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On the social welfare function and the parameters of income distribution
JOURNAL
OF ECONOMIC
THEORY
5, 317-382 (1972)
On the Social Parameters
Welfare Function and the of income Distribution AMOZ
Department
of
Economics, Cornell
KATS University, Ithaca, New York 14850
Received March 9, 1972
In a paper published recently in this journal Eitan Sheshinski [I] presented the following situation: Consider a vector Z = (II , Z2 ,..., ZJ E Iw” of income distributed among n individuals with Ii > 0 (i = 1, 2,..., n), n > 2 (see footnote I), and let each individual have a utility of income function U,(Z,), Vi : D + R (where Qn is the nonnegative orthant of UP). Let all those utility functions be identical. Let there be an individualistic social welfare function W: BP -+ R expressed as W(U(ZJ, U(Z,),..., U(l,)) = W(U(Z)) where U(ZJ). Consider the following possible indicators w = twzl), mh..., of income distributions:
i = i f zi,
(1)
2=1
-q f lzi-zjl, G(z) = 2n2f i=l j=l
(2)
where (1) is the average income and (2) is the Gini index. We want to express Was wwl~,
w2L
WJ)
= mxzl)>
w2L
WJ)
= w,
G),
such that Z? is strictly increasing in
i
and strictly decreasing in G(Z).
(3)
In his paper Sheshinski gives an example of a function that satisfies (3), namely, w(wl),...,
UK8 = H 15 i U-lbWV~~, &l j-1
1 The case n = 1 is trivial and will be commented upon later.
377 Copyright AI1 rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
W&l/
(4)
378
KATS
for any invertible function U and any arbitrary strictly increasing transformation H. He then asks and leaves open the following two questions: 1. Is the example given in (4) the only admissible form of welfare function that satisfies (3) ? 2. Can one find a set of necessary and/or sufficient conditions for H and U such that (3) will be satisfied ? In this note we intend to answer partially both those questions. Let us first indicate that the following can be an example of a function that satisfies (3): i = 1) 2)...) n, U(ZJ = zi ) WW,..., WZnN (5)
To see that the function H in (5) satisfies condition (3) it is sufficient to observe that it can be reduced to H = logf - G(Z). Moreover, it is not a monotone transformation of the function in (4). To find the set of necessary conditions for a function to satisfy (3) we shall make a series of observations which will be incorporated later into a proposition. Since the example in (4) is for any invertible U, the only possible necessary condition on tJ might be invertibility. As it turns out, this is the case. LEMMA 1. Zf H( iI(Z,
., U(Z,)) satisfies (3), then U is invertible.
Proof. We shall prove by contradiction. Let Z = (Z, , Zzo,Zao,..., Zno) and I’ = (I,‘, Zzo,Z30,..., Z,O) be two income distributions which differ only in the first coordinate and such that ZI > II’ and U(Z,) = U(Z,‘). By definition, H(U(Z))
= H(U(Z’)).
(6)
Also, 1 > I’. Therefore, for (6) to satisfy (3) we have to have G(Z) > G(Z’) for all (Zzo,..., Z,O). In particular, pick min(Z,O,..., Z,O) 3 z, > Zl’ >, 0, which implies G(Z) < G(Z’) and the lemma follows by the established Q.E.D. contradiction. This result does not, of course, justify, but, at least, can add one more argument for the use of strictly increasing utility-of-income functions. Also it states that if U is increasing, then there is no satiation point.
379
THE SOCIAL WELFARE FUNCTION
Let us now restrict ourselves to a discussion of the function HK ,..., In> = H(Z) instead of H(U(Z)) (i.e., let us assume that U(ZJ = Zi for all i = 1, 2,..., n). We shall return later to the more general case. 1. Let Z, I’ E fP then Z is symmetric of the coordinates of (I,‘,..., I,‘).
DEFINITION
is a permutation
to I’ iff (Z, ,..., Z,J
LEMMA 2. Zf Z is symmetric to Z’ and H( .) sutisjies (3) then H(Z) = H(Z’). In this case, we shall say that “H is symmetric in I.”
The proof is straightforward Z = f’ and G(Z) = G(Z).
by noticing that Z symmetric to Z’ implies
LEMMA 3. For Z E @, G(Z) = G(U) for all h > 0. And Z symmetric to I’ impIies H(M) = H(XZ’) for all h > 0.
The proof is immediate. COROLLARY 1. Zf H(e) satisfies condition (3), then it is strictly increasing along rays from the origin.
The proof follows from the fact that f is strictly increasing along rays from the origin. Since the Gini index is fixed for all income distributions characterized by vectors along a given ray from the origin, we shall distinguish among rays, and therefore (as will be shown) among different Gini indices, by comparing their projections on a given simplex. In order to do so, we will make the following two definitions: DEFINITION 2. Let ei(r) = (0, 0 ,..., 0, r, 0 ,..., 0) be a vector in 9” whose components are all zero, except for the i-th component, ifs{I, 2,..., n> whose value is r, r > 0. Then the (n-dimensional) r simplex is
I
I: Z = i
DEFINITION LEMMA
clliei(r), f 01~= 1 and q 3 0 for all i = 1, 2 ,..., n
i=l
i-l
3. Q(r) = (r/n, r/n ,..., r/n)
r > 0.
