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DEWEY
,01-1
Technology
Department of Economics
Working Paper Series
Massachusetts
Aggregating
Institute of
Infinite Utility
Streams
with Inter-generational Equity
Kaushik Basu
Tapan Mitra
Working Paper 02-19
April 26, 2002
Room
E52- 251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn. com/abstractJd=xxxxx
Technology
Department of Economics
Working Paper Series
Massachusetts
Institute of
Aggregating Infinite Utility Streams
with Inter-generational Equity
Kaushik Basu
Tapan Mitra
Working Paper 02-19
April 26, 2002
Room
E52- 251
50 Memorial Drive
Cambridge, MA 02142
This paper can be downloaded without charge from the
Social Science Research Network Paper Collection at
http://papers.ssrn. com/abstract_id=xxxxx
^^i^ss""''
LIBRARIES
Aggregating Infinite Utility Streams with
Inter-generational Equity*
Kaushik Basu^and Tapan Mitra^
April 26, 2002
Abstract
It
has been
known
that, in aggregating infinite utility streams, there does not exist
any social welfare function, which
and
continuity.
We show
continuity axiom.
satisfies
that the impossibility result persists even without imposing the
Hence, the problem of accommodating inter-generational equity
obstinate than previously supposed.
possibility results
the axioms of Pareto, inter-generational equity
The paper goes on
is
more
to explore the scope for obtaining
by weakening the Pareto axiom and placing
restrictions
on the domain
of utilities.
Journal of Economic Literature Classification Nmnbers: DOO, D70, D90.
*The authors
^Department
are grateful to Jorgen Weibull for helpful suggestions.
of Economics, M.I.T.,
nomics, Cornell University, Ithaca,
NY
Cambridge,
MA
02142, Email: kbasu@mit.edu and Department of Eco-
14853.
^Department of Economics, Cornell University, Ithaca,
NY
14853, Email: tml9@cornelI.edu.
Introduction
1
The
subject of inter-generational equity in the context of aggregating infinite utility streams has
been of enduring interest to economists.
Ramsey
(1928) had maintained that discoimting one
generation's utihty or income vis-a-vis another's to be "ethically indefensible",
that "arises merely from the weakness of the imagination".
any such process of aggregation should
satisfy the
Since
it is
and something
generally agreed that
Pareto principle, Ramsey's observation raises
the question of whether one can consistently evaluate infinite utility streams while respecting
intergenerational equity,
and
some form
at least
of the Pareto axiom.
In an important contribution to this literature, Svensson (1980) established the general possibiUty result (for a social welfare relation
axioms of Pareto and inter-generational
plete) ordering
(SWR)) that one can
equity. It is
by non-constructive methods;
find
an ordering, that
satisfies
the
worth noting here that he obtains the (com-
specifically,
he defines a partial order satisfying the
two axioms, and then completes the order by appealing to Szpilrajn's lemma.
Svensson's paper builds on the earlier seminal contribution to this Hterature by
Diamond
(1965). In this paper,
social welfare fimction
(SWF)
Diamond
presents the celebrated theorem^ that there does not exist any
(that
nimiber) satisfying three axioms:
is,
a function that aggregates an infinite stream into a real
Pareto, intergenerational equity and continuity (in the sup
metric).
Neither contribution addresses the following question: does there exist a social welfare function satisfying intergenerational equity, and the Pareto axiom? Since continuity of the
Diamond's exercise
^Diamond
is
a technical axiom
attributes the result to Yaari.
(in contrast to
the other two axioms),
we think
SWF
in
that this
is
an issue worth
investigating.''
The main
question.
The
principal task of this paper
result that motivates this
paper
is
to provide an answer to this
is
the finding that the continuity axiom
is
not needed for the Diamond- Yaari impossibihty theorem.
If
we denote by
and by
X
Y
(a subset of the reals) the set of admissible utility levels of each generation,
the set of infinite streams of these utility levels, an
SWF
is
a function from
X
to the
turns out that the answer to the question posed in the previous paragraph depends on
reals. It
the nature of the utility space (that
is,
the set, denoted by
y
),
and on the form of the Pareto
axiom. As a consequence, our results can be classified conveniently into four parts.
First,
if
one adopts the standard form of the Pareto Axiom, then regardless of the
specific
nature of the utility space, one obtains the general impossibihty theorem mentioned above:
there
is
SWF
is
no
SWF
which
satisfies
the Pareto and equity axioms.
necessarily inequitable. In particular, this
In other word, any Paretian
means that the complete
orderings, satisfying
the Pareto and equity axioms, obtained by Svensson, do not have real-valued representations.
