Di¤erentiable Welfare Theorems
Existence of a Competitive Equilibrium
Econ 2100
Fall 2015
Lecture 19, November 9
Outline
1
Welfare Theorems in the di¤erentiable case.
2
Aggregate excess demand and existence.
3
A competitive equilibirum is a …xed point.
4
Brouwer’s Theorem.
Welfare and Equilibrium In the Di¤erentiable Case
A Pareto optimal allocations (x ; y ) solves the following single optimization
problem
max
(x ;y )2RL(I +J )
u1 (x1 )
ui (xi ) ui
Fj (yj ) 0
P
PI
x
! l + j ylj
i =1 li
s:t:
i = 2; 3; :::; I
j = 1; 2; :::; J
l = 1; 2; :::; L
A competitive equilibrium with transfers x ; y ; p solves the following I + J
optimization problems (with wi = p xi for all i):
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
collections of optimization problems?
Welfare and Equilibrium 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
Fj (yj ) 0
PJ
PI
! l + j =1 ylj
i =1 xli
The …rst order conditions for a Pareto optimum are:
@ui (xi )
0
xli : i
l
= 0 if xli > 0
@xli
@Fj (yj )
=0
@ylj
where, to make the notation more compact,
ylj
:
l
i = 2; 3; :::; I
j = 1; 2; :::; J
l = 1; 2; :::; L
for all i and l
for all j and l
j
1
= 1.
With convexity assumptions, these conditions are also su¢ cient.
Pareto Optimality In The Di¤erentiable Case
The …rst order conditions for a Pareto optimum are:
@ui (xi )
0
xli : i
l
= 0 if xli > 0
@xli
@Fj (yj )
=0
ylj :
l
j
@ylj
where, to make the notation more compact, 1 = 1.
for all i and l
for all j and l
Interpret the multipliers as follows:
l is the change in consumer 1 utility from a marginal increase in the
aggregate endowment of good l (this the “shadow price” of good l).
i is the change in consumer 1 utility from a marginal decrease in the utility of
consumer i.
j
is the marginal cost of tightening production constraint j.
@F
Thus j @yljj is the marginal cost of increasing ylj ; it must be equal to the
marginal bene…t of good l (measured in consumer 1 utility).
“Marginal” Conditions for Pareto Optima
At an interior solution (xli > 0) we can rewrite these …rst order conditions as
the “usual” equalities between marginal rates of substitution and
transformation.
MRS between any two goods must be equal across any two consumers:
@u i
@xli
@u i
@xl 0 i
=
@u i 0
@xli 0
@u i 0
@xl 0 i 0
for all i; i 0 and l; l 0
MRT between any two goods must be equal across any two …rms:
@F j
@ylj
@F j
@yl 0 j
=
@F j 0
@yl 0 j
@F j 0
@yl 0 j 0
for all j; j 0 and l; l 0
MRS must be equal to MRT:
@u i
@xli
@u i
@xl 0 i
=
@F j
@ylj
@F j
@yl 0 j
for all i; j and l; l 0
Welfare Theorems in the Di¤erentiable Case
An allocation is Pareto optimal if and only if it is a solution to:
max
(x ;y )2RL(I +J )
u1 (x1 )
s:t:
ui (xi ) ui
Fj (yj ) 0
PJ
PI
! l + j =1 ylj
i =1 xli
i = 2; 3; :::; I
j = 1; 2; :::; J
l = 1; 2; :::; L
The …rst order conditions evaluated at an optimum are (with
0
@u i
xli : i @x
8i; l
and
ylj : l
l
li
= 0 if xli > 0
1
= 1):
@F j
j @ylj
= 0 8j; l
A price equilibrium with transfers solves
max ui (xi ) s.t. p xi
xi
0
wi i = 2; 3; :::; I
and
max p yj s.t. Fj (yj )
yj
0 j = 1; :::; J
The …rst order necessary (and su¢ cient) conditions evaluated at (x ; y ) yield:
0
@F j
@u i
xli : @x
8i; l
ylj : pl
i pl
j @ylj = 0 8j; l
li
= 0 if xli > 0
Let
l
= pl ,
i
=
1
i
, and
j
=
j
Then, any solution to the …rst optimization problem solves the others.
Welfare Theorems in the Di¤erentiable Case
Summary
In the case of a di¤erentiable economy (see all assumptions we listed last class), we
can use the …rst order conditions to prove the following.
Every Pareto optimal allocation is a price equilibrium with transfers for some
appropriately chosen price vector p and welfare transfers w (Second Welfare
Theorem);
Any allocation corresponding to a price equilibrium with transfers must be
Pareto optimal (First Welfare Theorem).
