Kobe University Repository : Kernel
Title
A NOTE ON THE MONOTONE TRANSFORMATION
OF CES SOCIAL WELFARE FUNCTIONS
Author(s)
Iritani, Jun / Miyakawa, Toshiji
Citation
Kobe University Economic Review,48:19-28
Issue date
2002
Resource Type
Departmental Bulletin Paper / 紀要論文
Resource Version
publisher
DOI
URL
http://www.lib.kobe-u.ac.jp/handle_kernel/81000934
Create Date: 2017-07-29
19
Kobe University Economic Review 48 (2002)
A NOTE ON THE MONOTONE TRANSFORMATION
OF CES SOCIAL WELFARE FUNCTIONS
By JUN IRITANI AND TOSHIJI MIYAKAW A
The conventional function C2::;~=l(Uh)l-P-a)/(1-p) has been frequently used in
the fields where economists look for policy implications on the choice of institutions,
e.g., the choice of tax functions. The function is expected to represent various types of
functions such as utilitarian or Rawlsian functions by the suitable choice of the parameter. In this paper we point out that a serious problem may exist in using the function.
1. Introduction
Much literature on the optimal tax has made use of the following social welfare function:
. ~f "n
( )l- p - a 1)
k.Jh = 1 Uh
f CON (U 1, ... , Un' p) 1 -p
'
where U h is a utility level of agent h(h = 1, ... , n) and where a is a positive constant
(see Atkinson (1973), Stern (1976), Harris-MacKinnon (1979), Atkinson-Stiglitz (1980,
page 340, 402), Tuomala (1984)). We call this form of conventional function a 'CON
function'. The CON function is very useful because this function can represent not only
the utilitarian social welfare function but also the Rawlsian social welfare function by
choosing the parameter p. In effect, the CON function represents the purely utilitarian social welfare function ~ u h when p is nil. Furthermore, it is said to represent the
Rawlsian social welfare function min {u 1> ••• , un} when p is a positive infinite.
On the other hand, we know another functional form that represents the purely utilitarian or Rawlsian social welfare function according to the proper choice of a parameter.
It is a CBS function:
E
(
lCES
U 1,
••• ,
Un' P
)
def [ (
=
U1
It is well known that fCES(U 1,
) 1-.0
••• ,
+ ... + (Un) I- PJ l/(l-p) .
un' 0) = ~h u h and f CES(U 1,
••• ,
un. 00) = min {U 1•
.•. , Un}·
The CON function is closely related to the CBS function since the former is obtained by a monotone transformation of the latter. This fact seemingly gives a justification for economists to exploit the CON function as a social welfare function in practical models. However, the existence of a monotone transformation between CON and
CES functions does not necessarily imply that the CON functions tend to the Rawlsian
1)
The symbol " ~ " indicates that the left hand side is defined by the right hand side.
20
JUN IRITANI AND TOSHIJI MIY AKAWA
function as p tends to infinity. Motivated thus, we re-examine in this note whether the behavior of the CON function is pertinent to the objective for economists or not. In this
event we are faced with three kinds of continuity problems. One is whether the limit of
CON functions is equal to the Rawlsian function or not as p tends to infinity. Another
is whether the limit of preference relations generated by CON functions is equal to that
generated by the Rawlsian function or not. The other is whether the limit of the solutions of maximization problems under CON functions is identical to that under the
Rawlsian objective function.
In Section 2, we establish the property of CON functions that JeoN ( U l' "', Un' p)
does not converge to the Rawlsian function min {U 1, " ' , un} as p tends to infinity. This
property may cause fatal errors in calculations when we use CON functions with large
p to obtain an approximate solution to a Rawlsian maximization problem. We will provide simple numerical examples concerning the problem. The numerical results do not
necessarily confirm that the solutions under CON functions are approximate solutions
to the Rawlsian problems even when p is very large. On the other hand, the CES functions exhibit good behavior irrespective of methods of simulations and values of p.