Indeed, Svensson (1980) goes on to
show that there are complete
orderings, which satisfy the
Pareto and equity axioms, and in addition are continuous (in a topology weaker than the discrete
topology). Clearly, the continuity of the ordering (obtained by him)
representation problem.
The proof
of our result has
an
affinity
is
unable to overcome the
with the demonstration that
lexicographic preferences do not have a real- valued representation: one "runs out of real numbers"
in
a similar way.
Suppose we do want to use a (real-valued) SWF.
How
does one get around this problem of
"•in this connection, we may note that the Hne of research, initiated by Koopmans (1960), and continued
by Koopmans, Diamond and Wilhamson (1964) and Diamond (1965) estabhshes that Paretian social welfare
relations, continuous in suitably defined metrics, necessarily exhibit "impatience" in the sense that the current
generation receives more favorable treatment than future generations. Burness (1973) explores the impatience
implications of continuously different iable Paretian social welfare functions.
One way
respecting intergenerational equity?
axiom.
If
we use the weak Pareto axiom, and
set of non-negative integers,
and
an
this is to use
restrict
weak Pareto axioms.
two elements (corresponding
respectively, so that
do
Y =
to, say,
{0, 1}
),
weaker forms of the Pareto
the utihty space to be a subset of the
then one can estabhsh a possibiHty
fimction satisfying the equity and
of just
to
a social welfare
result: there is
Specifically, if the utility space consists
the "good" and "bad" states, and represented by
1
the above two results would indicate that there exists
SWF satisfying equity and the weak Pareto axiom,
but there
is
no
SWF satisfying equity and
the standard Pareto^ axiom.
Encouraged by our second
result,
one might ask whether
SWF satisfying the equity aad weak Pareto axioms
Y
).
[0, 1]
Our
third result indicates that the answer
(the utility space chosen
their results,
is
always possible to obtain a
(regardless of the nature of the utility space,
in the negative.
If
Y
result
is
is
no
SWF
full
the closed interval
a generalization of Diamond's since
strength of the Pareto axiom;
weak Pareto (and
it
suffices].
this impossibility result.
The
Of
course, continuity
interesting observation
the Diamond- Yaari technique to work
utihty space case,
argument
is
that
is
is still
is
One
equity) suffices.
an
SWF
were to
also does not
[Indeed, as
we
which we
call
it,
at the heart of the demonstration of
that the extent of continuity needed for
already implied by the weak Pareto axiom in the
making a separate continuity axiom superfluous.
if
exist,
and
shows that one does
demonstrate, one does not even need the weak Pareto axiom; a weaker form of
the dominance axiom,
in estabhshing
satisfying the equity
not need to impose the continuity axiom to get his impossibihty result.
need the
is
by both Diamond (1965) and Svensson (1980)
mentioned above), one can show that there
weak Pareto axioms. This
it is
[Specifically,
[0, 1]
the underlying
the weak Pareto axiom would imply that
it
would
have some monotonicity properties, and monotone functions on
[0, 1]
are continuous almost
everywhere]
Finally, as
we
we note
"weak dominance", a general
call
satisfying the equity
Y.
in our fourth result,
It is
we weaken the Pareto axiom
if
what
an
SWF
can be obtained: there
possibility result
and weaJc dominance axioms,
sufficiently to
exists
regardless of the nature of the utility space,
worth remarking that, when the utihty space,
Y
and weak dominance axioms, obtained by applying our
,
is [0, 1],
any
SWF satisfying the equity
last result, necessarily violates
the weak
Pareto axiom (given our third result).
Paretian Social Welfare Functions are Inequitable
2
Let
R
be the
numbers,
N
the set of positive integers, and
F C R is the set
of
all
set of real
integers.
Suppose
X
= Y^
is
all
teN, xteY represents the amount of
period
y
T^ z;
If
t)
the set of
all
earns. For all
and we write y
Y
>>
z, ii yi
>
streams
is trivial.
>
Zi, for all
X
is
z
\i yi
social welfare function
{xt}
eX, then
>
Zi,
{xt}
is
(xi,
generation (that
for all zeN;
is,
a;2,
),
where, for
the generation of
we write y >
z
ii
y
>
z
and
a singleton, and the problem of ranking or evaluating
Thus, without further mention, the set
(SWF)
=
Then
ieN.
to haveat least two distinct elements.
