto efficient allocation
{ }h hx His Pareto efficient if there is no feasible Pareto preferred allocation.
Microeconomic Theory -4- Equilibrium and efficiency
John RileyOctober15, 2018
Definition: Pareto preferred allocation
The allocation { }
h
hx His Pareto preferred to { }
h
hx Hif all consumers weakly prefer{ }
h
hx Hover
{ }h hx H and at least one consumer strictly prefers { }
h
hx H .
Definition: Pareto efficient allocation
{ }h hx His 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 His a Walrasian
Equilibrium allocation, then { }
h
hx His Pareto Efficient.
Microeconomic Theory -5- Equilibrium and efficiency
John RileyOctober15, 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 RileyOctober15, 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 RileyOctober15, 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 xis the slope of the indifference curve.
slope =
Microeconomic Theory -8- Equilibrium and efficiency
John RileyOctober15, 2018
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation Ax and Bxis not a PE allocation if there
is an exchange of commodities 1 2( , )e e esuch that
( ) ( )A AA AU x e U x and ( ) ( )
B B
A AU x e U x
**
Microeconomic Theory -9- Equilibrium and efficiency
John RileyOctober15, 2018
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation Ax and Bxis not a PE allocation if there
Is an exchange of commodities 1 2( , )e e esuch that
( ) ( )A AA AU x e U x and ( ) ( )
B B
A AU x e U x
Proposition:If 0Ax and 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 RileyOctober15, 2018
Pareto Efficient allocation in a 2 person 2 commodity economy
An allocation Ax and Bxis not a PE allocation if there
Is an exchange of commodities 1 2( , )e e esuch that
( ) ( )A AA AU x e U x and ( ) ( )
B B
A AU x e U x
Proposition:If 0Ax and 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,
( ) ( )
A B
A BMRS x MRS x
Consider a proposal by Alex of 1 2( , )e e ewhere
1 20e e
and the exchange rate lies between the
two marginal rates of substitution
slope =
slope =
Microeconomic Theory -11- Equilibrium and efficiency
John RileyOctober15, 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 RileyOctober15, 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 emust raise the utility of both consumers.
So the initial allocation ,A Bx xis not a Pareto efficient
allocation.
Microeconomic Theory -13- Equilibrium and efficiency
John RileyOctober15, 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 RileyOctober15, 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.
{ ( ) | ( ) ( )}B BA B B
e
Max U x e U x e U x
Class Exercise
What exchange *esolves this problem if the allocation
,A Bx x is Pareto efficient?
Pareto preferred allocation
Pareto efficient
allocation
Microeconomic Theory -15- Equilibrium and efficiency
John RileyOctober15, 2018
3. Efficiency in an Edgeworth-Box diagram
Consider Alex and Bevwith endowments
Aand B .
( ) ( )A BMRS MRS so there are gains from exchange.
Microeconomic Theory -16- Equilibrium and efficiency
John RileyOctober15, 2018
Efficiency in an Edgeworth-Box diagram
If the endowments are
Aand 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 RileyOctober15, 2018
Box rotated 180
Microeconomic Theory -18- Equilibrium and efficiency
John RileyOctober15, 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 RileyOctober15, 2018
The allocation Axand B Ax x
is Pareto- efficient since the
marginal rates of substitution are equal.
Microeconomic Theory -20- Equilibrium and efficiency
John RileyOctober15, 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 RileyOctober15, 2018
4.Properties of a consumers choice
Non satiation property
For every x , there is at a commodity jsuch 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 hxbe the choice of consumer h .
(i) If ( ) ( )
h h h hU x U xthen
h hp x p x and (ii)if ( ) ( )
h h h hU x U xthen
h hp x p x .
*
Microeconomic Theory -22- Equilibrium and efficiency
John RileyOctober15, 2018
2.Properties of a consumers choice
Non satiation property
For every x , there is at a commodity jsuch 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 hxbe the choice of consumer h .
(i)If ( ) ( )
h h h hU x U xthen
h hp x p x (ii)If ( ) ( )
h h h hU x U xthen
h hp x p x
Proof of (i):If ( ) ( )
h h h hU x U xthen
h hp x p x
Suppose instead that ( ) ( )
h h h hU x U xand
h hp x p x .
Then hxis 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 RileyOctober15, 2018
(ii)If ( ) ( )
h h h hU x U xthen
h hp x p x
Proof:If ( ) ( )
h h h hU x U xthis follows from (i)
Suppose instead that ( ) ( )
h h h hU x U xand
h hp x p x .
Define
1 1 1
( ,. , , ,, )h j j j nx x x x x x .
Then for some jand all small 0
1 1 1 1 1 1
( ) ( ,. , , ,, ) ( ,. , , ,, )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 RileyOctober15, 2018
(ii)If ( ) ( )
h h h hU x U xthen
h hp x p x
Proof:If ( ) ( )
h h h hU x U xthis follows from (i)
Suppose instead that ( ) ( )
h h h hU x U xand
h hp x p x .
Define
1 1 1
( ,. , , ,, )h j j j nx x x x x x .
Then for some jand all small 0
1 1 1 1 1 1
( ) ( ,. , , ,, ) ( ,. , , ,, )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
h hp x p x we can choose 0 such that hxis in the budget set and ( ) ( )h h hU x U x.
Then again hxis 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 RileyOctober15, 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 His a Walrasian
Equilibrium allocation, then { }
h
hx His Pareto Efficient.
Proof:
Remember that { }
h
hx His an equilibrium allocation.
Consider any Pareto preferred allocation { }
h
hx H
Step 1:
For some h ,( ) ( )
h h hU x U x.
By the non-satiation property (i)
h hp x p .
*
Microeconomic Theory -26- Equilibrium and efficiency
John RileyOctober15, 2018
First welfare theorem for an exchange economy
If ( ), {1,, }
h hU x h HH =satisfies the non-satiation property and { }h hx His a Walrasian
Equilibrium allocation, then { }
h
hx His Pareto Efficient.
Proof:
Remember that { }
h
hx His an equilibrium allocation.
Consider any Pareto preferred allocation { }
h
hx H .
Step 1
For some h ,( ) ( )
h h hU x U x.
By the non-satiation property (i)
h hp x p .
Step 2
For all h ,( ) ( )
h h hU x U x.
By the non-satiation properties (i) and(ii)
h hp x p
Microeconomic Theory -27- Equilibrium and efficiency
John RileyOctober15, 2018
Summarizing,
(a)
h hp x p for some hH
(b)
h hp x p for all hH
Summing over consumers,
h h
h h
p x p x p p
H H
Step 3:
For any feasible allocation { }
h
hx Hthe total consumption vector must satisfy
h
h
x x
H
.
Since 0p it follows that for any feasible allocation
p x p .
Since p x p it follows that xis 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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