Existence of Walrasian Equilibrium
Econ 2100
Fall 2015
Lecture 20, 11 November
Outline
1
Existence of a competitive (Walrasian) equilibirum
Existence: Preliminaries
We want conditions on preferences
PL such that an equilibirum exists in an
exchange economy with ! = i =1 ! i >> 0.
Consumer i’s Walrasian demand function xi (p) is de…ned as:
xi (p) %i x
for all x 2 fx 2 Xi : p x p ! i g :
REMARK: Walrasian demand must be well de…ned for an equilibirum to exist.
The excess demand function of consumer i is zi (p) = xi (p) ! i
PL
The aggregate excess demand function is z(p) = i =1 zi (p):
If X i = RL+ and %i is continuous, strictly convex, and locally non-satiated for
each i, and ! >> 0, then z (p) satis…es Walras’Law: p z(p) = 0 for all p.
Under the same assumptions, a price vector p
0 forms a Walrasian
equilibrium if and only if z(p ) 0.
In an Edgeworth box, this reduces to …nding a zero for the excess demand of
one good (by Walras’law), and one uses the intermediate value theorem.
Let g : RL ! RL be de…ned by: gl (p) = maxfpl + zl (p) ; 0g for l = 1; 2; ::; L.
An equilibrium is a p
0 such that g (p ) = p . Why?
either gl (p ) = 0 and thus pl = 0 and zl (p ) 0, or
gl (p ) 6= 0 and thus pl + zl (p ) = pl which implies zl (p ) = 0.
When does a function have a …xed point? Brower’s theorem.
Brouwer’s Theorem and Existence
Theorem
A continuous f : S ! S, where S
RL is convex and compact, has a …xed point.
How can we use this theorem to show that an equilibirum exist?
Let g : RL ! RL be de…ned by: gl (p) = max fpl + zl (p) ; 0g for l = 1; 2; ::; L. Can
we apply Brower?
1
2
Need domain and range to be the same convex and compact subset of RL .
They are not, but since only relative prices P
matter we can normalize them.
L
Take the domain to be L 1 = fp 2 RL+ : l =1 pl = 1g, and divide g ( ) by
the sum of its elements so that the range is also L 1 .
X
g ( ) needs to be continuous.
Assume preferences that yield continuity of excess demand.
X
3
g ( ) needs to be a function.
Assume strictly convex preferences (there is a theorem for correspondences). X
4
g ( ) must be well de…ned even if some prices are zero.
If preferences are monotone, excess demand can blow up. But we need
monotonicity for other properties. Assume this away (for now).
‘Easy’Existence Theorem
Theorem
Assume the aggregate excess demand function z : L 1 ! RL is continuous and
such that p z (p) = 0 for all p. Then, there exists p 2 L 1 such that z (p ) 0.
If excess demand is a continuous function that satis…es Walras’Law, an
equilibrium exists.
Remark
We should prove existence from assumptions on the primitives of the economy,
i.e.preferences and endowments, not on excess demand.
Although many assumptions in this theorem can be written from preferences
way (use the results from last class), the fact that excess demand is well de…ned
and continuous over its entire domain cannot (need to rule out zero prices).
The proof goes as follows
1
De…ne a function of excess demand (as described last class) so that...
2
... Brouwer’s …xed point theorem applies, and a …xed point of this function
exists.
3
Complete the proof by showing this …xed point is an equilibrium.
‘Easy’Existence Proof: Step 1a
Let g :
L 1
! RL be de…ned by
gl (p) = max fpl + zl (p) ; 0g
with l = 1; 2; ::; L:
Remark
Claim: g (p) 6= 0.
By de…nition, gl (p)
pl + zl (p).
Thus
p g (p)
p (p + z (p))
=
p p + p z (p)
=
p p+
0
|{z}
>0
|{z}
By Walras’Law by p2
L
1
Since there must be some good that has a positive price, gl (p) cannot be
equal to zero for all l.
Thus max fpl + zl (p) ; 0g > 0 for at least one l .
‘Easy’Existence Proof: Step 1b
De…ne h :
L 1
!
L 1
as
g (p)
h (p) = PL
l =1 gl (p)
h ( ) is well de…ned.
h ( ) is continuous because z ( ) is continuous and thus g ( ) is continuous.
h ( ) maps from
L 1
, a convex and compact set, to itself.
Therefore
h ( ) is a continuous function from a compact convex set to itself; by Brouwer’s
theorem, it has a …xed point.
‘Easy’Existence Proof: Step 2
By Brouwer’s …xed point theorem, there exists a p such that
g (p )
p = h (p ) = PL
l =1 gl (p )
Rewrite this as
g (p ) = p
L
X
l =1
for some real number .
Observe that
6= 0 because g (p) 6= 0.
Next, we show that
= 1.
