Download First Welfare Theorem

First Welfare Theorem
Econ 2100
Fall 2015
Lecture 17, November 2
Outline
1
First Welfare Theorem
2
Preliminaries to Second Welfare Theorem
Last Class De…nitions
A feasible allocation (x; y ) is Pareto optimal if there is no other feasible
allocation (x 0 ; y 0 ) such that
xi0 %i xi
for all i
and
xi0 i xi
for some i:
L
An allocation (x ; y ) and a price vector p 2 R are a competitive
equilibrium if
1
2
3
for each j = 1; :::; J: p yj p yj for all yj 2 Yj ;
for each i = 1; :::; I :
P
xi %i xi for all xi 2 Bi (p ) = fxi 2 Xi : p xi
p !i + j
and
P
P
P
i xi =
i !i +
j yj .
ij
p
yj g ;
What is the relationship between competitive equilibrium and Pareto
e¢ ciency?
Is any competitive equilibirum Pareto e¢ cient? First Welfare Theorem.
This is about excluding something can Pareto dominate the equilibrium
allocation.
Is any Pareto e¢ cient allocation (part of) a competitive equilibrium? Second
Welfare Theorem.
This is about …nding prices that make the e¢ cient allocation an equilibirum.
First Welfare Theorem: A Picture
Things seem easy
O2
ω
x*
O1
First Welfare Theorem: Counterexample
An Edgeworth Box Economy
Consider a two-person, two-good exchange economy.
Person a has utility function Ua (x1a ; x2a ) = 7 and person b has utility function
Ub (x1b ; x2b ) = x1b x2b .
The initial endowments are ! a = (2; 0) and ! b = (0; 2).
CLAIM: xa = (1; 1), xb = (1; 1), and prices p = (1; 1) form a competitive
equilibirum.
a’s utility is maximized.
b’s utility when her income equals 2 is maximized (this is a Cobb-Douglas utility
function with equal exponents, so spending half her income on each good is
optimal).
xa + xb = (2; 2) = !.
Is this allocation Pareto optimal? No:
x^a = (0; 0) and x^b = (2; 2) Pareto dominates xa , xb since consumer a has the
same utility while consumer b’s utility is higher.
How do we rule examples like this out?
Need consumers preferences that are locally non satiated: there is always
something nearby that makes a consumer better o¤.
Local Non Satiation
De…nition
A preference ordering %i on Xi is satiated at y if there exists no x in Xi such that
x i y.
De…nition
The preference relation %i on Xi is locally non-satiated if for every x in Xi and for
every " > 0 there exists an x 0 in Xi such that kx 0 xk < " and x 0 i x.
Remember: ky
two points.
zk =
qP
L
l =1
(yl
2
zl ) is the Euclidean distance between
Remark
If %i is continuous and locally non-satiated there exist a locally non-satiated
utility function; then, any closed consumption set must be unbounded (or
there would be a global satiation point).
Local Non Satiation and Walrasian Demand
Lemma
Suppose %i is locally non-satiated, and let xi be de…ned as:
xi %i xi
Then
for all xi 2 fxi 2 Xi : p xi
x i %i x i
implies
p xi
xi
implies
p xi > wi
wi g :
wi
and
i
xi
In words, if a consumption vector is preferred to a maximal consumption
bundle (i.e. an element of the Walrasian demand correspondence) it must cost
more.
Something strictly preferred to a maximal bundle must not be a¤ordable (or
the consumer would have chosen it).
First Welfare Theorem
Theorem (First Fundamental Theorem of Welfare Economics)
Suppose each consumer’s preferences are locally non-satiated. Then, any allocation
x ; y that with prices p forms a competitive equilibrium is Pareto optimal.
There are almost no assumptions...
Local non-satiation has bite: there is always a more desirable commodity
bundle nearby.
Among the economic assumptions implicit in our de…nition of an economy one
is important for this result: no externalities.
Externalities are present if one person’s consumption in‡uences another person’s
preferences
For example: in a two-person, two-good exchange economy person a has utility
function U a (xa1 ; xa2 ) = xa1 xa2 xb1 .
Proof of the First Welfare Theorem (by contradiction)
Suppose not: there exists a feasible allocation x; y such such that xi %i xi for all i,
and xi i xi for some i.
