1
ECON 301 – LECTURE #12
WELFARE THEOREMS
STATEMENT:
“A competitive equilibrium is Pareto Optimal”.
PROOF: (by contradiction)
Suppose that an allocation bundle
x
(i.e.
x
= (
x
1
A
, x
2
A
, x
1
B
, x
2
B
)) is a competitive
equilibrium that is not Pareto Optimal.
Thus, there exists an allocation bundle
y
(i.e.
y
= (
y
1
A
, y
2
A
, y
1
B
, y
2
B
)) such that
y
is
feasible…
y
1
A
+ y
1
B
=
!
1
A
+
!
1
B
(1)
and
U
A
(
y
)
!
U
A
(
x
) but U
B
(
y
) > U
B
(
x
)
y
2
A
+ y
2
B
=
!
2
A
+
!
2
B
(2)
However, since
x
was the allocation bundle chosen for utility maximization at the
equilibrium price vector, we have…
P
1
x
1
A
+ P
2
x
2
A
= P
1
!
1
A
+ P
2
!
2
A
Budget Constraints are satisfied.
P
1
x
1
B
+ P
2
x
2
B
= P
1
!
1
B
+ P
2
!
2
B
meaning that for allocation bundle
y
to be “better” than allocation bundle
x
the following
must be true…
P
1
y
1
A
+ P
2
y
2
A
"
P
1
!
1
A
+ P
2
!
2
A
P
1
y
1
B
+ P
2
y
2
B
> P
1
!
1
B
+ P
2
!
2
B
which implies,
P
1
(y
1
A
+ y
1
B
) + P
2
(y
2
A
+ y
2
B
) > P
1
(
!
1
A
+
!
1
B
) + P
2
(
!
2
A
+
!
2
B
)
(3)
but the allocation bundle
y
must be feasible.
..subbing (1) and (2) into (3), we get
P
1
(
!
1
A
+
!
1
B
) + P
2
(
!
2
A
+
!
2
B
) > P
1
(
!
1
A
+
!
1
B
) + P
2
(
!
2
A
+
!
2
B
)
the above says that, at allocation bundle
y
, the value of the individual endowments for
good 1 and good 2 exceeds the value of the individual endowments of goods 1 and 2 in
the economy.
So at this equilibrium price vector the allocation bundle
y
is not
feasible…we have our contradiction!
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
We derived this contradiction by assuming that the competitive equilibrium solution
(allocation bundle
x
) was not Pareto efficient.
Therefore, this assumption must be wrong.
THE FIRST FUNDAMENTAL THEOREM
The First Fundamental Theorem of Welfare Economics provides the link which connects
the Walrasian General Equilibrium Theory to the Pareto Optimal Theory.
On one hand,
Walrasian Theory deals with the various concepts of equilibrium.
On the other hand,
Pareto Theory deals with the various concepts of efficiency.
The First Fundamental Theorem of Welfare Economics essentially says that “the
Walrasian concept of general equilibrium is consistent with the Pareto concept of
efficiency”.
So what does this mean in the context of our Pure Exchange Economy?
We have studied both the Walrasian general equilibrium pure exchange economy and the
Pareto optimal pure exchange economy with two goods (X,Y) and two consumers (A,B).
Let’s summarize what we know so far…
WALRASIAN PURE EXCHANGE
PARETO OPTIMAL PURE EXCHANGE
The Walrasian pure exchange model
The Pareto Optimal pure exchange model
provides equilibrium prices (P
X
, P
Y
)
provides optimal allocations (X
A
, X
B
, Y
A
,
and allocation (X
A
, X
B
, Y
A
, Y
B
) such
Y
B
) such that
that
(a) no one can be made better off without
(a) both consumers are in equilibrium
making the other worse off (indifference
(indifference curves are tangent to
curves are tangent to each other)
budget lines)
(b) both goods are fully allocated (quantities
This is the end of the preview.
Sign up
to
access the rest of the document.
 Winter '09
 COREYVANDEWAAL
 Economics, Equilibrium, KaldorHicks efficiency, Pareto Optimal

Click to edit the document details