Second Welfare Theorem
Econ 2100
Fall 2015
Lecture 18, November 4
Outline
1
Second Welfare Theorem
Quasi-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 quasi-equilibrium with transfers if there exists a vector of
wealth levels
X
X
w = (w1 ; w2 ; :::; wI )
with
wi = p ! +
p yj
I
J
i
j
such that:
1
For each j = 1; :::; J:
2
For every i = 1; :::; I :
if
p
x
i
yj
xi
p
yj
then
for all yj 2 Yj .
p
x
3
wi
P
i
xi = ! +
P
j
yj
Aggregate wealth is divided among consumers as to satisfy their budget
constraint.
In a competitive equilibrium wi = p
!i +
P
j
ij (p
yj ).
The maximzing condition for consumers is weakened.
In an equilibrium with transfers xi %i xi
for all xi 2 fxi 2 Xi : p
xi
wi g
Second Welfare Theorem: Preliminaries
Last class, we have seen a few counterexamples to a possible second welfare
theorem.
This is what we need to avoid those issues.
To show that for any Pareto optimal allocation one can …nd prices that
make it into a competitive equilibrium requires a few assumptions
We need to transfers to overcome the limitations imposed by private
ownership.
We need convexity of production sets.
We need convexity and local non-satiation of preferences.
We need to eliminate boundary issues.
Second Welfare Theorem
Theorem (Second Fundamental Theorem of Welfare Economics)
Consider an economy fXi ; %i gi =1 ; fYj gj =1 ; ! and assume that Yj is convex for
all j = 1; :::; J, and %i is convex and locally non-satiated for all i = 1; :::; I .
I
J
Then, for each Pareto optimal allocation x ; y there exists a price vector p 6= 0
such that (x ; y ; p ) form a quasi-equilibrium with transfers.
The proof uses the separating hyperplane theorem.
If an allocation is Pareto optimal there is an hyperplane that simultaneously
supports the better-than sets of all consumers and all producers.
That hyperplane yields a candidate equilibrium price vector.
The proof is in three parts: aggregation, separation, and decentralization.
We start with a Pareto optimal allocation and construct the corresponding
quasi-equilibrium with transfers.
Proof of the Second Welfare Theorem: Aggregation
First, we aggregate all consumers preferences when evaluating the Pareto
e¢ cient consumption bundle x .
De…ne the following sets:
Vi = fxi 2 Xi : xi
i
xi g
RL
and
V =
X
Vi
i
V is the set of all bundles strictly preferred to x by every consumer.
Claim: V is convex.
Take x 0 ,x 00 2 Vi (so both are strictly preferred to xi ) and w.l.o.g. assume
x 0 %i x 00 .
Since preferences are convex, for any 2 [0; 1]
x 0 + (1
) x 00 %i x 00
By transitivity, we have
x 0 + (1
) x 00 %i x 00 i xi
0
00
Therefore, x + (1
) x is an element of Vi and therefore each Vi is convex.
V is convex because it is the sum of I convex sets.
Proof of the Second Welfare Theorem: Aggregation
Second, we aggregate all …rms and de…ne the set of attainable consumption
bundles.
De…ne the aggregate production set as
8
9
<X
=
X
Y =
Yj =
yj 2 RL : y1 2 Y1 ; :::; yJ 2 YJ
:
;
j
j
The set of consumption bundles that can be allocated to consumers is
Y + f!g
This set is convex since it is the sum of J + 1 convex sets.
Geometry of the Proof
Draw V and Y + f!g.
This is the set of
consumption bundles
strictly preferred to x*
by all consumers
V=ΣiVi = Σi {xi in Xi : xi>xi*}
Y+ω
Σ i xi*
This is the set of ‘attainable’
consumption bundles given
the aggregate production set
and the aggregate endowment
x* is a Pareto optimal
consumption bundle
Proof of the Second Welfare Theorem: Separation
Next, we separate the sets V and Y + f!g.
Since (x ; y ) is a Pareto optimal allocation, V \ Y + f!g = ;.
If not, some consumer can obtain a consumption bundle preferred to what she
gets in x , contradicting the assumption that x is Pareto optimal.
Since V and Y + f!g and two disjoint convex sets, one can apply the
Separating Hyperplane Theorem.