By local non satiation,
xi %i xi implies p
xi i xi implies p
xi
p
xi > p
P
! i + Pj
!i + j
Therefore, summing over consumers
I
I
I X
J
X
X
X
p xi >
p !i +
i =1
i =1
ij (p
ij (p
ij (p
yj )
yj )
yj )
accounting
=
p
!+
Since each yj maximizes pro…ts at prices p , we also have
J
J
X
X
p yj
p yj :
j =1
Substituting this into the previous inequality:
I
J
X
X
p xi > p ! +
p yj
i =1
p
j =1
i =1 j =1
j =1
J
X
j =1
p
!+
J
X
p
yj
j =1
But then x; y cannot be feasible since if it were we would have
I
J
I
J
X
X
X
X
xi = ! +
yj
)
p xi = p ! +
p yj
i =1
j =1
which contradicts the expression above.
i =1
j =1
yj
Competitive Equilibrium and the Core
Theorem
Any competitive equilibrium is in the core.
Proof.
Homework. This is very very similar to the proof of the First Welfare Theorem.
A ‘converse’can be established in cases in which the economy is “large”, that
is, it contains many individuals.
That is called the core convergence theorem and I do not think we will have
time for it.
Second Welfare Theorem: Preliminaries
This is a converse to the First Welfare Theorem.
The statement goes: under some conditions, any Pareto optimal allocation is
part of a competitive equilibrium.
Next, we try to understand what these conditions must be. We state and prove
thw theorem next class.
In order to prove a Pareto optimal allocation is part of an equilibrium one
needs to …nd the price vector that ‘works’for that allocation, since an
equilibrium must specify and allocation and prices.
First, we see a simple sense in which this cannot work: Pareto optimality
disregards the budget constraints.
This is …xed by adjusting the de…nition of equilibrium.
Then we see two counterexamples that stress the need for convexities.
These are …xed by assuming production sets and better-than sets are convex.
Finally, we see an example showing that an equilibrium may not exist, and
therefore there is no way the theorem holds.
This is …xed by, again, adjusting the de…nition of equilibrium.
O2
The way to make the Pareto optimal
allocation an equilibirum is to stick a price
vector tangent to both indi¤erence curves.
ω
Initial
Endowment
This is not enought, however, since the
Pareto optimal allocation violates the budget
constraint of consumer 2.
Pareto
Optimal
Allocation
From the Edgeworth box, you can see that
there is no hope of going from any Pareto
optimal allocation to prices compatible with
maximization and initial endowment.
x
O1
O2
ω
Initial
Endowment
Income Transfers
Pareto
Optimal
Allocation
x
O1
There is hope, however, to keep the slope of
the budget constraint …xed and adjust income
so that things work out.
One needs to adjust income appropriately.
We achieve this by modifying the de…nition of
equilibrium.
Equilibrium With Transfers
De…nition
Given an economy fXi ; %i gi =1 ; fYj gj =1 ; ! , an allocation x ; y and a price
vector p constitute a price equilibrium with transfers if there exists a vector of
wealth levels
I
J
X
X
w = (w1 ; w2 ; :::; wI )
with
wi = p ! +
p yj
I
J
i =1
j =1
such that:
1
For each j = 1; :::; J:
2
For each i = 1; :::; I :
3
p
xi %i xi
I
P
i =1
xi =
yj
p
yj
for all yj 2 Yj .
for all xi 2 fxi 2 Xi : p
I
P
i =1
!i +
J
P
j =1
xi
wi g
yj
Aggregate wealth is divided among consumers so that the budget constrants
are satis…ed.
How is the wealth of each consumer e¤ected? They get a positive or negative
transfer relative to the value of their endowment at the equilibrium
prices.
P
A competitive equilibrium satis…es this: set wi = p ! i + j ij (p yj ).
Equilibrium With Transfers
Remark
The income transfers (across consumers) that achieve the budget levels in the
previous de…nition are:
3
2
J
X
yj 5
Ti = wi 4p ! i +
ij p
j =1
Summing over consumers, we get
2
I
X
X
X
Ti =
wi 4
p
i
i =1
=
I
X
i =1
=
0
wi
2
4p
!i +
i
!+
I X
J
X
i =1 j =1
J
X
j =1
p
3
ij
p
3
yj 5
yj 5
Transfers redistribute income so that the ‘aggregate budget’balances.
This is important: in a general equilibirum model nothing should be ‘outside’
the economy
Counterexample I to Second Welfare Theorem
Need convex preferences for the Second Welfare Theorem
x*
p
Í2
Í1
x* is Pareto optimal, but one can see it is not an equilibrium at prices p
Counterexample II to Second Welfare Theorem
One Producer One Consumer Economy (Robinson Crusoe)
x is the preferences maximizing
bundle at the given prices
x does not maximizes profits in
production set Y at the given prices
(not even locally)
x
Í1
Y
Need convex production sets for Second Welfare Theorem
Convexity
De…nition
A preference relation % on X is convex if the set % (x) = fy 2 X j y % xg is
convex for every x.
If x 0 and x 00 are weakly preferred to x so is any convex combination.
Convexity implies existence of an hyperplane that ‘supports’a consumer’s
better than set.
