Market Equilibrium Price: Existence, Properties and
Consequences
Ram Singh
Lecture 5
September 22, 2016
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Questions
Today, we will discuss the following issues:
How does the Adam Smith’s Invisible Hand work?
Is increase in Prices bad?
Do some people want increase in prices?
If yes, who wants increase in prices and of what type?
Does increase in prices have distributive consequences?
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Market Equilibrium Price I
Definition
Walrasian Equilibrium Price: A price vector p∗ is equilibrium price vector, if for
all j = 1, ..., J,
zj (p∗ ) =
N
X
xji (p∗ , p∗ .ei ) −
i=1
N
X
eji = 0, i.e., if
i=1
∗
z(p ) = 0 = (0, ..., 0).
Proposition
If p∗ is equilibrium price vector, then p0 = tp∗ , t > 0, is also an equilibrium
price vector
If p∗ is equilibrium price vector, then p0 6= tp∗ , t > 0, may or may not be an
equilibrium price vector
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Market Equilibrium Price II
Two goods: food and cloth
Let (pf , pc ) be the price vector. We can work with p = ( ppcf , 1) = (p, 1). Since,
we know that for all t > 0:
z(tp) = z(p)
Therefore, we have pf zf (p) + pc zc (p) = 0, i.e.,
pzf (p) + zc (p) = 0.
Let:
zi (p) is continuous for all p >> 0, i.e., for all p > 0.
Note
When utility function is monotonic, xf (p) will explode as pf = p → ∞.
Therefore,
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Market Equilibrium Price III
there exists small p = > 0 s.t. zf (p, 1) >> 0 and zc (p, 1) < 0 (Why?).
there exists another p0 >
1
s.t. zf (p0 , 1) < 0 and zc (p, 1) > 0. (Why?).
Therefore, for a two goods case we have:
There is a value of p such that zf (p, 1) = 0 and zc (p, 1) = 0
That is, there exists a WE price vector.
In general, Equlibrium price is determined by Tatonnement process
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WE: General Case I
References: Jehle and Reny (2008).
Also see: Arrow and Debreu (1954), and McKenzie (2008); Arrow and Hahn
(1971).
Recall, we have
For all i = 1, 2, ..., N, we have: xi (tp) = xi (p), for all t > 0.
z(tp) = z(p), for all t > 0.
Let
p̂ = (p̂1 , ..., p̂M ) >> (0, ..., 0) be a price vector
t=
1
p̂1 ,...,p̂M
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WE: General Case II
Then p = t p̂ is such that
pj ≥> 0 and
M
X
pj = 1
j=1
Without any loss of generality, we can restrict attention to the following set


M


X
P = p = (p1 , ..., pM )|
,
pj = 1 and pj ≥

1 + 2M 
j=1
where ≥ 0.
Remark
Note that P is non-empty, bounded, closed and convex set for all ∈ (0, 1).
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WE: General Case III
Theorem
Suppose u i (.) satisfies the above assumptions, and e >> 0. Let {ps } be a
sequence of price vectors in RM
++ , such that
{ps } converges to p̄, where
p̄ ∈ RM
+ , p̄ 6= 0, but for some j, p̄j = 0.
Then, for some good k with p̄k = 0, the sequence of excess demand
(associated with {ps }), say {zk (ps )}, is unbounded above.
Theorem
Under the above assumptions on u i , there exists a price vector p∗ >> 0,
such that z(p∗ ) = 0.
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WE: Proof I
Let,
z̄j (p) = min{zj (p), 1}.
(1)
Note
z̄j (p) = min{zj (p), 1} ≤ 1. Therefore,
0 ≤ max{z̄j (p), 0} ≤ 1.
Next, let’s define the function f = (f1 (p), ..., fM (p)) such that: For j = 1, .., M,
fj (p) =
Note that
PM
j=1 fj (p)
fj (p) ≥
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Nj (p)
+ pj + max{z̄j (p), 0}
=
,
PM
D(p)
M + 1 + j=1 max{z̄j (p), 0}
= 1. Moreover, we have
Nj (p)
≥
≥
.
M + 1 + M.1
M + 1 + M.1
1 + 2M
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WE: Proof II
Therefore,
f (p) : P 7→ P .
Since D(p) ≥ 1 > 0, the above function is well defined and continuous over a
compact and convex domain. Therefore, there exists p such that
f (p ) = p , i.e.,
fj (p ) = pj , for all j = 1, .., M.
That is, for all j = 1, .., M,
Nj (p )
D(p )
pj [M +
M
X
max{z̄j (p ), 0}]
= pj , i.e.,
= + max{z̄j (p ), 0}.
(2)
j=1
Next, we let → 0. Consider the sequence of price vectors {p }, as → 0.
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WE: Proof III
0
Sequence {p }, as → 0, has a convergent subsequence, say {p }.
Why?
0
Let {p } converge to p∗ , as 0 → 0.
Clearly, p∗ ≥ 0. Why?
Suppose, pk∗ = 0. Recall, we have