A
If
utility that the t"'
write y
has only one element, then
infinite utihty
possible utilities that any generation can achieve.
possible utility streams.
y,zeX, we
M the set of non-negative
a mapping
W :X
Y will always be assumed
Consider
now some
we may want the
of the gixioms that
SWF
The
to satisfy.
axiom
first
is
the standard Pareto condition.
Pareto Axiom:
For
The next axiom
call
it
is
all
'finite
-
The
It is
{i,j}, Xk
W{x) > W{y).
We
=
For
yk,
all
shall
equivalent to the notion of 'finite equitableness' (Svensson,
anonymity' (Basu, 1994).
Anonymity Axiom:
A;eN
then
the one that captures the notion of 'inter-generational equity'.
the 'anonymity axiom'. ^
1980) or
for
x,yeX,iix>y,
x,yeX,
then W{x)
there exist
if
=
principal result of this section
'^
i,jeN such that
Xi
=
=
and Xj
yj
yi,
and
W{y).
that there
is
is
no
social welfare function
which
satisfies
both axioms.
Theorem
1
There does not exist any
SWF
satisfying the Pareto
Proof. Assume, on the contrary, that there
the Pareto and
satisfies
writing,
Let
Anonymity Axioms. Since
we may suppose that
Z
numbers
y^
=
denote the open interval
in Z.
a social welfare function,
is
>
there exist real numbers y° and y^ in Y, with y^
1
and
(0, 1),
y'^
=
and
=
W X—
>
:
R, which
V C M contains at least two distinct elements,
y".
Without
loss of generality,
and to ease the
0.
let ri, r2, r^,
We
Let p be an arbitrary number in Z.
E{p)
and Anonymity Aodoms.
{n G
N
:
r„
...
be an enumeriation of the rational
define the set:
< p}
.
In informal discussions throughout the paper, the terms "equity" and "anonymity" are used interchangeably.
^The Anonymity Axiom
the social ordering
ordering.
May
is
figures prominently in the social choice theory literature,
invariant to the information regarding individual orderings as to
where
who
it is
stated as follows:
holds which preference
Thus, interchanging individual preference profiles does not change the social preference
(1952) and Sen (1977).
profile.
See
Clearly, E{p)
is
an
We now
infinite set.
define a sequence a{p)
=
(a(p)i, 0(^)2, ••) as follows:
ifneEip)
1
a{p)n
=
<
otherwise
Note that the sequence
the sequence b{p)
=
will
have an
(fe(p)i, 6(^)2,...)
appearing in the a(p)sequence
W{a{p)).
Now,
for if
We
let
is
number
infinite
same
as the
replaced by a
of I's
and an
infinite
number
Define
of O's.
as the sequence a{p) except that the first
By
1.
the Pareto axiom,
we have W{b{p)) >
denote the closed interval [W{a{p)), W{b{p))] by I{p).
q be an arbitrary real
n e E{p), then
there are an infinite
r„
<
p,
number
number
in Z, with q
and since p <
of rational
q,
>
p-
we must have
numbers
Clearly,
r„
<
we must have E{p) C E{q),
so that
q,
n G E{q).
Further,
Thus, comparing the
in the interval {p,q)-
sequence a{p) with the sequence a{q), we note that:
(i)
i/n e N, and
a(p)„
=
1,
then a(g)„
=
1
(1)
and:
(n)
there are an infinite
for which a(p)„
We now
=
number 0/ n
and
a(g)„
=
£
1
(2)
proceed to compare the sequence a{q) with the sequence
index for which the seqence a{p)
differs
from the sequence
There are two cases to consider: (A) a{q)m
=
1,
b{p); that
(B) a{q)m
=
0.
N
is,
b{p).
a(p)m
Let
=
0,
In case (A),
m
G
N
be the
and b{p)m
we
=
1-
clearly have
a{q)
>
b{p),
and
so:
W{a{q)) > W{b{p))
In case (B),
while a{q)M
=
we proceed
By
1-
=
Now,
define b'{p) as follows:
n
while a{q)m
^ m,n^ M.
=
Since b{p)m
N
nG
b'{p)
such that n
6'(p)m
=
1,
=
0,
observation
(2),
= 0,
and 6(p)m
m,n ^ M, we
we must
have 6'(p)„
therefore have a{q)
>
W{a{q))
Combining
(15)
and
=
b'{p)M
(16),
=
we
=
=
[W{a{q)), W{b{q))]
lying entirely to the
left
is
b'{p)n
=
M
Also, clearly
b{p)n for
^ m,
we have 6(p)m
all
n G
N
for
=
0.
such that
W{b{p))
(4)
=
=
a(g)m, b'{p)M
b{p)n
=
a(p)„
b'(j)),
>
1,
<
a(g)„
=
1
=
a{q)M, and for aU
by observation
(1).
