Lecture Notes, May 20, 2011
Fundamental Theorems of Welfare Economics
Pareto Efficiency
Definition
:
An allocation
,
is attainable if there is
(note
x
i
,
i
H
y
j
Y
j
,
j
F
change in notation)
so that
.
(The inequalities hold
0
i
H
x
i
j
F
y
j
i
H
r
i
co-ordinatewise.)
Definition
:
Consider two assignments of bundles to consumers, v
i
, w
i
X
i
, i
H.
v
i
is said to be Pareto superior to w
i
if for each i
H, u
i
(v
i
)
u
i
(w
i
) and for some
h
H,
u
h
(v
h
)
u
h
(w
h
) .
Note that Pareto preferability is an incomplete ordering.
There are many allocation
pairs that are Pareto incomparable.
Definition
:
An attainable assignment of bundles to consumers,
, is said
w
i
,
i
H
to be Pareto efficient (or Pareto optimal) if there is no other attainable assignment
so that
v
i
is Pareto superior to
w
i
.
v
i
Definition
:
, x
0i
R
N
, y
0j
R
N
, is said to
p
0
,
x
0
i
,
y
0
j
,
p
0
R
N
,
i
H
,
j
F
be a competitive equilibrium in a private ownership economy if
(i)
y
0
j
Y
j
and
p
0
y
oj
p
0
y
for all
y
Y
j
,fo
ra
l
l
j
F
(ii)
x
0
i
X
i
,
M
i
p
0
p
0
r
i
j
F
ij
p
0
y
0
j
p
0
x
0
i
M
i
p
0
and
u
i
u
i
(x)
for all
with
for all
,
and
x
0
i
x
X
i
p
0
x
M
i
p
0
i
H
(iii)
0
i
H
x
0
i
j
F
y
0
j
i
H
r
i
(co-ordinatewise) with
= 0
for co-ordinates k so that the strict inequality holds.
p
k
0
This definition is sufficiently general to include the equilibrium developed in each
of
Theorems
14.1, 18.1, and 24.7.
Properties (i) and (ii) embody
decentralization.
Property (iii) is market clearing.
Economics 113
Spring 2011
University of California, San Diego
Prof. R. Starr, Ms. S. Fried
May 20, 2011
1