As an exercise, write these conclusions like formal ‘theorems’rather than
‘observations’.
Note: the price of each good corresponds to its marginal value in the Pareto
optimality problem.
The Other Planner’s Problem
An equivalent maximization problem (also called the "Planner’s problem")
I
X
Fj (yj ) 0
j = 1; 2; :::; J
PJ
PI
max
s:t:
i ui (xi )
L(I
+J
)
y
l = 1; 2; :::; L
x
!
+
(x ;y )2R
l
j =1 lj
i =1 li
i =1
The …rst order conditions for this problem are:
@ui
0
ylj :
xli : i
8i; l
l
= 0 if xli > 0
@xli
Connect to Pareto optimality:
l
=
l
1
,
i
=
i
1
, and
j
j
@Fj
= 0 8j; l
@ylj
Conect to equilibrium:
1
pl = l ,
i = i , and
j
=
l
1
j
=
j
This yields a new interpretation of equilibrium prices at an interior optimum
PL
l =1
@u i
i @xli
=
PL
l =1
l
and thus
PL
@u i
i @x
li
l =1
For any two consumers i and j:
@u i
i @x
li
=
@u i
@x
PL li @ui
l =1 @xli
PL
=
for all l (r.h.s. same for all i).
l
l =1
l
@u j
@xlj
PL
@u j
l =1 @xlj
=
p
PL l
l =1
pl
for all l
Prices re‡ect the common marginal utilities of consumption.
Similar to the ‘separating’step in the proof of the Second Welfare Theorem:
price equals the slopes of all better-than sets, it supports them all at once.
Existence of A Competitive Equilibrium
The Existence Problem
We have theorems about the welfare properties of a competitive equilibrium, but no
result stating that such a thing exist.
Is it hard to prove existence? What assumptions do we need?
To answer these questions, we study the conditions that characterize an equilibrium.
Assumptions and notation
Consider an exchange economy with strictly positive aggregate endowment;
preferences are represented by continuous and quasi-concave utility functions.
P
Formally: J = 1, Y1 = RL+ , and Xi = RL+ , Ii =1 ! i >> 0, and each %i is
continuous, convex, and locally non-satiated.
Let xi (p; p ! i ) be i’s Walrasian demand:
xi (p; p ! i ) 2 arg
max
x 2fx 2RL+ :p x
p !i g
ui (x)
If the preferences are also strictly convex, the utility function is strictly
quasi-concave and Walrasian demand is a function.
In more general cases, it is a correspondence.
Excess Demand
De…nitions
The individual excess demand function of consumer i, zi : RL+ ! RL , is de…ned by
zi (p) = xi (p; p ! i )
!i
The aggregate excess demand function, z : RL+ ! RL , is de…ned by
z (p) =
I
X
zi (p)
i =1
If Walrasian demand is not single valued, aggregate excess demand is not
single valued either; in that case, one can de…ne individual and aggregate
excess demand correspondences.
Soon, we will use aggregate excess demand to …nd an equilibium.
Suppose you …nd a price vector p^ such that z (^
p ) = 0; is this an equilibirum
price vector?
First, some properties of aggregate excess demand.
Properties of Aggregate Excess Demand
Proposition
Suppose that for each i, X i = RL+ and %i is continuous, strictly convex, and
PI
strongly monotone. Suppose also that i =1 ! i >> 0. Then, the aggregate excess
demand function z (p) satis…es the following properties:
1
z (p) is continuous;
2
z (p) is homogeneous of degree zero;
3
p z (p) = 0 for all p (Walras’Law);
4
there exists an s > 0 such that zl (p) >
5
s for each l and for each p; and
if p n ! p, with p 6= 0 and pl = 0 for some l, then
max fz1 (p n ) ; z2 (p n ) ; :::; zL (p n )g ! 1:
For the …rst four properties local non satiation is enough.
The last property says that demand for some good explodes as one price
approaches zero.
This is not necessarily the good that has zero price.
Walras’Law
Walras’Law
In an exchange economy with locally non satisted preference
p z (p) = 0
for all p
The value of aggregate excess demand is zero for any price vector.
Proof.
p z (p)
= p
I
X
zi (p) = p
i =1
=
I
X
I
X
[xi (p; p ! i )
!i ]
i =1
[p xi (p; p ! i )
i =1
p ! i ] |{z}
= 0
by l :n:s :
If L 1 markets clear, the L-th market also clears
By Walras’Law:
L
X
X
0=
pl zl (p) =
pl zl (p) + pk zk (p) = 0 + pk zk (p)
l =1
l 6=k
Equilibrium in an Edgeworth Box
Assume strictly convex preferences: all excess demands are functions.