By
so that by the Pareto Axiom:
W{b'{p))
(5)
Wia{q)) > W{b{p))
(6)
get:
Thus, in both cases (A) and (B), we have W{a{q))
/(g)
exists.
—
which a{p)M
the Anonymity axiom implies that:
with a{q), we note that b'{p)m
y^
M
for
Since b{p) differs from a{p) in only the index m,
0.
W{b'{p))
Comparing
be the smallest integer
observation (2) above, such an
a{q)M
1
M
Let
as follows.
(3)
disjoint
>
W{b{p)).
This means that the interval
from the interval [W{a{p)), W{b{p))], the
of the former interval on the real
To summarize, we have shown that the
latter interval
line.
intervals I{p) associated with distinct values of
p G
(0, 1)
But, this
are non-overlapping.
a distinct rational number
(in
means that
to each real
number p G
(0, 1),
we can
associate
the interval /(p)), contradicting the countability of the set of
rational numbers.
Remark:
The construction (used
in the
above proof) of an uncountable family of distinct nested sets
number
E{p), with each set containing an infinite
of positive integers, can
be found
in Sierpinski
(1965, p. 82).
now suppose
Let us
welfare relation
which
satisfies
(SWR),
variant of Szpilrajn's
[0, 1];
it.
we
~
we
Theorem (due
more general
SWR is
We
result.
is
a social
an
SWR
do
this because, the use of a
to distinguish
^-Anonymity Axiom:
For
{i,j], Xk
=
t/fe,
is
the closed interval
setting.
is
complete and transitive.
We
use
and symmetric parts of ^. The properties of
SWR are easy to define. We shall call these axioms ^-Pareto and
them from the axioms applied
all x,
Y
which apphes to any utihty space y.His proof, as well as
a binary relation, ^, on X, which
^-Pareto Axiom: For
-
for
to Suzumura, 1983) allows us to give a particularly easy
to denote, respectively, the asymmetric
^-Anonymity
fceN
review Svensson's
briefly
Paxeto and Anonymity for an
for
and instead look
then have the result due to Svensson (1980) that there
state the version of his result
Formally, an
and
SWF
an
Also, Svensson (1980) restricts his exercise to the case where
ours, applies to this
>-
We
for
the (appropriate relational versions of the) Pareto and Anonymity axioms. For
reasons of completeness
proof of
we abandon the search
that
ye
X
all x,
then x
,
x
>
y eX,
~ y.
y implies x
if
to
an SWF.
>- y.
there exist
i,
j e N, such that Xj
=
yj
and
Xj
=
yt
and
First, let us give a
is
a binary relation on
of i?
all
x^
for all
if,
x,y eO,
statement of Suzumura's theorem.
Q and R* an
a; /f!t/
x^,...,x*eQ, [x^Rx"^
,
ordering on
xR*
implies
fi,
We
y.
and not x^Rx^, and
we
Let
il
be a
shall say that
say that
R is
set of alternatives.
R*
is
consistent
for all A;e {2, 3, ...,t
—
If
R
an ordering extension
if,
for all
teN, and
1}, x^Rx'^^^] implies
for
not
x^RxK
Lemma
1 Szpilrajn's Corollary [Suzumura, 1983,
has an ordering extension
In contrast to
Theorem
if
Theorem
and only
1,
we now
2 (Svensson, 1980)
;
Theorem
A (5)]: A
binary relation
R
on
fl
if it is consistent.
have:
There exists a social welfare relation satisfying the ^-Pareto
and "^-Anonymity Axioms.
Proof. Define two binary
And
xPy.
xly.
Now
It is
if
relations,
there exists i,j such that Xi
define the binary relation
easy to verify that
extension ^.
P and /, on X, as follows.
Clearly
^
R
is
satisfies
R
=
yj
and
as follows:
=
Xj
xRy
<^
yt
For
and Xk
xPy
all
=
x,y eX, x
>
y implies
k
^
i,j,
Vk, for all
or xly.
Hence, by Szpilrajn's Corollary,
consistent.
Axiom and
the ^-Pareto
the
then
it
has an ordering
^-Anonymity Axiom.
