Market Equilibrium
Ram Singh
Lecture 3
September 22, 2016
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Market Exchange: Basics
Let us introduce ‘price’ in our pure exchange economy. Let,
There be N individuals and M goods
i
ei = (e1i , ..., eM
) denote endowment for individual i
pi denote the ‘price’ of ith good; pi > 0 for all i = 1, .., M.
So the price vector is
p = (p1 , ..., pM ) >> 0.
Assume
each good has a market and each individual is ‘price-taker’.
For each individual,
Total value of the initial endowment depends on the price vector
However, the total value of the bundle bought cannot exceed the total
value of her endowment.
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Market Exchange: 2 × 2 economy I
For person 1, the set of feasible allocations/consumptions is the set of
y1 = (y11 , y21 ) such that:
p1 y11 + p2 y21 ≤ p1 e11 + p2 e21 .
Assuming monotonic preferences, Person 1 maximizes utility by choosing
bundle x1 = (x11 , x21 ) s.t.
p1 x11 + p2 x21 = p1 e11 + p2 e21
Person 2 maximizes utility s.t.
p1 x12 + p2 x22 = p1 e12 + p2 e22 .
Recall, within the Edgeworth box, for each allocation (x1 , x2 ), we have
x11 + x12 = e11 + e12 , and x21 + x22 = e21 + e22 .
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Market Equilibrium
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Market Exchange: 2 × 2 economy II
Note: The budget line for person 2 is: p1 x12 + p2 x22 = p1 e12 + p2 e22 . However,
p1 e12 + p2 e22
= p1 x12 + p2 x22 , i.e.,
p1 e12 + p2 e22
= p1 (e11 + e12 − x11 ) + p2 (e21 + e22 − x21 ), i.e.,
0
p1 x11 + p2 x21
= p1 (e11 − x11 ) + p2 (e21 − x21 ), i.e.,
= p1 e11 + p2 e21 ,
which is the budget line for the 1 person.
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Preferences and Utilities: Assumptions
We assume:
Preference relations to be continuous, strictly monotonic, and strictly
convex
The utility functions to be continuous, strictly monotonic and strictly
quasi-concave
However, several of the results will hold under weaker conditions.
Question
What is the role of assumption that the preferences are ‘strictly
quasi-concave’?
Let
u 1 (.) denote the utility function for person 1
u 2 (.) denote the utility function for person 2
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Competitive Equilibrium: 2 × 2 economy I
An allocation is x̂ = (x̂1 , x̂2 ) along with a price vector p = (p1 , p2 ) is
competitive equilibrium, if
1
x̂1 = (x̂11 , x̂12 ) maximizes u 1 (.) subject to p1 x11 + p2 x21 = p1 e11 + p2 e21
2
x̂2 = (x̂21 , x̂22 ) maximizes u 2 (.) subject to p1 x12 + p2 x22 = p1 e12 + p2 e22
3
x̂11 + x̂12 = e11 + e12
4
x̂21 + x̂22 = e21 + e22
For ‘well-behaved’ utilities:
1. Implies : In equi. IC of person 1 will be tangent to her budget line.
2. Implies : In equi. IC of person 2 will be tangent to his budget line
We know that: both of the demanded bundles, i.e., x̂1 and x̂2 lie on the
same line. Why?
3 and 4 imply that the demanded bundles, i.e., x̂1 and x̂2 coincide. Why?
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Competitive Equilibrium: 2 × 2 economy II
Therefore, the ICs are tangent to each other
Therefore, the equi. allocation x̂ = (x̂1 , x̂2 ) is Pareto Optimum.
Question
Suppose, x̂ = (x̂1 , x̂2 ) is a Competitive (market) equilibrium allocation
Does the eq. allocation x̂ = (x̂1 , x̂2 ) belong to the core?
Are unilateral deviations from x̂ = (x̂1 , x̂2 ) profitable?
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Competitive Equilibrium: N × M economy I
Consider a N × M economy denoted by (u i (.), e).
An allocation x̂ = (x̂1 , ..., x̂N ) along with a price vector p = (p1 , ..., pM ) is a
competitive equilibrium, if the following conditions are satisfied:
First: x̂i maximizes u i (.). That is, x̂i solves
max{u i (xi )}
(1)
xi
i
subject to p.xi = p.ei , i.e., p1 x1i + ... + pM xMi = p1 e1i + ... + pM eM
.
Second: For all j = 1, ..., M
N
X
i=1
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x̂ji =
N
X
eji
(2)
i=1
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Competitive Equilibrium: N × M economy II
Definition
(x̂; p), i.e., (x̂1 , ..., x̂N ; p) is called a Competitive or Walrasian equilibrium, if
(x̂i , p) together satisfy (1) and (2) simultaneously, for all i = 1, ..., N.
