Lecture Notes
Econ 702
Spring 2004
1
Jan 27
•
What is an equilibrium?
An equilibrium is a statement about what the outcome of an economy is.
Tells us
what happens in an economy
.
An equilibrium is a mapping from environment (preference, technology, information,
market structure) to allocations where,
1. Agents maximize
2. Agents’ actions are compatible.
•
One of the important questions is, given the environment what type of equilibria we
should look at. The economist doesn’t have the right to choose what happens. For
example, for an economy with two people it might not be appropriate to think of
ArrowDebreu (maybe they are not price takers, they might not behave competitively).
•
In order something to be predictable by theory it needs to exist and be unique. For
this reason, existence and uniqueness are properties that we want the equilibrium to
have.
•
Optimality is a property of an allocation. An equilibrium allocation might or might
not have this property.
•
In this class, we will go over some of the ’popular’ notions of equilibrium in macroeco
nomics that might be different from Walrasian equilibrium.
1
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1.1
Valuation Equilibrium
•
In macroeconomics, we are interested in infinite dimensional commodity spaces. We
want to look at the relationship between competitive equilibrium and Pareto optimal
ity in models with infinitedimensional spaces. You looked at competitive equilibrium
and Pareto optimality in 701, but the proofs of the FBWT and SBWT were done in
the context of finitedimensional commodity spaces. Here we want to show that the
welfare theorems hold for economies with infinite dimensional spaces. To do this, we
introduce the equilibrium concept ’valuation equilibrium’.
•
Before defining valuation equilibrium, we first need to define the environment:
1.
L
, Commodity space:
L
is a topological vector space.
Definition 1 (Vector Space)
A vector space is a space where the operations addition
and scalar multiplication are defined, and where the space is closed under these two
operations. i.e. If we take two sequences
a
=
{
a
i
} ∈ L
and
b
=
{
b
i
} ∈ L
, it must be that
a
+
b
∈ L
. And if we take
k
∈ R
+
, k >
0
, it must be that
a
∈ L ⇒
d
=
ka
∈ L ∀
k >
0
.
Definition 2 (Topological Vector Space)
A topological vector space is a vector
space which is endowed with a topology such that the maps
(
x, y
)
→
x
+
y
and
(
λ, x
)
→
λx
are continuous. So we have to show the continuity of the vector operations addition
and scalar multiplication.
2.
X
⊂ L
, Consumption Possibility Set:
Specification of the ’things’ that people could do (that are feasible to them).
X
con
tains every (individually) technologically feasible consumption point.
3.
U
:
X
→ R
, Specifies the preference ordering.
4.
Y
, Production possibility set.
What is an allocation in this environment? An allocation is a pair (
x, y
). On the other
hand, a feasible
allocation is (
x, y
) such that x=y (agents’ actions need to be compatible).
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 Spring '08
 Staff
 Economics, Equilibrium, Hilbert space, Economic equilibrium, ADE

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