Remcirks:
(i)Theorem 2 shows that there
equity and the Pareto principle.
an evaluation of
(ii)
A
utility
is
no inherent
Theorem
1
conflict
between the concept of intergenerational
shows that a
conflict arises
when we
try to obtain
streams in terms of real numbers, while respecting these two properties.
simple example of our set-up
is
interpret this as follows: there are precisely
one where the
two
states in
10
utility space,
Y =
{0,
l}.One might
which each generation might
find itself,
a good state and a bad
in the
respects
3
If
bad
state.
Theorem
state.
The
utility
1 tells
obtained by each generation
in the
us that even in this simple set-up, there
is
good
no
and
state,
SWF
which
Anonymity and the Pareto Axiom.
Relaxing Pareto
we want
work with a
to
this section
is
and
social welfare function
anonymity, what possibilities do we have?
respect inter-generational equity or
In the light of our
Theorem
1,
full-brunt of the Pareto condition (even
if
possibilities that are technically allowed in
any eventuaUty.
Indeed
we
are
armed with the
following
weak
Note that
we
all x,
recognize that
y e X,
if
all x,
yeX,ii x
human
sufficiently small changes in well-being
set of feasible utilities will
^Our Weak Pareo Axiom
where typically it is stated as
is
the
not arise under
the comparison of the moral
Basu, 1994)
(see
all
it
may be enough
to be
Paretianism.*^
W{x) > W{y). For
if
may
our specification of the domain,
for certain ethical discourses involving
Wccik Pareto Axiom: For
then
in
we do not need the
committed Paretians) simply because
worth of individual actions and universalizable rules
= Vk,
what we explore
to relax the Pareto axiom.
arguable that for certain philosophical and even policy purposes
It is
Xk
is 1
somewhat
if
>>
e
y,
N
such that Xj
then
set.^
different
it
The same
from the
>
yj,
is
not endlessly
all
k
y^ j,
fine,
so that
seems reasonable to suppose that the
is
true
if
the benefits axe measured
Weak Pareto Axiom used
every individual in a society
and, for
W{x) > W{y).
perception or cognition
go imperceived,
be a discrete
follows:
there exists j
is
better
oif,
in social choice theory,
then society
is
better off
.
See,
example, Sen(1977). Sometimes, though, in the social choice theory literature, this axiom is called the Pareto
Axiom, and our Pareto Axiom is called the Strong Pareto Axiom. See, for example, Arrow (1963) and Sen (1969).
for
*The
limits
on human perception have been used in a different context by Armstrong(1939) to argue that it is
is a transitive relation. For a discussion of this issue in individual choice
implausible to suppose that indifference
11
in
money and
there
Akademi, 1976).
a well-defined smallest unit, as
is
Thus,
is
true for
currencies (Segerberg
all
this very reasonable possibility), there
is
X
a social welfare function (on
yC
3 Assume
M
SWF satisfying the
There exists an
.
(which captures
respecting
)
and the Weeik Pareto Axioms.^ Our next theorem provides an interesting
Theorem
M
y C
seems worthwhile to explore whether with
it
and
Anonymity
possibility result.
Weak Pareto and Anonymity
Axioms.
Proof. For each
A;eN, which are
partition of
further,
X.
xeX,
> N}.
That
Let
is, if
there
:
be the collection
F
and
belong to
{E
$5,
:
N eN,
some
E=
E{x)
for
E=
then either
for all
ieN, the
:
X — M as follows.
*•
set {a;i,X2,
...} is
Given any x eX,
such that yk
some xeX}.
E
F, oi
is
=
Xk for
Then ^
disjoint
all
is
a
from F;
the axiom of choice, there
is
Given any xeX, we can denote
a function, g
for
let
f{x)
=
min{xi,
2:2, ...}.
Since
a non-empty subset of the set of non-negative integers
and therefore has a smallest element [Munkres, 1975,
By
is
UecqE = X.
Define a function, /
XjcM
3
E
= {yeX
E{x)
let
each
N>
:
Q
p. 32].
^^
1, {xi,
X
,
Thus, /
is
well-defined.
such that g{E)
...,xn)
e
E
by x{N), and
for
(xi
each
E eQ.
-I- ... -I-
x^) by
I{x{N)). Now, given any x,t/ in £^e^, define /i(x,y) =\imj^^oo[I{x{N)) — I{y{N))]. Notice that
h
is
is
well-defined, since given
a constant
> M.