In Section 3, we investigate the convergence of preference "relations" induced by a sequence of the welfare functions, where the closed convergence topology defines the concept of convergence. Surprisingly enough, the sequence of preference relations given by
the CON functions converges to the Rawlsian preference relation as far as the preference relations are concerned. This is also the case for the preference relations by CES
functions. Thus, a sequence of the preference relations represented by the CON functions does not converge to that by the limit of the CON functions. This implies that
the mapping on the set of social welfare functions to the set of preference relations is
not necessarily continuous. Therefore we must be very careful in dealing with the CON
functions by which we are to represent a Rawlsian social welfare function when p is
very large.
In Section 4, we examine the limits of a sequence of solutions. Economists often consider a sequence of social welfare maximization problems, each objective function of
which is either a CON function or a CES function. They want to know the behavior of
solutions when the social preference varies from utilitarian to Rawlsian. Here we shall
give an answer to the question whether the sequence of solutions, each of which attains
a maximum under the associating social welfare function, tends to that under the limit
function of social welfare functions. Many researchers (for example, Atkinson (1973),
Stem (1976), Tuomala (1984» have thought that the solution is near a Rawlsian solution when the value of p in the CON function is very large. In order to justify their reasoning, we give a proof of the convergence of the solutions in this paper.
21
A NOTE ON THE MONOTONE TRANSFORMATION OF CES SOCIAL WELFARE FUNCTIONS
2. Properties of the CON functions
Here, we ask a question whether !CON(U 1• '''' un. p) converges to the Rawlsian social
welfare function min {U l , ... , un} or not as p ~
The answer is negative. Results are
summarized in the following property.
(x).
Property 1. It holds that:
lim !CON(U 1, "', un' p) =
P~OO
{O
-(X)
if
Uh ~
if 0
<
Proof It is trivial that the result holds when u h
some h such that 0
(~h2~P
=
< uh <
1, for all h,
Uh
<
1, for some h.
> 1 for all h. Suppose that there exists
1. We can write
[(Uh;I~P ]/[ ~ -\ J.
It is clear that the denominator [( 1/p) -1] converges to - 1 as p ~
Let us focus
on the numerator and define 0 def (1/ Uh) -1 > O. Then, we can rewrite the numerator
as:
(X).
By Taylor's series expansion, we have (1 +o)P ~ 1 +op+ (o2/2)p(p-1) for p
This leads us to the inequality
(Uh)l-P >
------_
P
> 2.
1+op+ (o2/2)p(p-1)
UhX----~------~~---
P
Because the right hand side converges to infinity as p ~ (X), the numerator also conD
verges to infinity. Therefore, lim p--7oo!CON(U l, ... , un' p) = - ( X ) .
On the other hand, the CBS function satisfies the following property:
Property 2. It holds that:
lim !CES(U I.... , un' p) = min {u l• ... , Un}.
P~ 00
Proof. Without loss of generality, we can assume that 0 < U l <
for sufficiently large p that (U1)1-P > (U 2)1-P ~ ... > (Un)l-P
(n)l!Cl-p)u l :::;; !CES(U l, ... , un' p)
<
U l·
U 2 :::;; ...
<
un' It holds
> O. This leads us to
(1)
Therefore, we have limp.-oo!CES(U l, ... , un' p) = min {U l , ... , un}.
D
This property shows that the CBS function at p = (X) is equal to the Rawlsian social
welfare function. By Property 1, ICON never converges to the Rawlsian social welfare function as p ~
It is, however, widely admitted by many authors that the CON function with large p represents the Rawlsian social welfare function (e.g., Atkinson and
Stiglitz (1980), Tuomala (1984), Mas-Colell, Whinston and Green (1994, p. 828)). The
(x).
22
JUN IRITANI AND TOSHIJI MIYAKAWA
rationale comes from the following property.
Property 3. The CON function fCON(U l , ... , Un' p) is a monotone transformation of the
CESfunction fCES(U l , ... , un' p) for any p.
Proof Let us define the function g by
g
(
y)
~f
-
Y
l-p
1
-a
.
