Economics 201B–Second Half
Lecture 5
The Welfare Theorems in the Arrow Debreu Economy
•
Local nonsatiation
: The preference relation
i
on the con
sumption set
X
i
is locally nonsatiated if, for every
x
i
∈
X
i
and every
ε >
0, there exists
x
i
∈
X
i
such that

x
i
−
x
i

< ε
and
x
i
i
x
i
.
–
Note that this is a substantial weakening of monotonicity
–
Important especially with production, since we want to al
low for input goods which provide no direct consumption
utility
Theorem 1 (First Welfare Theorem)
If preferences are lo
cally nonsatiated and
(
x
∗
, y
∗
, p
∗
, T
)
is a Walrasian Equilibrium
with Transfers, then
(
x
∗
, y
∗
)
is Pareto Optimal.
Proof:
Let
W
i
=
p
∗
·
ω
i
+
J
j
=1
θ
ij
p
∗
·
y
∗
j
+
T
i
W
i
is the income available to person
i
. Observe that
I
i
=1
W
i
=
I
i
=1
p
∗
·
ω
i
+
I
i
=1
J
j
=1
θ
ij
p
∗
·
y
∗
j
+
I
i
=1
T
i
=
p
∗
·
⎛
⎜
⎝
I
i
=1
ω
i
⎞
⎟
⎠
+
J
j
=1
⎛
⎜
⎝
I
i
=1
θ
ij
⎞
⎟
⎠
p
∗
·
y
∗
j
+ 0
=
p
∗
·
¯
ω
+
J
j
=1
p
∗
·
y
∗
j
1
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By the definition of Walrasian Equilibrium with Transfers, (
x
∗
, y
∗
)
is a feasible allocation.
Suppose (
x
∗
, y
∗
) is not Pareto Optimal. Then there is a feasible
allocation (
x , y
) such that
x
i
i
x
∗
i
(
i
= 1
, . . . , I
)
x
i
i
x
∗
i
for some
i,
WLOG
i
= 1
x
1
1
x
∗
1
⇒
x
1
∈
B
1
(
p
∗
, y
∗
, T
)
⇒
p
∗
·
x
1
> W
1
We claim that
p
∗
·
x
i
≥
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 Spring '06
 ANDERSON
 Economics, Fundamental theorems of welfare economics

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