Introduction to Equilibrium Analysis
Ichiro Obara1
University of Minnesota1
October 25, 2007
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Existence
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Introduction
Introduction
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Existence
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Introduction
Motivation
Why do we study competitive (Walrasian) equilibrium? How are prices determined? Need a theory to pin down prices (”Theory of value”). To analyze interactions between different markets (general equilibrium analysis vs partial equilibrium analysis). To reduce the number of exogenous variables and derive prices & allocations from the primitives (preference and technology) of the economy.
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Existence
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Introduction
Questions
Some theoretical questions we may ask: Does equilibrium satisfy any nice property? (Efficiency) Is there any equilibrium? (Existence) Is equilibrium unique? (Uniqueness)
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Existence
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Notations and Definitions
Notations and Definitions
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Existence
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Notations and Definitions
Notations
Inequalities For any x, y ∈
L,
x ≥ y ⇔ xl ≥ yl for l = 1, ..., L. x > y ⇔ xl ≥ yl for l = 1, ..., L and xl > yl for some l. x >> y ⇔ xl > yl for l = 1, ..., L.
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Existence
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Notations and Definitions
Notations
Derivatives Dl ui (x) ≡
∂ui (x) ∂xl . L.
Dui (x) ≡ (D1 ui (x), ..., DL ui (x))) ∈
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Existence
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Notations and Definitions
Consumer
I consumers indexed by i = 1, ..., N. Consumer i’s consumption vector xi = (xi,1 , xi,2 , ..., xi,L ) ∈ commodities indexed by l = 1, ..., L). Consumption set Xi ∈
L L
(L
for consumer i is the set of feasible
L +
consumption vectors for consumer i. Assume Xi = this course. Consumer i’s preference
i
throughout
defined on Xi .
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Existence
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Notations and Definitions
Maintained Assumptions
i i
is complete and transitive. is continuous, thus there exists a continuous utility function that represents
i
ui : Xi →
(Proposition 3.C.1,MWG). We use
i
and ui interchangeably.
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Existence
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Notations and Definitions
Other properties
i
is locally nonsatiated, i.e. ∀i, ∀xi ∈ Xi , ∀ > 0, there exists xi such and xi
i
that xi − xi <
i i i
xi .
i
is monotone if xi >> xi → xi
xi for any xi , xi ∈ Xi .
i
is strongly monotone if xi > xi → xi is convex if xi xi , xi xi for
i i
xi for any xi , xi ∈ Xi .
xi , xi , xi ∈ Xi → αxi + (1 − α)xi
i
xi for any α ∈ [0, 1]. xi for
i
is strictly convex if xi
i
xi , x i
xi , xi = xi ∈ Xi → αxi + (1 − α)xi
Obara (U of M) Existence
xi for any α ∈ (0, 1).
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Notations and Definitions
Economy
Let r ∈
L +
be the endowments for this economy. The basic data for {(Xi ,
I i )}i=1 , r
the economy are summarized by E =
. . In
We focus on pure exchange economy E pure = {(Xi , this economy, consumers:
are given initial endowments ei ∈
L +, i
I i , ei )}i=1
= 1, ..., N. (r =
i
ei )
take price as given and trade to maximize their utility.
Given p ∈
L
and ei , consumer i’s budget set is denoted by Bi (p, ei ) = {xi ∈ Xi |p · xi ≤ p · ei }.
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Existence
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Notations and Definitions
Feasible Allocations
An allocation x is feasible for E if (1) ∀i, xi ∈ Xi and (2)
i
xi ≤ r .
Feasible Allocations
A = {x ∈
N×L + |xi
∈ Xi ,
i
xi ≤ r }
We implicitly assume a “free disposal” technology (Y =
L) −
here.
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Existence
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Notations and Definitions
Competitive Equilibrium
Definition (x ∗ , p ∗ ) ∈ X × xi∗
i L +
is a competitive (Walrasian) equilibrium for E pure if
xi for all xi ∈ Bi (p ∗ , ei ) for i = 1, ..., N (consumers maximize
their utility) and
i
xi∗ ≤
i ei
i
∗ xi,l =
i ei,l
if pl > 0 .
(X = X1 ×, ..., ×XN ).
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Existence
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Notations and Definitions
Remark p∗ ∈
L +
is without loss of generality if the preferences are locally
nonsatiated and the free disposal technology is available. We can guarantee p ∗ >> 0 if there is any consumer with strongly monotone preference (x > x ⇒ ui (xi ) > ui (xi )) If there is any consumer with strongly monotone preference, then the second condition can be replaced by conditions).
i
xi∗ = r (Markets clearing
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Existence
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Examples
Examples
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Existence
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Examples
Edgeworth Box
Here is a typical CE in a Edgeworth box.
2
x*
e
1
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Existence
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Examples
Edgeworth Box
Here is a typical CE in a Edgeworth box.
2
x* Offer curves e
1
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Existence
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Examples
Edgeworth Box
This is not a CE. Excess demand for good 1 and excess supply for good 2.
2
x2(p) x* x1(p) e
1
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Existence
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Examples
Edgeworth Box
This is a CE with free disposal. p1 > 0, p2 = 0.
e
x*
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Existence
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Examples
Edgeworth Box
There may be multiple CE.
2
z* y* x*
e
1
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Existence
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Examples
Edgeworth Box
...or maybe none.
2
e
1
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Existence
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Examples
Exercise Two consumers with Cobb-Douglas preference. u1 (x) = x α y 1−α , u2 (x) = x β y 1−β , α, β ∈ (0, 1). e1 = (1, 0), e2 = (0, 1). Find a competitive equilibrium.
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Existence
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Examples
Consumers maximize their utility,
Dl ui (x) = λi pl for l = 1, 2 and i = 1, 2, and the markets clear: x1,l + x2,l = e1,l + e2,l for l = 1, 2.
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Existence
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Examples
We know that:
1 x1,1 (p) = α ppe1 = α. 1 1 x1,2 (p) = (1 − α) ppe1 = (1 − α) p1 . p2 2 2 x2,1 (p) = β ppe2 = β p2 . p1 1 2 x1,2 (p) = (1 − β) ppe2 = (1 − β). 2
(Offer curves are vertical). Combined with the market clearing conditions, we can derive
∗ ∗ ∗ ∗ x1 (p) = (α, β) , x2 (p) = (1 − α, 1 − β), p1 /p2 = 1−α β .
Note that only the price ratio can be determined.
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Existence
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Walras’ Law
Walras’ Law
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Existence
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Walras’ Law
Walras’ Law
Note that one market clearing condition is redundant in the example. This is true in general.
Walras’ Law
Suppose that ui is locally nonsatiated and xi maximizes ui in Bi (p, ei ) for i = 1, ..., N. If L − 1 markets clear and p >> 0, then the last market must clear as well.
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Existence
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Walras’ Law
Proof Step 1 p · xi = p · ei . Suppose not (p · xi < p · ei ). Then we can find xi ∈ Xi in a close neighborhood of xi such that p · xi < p · ei and ui (xi ) > ui (xi ) (by local nonsatiation). Contradiction. Step 2
i
p · xi =
i
p · ei → p ·
i
xi = p ·
i ei
Step 3 Suppose that the markers clear for (say) l=1,...,L-1. Then pL ·
i
xi,L = pL ·
i ei,L .
Since pL > 0,
i
xi,L =
i ei,L .
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Existence
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