-p
g( y) is a monotone, concave function, since g' (y) = y -p > 0 and g" (y) = ( - p)
y-p-l < O. Using this monotone function g, we can represent the CON function as follows:
fCON(U l , ... , un' p) = g(fCES(U I, ... , Un' p)).
Therefore, the CON function is a monotone transformation of the CES function.
0
By Property 3, solutions to maximization problems of the social welfare function fCES
under some constraint set coincide with solutions to problems of fCON under the same constraint set for any p. This together with Property 2 leads us to the conclusion that a solution to the maximization problem of JCON ( • , p) with large p approximates a solution to
the Rawlsian social welfare maximization problem under the same constraint set. From
the theoretical point of view, there are no problems in using the CON function with
large p for obtaining the Rawlsian social welfare maximum.
However, Property 1 of the CON function may cause a problem in the numerical simulation of a social welfare maximization, for example, the optimal income tax simulation
(Stern (1976), Tuomala (1984)). We give a simple example in order to show that the
CON function is not necessarily appropriate for deriving the Rawlsian social welfare
maximum.
Suppose that an economy consists of two persons. We consider a problem for finding
an income distribution that maximizes the value of the social welfare function fCON or
fCES with a given total income i. We denote the income of person i by Ii' i = 1,2. A feasible distribution is a pair (II' 12 ) satisfying 11 +12 = i. For the sake of simplicity, we assume that both persons have the same utility function u i = (Ii )o.5, i = 1, 2, and the total income is i = 100.
We know that the solution to this problem is (11,12 ) = (50,50), irrespective of the
form of social welfare functions. For numerical simulations, we adopt the following
two procedures to find a solution to the problem.
[Procedure A]
(A. i) We start from a feasible distribution (II' 12 ) = (0.1, 99.9) and obtain the value
of social welfare.
(A. ii) We increase the first person's income by !::t.I = 0.1 and decrease the income of
the second person by !::t.l. We calculate the new value of social welfare.
A NOTE ON THE MONOTONE TRANSFORMATION OF CES SOCIAL WELFARE FUNCTIONS
23
(A. iii) We stop calculating the value of social welfare and regard the present pair
(11' 12 ) as the solution if the ratio of the marginal increment of welfare 11 W /111 to the
value of welfare W is less than 0.05 percent, i.e.,
1 11 W
W ----;;:J < 0.0005.
Otherwise we return to step (A. ii).
[Procedure B]
(B. i) We start from a feasible distribution (11,12) = (0.1,99.9) and obtain the value
of social welfare.
(B. ii) We increase the first person's income by 111 = 0.1 and decrease the income of
the second person by 111. We calculate the new value of social welfare.
(B. iii) We stop calculating the value of social welfare and regard the present pair
(11' 12) as the solution if the marginal increment of welfare 11 W /111 is less than 0.05 percent, i.e.,
I1W
----;;:J < 0.0005.
Otherwise we return to step (B. ii).
The difference between the two procedures lies in the convergence criteria to the solution. Procedures A and B depend on (1/ W) (11 W /111) and (11 W /111) respectively.
First, we calculate approximate solutions for the cases (p=2, 5, 10, or 15) and (a=O,
1, 5, or 10) under Procedure A. They are listed in Table 1:
Table 1. Solutions in Procedure A Applying to JCON
p=2
p=5
p=lO
p=15
a=O
(48.4, 51.6)
(49.9, 50.1)
(50.0, 50.0)
(50.0, 50.0)
a=1
(46.0, 54.0)
(16.0, 84.0)
(5.3,94.7)
(3.4, 96.6)
a=5
(29.2, 70.8)
(9.4,90.6)
(4.0,96.0)
(2.8, 97.2)
a=10
(20.2, 79.8)
(7.5,92.5)
(3.5,96.5)
(2.6, 97.4)
The value of parameter a affects the precision of calculated solutions drastically. Thedifference between the calculated solution and the true solution (50, 50) becomes large as
the value of a increases. In particular, when p is large, i.e., when the social welfare is
deemed to be close to Rawlsian, the distance between them becomes terribly large.