Definition
The set of Walrasian/Competitive Equilibria, W (u i (.), ei )N×M , is given by
W (u i (.), ei )N×M = {x = (x1 , ..., xN ) | ∃p such that (xi , p) satisfy (1) and (2), }
simultaneously, for all i = 1, ..., N.
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Competitive Equilibrium: N × M economy III
Remark
We will show that:
Walrasian/Competitive equilibrium may not exist. However,
If utilities fns are continuous, strictly increasing and strictly
quasi-concave, there does exist at least one equilibrium.
In general there can be more than one Competitive equilibrium.
Walrasian/Competitive equilibrium depends on the vector of initial
endowments, i.e., e.
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Some Observations I
Let x̂ = (x̂1 , ..., x̂N ) be a Competitive equilibrium allocation.
Proposition
Suppose, (x̂, p) is a competitive equilibrium. Then, x̂ = (x̂1 , ..., x̂N ) is a
feasible allocation.
Proposition
Suppose, (x̂, p) is a competitive equilibrium. Take a bundle yi . If
u i (yi ) > u i (x̂i ), then p.yi > p.ei . Formally,
u i (yi ) > u i (x̂i ) ⇒
u i (yi ) > u i (x̂i ) ⇒
p.yi > p.ei


J
J
X
X

pj yji >
pj eji 
j=1
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Market Equilibrium
j=1
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Some Observations II
Proposition
Suppose, (x̂, p) is a competitive equilibrium, and the individual preferences
are monotonic, i.e., u i is increasing. Take a bundle yi . If u i (yi ) ≥ u i (x̂i ), then
p.yi ≥ p.ei . Formally,
u i (yi ) ≥ u i (x̂i ) ⇒ p.yi ≥ p.ei i.e.,
p.yi < p.ei
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⇒ u i (yi ) < u i (x̂i )
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Competitive Equilibrium and Core I
Let
W (u i (.), ei )N×M denote the set of Walrasian/competitive allocations.
C(u i (.), ei )N×M denote the set of Core allocations.
For a 2 × 2 economy, suppose an allocation x̂ = (x̂1 , x̂2 ) along with a price
vector p = (p1 , p2 ) is competitive equilibrium. Then,
Individual i prefers xi at least as much as ei
Indifference curves of the individuals are tangent to each other
Allocation x̂ = (x̂1 , x̂2 ) is Pareto Optimum
In view of the above, allocation x̂ = (x̂1 , x̂2 ) is in the Core.
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Competitive Equilibrium and Core II
So, for a 2 × 2 economy,
x ∈ W (u i (.), ei ) ⇒ x ∈ C(u i (.), ei ).
Theorem
Consider an exchange economy (u i (.), ei )N×M , where individual preferences
are monotonic, i.e., u i is increasing. If x is a WEA, then x ∈ C(u i (.), ei )N×M .
Formally,
W (u i (.), ei )N×M ⊆ C(u i (.), ei )N×M .
Proof: Take any x WEA. Let, x along with the price vector p be a WE.
Suppose
x 6∈ C(e).
Therefore, there exists a ‘blocking coalition’ against x. That is,
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Competitive Equilibrium and Core III
there exists a set S ⊆ N and an ’allocation’ say y, s.t.
X
X
yi =
ei
i∈S
(3)
i∈S
Moreover,
u i (yi ) ≥ u i (xi ) for all i ∈ S
and for some i 0 ∈ S
0
(4)
0
u i (yi ) > u i (xi ).
(5)
(3) implies
p.
X
i∈S
yi = p.
X
ei
(6)
i∈S
(4) implies
p.yi ≥ p.xi = p.ei , for all i ∈ S
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Market Equilibrium
(7)
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Competitive Equilibrium and Core IV
(6) implies: for some i 0 ∈ S
0
0
0
p.yi > p.xi = p.ei .
(8)
(7) and (8) together give us:
p.
X
i∈S
yi > p.
X
ei
(9)
i∈S
But, (4) and (9) are mutually contradictory. Therefore,
x ∈ C(e).
Theorem
Consider an exchange economy (u i , ei )i∈{1,..,N} , where u i is strictly
increasing, for all i = 1, .., N.
Every WEA is Pareto optimum.
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Market Equilibrium
September 22, 2016
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Competitive Equilibrium: Merits and Demerits I
Question
Is the price/market economy better than the barter economy, in terms of
its functioning?
Is the price/market economy better than the barter economy, in terms of
the outcome achieved?
Question
What are the limitations of a market economy?
Can these limitations be overcome?
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