De…nition
In an exchange economy, an allocation x is supported by a non-zero price vector p
if: for each i = 1; :::; I
x 0 %i x i
=)
p x0
p xi
Convexity also yields an hyperplane that ‘supports’all producers’better than
set at the same time.
This hyperplane is the price vector that makes a Pareto optimal allocation an
equilibirum.
Counterexample III to Second Welfare Theorem
!
is the unique Pareto
optimal allocation, hence the
only candidate for an
equilibrium.
ω is the initial endowment
Consumer 1 owns all of good 2
Consumer 2 owns all of good 1
Consumer 2 only
For any p≥0,
cares about good 1
- 2 demands ω2 when her wealth is w2;
- but ω1 is never optimal for 1 if
p≥0… she would demand an infinite
quantity of good 1
Í2
The unique price vector that
can support it as equilibrium
(normalizing the price of the
…rst good to 1) is p = (1; 0)
.
The corresponding wealth is
p=(1,0)
is the candidate
Equilibrium
price vector
Consumer 1 preferences have
infinite slope at ω1
Í1
There is no Walrasian equilibrium in this case
w1 = (1; 0) (0; ! 2 ) = 0
.
However, ! 1
is not
maximial for consumer 1
at
p
since she would like more
of good 2
(it has zero price,
hence she can a¤ord it.
There is no competitive
equilibrium in this example.
Quasi-Equilibrium
To …x the ‘existence at the boundary’problem, with make a small change to
the de…nition of equilibrium.
De…nition
Given an economy fXi ; %i gi =1 ; fYj gj =1 ; !, an allocation x ; y and a price vector
p constitute a quasi-equilibrium with transfers if there exists a vector of wealth
levels
I
J
X
X
w = (w1 ; w2 ; :::; wI )
with
wi = p ! +
p yj
I
J
i =1
j =1
such that:
1
For each j = 1; :::; J:
2
For every i = 1; :::; I :
p
if
3
I
P
i =1
xi =
I
P
i =1
!i +
J
P
j =1
x
i
xi
yj
p
then
yj
for all yj 2 Yj .
p
x
wi
yj
Make sure you see why this deals with the problem in the previous slide.
Any equilibrium with transfers is a quasi-equilibrium (make sure you check
Problem Set 10 (Incomplete)
Due Monday 9 November
1
2
3
Consider a competitive
model with onepinput, two outputs, and two …rms with production functions
p
y1 = f1 (`1 ) = 2`1 and y2 = f2 (`2 ) = 2`2 (where `j denotes the amount of the input ` used by …rm j, and
…rm j produces only good j). Consumer a is endowed with 25 units of the input ` and owns no shares in the
…rms, and has utility function Ua (xa1 ; xa2 ) = xa1 xa2 . Consumer b owns both …rms, but has zero endowment,
and has utility function Ub (xb1 ; xb2 ) = xb1 + xb2 . Find a competitive equilibrium in this model. Is it unique?
Prove that any competitive equilibrium is in the core (for an exchange economy).
Consider a two-person, two-good exchange economy in which person a has utility function
Ua (xa1 ; xa2 ) = xa1 xa2 xb1 and person b has utility function Ub (xb1 ; xb2 ) = xb1 xb2 . The initial endowments
are ! a = (2; 0) and ! b = (0; 2).
1
2
4
Consider a two-person, two-good exchange economy in which person a has utility function Ua (xa1 ; xa2 ) = 1
if xa1 + xa2 < 1 and Ua (xa1 ; xa2 ) = xa1 + xa2 if xa1 + xa2 1. Person b has utility function
Ub (xb1 ; xb2 ) = xb1 xb2 . The initial endowments are ! a = (1; 0) and ! b = (0; 1).
1
2
3
5
Show that p = (1; 1) and xa = (1; 1), xb = (1; 1) is a competitive equilibrium.
Prove or provide a counterexample to the following statement: in this economy any competitive equilibrium is
Pareto optimal.
Show that p = (1; 1) and xa = ( 21 ; 12 ), xb = ( 12 ; 12 ) is a competitive equilibrium.
Is this allocation in the core? Explain your answer.
Does the …rst welfare theorem hold for this economy? Explain your answer.
Consider a two-person, two-good exchange economy where the agents’utility functions are
Ua (xa1 ; xa2 ) = xa1 xa2 and Ub (xb1 ; xb2 ) = xb1 xb2 , and the initial endowments are ! a = (1; 5) and ! b = (5; 1).
1
2
3
Find the Pareto optimal allocations and the core. Draw the Edgeworth box for this economy.
Find the individual and market excess demand functions. Find the equilibrium prices and allocations.
Show directly that every interior Pareto optimal allocation in this economy is a price equilibrium with transfers
by …nding the associated prices and transfers.