4. Z and I’ are on the r-simplex for some r > 0 ly
i = i’.
The proof is immediate. In order to compare the Gini indices of vectors on the same r-simplex we shall use the fact that Z symmetric to I’ implies G(Z) = 41’). We shall,
380
KATS
therefore, restrict ourselves, without vectors that satisfy the following:
loss of generality,
z1 I>I 2 I>I 3 ,,‘.‘, > LEMMA
1>I,.
only to those (7)
5. Zf Isatisfies (7), then G(Z) = -$ i
(n - 2i + 1) Zi .
a=1
Proof.
Assuming that (7) is satisfied, we can write
-fl g 1zi - zj I = 2 ii Ii (4 - zJ i=l j=i Q.E.D.
and (8) follows.
The reader might notice that Eq. (8) gives the Gini index of an income distribution not in terms of absolute differences between the incomes of any two individuals but as a weighted sum of the income of each individual (when these are ordered from the greatest to the smallest), and the weights depend only on n, the number of individuals. Notice also that the weights sum up to zero, i.e., t (n - 2i + 1) = 0. i=l
It follows directly that G(Q(r)) = 0. We can now state 6. Zf H(a) satisfies condition (3) then for Z, I’ on an r-simplex b HV’) ifs
LEMMA
WI
i (n - 2i + l>(fi - Ii’> < 0, i=l
where 1, p are those permutations condition (7).
of Z and I’, respectively, that satisji
The proof follows from lemma 5. At this point we can specify another property of H. LEMMA
convex.
7.
On a given r-simplex the Gini-lower-contour
sets are strictly
381
THE SOCIAL WELFARE FUNCTION
Proof. Let Z, I’ be on the same r-simplex with G(Z) -=cG(Z). Let I” = hZ + (1 - X) I’ for 0 -C h < 1. Without loss of generality, assume that Z, I’ satisfy condition (7). It follows that I” also satisfies this condition. G(Z”) - G(Z’) = i
(n - 2i + l)(Zt” - Zi’)
i=l
= il (n - 2i + l)(Ui
+ (1 - A) Zi’ - li’)
= il (n - 2i + I) A(z~ - Ii’) = h(G(Z) - W’))
< 0. Q.E.D.
We can combine now the previous results into PROPOSITION 1. Let H(Z): Qn + R, then H(.) satisfies condition ifs the following hold simultaneously:
(3)
1. H is symmetric. 2. H is strictly increasing along rays from the origin. 3. Zf Z, I’ are on an r-simplex then H(Z) > H(Z’) QT tl (n - 2i + l)Cfi - Ii’) < 0 Proof: condition
.for all
r 3 0.
The necessity has been shown above. Sufficiency follows from (3) and the observations in Lemmas 3-6. Q.E.D.
At this point it might be of some interest to consider the following example: U(Zi) z Zi i = 1,2, (9) WW,), U(h)) = ZJs . Notice that this example satisfies all the conditions in Proposition 1 except for the fact that His not strictly increasing along rays that are not interior to Gn (see footnote 2). In other words, the social welfare does not change, and in fact remains in its lowest level, as long as there is at least one individual in the society that has no income whatsoever. To resolve this incompatibility between condition (3) and example (9) one can either restrict example (9) to all income distributions Z > 0 (i.e., a A ray j3 from the origin is interior to 52” iff for every I0 E 8, I0 # 0, there exists c > 0 such that {Z : 11Z - I0 11< C} C R”.
382
KATS
Zi > 0 for i = 1,2) or redefine condition (3) in such a way that it will take into account not only the Gini index per se but also will assume a minimum level of income to all members of society. If we restrict ourselves to only strictly increasing utility functions U (and as was indicated above there is enough reason, both economically and mathematically, to do so) then by reproducing the same analysis carried out previously one can state PROPOSITION 2. Let H(U(Z,), U(Z,),..., U(Z,)): EP -+ Iw, then H(.) satisfies condition (3) if the following hold simultaneously:
1. H is symmetric in U, i.e., Zsymmetric to I’ =>H( U(Z)) = H( U(Z’)). 2. H is strictly increasing along rays from the origin. 3. Zf U(Z), U(Z’) are on an r-simplex, then H( U(Z)) 3 H(U(Z’)) 23 i
(n - 2i + l)(U(Ii)
- U(Ii’))
< 0
for all
r 3 0.
i=l
Notice that the assumption that U is increasing implies that His defined on a domain which is bounded from below. Remark. Notice that for n = 1 we get Z-Z= H(U(Z,)), G(Z,) = 0 for all ZI >, 0 and H will satisfy condition (3) iff H is strictly increasing in Z, .
