Download Lecture 6: General Equilibrium - Existence

Lecture 6: General Equilibrium - Existence
HS 12
Overview
1
Setting the Stage
2
Result
3
Proof
4
Remarks
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
2/13
Setting the Stage
We present the argument for the existence of a Walrasian
equilibrium in the context of an Exchange economy.
This is a special case of the model from the previous
lecture in which no production is possible: Y j = {0} for all
j ∈ J . In effect: There are no firms, only consumers.
In this case
an allocation is given by x = (x1 , . . . , xI ).
feasibility requires xi ∈ Rn+ and
X
xi =
i∈I
X
ei .
i∈I
the incomes are given by mi (p) = p · ei .
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
3/13
Setting the Stage
The condition for p∗ 0 to be a Walrasian equilibrium
price vector can be written as
z(p∗ ) = 0.
Here z : Rn++ Rn is the aggregate excess demand function
given by
X
X
zk (p) =
xki (p, p · ei ) −
eki .
i∈I
i∈I
Answering the question “Does a Walrasian equilibrium
exist?” is thus the same as answering the question “Does
there exists p∗ 0 such that z(p∗ ) = 0?”
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
4/13
Result
Theorem 5.5: Consider an exchange economy in which the
utility functions u i of all consumers are continuous, strongly
increasing, and strictly quasiconcave. Assume, in addition, that
the aggregate endowment of each good is strictly positive, that
is
X
ei 0.
i∈I
Then a Walrasian equilibrium price vector p∗ 0 exists.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
5/13
Structure of the Proof
1
Derive properties of individual demand from assumptions
on preferences.
2
Derive properties of excess demand from properties of
individual demand.
3
Show that these properties of excess demand imply the
existence of an equilibrium.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
6/13
Step 1 of the Proof
We already know that under the assumptions, each
consumer’s problem has a unique solution xi (p, y ) for
given income y if p 0.
We simply use p · ei as income (each individual sells its
initial endowment at given prices).
Furthermore, the theorem of the maximum assures that
xi (p, p · ei ) is continuous in p for p 0.
Thus we know:
Individual demand xi (p, p · ei ) is well-defined and
continuous in p whenever p 0.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
7/13
Step 2 of the Proof (I)
This structure of individual demand has implications for
aggregate excess demand.
For p 0: Aggregate excess demand z(p) is
continuous in p,
homogeneous of degree zero in p
(i.e., z(t p) = z(p), for all t > 0),
for all p 0, we have p · z(p) = 0 (“Walras Law”).
Continuity and homogeneity are rather obvious
implications of the corresponding properties of individual
demand functions; Walras Law needs a detailed proof . . .
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
8/13
Step 2 of the Proof (II)
There is a further (technical) property of aggregate excess
demand.
Let {pm } be a sequence of price vectors with pm 0 that
converges to some p̄ 6= 0 with p̄k = 0 for some good k .
Then for some good k 0 with p̄k0 = 0, the associated
sequence of excess demands for good k 0 is unbounded
above.
This property simply says that if the prices of some but not
all goods come arbitrarily close to zero, then the excess
demand for at least one of these goods goes to infinity.
I am not going to prove this property here (see Theorem
5.4.); but note that this is where the assumption that the
aggregate endowment is strictly positive is used.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
9/13
Step 3 of the Proof (I)
For the final part of the proof, we need a fixed-point
theorem.
A vector x∗ is called a fixed point of the function f if
f(x∗ ) = x∗ .
Brouwer’s Fixed-Point Theorem:
Let S ⊂ Rn be a non-empty, compact, and convex set. Let
f : S → S be a continuous function. Then there exists a
point x∗ ∈ S, so that f(x∗ ) = x∗ .
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
10/13
Step 3 of the Proof (II)
Theorem 5.3 finishes the proof by combining the properties
of the aggregate excess demand function established in
Step II with Brouwer’s fixed point theorem to demonstrate
that there exists p∗ 0 with z(p∗ ) = 0.
The key idea is to construct a function f mapping price
vectors into price vectors, such that p∗ is a fixed point of
this mapping if and only if aggregate excess demand is
equal to zero.
Constructing such a function isn’t very difficult.
Nevertheless, the proof is rather messy because there is no
straightforward way to meet the compactness requirement
in Brouwer’s fixed point theorem.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
11/13
Remarks: Uniqueness of Walrasian Equilibrium
The above argument establishes fairly general conditions
under which a Walrasian equilibrium in an exchange
economy exists.
From the homogeneity of aggregate excess demand, it is
clear that there is no hope for the uniqueness of Walrasian
equilibrium prices – the best we can hope for is that a
Walrasian equilibrium allocation and “relative prices” are
uniquely determined.
Alas, it can be shown that unless much more stringent
assumptions on preferences and/or endowments are
imposed, Walrasian equilibrium allocations are, in general,
not unique.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
12/13
Remarks: Existence in Production Economies
Existence of Walrasian equilibrium in the model with
production can be proven following the same logic as in the
case of an exchange economy.
1
Continuity properties of individual demand and supply
functions.
This requires the assumptions on Y j introduced in the
previous lecture.
2
Establish same properties of the aggregate excess
demand function as above. Aggregate excess demand is
now given by
X
X
X
X
z(p) =
xi (p, p · ei +
θij Πj (p)) −
ei −
yj (p).
i∈I
j∈J
i∈I
j∈J
Rather than assuming a strictly positive aggregate
endowment, a weaker assumption will do.
3
Use Brouwer’s fixed point theorem in the same way as
above.
Advanced Economic Theory
Lecture 6: General Equilibrium - Existence
13/13