!
gl (p )
=p
‘Easy’Existence Proof: Step 3a
Claim: for each l = 1; ::; L
pl gl (p ) = pl (pl + zl (p ))
This is easy to show:
if gl (p ) 6= pl + zl (p )
)
|{z}
by de…nition
gl (p ) = 0 |{z}
) pl = 0
by
6=0
Thus: either gl (p ) = pl + zl (p ) or pl = 0; in both cases the claim holds.
Summing over l, we obtain:
p g (p ) = p (p + z (p )) = p
p +p
z (p ) = p
p +
0
|{z}
by Walras’Law
By the existsence of a …xed point (last slide), we know that g (p ) = p
taking the inner product with p on both sides:
p g (p ) = p p
Since we have just shown that p g (p ) = p p , we have
p p =p p
and thus = 1 as desired.
;
‘Easy’Existence Proof: Step 3b
Summary
By the …xed point theorem, g (p ) = p
, and since
= 1: g (p ) = p .
pl = gl (p ) = max fpl + zl (p ) ; 0g
|
{z
}
for l = 1; 2; ::; L
The …xed point is an equilibrium
Claim: p is an equilibrium.
Proof.
The equation above implies:
by de…nition of g ( )
Therefore:
zl (p )
0
for l = 1; 2; ::; L
If not, there exists some k such that
zk (p ) > 0
and
pk = max fpk + zk (p ) ; 0g = pk + zk (p )
an impossibility.
Since p is an equilibirum if and only if z (p )
0, we are done.
What Took So Long
Before 1950 economists focused on proving existence by showing that the
system of L 1 equations in L 1 unknowns given by the condition that
excess demand equals zero had a solution.
In 1950, John Nash proved existence of the (Nash) equilibrium of a game.
This was the breakthrough needed by Arrow, Debreu, and McKenzie to prove
existence of a competitive equilibrium shortly thereafter.
A game is a de…ned by describing choices and payo¤s for each player;
each i = 1; :::; I chooses a strategy in the set Si ; let S = S1
S2
::
SI ;
s = (s1 ; ::; sI ) 2 S a strategy pro…le (an action for each player);
ui (s) is player i ’s payo¤ function from s.
A Nash equilibrium is a strategy pro…le such that: each player maximizes her
payo¤ given the other players’strategies.
Thus, a Nash Equilibrium is an s such that for all i
ui (s1 ; ::; si ; ::; sI ) ui s1 ; ::; si 1 ; t; si +1 ; ::; sI
for any t 2 Si :
or
ui si ; s i
ui t; s i
for any t 2 Si :
Nash proved that such a strategy pro…le exists in any game when (i) the
strategy sets are compact and convex, and (ii) payo¤ are strictly quasi-concave
and continuous functions. How? Using Brower’s Theorem.
Nash’s Existence Theorem
A Nash equilibirum is a …xed point
For each i, let BRi : S
i
! Si be de…ned as follows:
BRi (s i ) = fsi 2 Si : ui (si ; s i )
ui (t; s i )
8t 2 Si g
this function describes i ’s ‘best response’to the other players strategy s i .
De…ne the mapping BR : S ! S as
BR (s) = BR1 (s
1)
:::
BRI (s I )
s is a Nash equilibrium if and only if s = BR (s ): a …xed point.
In a Nash equilibrium every player chooses a best response.
Assumptions (i) and (ii) from the previous slide are enough to use Brouwer’s
Theorem.
we need S closed, bounded, and convex; so, assume each Si has those proerties.
we need BR (s) to be a continuous function; so, assume each ui (s) is continuous
and strictly quasi-concave, so that each BRi (s) is a continuous function.
Under these assumption, every game has a Nash equilibrium.
From Nash to Walras
Arrow, Debreu, and McKenzie saw Nash’s paper and used it to solve the
existence problem. How?
Build a ‘game’that describes a Walrasian (competitive) equilibirum
The players are the consumers and the ‘Price Player’.
Each i’s payo¤ function is ui (xi ), consumer i’s utility function;
this does not depend on others’consumption (di¤erent from a game);
Consumer i’s best response to a price chosen by the price player is her
Walrasian demand xi (p).
Consumers best responses are summarized by aggregate excess demand.
The Price Player’s payo¤ is the value of the aggregate excess demand:
X
uPP (q; x) = q
(xi (p) ! i ) = q z(p)
i
The price player best response to an excess demand chosen by consumers is
the price that maximizes the value of that excess demand.
The best response mapping of this game is given by consumers’Walrasian
demands and a price that maximizes the value of aggregate excess demand.
A Nash equilibrium is given by a price vector and consumption choices such
that all players choose a best response.
From Nash to Walras
A Nash equilibirum is a …xed point
By Nash’s existence theorem, there exist a p 2
i such that
L 1
and an xi (p ) for each
each xi (p ) maximizes individuals’utility (given p ), and
p maximizes the value of aggregate excess demand (given xi (p )).
Claim: this is a Walrasian equilibrium
Why is p a competitive equilibrium?