These results show that we had better set a = 0 in this type of situation.
Table 2. Solutions in Procedure A Applying to fCEs
p=2
p=5
p=10
p=15
(48.4, 51.6)
(49.3, 50.7)
(49.6, 50.4)
(49.8, 50.2)
24
JUN IRITANI AND TOSHIJI MIY AKAW A
On the other hand, the results in Table 2 are obtained when we use the CES social welfare function iCES under Procedure A. This case exhibits good results.
Next, let us show the examples obtained under Procedure B. The use of two types of social welfare functions produces the following solutions for p=2, 5, 10, 15 in Table 3.
Table 3. Solutions in Procedure B Applying to fcoN and fcEs
p=2
p=5
p=lO
p=15
CON
(44.2, 55.8)
(10.0, 90.0)
(3.5,96.5)
(2.4,97.6)
CES
(49.5, 50.5)
(49.9, 50.1)
(49.9, 50.1)
(50.0, 50.0)
In this case, much more contrastive results are obtained. We must say that the CON
function is not an appropriate objective function to represent Rawlsian social welfare under Procedure B. On the other hand, the CES function still exhibits a good performance.
We can say from above that we must be very careful when we make use of the CON
function. It is true that there are cases of a = 0 in Procedure A where calculated solutions are very near the real solution. But in all many cases, the calculation under CON
functions fails to reach the real solution. On the other hand, the CES social welfare function exhibits a stable and good performance without exceptions. We must conclude that
the CES function is superior to the CON function as far as numerical calculations are
concerned.
3. Convergence of the preferences
In this section, we shall show that a sequence of the preference relations induced by
iCON' P = 1, 2, ... converges to the Rawlsian preference. The closed convergence topology is a standard topology on the set of preference relations (see Hildenbrand (1974».
And thus, we discuss the problem on the convergence of the preference relations in the
closed convergence topology.
Let us denote two utility profiles by U = CUI' ... , Un) and u' = Cu~, ... , u~). The social preference relations» v and» on the utility profiles is defined by
def
»v
{CU, u') E12!nlicONCU, v)
> iCONCU',
»~ {CU, u') E 12!n I min {UI, "', un}
v)},
> min {u~,
V
= 1,2, ...
... , u~}}.
Thus, the set » v is the social preference relation induced by iCON C., v) and » is the preference relation induced by the Rawlsian social welfare function min {u I , ... , Un}. Note
that the preference relation by iCON C" v) is equal to the preference relation by
iCESC " v) for each v. Thus,
» v= {CU, u') E 12!n licESCu, v) > iCESCu', v)}.
Let Xv be a preference field of iCESCUl! ... , un' v) and let X be a preference field of
A NOTE ON THE MONOTONE TRANSFORMATION OF CES SOCIAL WELFARE FUNCTIONS
25
min {U 1, ••• , Un}. Here, we assume XII = X = 12: for all v. A topological space
(P, ~) is the set of preference relations P endowed with the closed convergence topology ::re. We define
F
def
=
{(x, y) E XXXix ':f y},
def
FII = {(x, y) E XII X XII Ix ':f II y}.
Moreover, let Ls(FJ be the topological limit superior of F/s and Li(FJ be the topological limit inferior of F/s. The following theorem on the convergence of preference relations is available:
Theorem (Hildenbrand (1974, p.96)) The sequence (XII'
in (P,::rJ if and only if Li(FJ = F = Ls(FJ.
>- J
converges to (X,
>-)
Using the above theorem, we can prove the following proposition.
Proposition 1 (Convergence of Preferences) The sequence of the social preference relations (XII' >- J converges to the Rawlsian social preference relations (X, >-) in the
closed convergence topology.
Proof. It is obvious that Li(FJ C Ls(FJ. Th~refore, let us prove Li(FJ =:J F =:J
Ls (FJ. The proof consists of two steps. Step 1 proves Ls (FJ C F, and Step 2 proves
Fe Li(F).