Then i/(x,y)e
\h{x,y)\]].
theory, see
for all A^
Majumdar
any x,y
in
E eQ, there is some M e N, such that
Now, given any x,y
in Ee'ii, define
[
H{x,y)
I {x{N))
=
— I {y{N))]
0.5[h{x,y)/[l
+
(-0.5, 0.5).
(1962).
While our choice of F as a subset of the set of non- negative integers is motivated by the imprecision of
human perception, the mathematical technique used to obtain our possibility result applies also to the case where
Y = {(1/n) n € N},where clearly human perception has to be considered to be sufficiently refined.
^
:
12
We now
class,
define VF
:
J'C
^ M as follows.
Then, using the fimction
E{x).
Given any
we
g,
xeX, we
associate with
get g{E{x))e E{x).
Next, using the functions, h
and H, we obtain h{x,g{E{x))) and H{x,g{E{x))). Now, define W{x)
The Anonymity Axiom can be
such that
Xi
—
yj
and Xj
=
yi,
Furthermore, denoting this
H{x,g{E))
f(x)
=
=
H{y,g{E)).
while Xk
common
Xi
>
yi,
Furthermore, denoting the
If
x,y are
...} is
verified as follows.
W{x) =
fix)
+
i,j,
=
then E{x)
=
E{y).
h{y,g{E)), and so
the same as the set {yi,y2,
If
•••},
so that
x,y are in X, and there exists
while Xk
=
yk for
common
set
by E, we see that h{x, g{E))
all
keN, such
so that
2/2, •••},
»
y,
then E{x)
^
that k
>
h{y, g{E)).
set {xi, X2,
we have f{x) >
then E{x)
i,
...} is
=
E{y).
This implies
at least as large
Thus, we obtain the
f{y).
0.5.
Further, since x
+
H{x,g{E)) > f{y)
E{y).
7^
Thus, we
will
not be able to compare
However, we do know that H{x,g{E{x)))
H{x,g{E{X))) with H{y,g{E{y))).
<
^
exist i,j in N,
W{x) > W{y).
x,yeX, and x
H{y,g{E(y)))
+ H{x,g{E{x))).
W{x) = W{y).
as the smallest element of the set {yi,
If
f{x)
X, and there
yk for al^Ai^N, such that k
H{x, g{E)) > H{y, g{E)). Further, the smallest element of the
desired inequality:
in
=
by E, we see that h{x,g{E))
set
The Weak Pareto Axiom can be
ieN, such that
=
Further, the set {xi,X2,
Thus, we obtain:
f{y).
verified as follows.
equivalence
it its
1
-
>>
0.5
y,
>
we have f{x) >
/(y)
f{y)
+ H{y,g{E)) =
+
1.
>
-0.5, and
Thus, we obtain:
W{y). U
Remarks:
(i)
The important
point to note
positively sensitive to
that our
is
an increase in
utility of
being unchanged (and therefore that
it
Weak
Pareto Axiom demands that the
a single generation, the
utilities of
be positively sensitive to increases
13
SWF
be
other generations
in utilities of
any
finite
number
SWF
of generations, the utilities of other generations being
be positively sensitive to an increase in
be positively sensitive to an increase
Weak
is
an
infinite
number
unchanged. This
is
and
),
also that the
However,
it
of generations,
need not
when
the
the principal difference of our
Pareto Axiom from our Pareto Axiom.
Consider the simple set-up, introduced in the previous section, where
(ii)
Theorem
1
Theorem
3 implies that there
implies that there
follows that for
alternatives
no
is
is
an
SWF
respecting the Pareto and
SWF satisfying the Weak Pareto
any social welfare function,
x,y E
X
such that x
>
W,
so obtained,
it
Y =
{0,1}. Now,
Anonymity Axioms. And,
and Anonymity Axioms.
must be the case that there
It
exist
W{x) < W{y).
y,but
Generalizing the Diamond- Yaari Result
4
The
it
utilities of all generations.
in utiUties of
utihties of a (non-empty) set of generations
unchanged
possibility result of the previous section
can be extended to
possible
all utility
spaces, Y.
by demonstrating that when Y
the Anonymity and
Weak
is
is
reason for optimism, and
We show in this
we might ask whether
section that such
the closed interval
[0,
l],then there
Pareto axioms. This generalizes the result of
an extension
is
no
is
not
SWF satisfying
Diamond
(1965) in two
respects.
Diamond had shown
that (when the utility space
Anonymity, Pareto, and a continuity axiom
the continuity axiom
full
is
redundant
power of the Pareto axiom
suffices for this
is
(in the
Y
is
[0,1]),
there
sup metric on X).
for this impossibility result.
is
no
Owe
In addition,
SWF
result
it
satisfying
shows that
shows that the
not needed for the impossibility result; the weaJi Pareto cixiom
purpose.