Walras’Law (0 = p z (p)) implies p z (p ) = 0.
The price player maximization implies p z (p ) p z (p ) for all p 2 L 1 .
These together imply z (p ) 0 (make sure you convince yourself of this).
Last class we proved that z (p ) 0 and p
0 mean p is an equilibrium.
Remarks
To use Brouwer’s theorem, best responses must be functions.
even if aggregate excess demand is a function, BRPP ( ) can be multi-valued.
We need …xed point existence for correspondences: Kakutani’s theorem.
We can add production by having J more players (the …rms) that maximize
pro…ts.
An General Existence Theorem
Theorem
Suppose an economy satis…es the following properties.
1
For each i:
2
For each j:
3
Xi RL is closed and convex
%i satis…es local non-satiation, and convexity
!i b
xi for some b
xi 2 X i .
Yj RL is closed, convex,
includes the origin, and satis…es free-disposal.
The set of feasible allocations is compact.
Then a Walrasian quasi-equilibrium exists (if ! i
exists).
b
xi for all i then an equilibrium
Issues a proof needs to take care of.
When some prices are zero and individual’s preferences are locally non satiated,
their demand can explode.
This is because the budget set is not compact.
Demand (and therefore excess demand) is not necessarily continuous at zero
prices.
Think about how could this happen.
We need all prices to be strictly positive.
Proving a Not So Simple Existence Theorem
This is a sketch of the proof for existence of a competitive equilibrium.
1
Truncate the economy, so that all choices must belong to a compact set.
2
Construct a ‘game’with I + J + 1 players: consumers, …rms, and the ‘price
player’.
3
Show that each player’s best response is a non-empty, convex, and upper
hemi-continuous correspondence.
4
Hence the ‘product’best-response correspondence that describes this game
inherits those properties.
5
Use Kakutani’s …xed point theorem to show this correspondence has a …xed
point.
6
This …xed point is an equilibrium of the truncated economy.
7
Prove that the truncation does not matter: an equilibrium of the truncated
economy is a quasi-equilibrium of the whole economy.
8
DONE.
Some of the tricky issues have to do with getting strictly positive prices so that
the correspondences are upper hemi-continuous. See a book for the proof.
Econ 2100
Fall 2015
Problem Set 11
Due 17 November, Monday, at the beginning of class
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 (x1 ; : : : ; xm ), and rebates (R1 ; : : : ; RN )
such that each consumer
i is maximizing utility over the budget set at the bundle xi , all markets
P
1
clear, and Ri = N i;j tij pj xij for each i = 1; : : : ; N .
(a) Write down the budget set for a typical consumer in this economy.
(b) Illustrate an equilibrium with taxes in an Edgeworth Box. Does the First Welfare Theorem hold
for equilibria with taxes?
(c) 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 ).
2. Consider an exchange economy with L goods and N consumers.
Assume each consumer i’s utility
P
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.
P
P
(a) P
Show that if the aggregate endowment of each good is constant ( i ! i1 =
=
i ! i2 =
!
)
the
economy
has
at
most
one
equilibrium.
i iL
P
P
P
(b) Show that if i ! i1 > i ! i2 >
> i ! iL any competitive equilibrium price vector p must
satisfy p1 < p2 <
< pL .
(c) Suppose that vi (c) = ln(c) for each i. Fix the aggregate endowment ! 2 RL++ . 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 also a competitive
price vector for any other
P equilibrium
0
0
0
initial endowment allocation (! 1 ; : : : ; ! N ) such that i ! i = !.
i
L
3. Consider an exchange economy where for each i we have
P that X = R+ , and %i is continuous, strictly
convex, and locally non-satiated. Suppose also that i ! i >> 0. Prove that the aggregate excess
demand function z (p) satis…es the following properties:
1
(a) z (p) is continuous;
(b) z (p) is homogeneous of degree zero;
(c) there exists an s > 0 such that zl (p) >
s for each l and for each p.
(d) Now also assume that for each i %i is strongly monotone and show that if pn ! p, with p 6= 0
and pl = 0 for some l, then max fz1 (pn ) ; z2 (pn ) ; :::; zL (pn )g ! 1 .
4. 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) .
(a) Find the competitive (Walrasian) equilibirum, or argue this economy does not have one, when
both utility functions are strictly decreasing everywhere.
(b) 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 .
(c) 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 .
5. Consider a two goods two consumers economy where the utility functions are u1 (x11 ; x21 ) = x11
1
(x21 ) 8 and u2 (x12 ; x22 ) = 81 (x12 ) 8 + x22 , and the individual endowments are ! 1 = (2; r) and
8
! 2 = (r; 2), r = 28=9 21=9 .
(a) Find Walrasian demands of both consumers.
(b) Let p2 = 1, write the aggregate excess demand function for the two goods, and use the aggreate
excess demand of good 1 to …nd the three equilibria of this economy.
2