Problem Set 1
Econ 702, Spring 2004
Prepared by Ahu Gemici
February 7, 2004
Problem 1
Show that without the assumption of continuity of the price func
tion, the proof of First Basic Welfare Theorem doesn’t go through.
Solution:
I will write down the proof of FBWT with the assumption that the price
function is continous and point out to the places we need this property for the
proof to go through.
Theorem 1 (First Basic Welfare Theorem)
Suppose that for all
i
, all
x
∈
X
i
there exists a sequence
{
x
n
}
∞
n
=0
in
X
i
converging to
x
with
u
(
x
n
)
≥
u
(
x
)
for all n (local nonsatiation). If an allocation
[(
x
*
i
)
j
∈
J
,
(
y
*
i
)
i
∈
I
]
and a conti
nous linear functional
ν
constitute a competitive equilibrium, then the allocation
[(
x
*
i
)
j
∈
J
,
(
y
*
i
)
i
∈
I
]
is Pareto optimal.
Proof.
As you will remember from your 701, the ﬁrst thing before we prove
FBWT is to show that if
x
*
i
solves agent i’s optimization problem, and if there
is another allocation that gives him at least as much (or strictly more) utility,
then it must be that that allocation is at least as (or strictly more) expensive
as
x
*
i
. In other words,
∀
i
, all
x
i
∈
X
,
u
(
x
i
)
≥
u
(
x
*
i
)
⇒
ν
(
x
i
)
≥
ν
(
x
*
i
)
(1)
u
(
x
i
)
> u
(
x
*
i
)
⇒
ν
(
x
i
)
> ν
(
x
*
i
)
(2)
First, showing (1):
Suppose not, so that,
u
(
x
i
)
≥
u
(
x
*
i
)
and
ν
(
x
i
)
< ν
(
x
*
i
)
By local nonsatiation of
u
, for every
±
≥
0, there is
x
0
i
∈
X
i
∩{
˜
x
i
:
k
˜
x
i

x
i
k
< ±
}
with
u
(
x
0
i
)
> u
(
x
i
)
≥
u
(
x
*
i
) But then for suﬃciently small
±
,
ν
(
x
0
)
< ν
(
x
*
i
).(Note
that without the continuity we would not be able to show the last point (i.e.
that
ν
(
x
0
)
< ν
(
x
*
i
))). This contradicts with
x
*
i
being the optimal solution to
1
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Documentthe agent’s problem.
Showing (2):
This follows from the fact that
x
*
i
solves the agent’s problem.
Now consider an allocation [(
x
i
)
,
(
y
i
)] and suppose that each consumer prefers
x
i
to
x
*
i
,with strict preference for at least one consumer. From (1) and (2), it
must be that,
ν
(
x
i
)
≥
ν
(
x
*
i
)
∀
i
(3)
ν
(
x
i
)
> ν
(
x
*
i
)
for some i
(4)
Sum over i,
X
i
ν
(
x
i
)
>
X
i
ν
(
x
*
i
)
(5)
By the linearity of
ν
we can write
ν
(
x
)
> ν
(
x
*
) where
x
=
∑
i
x
i
andy
=
∑
i
y
i
Moreover, proﬁt maximization implies
ν
(
y
)
≤
ν
(
y
*
)
⇒
ν
(
x
)

ν
(
y
)
> ν
(
x
*
)

ν
(
y
*
)
(6)
⇒
ν
(
x

y
)
> ν
(
x
*

y
*
) (
By linearity of ν
)
(7)
Feasibility implies
x
*

y
*
= 0. By (7), it must be that
x

y
6
=
x
*

y
*
which in turn implies
x

y
6
= 0. Hence, the allocation [(
x
i
)
,
(
y
i
)] is not feasible.
Contradiction.
You can see where we need the continuity of
ν
: Supposing that (
x
*
, y
*
) is
not PO, we take another feasible allocation that gives some of the households
more utility. Then we arrive at the contradiction through showing that such an
allocation is not feasible. To conclude that such an allocation is not feasible,
we need to use (1) and (2) which holds under the assumption of continuity of
the price functional. Hence with no continuity, we have no (1) and (2) and the
proof doesn’t go through.
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '08
 Staff
 Optimization, Trigraph, Convex function, Topological vector space

Click to edit the document details