Separate V and Y + f!g
By the Separating Hyperplane Theorem, there exist a p 2 RL with p 6= 0 and an
r 2 R such that
p z
r
and
p z
r
for all z 2 V ,
for all z 2 Y + f!g
Proof of the Second Welfare Theorem: Separation
Next, we look at the implication of separation for consumers. By separation,
p z r
for all z 2 V ,
Claim: if xi %i xi for all i, then p (
Take any xi %i xi for all i.
P
i
xi )
r
remember this as
we will use it later
By local non-satiation, for each i there exists an x^i (near xi ) such that x^i
P
Hence, x^i 2 Vi for all i, and i x^i 2 V .
P
So, p ( i x^i ) r (by separation);
P
Take a sequence of x^i that goes to xi (check how this works): p ( i xi )
i
xi .
r.
Applying this result to xi %i xi , separation
! tells us that
X
p
xi
r
i
Geometry of the Proof
P
We have shown that i fxi 2 Xi : xi %i xi g belongs to the closure of V which
is contained in the half-space z 2 RL : p z r .
p
This is the set of
consumption bundles
strictly preferred to x*
by all consumers
Σ i xi*
V
Y+ω
This is the set of ‘attainable’
consumption bundles given
the aggregate production set
and the aggregate endowment
x* is a Pareto optimal
consumption bundle
Proof of the Second Welfare Theorem: Separation
Next, use the implication of separation for consumers …rms. By separation
p z r
for all z 2 Y + f!g
P
Choosing z = j yj + ! 2 Y + f!g one gets
X
p (
yj + !) r
j
Next, put together the implications of separation for consumers and …rms.
The Pareto optimal allocation
is X
feasible, and therefore:
X
xi =
yj + ! 2 Y + f!g
i
j
Hence we have
p
X
i
xi
!
r
Putting together this inequality and the opposite
one from the previous slide:
!
X
p
xi = r
i
Geometry of the Proof
P
L
i xi belongs to Y + f!g and it lies in the half-space z 2 R : p z
r .
p
This is the set of
consumption bundles
strictly preferred to x*
by all consumers
Σ i xi*
V
Y+ω
Σ j y j* + ω
This is the set of ‘attainable’
consumption bundles given
the aggregate production set
and the aggregate endowment
x* is a Pareto optimal
consumption bundle
Second Welfare Theorem Proof: Decentralization
We have shown the following holds
0
1
!
X
X
p
xi = p @! +
yj A = r
i
j
Claim: x satis…es the consumers’condition in a quasi-equilibrium with transfers at
prices p = p.
For some consumer i, take an x such that x
p x wi for some wi .
As shown previously,
0
p
@x +
|
X
n6=i
{z
1
xn A
i
r =p
}
this satis…es xi %i xi for all i
Hence:
p
Set wi = p
xi so that we have p
x
p
x
xi . We need to show that
0
@xi +
xi
wi as desired.
X
n6=i
1
xn A
Second Welfare Theorem Proof: Decentralization
We have shown the following holds
1
0
!
X
X
yj A = r
p
xi = p @! +
j
i
Claim: y maximizes pro…ts at prices p .
For any …rm j and any yj 2 Yj , we have
X
yj +
yk 2 Y
k 6=j
Hence, by separation and the equation above we have
1
0
1
0
X
X
yk A r = p @! + yj +
p @! + yj +
yk A
k 6=j
k 6=j
Hence,
p
yj
p
Therefore, yj maximizes pro…ts at prices p .
yj
Proof of the Second Welfare Theorem: End
Summary
We have shown that x satis…es the consumers’condition in a
quasi-equilibrium with transfers at prices p and income wi = p
xi .
We have also shown that y maximizes pro…ts at prices p .
Therefore
We have shown that the Pareto optimal allocation (x ; y ) and the prices p form
a quasi-equilibrium with transfers.
The equilibirum prices are given by an hyperplane that simultaneously supports
all consumers better-than set and the aggregate production set.
Proof of the Second Welfare Theorem: Coda
The last step is to show that a quasi-equilibrium with transfers is also an
equilibrium with transfers.
You will prove this in Problem Set 10.
First, you have to show that, under local non satiation, if there is a consumption
bundle cheaper than a consumer’s wealth, condition 2. of a quasi-equilibrium
with transfers is equivalent to condition 2. of an equilibrium with transfers
(there is nothing strictly cheaper than ! i in our counterexample from last class).
Then, add strict monotonicity (something else violated by that counterexample)
and show that a quasi equilibrium with transfers which has strictly positive
wealth for all consumers is an equilibrium with transfers.