14
Theorem 4 Assume Y =
the
There does not exist any
[0, 1].
Weak Pareto and
Anonymity Axioms.
we
Instead of proving this theorem directly,
Theorem 4
is
an obvious
form of the Pareto
=
Uk,
then
since
not as
we
if
For
all
Weak
we wish
to
call
x,y eX,
all
last inequality in
the definition of the
It is
which we
W{x) > W{y). For
Note that the
To prove
corollary.
criterion,
Dominance Axiom:
Xk
SWF satisfying the
if
more powerful theorem,
shall establish a
of which
this stronger result, let us next define a very
weak
the Dominance Axiom.
there exists j
x,yeX,
if
x
>>
e
y,
N
such that Xj
then
Hence,
recommend the use
Weak
of such a
as one can. Further, our proof indicates that
it is
and, for
all
k
^ j,
a weak inequality, unlike in
is
Pareto
is
weak form
are going to prove an impossibihty result, clearly
yj,
W{x) > W{y).
the statement of this axiom
Pareto Axiom.
>
it is
stronger than Dominance.
of the Pareto condition, but
better to use as
weak an axiom
Dominance axiom that
precisely the
is
needed
to obtain the impossibility result.
Theorem
Axiom and
Assume
5
the
Y =
There does not exist any
[0,1].
W X ^M, which
:
Denote the vector
satisfies
(1, 1, 1,
...)
u
This sequence
satisfying the
Dominance
Anonymity Axiom.
Proof. To establish the theorem, assume
function,
SWF
is
=
in
X
Y=
[0, 1]
and that there
exists
a social welfare
the Dominance and Anonymity Axioms.
by
(1, 1, 0,
e.
1/2,
Define the sequence u in
1, 0,
1/4, 2/4, 3/4,
X
1, ...)
best understood as sequence u, defined below, with the
15
as follows:
(7)
first
term changed from
r
to
1.
For
s e7
=
(-0.5, 0.5), define:
x(5)
Then
(l/8)e
<
x{s)
<
(7/8)e,
Define the function, /
:
7
=
and so x{s)
eX
number of points
ti
in
X
ti
and then
=
W{x{s))
Let a
e
in s
on
7.
Thus / has only a countable
7 be a point of continuity of the function /.
as follows:
=
(0, 1, 0, 1/2, 1, 0, 1/4,
2/4, 3/4,
1, ...)
(9)
define:
a;(a)
Clearly,
(8)
each sel.
monotonic non-decreasing
is
of discontinuity in 7.
Define the sequence
for
s)e
^ R by:
f{s)
By the Dominance Axiom, /
+ 0.25(1 +
0.512
x{a)eX, and Xi{a) >
=
0.5w
+ 0.25(1 + a)e
Xi(a), while Xk{a)
=
(10)
Xk{a) for each
keN, with
k
^
1.
Thus, by
the Dominance Axiom, we have:
W{x{a)) > W{x{a))
We
denote [W{x{a))
Denote max(0.5
-
—
W{x{a))] by
a, 0.5
+
a)
9;
then ^ >
by A; then,
0.
A
16
>
0.
Since /
is
continuous at
a,
given the 9
defined above, there exists 6
e (0,
A), such that:
0<\s-a\<6
Note that
We
for
<
|s
—
a|
define (following
<
6,
\f{s)-f{a)\<9
(11)
we always have sel.
Diamond),
u, interchanging the initial
implies
keN, a sequence
for each
+
with the {k
l)st 1
u''
by starting with the sequence
appearing in «, and then interchanging the
sequence:
wuh
(l/2^2/2^...,(2'=-l)/2^o)
(o,l/2^2/2^...,(2'=-l)/2*=)
(12)
so that:
u''
Now,
for each A;eN,
=
we
(1, 1, 0, 1/2, 1,
...,
respectively,
eX
for
each keN.
...,
(2^
=
0.51*'=
-
1)/2^
0,
(13)
...)
+ 0.25(1 + a)e
Comparing the expressions
and using the expressions
Anonymity Axiom
l/2^ 2/2^
use u^ to define x''{a) as follows:
x'=(a)
Clearly, x''(a)
0, 0,
for
u and
u''
in (9)
(14)
for
and
x
(a)
and
x''{a) in (10)
(13) respectively,
we
and
(14)
see that the
yields:
W{x''{a))
=
W{x{a))
17
for
all
keN
(15)
Choose
that
<
{a
K N with K >2
e
—
S)
<
6,
such that (1/2^-2)
6,
and define S
=
{a
-
(1/2^-2)).