Let the p2 = 1, so we need only …nd p1 , and write market excess demand as
z (p1 ; 1) = (z1 (p1 ; 1) ; z2 (p1 ; 1))
By Walras’Law, an equilibrium satis…es z1 (p1 ; 1) = 0 (the other market also
clears).
An equilibrium exists if for some price of good one z1 (p1 ; 1) = 0.
z1 (p1)
The aggregate excess demand function
z1 (p1,1)
A zero of the excess demand
function is an equilibrium
p*1
p1
In an Edgeworth Box Economy
Showing an equilibrium exists is like showing that some function has a zero.
Use the Intermediate Value Theorem: any continuous function over an interval
that has negative and positive values must have (at least) one zero.
With more than 2 goods we need a more general version of this idea.
Excess Demand and Competitive Equilibirum
In an exchange economy, a Walrasian equilibrium is de…ned as follows.
De…nition
x ; y and p 2 RL form a Walrasian (competitive) equilibrium of an exchange
economy if and only if
1
y1
2
xi 2 arg
3
0, p
PI
i =1 xi
y1 = 0, and p
max
x 2fx 2RL+ :p
PI
i =1
x
p
!i g
0.
ui (x)for all i
! i = y1
The next result shows that a competitive equilibrium requires aggregate excess
demand to be non-positive.
Proposition
Consider an exchange economy where each consumer’s preferences are continuous,
strictly convex, and locally non-satiated. In this economy, a price vector p
0 is a
competitive equilibrium price vector if and only if
z (p )
0
p
0 is a competitive equilibrium price vector if and only if z (p )
0.
Proof.
One direction is trivial; substituting 1. and 2. into 3:
I
I
X
X
[xi (p ; p ! i ) ! i ] = y1
0
)
z (p) =
zi (p)
i =1
0
i =1
The other is also easy. If p is such that z (p )
y1 =
I
X
[xi (p ; p
!i )
!i ]
and
0 then let
xi = xi (p ; p
!i )
i =1
Therefore
p y1 = p
I
X
i =1
[xi (p ; p
!i )
!i ] =
I
X
[p
xi (p ; p
!i )
p
i =1
where the last equality follows by local non satiation.
Thus, y1 , xi and p satisfy the de…nition of competitive equilbrium.
!i ] = 0
Zero Prices In Equilibrium?
What happens if some prices are equal to zero in equilbirum?
Remark
p is an equilibrium i¤
p
0
and
z (p )
p
By Walras’law, the following must also hold
(it holds for any p, hence also at p ).
0
z (p ) = 0
Look at these expressions for some good l: equilbrium implies
zl (p )
0
and since Walras’law holds we also have:
zl (p ) = 0
if
pl > 0
Conclude that in equilibrium a good can be in excess supply only if it is free:
zl (p ) < 0
only if
pl = 0
With locally non satiated preference, some goods can have a zero price in
equilibrium but they will be in excess supply.
Monotonicity and Zero Prices
If preferences are strongly monotone (this is more restrictive than local non
satiation), then an equilibrium price vector must be strictly positive: if not,
consumers would demand an in…nite amount of any free good.
With strong monotonicity, a price vector is an equilibrium if and only if it
‘clears all markets’
Formally, a price vector p is an equilibrium if and only
zl (p ) = 0
for all l;
this is a system of L equations in L unknowns.
This system, however, has redundancies: by Walras’law one of these
equations is implied by the others.
On the other hand, only price ratios matter (one price is undetermined), so we
really have L 1 equations in L 1 unknowns.
Equilibrium As A Fixed Point
Summary
An equilibrium price vector p must satisfy:
zl (p ) 0 for all l, and
z (p ) 0
or
whenever zl (p ) < 0 then pl = 0
A useful observation
Let the function g : RL ! RL be de…ned by
gl (p) = max fpl + zl (p) ; 0g
CLAIM: An equilibrium is a p
g (p ) = p
for l = 1; 2; ::; L
0 such that
or
gl (p ) = pl for all l
At an equilibrium price vector p :
1
either gl (p ) = 0 and thus pl = 0 and zl (p ) < 0, or
2
gl (p ) 6= 0 and thus pl + zl (p ) = pl which implies zl (p ) = 0.
In both cases we have an equilibrium thus establishing the claim.
Brouwer’s Fixed Point Theorem
Summary
The L-dimensional mapping g (p) is de…ned by
gl (p) = max fpl + zl (p) ; 0g
for l = 1; 2; ::; L
We conclude an equilibirum exists if we prove that there exists a p such that
g (p ) = p
This is a …xed point: we want to prove that g (p) has …xed point.
The following theorem gives conditions for functions to have a …xed point.