Step 1 Let (u, u') E Ls(FJ. By definition, {v IN((u, u'), E) nFII =1= 0} is an infinite
set for every E > 0, where N((u, u'), E) is an E-neighborhood of (u, u') E 122n.
We can assume (u, u') E 12!n+. For each positive integer v, there exist a positive
integer mil and a pair of utility profiles ( u (mJ, u' (m)) satisfying mil 2 v and
(u(mJ, u'(mJ) E N((u, u'), l/v) nFm • This implies
"
fCES(u(mJ, mJ :s;; fCEs(u'(mJ, mil)
(2)
By the mean value theorem, we have
fCES(u(m), mJ = fCES(u+u(mJ -u, mJ
= fCES(U,
n
fJfCES
mJ+ i~l(u(mJi-ui)~(U[A], mJ
(3)
t
for some A (0
tain
< A < 1), where u [A] ~f u+ A (u(m
8fcES ( [ ]
)
fJu i uA,m ll
ll
)
-u). By definition of fCES' we ob-
= (fCES(U[A], mJ )m"
U[AJ
'
where U[A]i is the i-th element ofu[A] E 12n. Let us define U[A]rnin = min{u[A]klk
= 1, ... , n}. Then, by inequality (1), we have
26
JUN IRITANI AND TOSHIJI MIY AKAWA
(n ) m'/(l-m~)( Uu[}.J[}.J
rnin
i
)m~ <
-
81eEs ( [1J
) <
8u
U II. ,mil i
(U [}.J rnin )m~ <
u [}.J
i
1
_.
This implies that the second term in the right hand side of (3) converges to zero since
the sequence u(mJi converges to u i. Therefore,
if v ~ 00, then leEs(u(mJ, mJ ~ min {u l , "', un}.
(4)
Similarly, we have for the utility profile u ' = (u~, ... , u~) that
if v ~ 00, then leEs(u'(mJ, mJ ~ min {u~, ... , u~}.
(5)
Combining (2) and (4) with (5), we obtain min {u l , ••• , un} < min {u~, "', u~}. This
implies (u, u ' ) E F.
Step 2 Let (u, u ' ) E F. We have min {u l , ... , un} ::;: min {u~, ... , u~}. Let V be a neighborhood of (u, u'). There exists a point (v, v') E V such that Vj ~f min {VI' ... , V n }
• { I
'}
A n d t h ere eXIsts
. a posItive
. . .mteger Vo suc h t h at t h
= mIn
VI;"" V n •
e 'mtersec< V kI def
tion of real intervals [(n)I/O-II)v j , v) n [(n)l/O-II)v~, v~J is empty for all v > vo. By
the inequality (1), we have (v, v') E VnFII for all v > Vo' Thus (u, u' ) E Li(FJ.
This completes the proof.
D
We should stress the asymmetric properties of the CON functions. Proposition 1
shows that the sequence of social preference relations (XII' >- J induced by ICON ( " v)
converges to (X, >-) induced by the Rawlsian social welfare function min { .} in the
standard closed convergence topology. At the same time, the sequence of functions
IcoN ( " v) does not converge to the function min { . }. On the other hand the CES function has favorable properties not only as a limit of functions (Property 2) but also as a
limit of preference relations (Proposition 1).
4. Convergence of solutions
Finally we shall prove that the sequence of the solutions uP of the social welfare maximization problem under IcoN ( " p), p = I, 2, ... converges to the solution of the
Rawlsian problem when the constraint sets are identical.
Let us consider an optimal income tax problem. We define U as a set of feasible utility profiles. We say a utility profile is feasible if there exists an income tax schedule under which a competitive equilibrium yields the utility profile. Here, a choice of an income tax schedule is equivalent to a choice of a feasible utility profile. Therefore, a typical optimal income tax problem with social welfare function ICON ( . , p) is represented by
max leoN(u, p) subject to u E u.
(6)
Let uP be the solution of this problem.