We
note
and so S el, and:
W{x{S))
We now
<
compare the welfare
x^{a)
=
f{S) > f{a)
levels associated
-9 =
W{x{a))
-
9
with x^{a) and x{S) as follows.
(16)
Notice that:
= 0.5m^ + 0.25(1 + a)e
= 0.5u + 0.25(1 + a)e -
0.5(u
- u^)
= x(a)-0.5(u-u^)
>
x{a)
-
0.5(l/2^)e
= 0.512 + 0.25(1 + a)e -
0.5(l/2^)e
= 0.5u + 0.25(1 + a -
{l/2^-'^))e
=
{l/2^-^))e
0.5u
+ 0.25(1 + a -
»0.5f2 +
0.25(1
+
+ 0.25(1/2^-^)6
5)e
= x{S)
Thus, by the Dominance Axiom, we have:
W{x''{a))>W{x{S))
18
(17)
Using
(15), (16)
and
(17),
W(x{a))
and the
-9 =
definition of 6,
The
intuitive idea
obtain:
= W{x^{a)) > W{x{S)) >
W{x{a))
a contradiction which estabHshes our
we
W{x{a))
-
9
result.
behind our proof
is
Diamond had used the axioms
straight-forward.
Pareto and Anonymity in conjunction with another axiom concerning the continuity of
precipitate
an impossibihty
Dominance Axiom), by a standard
everywhere. In other words,
Now,
result.
as soon as
we impose the Weak Pareto Axiom
result in analysis,
we know
W must have points of continuity.
to give us the impossibility result.
minimal amount of continuity that
In other words, there
is
is
of
W to
(or the
W cannot be discontinuous
that
And we show
that that
no need to assume
is
enough
continuity.
The
needed to create the impossibility arises naturally as a
consequence of the Dominance Axiom.
5
Weak Domincince
What we
have not yet found
Pareto criterion
is
satisfied.
is
an existence result where
Theorem
further to get such a result. In fact
the Dominance axiom
only a finite
number
is
5 indicates that
what we need
is
Y =
we need
to
and some version of the
weaken the Paxeto
a condition in which only the
asserted. If each policy question that
of generations, then this
[0, 1]
we
criterion
first
part of
consider affects the utihties of
"Weak Dominance" condition
suffices in
capturing
the idea of Pareto.
Weak Dominance Axiom:
For
all
x,yeX,
if
19
for
some
j
eN,
Xj
>
y^,
while, for
all
k
^
j,
Xk
=
yk,
then
Theorem
6
W{x)>W{y).
There
Proof. For each
keN, which
partition of
each
are
>
exists
xeX,
A''}.
an
SWF satisfying
E{x)
let
Let
$j
X. By the axiom
= {yeX
the
there
:
{E
be the collection
of choice, there
Weak Dominance and Anonymity Axioms.
is
is
some TVeN
E=
:
a function, 5
:
some xeX}.
for
Q'
—
>
Xk for
Then,
5
all
is
X, such that g{E) e E,
a
for
EeQ.
Given any x,y in
R as follows.
E e^, define h{x,y) = lim;v— oo[-^(3^(-^)) — ^(y(-^))]-We now define VT X —
:
Given any x e X, we associate with
get g{E{x)) eE{x), and, using h,
equivalence class, E{x).
it its
we obtain h{x,g{E{x))).
The Anonymity Axiom and the Weak Dominance Axiom
Now,
define
W{x)
Then, using
=
g,
we
h{x,g{E{x))).
are easily verified.
Conclusion
6
We summarize our
in
E{x)
=
such that y^
,
a
results,
marking out the terrain of what
is
possible
and what
is
not possible,
table:
N
[0,1]
Theorem^ 1^^
Pareto
Weak Pareto
Theorem
Dominance
Theorem' 5
Weak Dominance
Theorem
TABLE
It is
assumed
in Table 1 that the
4;:
Theorem
':.
6
1
Anonymity Axiom
weakening sequence of Pareto-type axioms.
The columns
20
3
is satisfied.
represent the
The rows
domain
represent a
restrictions, to
wit,
what
exists)
Y
is
equal
to.
The shaded area
and the unshaded area the zone of
represents the zone of impossibihty (where no
possibihty.
21
SWF
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23
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