Welfare Theorems in the Di¤erentiable Case
Question
What is the relationship between the …rst order conditions that correspond to a
competitive equilibrium and those that give Pareto optimality?
Assumptions needed for di¤erentiability
Consumers
Let Xi = RL+ and assume there exist ui (x ) representing %i that satisfy strong
monotonicity and convexity for each i .
Normalize things so that ui (0) = 0.
Assume each ui (x ) is twice continuously di¤erentiable, with rui (x )
0 for
any x , and also assume that ui (x ) is quasi-concave.
Producers
Production sets are Yj = y 2 RL : Fj (y ) 0 , where Fj (y ) = 0 de…nes the
transformation frontier.
Assume each Fj (y ) is convex, twice continuously di¤erentiable, with
rFj (y )
0 for any y , and also assume that Fj (0) 0.
Welfare Theorems in the Di¤erentiable Case
Given these assumptions, Pareto e¢ ciency solves the planners problem.
Remark
An allocation is Pareto optimal if and only if it is a solution to the following:
max
(x ;y )2RLI
RLJ
u1 (x11 ; x21 ; :::; xL1 )
subject to
ui (x1i ; x2i ; :::; xLi )
X
xli
ui
!l +
X
ylj
l = 1; 2; :::; L
j
i
Fj (y1j ; :::; yLj )
i = 2; 3; :::; I
0
j = 1; 2; :::; J
Welfare and Equilibirum In the Di¤erentiable Case
(x ; y ) is Pareto optimal if and only if it solves the following
max
(x ;y )2RL(I +J )
u1 (x1 )
s:t:
ui (xi ) ui
P Fj (yj ) 0P
! l + j ylj
i xli
i = 2; 3; :::; I
j = 1; 2; :::; J
l = 1; 2; :::; L
We can also write the maximization problems that must be solved by a
competitive equilibrium.
(x ; y ; p ) is a competitive equilibrium with transfers if x ; y solves:
max ui (xi )
s:t:
p
xi
wi
i = 2; 3; :::; I
max p
s:t:
Fj (yj )
0
j = 1; 2; :::; J
xi
0
and
yj
yj
What is the connection between the …rst order conditions of these two
optimization problems?
Econ 2100
Fall 2015
Problem Set 10
Due 10 November, Monday, at the beginning of class
1. Consider a competitive
model with one input,
two outputs, and two …rms with production functions
p
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?
2. Prove that any competitive equilibrium is in the core (for an exchange economy).
3. 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).
(a) Show that p = (1; 1) and xa = (1; 1), xb = (1; 1) is a competitive equilibrium.
(b) Prove or provide a counterexample to the following statement: in this economy any competitive
equilibrium is Pareto optimal.
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).
(a) Show that p = (1; 1) and xa = ( 21 ; 12 ), xb = ( 21 ; 12 ) is a competitive equilibrium.
(b) Is this allocation in the core? Explain your answer.
(c) Does the …rst welfare theorem hold for this economy? Explain your answer.
5. 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).
(a) Find the Pareto optimal allocations and the core. Draw the Edgeworth box for this economy.
(b) Find the individual and market excess demand functions (these are de…ned as the di¤erence
between demand and supply). Find the equilibrium prices and allocations.
(c) Show directly that every interior Pareto optimal allocation in this economy is a price equilibrium
with transfers by …nding the associated prices and transfers.
6. This exercise completes the proof of the second welfare theorem.
(a) Assume Xi is convex and %i is continuous. Let xi 2 Xi , p, and wi be such that xi
p xi
w. Prove that if there exists an x0 2 Xi such that p x0 < w, then xi
p xi > w.
1
xi implies
xi implies
(b) Now assume %i also satis…es local non satiation and prove that the following are equivalent:
1. xi minimizes p xi for xi 2 fxi 2 X : xi % x g
2. xi % xi for all xi 2 fxi 2 X : p xi p xi g
(c) Finally, consider an economy with continuous
P and strongly monotone preferences where Xi =
L
R+ . Suppose there exist y 2 Yj such that j yj + !
0. Prove that any quasi-equilibrium
with transfers that has w = (w1 ; :::; wI )
0 must be an equilibrium with transfers.
(d) Using these results and the theorem we proved in class, state all assumptions needed to prove
that any Pareto optimal allocation is (part of) a competitive equilibirum (you do not need to
prove anything).
2