M
X
0
0
0
pk M0 +
max{z̄j (p ), 0} = 0 + max{z̄k (p ), 0}.
(3)
j=1
as 0 → 0:
0
The LHS converges to 0, since lim0 →0 pk = 0 and term
PM
0
[M0 + j=1 max{z̄j (p ), 0}] on LHS is bounded.
However, the RHS takes value 1 infinitely many times. Why?
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WE: Proof IV
This is a contradiction, because the equality in (3) holds for all values of 0 .
Therefore, pj∗ > 0 for all j = 1, .., M. That is,
p∗ >> 0, i.e.,
lim p = p∗ >> 0.
→0
In view of continuity of z̄(p) over RM
++ , from (2) we get (by taking limit → 0):
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WE: Proof V
For all j = 1, .., M
pj∗
M
X
max{z̄j (p∗ ), 0}
=
max{z̄j (p∗ ), 0}, i.e.,
=
zj (p∗ ) max{z̄j (p∗ ), 0}, i.e.,
j=1

zj (p∗ )pj∗ 
M
X

max{z̄j (p∗ ), 0}
j=1
M
X

zj (p∗ )pj∗ 
j=1
M
X

max{z̄j (p∗ ), 0}
j=1
=
M
X
zj (p∗ ) max{z̄j (p∗ ), 0}, i.e.,
j=1
M
X
zj (p∗ ) max{z̄j (p∗ ), 0} = 0.
(4)
j=1
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WE: Proof VI
Recall, z̄j (p) = min{zj (p), 1}. Therefore,
zj (p∗ ) > 0
⇒ max{z̄j (p∗ ), 0} > 0;
zj (p∗ ) ≤ 0
⇒ max{z̄j (p∗ ), 0} = 0.
Therefore,
zj (p∗ ) > 0
⇒ zj (p∗ ) max{z̄j (p∗ ), 0} > 0;
zj (p∗ ) ≤ 0
⇒ zj (p∗ ) max{z̄j (p∗ ), 0} = 0.
Suppose, for some j, we have zj (p∗ ) > 0, then we will have
M
X
zj (p∗ ) max{z̄j (p∗ ), 0} > 0.
(5)
j=1
But, this is a contradiction in view of (4).
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WE: Proof VII
Therefore:
For any j = 1, .., M, we have
zj (p∗ ) ≤ 0.
(6)
Suppose, zk (p∗ ) < 0 for some k . From (4), we know
∗
p1∗ z1 (p∗ ) + ... + pk∗ zk (p∗ ) + ... + pM
zM (p∗ ) = 0.
Since pj∗ > 0 for all j = 1, .., M.
zk (p∗ ) < 0 implies pk∗ zk (p∗ ) < 0
Therefore, there must exist k 0 , such that
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WE: Proof VIII
zk 0 (p∗ ) > 0,
(7)
which is a contradiction in view of (6). Therefore,
For all j = 1, .., M, we have:zj (p∗ )
∗
z(p )
= 0, i.e.,
= 0.
That is,
p∗ is a WE price vector.
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