Atkinson (1973), Stem (1976) and Tuomala (1984) have computed the solution
uP E U of (6) and the associating optimal income tax schedule under the calibrated economic model with actual economic data. They have regarded the solution uP at large p
A NOTE ON THE MONOTONE TRANSFORMATION OF CES SOCIAL WELFARE FUNCTIONS
27
as an approximation of the solution u R to the maximization problem under the
Rawlsian social welfare function. But their assertions are justified only if uP at large p
is in a neighborhood of u R • Thus, we must show that the sequence of the solution
uP, p = 1, 2, ... converges to u R in order to justify their calculations. Let us answer this
question in the following.
Assume the set U to be a compact set. 2) Then there exists an upper bound K for all
(u l, ... , un) E U such that 0 < u h < K for any h = 1, ... , n. Let Z ~f II~ = 1[0, KJ be
the Cartesian product of the closed interval [0, KJ. We can establish next lemma.
V
Lemmal. (Continuity at Infinity) Let (U ):= 3 be a sequence which converges to
u* ~f (u;, "', u:), and U E U for all v = 3,4, .... Then it holds that
lim fCES(U v) = fCES(U*, 00).
(6)
V
V
,
v-
00
Proof We can assume u~::;; u~ ::;; ... ::;; u~ for every v, by choosing a suitable
subsequence from (U ):= 3 and renumbering individuals. By (1), we obtain
V
I fCEs ( u V , v) - fCES ( U*, 00) I
< max{lnl/(l-v)u~-u;l,
Inl/(l-v)u;-u~l, lu~-u;l}.
(7)
Since the right hand side of (7) converges to zero as v tends to infinity, we can conclude that the sequence (fCES (U v)):= 3 converges to fCES (u *, 00). Thus, fCES is continuous at J) = 00.
Note that problem (6) can be restated as follows:
maxfCES(U, p) subject to u E u.
(8)
We denote a solution to problem (8) by uP ~f (ui, "', u~). Let u* E U be an accumulation point of (uP),;= 3' and u be an arbitrary profile in U. Then it holds that
fCES(U, p) ::;; fCES(U P, p). Applying Lemma I to two sequences fCES(U, p), p = 3,4, ... ,
and fCES (uP, p), p = 3, 4, ... , leads us to
fCES(u, 00) < fCES(U*, 00) = min {u;, ... , u:}, for all u E U.
Thus, the profile u* is a solution of the Rawlsian social welfare maximization problem.
Let us summarize the above argument as a proposition.
V
,
Proposition 2 (Convergence) Let u * be an accumulation point of a sequence (uP),;= 3'
each of which is the solution of problem (6). For any u E U it holds that
fCES(U, 00) < fCES(U*, 00) = min{u; ... , u:}.
Proposition 2 also implies that the sequence uP of the solution under fCON ( ., p) converges to the solution u R with min { .}. Thus, the procedures used by Atkinson (1973),
Stern (1976), and Tuomala (1984) is mathematically correct in order to derive the
Rawlsian social welfare maximum by using the CON function. But problems still re2)
By this assumption, it is true that some class of non-progressive income tax functions are excluded.
28
JUN IRITANI AND TOSHIJI MIY AKAW A
main in using the CON function for numerical simulations, unless we can obtain the necessary calculation power that we need for approximating the solution.
REFERENCES
Atkinson, A. B. (1973), "How progressive should income tax be?", in M. Parkin and A. R. Nobay (eds.),
Essays in Modern Economics, Longman.
Atkinson, A. B. and J. E. Stiglitz (1980), Lectures on Public Economics, MacGraw-Hill.
Harris, R. G. and J. G. MacKinnon (1979), "Computing optimal tax equilibria", Journal of Public Economics
11, pp. 197-212.
Hildenbrand, W. (1974), Core and Equilibria of Large Economy, Prinston University Press.
Mas-Co1ell, A., Whinston, M. D. and J. R. Green (1994), Microeconomic Theory, Oxford University Press.
Stem, N. H. (1976), "On the specification of models optimum income taxation", Journal of Public Economics
6, pp. 123-162.
Tuomala, M. (1984), "On the optimal income taxation: Some further numerical results", Journal of Public Economics 23, pp. 351-366.