Theorem (Brouwer’s Fixed Point Theorem)
If S RL is convex and compact and the function f : S ! S is continuous, then
there exists an x 2 S such that f (x) = x (that is, f has a …xed point).
This is what we need to prove that a competitive equilibrium exists.
A general existence proof allows for correspondences (aggregate excess
demand is typically not a function); fortunately, there is similar theorem for
correspondences (Kakutani’s …xed point theorem).
Problem Set 11 (Incomplete)
Due Monday 16 November
1
Consider an exchange economy with N consumers and L goods in which each consumer has a utility function ui : RL+ ! R that is continuous, strictly
quasi-concave and strongly monotone, and an initial endowment ! i which contains a positive amount of each good. Each consumer must pay a tax on
the value of his or her gross consumption of each good, and these tax rates may vary across goods and consumers. Thus if consumer i buys xij units of
good j when the price for good j is pj , the tax she pays is tij pj xij . The tax rates ftij g are set exogenously by the tax authority; each consumer also receives
a lump-sum rebate Ri from the tax authority. In equilibrium, the total tax receipts must be rebated equally across the consumers in a lump-sum fashion.
An equilibrium with taxes is a price vector p , an allocation (x1P
; : : : ; xm ), and rebates (R1 ; : : : ; RN ) such that each consumer i is maximizing utility over
the budget set at the bundle xi , all markets clear, and Ri = N1 i ;j tij pj xij for each i = 1; : : : ; N.
Write down the budget set for a typical consumer in this economy.
Illustrate an equilibrium with taxes in an Edgeworth Box. Does the First Welfare Theorem hold for equilibria with taxes?
Show that an equilibrium with taxes exists. (This is hard; think about what is an equilibrium and what properties do consumers’demand functions satisfy in this
economy?)
Now consider the case in which taxes are applied to net consumption (the di¤erence between consumption and initial endowment). Thus, consumer i net trades are
xij ! ij units of good j and when the price for good j is pj the tax she pays is tij pj (xij ! ij ).
1
2
3
2
Consider P
an exchange economy with L goods and N consumers. Assume each consumer i’s utility function is additively separable; it has the form
Ui (x) = ` vi (x` ) for a strictly increasing, strictly concave, and di¤erentiable function vi : R++ ! R that satis…es limc !0 vi0 (c) = 1. Also assume each
consumer i’s initial endowment is strictly positive; that is, ! i
0.
1
2
3
3
Consider an exchange economy where for each i we have that X i = RL+ , and %i is continuous, strictly convex, and locally non-satiated. Suppose also that
P
i ! i >> 0. Prove that the aggregate excess demand function z (p) satis…es the following properties:
1
2
3
4
4
P
P
P
Show that if P
the aggregate
of each good is constant ( i ! i 1 = i ! i 2 =
= i ! iL ) the economy has at most one equilibrium.
P endowmentP
Show that if i ! i 1 > i ! i 2 >
> i ! iL any competitive equilibrium price vector p must satisfy p1 < p2 <
< pL .
L
Suppose that vi (c) = ln(c) for each i . Fix the aggregate endowment ! 2 R++ . Show that the set of competitive equilibrium prices does not depend on the
initial endowment allocation. That is, if p is a competitive equilibrium price vector in this economy with initial
endowment
allocation (! 1 ; : : : ; ! N ), then p is
P
also a competitive equilibrium price vector for any other initial endowment allocation (! 01 ; : : : ; ! 0N ) such that i ! 0i = !.
z (p) is continuous;
z (p) is homogeneous of degree zero;
there exists an s > 0 such that zl (p) > s for each l and for each p.
Now also assume that for each i %i is strongly monotone and show that if p n ! p, with p 6= 0 and pl = 0 for some l , then
max fz1 (p n ) ; z2 (p n ) ; :::; zL (p n )g ! 1 .
Consider an Edgeworth box economy with two individuals (A and B) and two goods (1 and 2). The consumers’utility functions are uA (x1 ; x2 ) and
uB (x1 ; x2 ); both utility functions are continuous and strictly quasi-concave. The initial endowments of the two consumers are
! A = (1; 0) and ! B = (0; 1) .
1
2
3
Find the competitive (Walrasian) equilibirum, or argue this economy does not have one, when both utility functions are strictly decreasing everywhere.
Find the competitive (Walrasian) equilibirum, or argue this economy does not have one, when both utility functions are decreasing with respect x1 and increasing
with respect to x2 .
Find the competitive (Walrasian) equilibirum, or argue this economy does not have one, when uA is decreasing with respect x1 and increasing with respect to x2
while uB is increasing with respect x1 and decreasing with respect to x2 .