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
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 Spring '08
 Staff
 Optimization, Convex hull, FBWT

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