One can ask also if any of the conditions or the results will have to be modified if we restrict ourselves to the class of strictly concave utility functions U. By Lemma 1 this class is reduced to either monotone increasing or monotone decreasing functions. Proposition 2 gives the conditions for monotone increasing functions. Conditions 2 and 3 in that proposition will have to be reversed if U is monotone decreasing. ACKNOWLEDGMENT I wish to thank Henry Wan and Ehud Kalai for their valuable comments. REFERENCE 1. E. SHESHINSKI, Relation between a social welfare function and the Gini index of income inequality, J. Icon. Theory 4 (1972), 98-100.
OF ECONOMIC
THEORY
5, 317-382 (1972)
On the Social Parameters
Welfare Function and the of income Distribution AMOZ
Department
of
Economics, Cornell
KATS University, Ithaca, New York 14850
Received March 9, 1972
In a paper published recently in this journal Eitan Sheshinski [I] presented the following situation: Consider a vector Z = (II , Z2 ,..., ZJ E Iw” of income distributed among n individuals with Ii > 0 (i = 1, 2,..., n), n > 2 (see footnote I), and let each individual have a utility of income function U,(Z,), Vi : D + R (where Qn is the nonnegative orthant of UP). Let all those utility functions be identical. Let there be an individualistic social welfare function W: BP -+ R expressed as W(U(ZJ, U(Z,),..., U(l,)) = W(U(Z)) where U(ZJ). Consider the following possible indicators w = twzl), mh..., of income distributions:
i = i f zi,
(1)
2=1
-q f lzi-zjl, G(z) = 2n2f i=l j=l
(2)
where (1) is the average income and (2) is the Gini index. We want to express Was wwl~,
w2L
WJ)
= mxzl)>
w2L
WJ)
= w,
G),
such that Z? is strictly increasing in
i
and strictly decreasing in G(Z).
(3)
In his paper Sheshinski gives an example of a function that satisfies (3), namely, w(wl),...,
UK8 = H 15 i U-lbWV~~, &l j-1
1 The case n = 1 is trivial and will be commented upon later.
377 Copyright AI1 rights
0 1972 by Academic Press, Inc. of reproduction in any form reserved.
W&l/
(4)
378
KATS
for any invertible function U and any arbitrary strictly increasing transformation H. He then asks and leaves open the following two questions: 1. Is the example given in (4) the only admissible form of welfare function that satisfies (3) ? 2. Can one find a set of necessary and/or sufficient conditions for H and U such that (3) will be satisfied ? In this note we intend to answer partially both those questions. Let us first indicate that the following can be an example of a function that satisfies (3): i = 1) 2)...) n, U(ZJ = zi ) WW,..., WZnN (5)
To see that the function H in (5) satisfies condition (3) it is sufficient to observe that it can be reduced to H = logf - G(Z). Moreover, it is not a monotone transformation of the function in (4). To find the set of necessary conditions for a function to satisfy (3) we shall make a series of observations which will be incorporated later into a proposition. Since the example in (4) is for any invertible U, the only possible necessary condition on tJ might be invertibility. As it turns out, this is the case. LEMMA 1. Zf H( iI(Z,
., U(Z,)) satisfies (3), then U is invertible.
Proof. We shall prove by contradiction. Let Z = (Z, , Zzo,Zao,..., Zno) and I’ = (I,‘, Zzo,Z30,..., Z,O) be two income distributions which differ only in the first coordinate and such that ZI > II’ and U(Z,) = U(Z,‘). By definition, H(U(Z))
= H(U(Z’)).
(6)
Also, 1 > I’. Therefore, for (6) to satisfy (3) we have to have G(Z) > G(Z’) for all (Zzo,..., Z,O). In particular, pick min(Z,O,..., Z,O) 3 z, > Zl’ >, 0, which implies G(Z) < G(Z’) and the lemma follows by the established Q.E.D. contradiction. This result does not, of course, justify, but, at least, can add one more argument for the use of strictly increasing utility-of-income functions. Also it states that if U is increasing, then there is no satiation point.
379
THE SOCIAL WELFARE FUNCTION
Let us now restrict ourselves to a discussion of the function HK ,..., In> = H(Z) instead of H(U(Z)) (i.e., let us assume that U(ZJ = Zi for all i = 1, 2,..., n). We shall return later to the more general case. 1. Let Z, I’ E fP then Z is symmetric of the coordinates of (I,‘,..., I,‘).
DEFINITION
is a permutation
to I’ iff (Z, ,..., Z,J
LEMMA 2. Zf Z is symmetric to Z’ and H( .) sutisjies (3) then H(Z) = H(Z’). In this case, we shall say that “H is symmetric in I.”
The proof is straightforward Z = f’ and G(Z) = G(Z).
by noticing that Z symmetric to Z’ implies
LEMMA 3. For Z E @, G(Z) = G(U) for all h > 0. And Z symmetric to I’ impIies H(M) = H(XZ’) for all h > 0.
The proof is immediate. COROLLARY 1. Zf H(e) satisfies condition (3), then it is strictly increasing along rays from the origin.
The proof follows from the fact that f is strictly increasing along rays from the origin. Since the Gini index is fixed for all income distributions characterized by vectors along a given ray from the origin, we shall distinguish among rays, and therefore (as will be shown) among different Gini indices, by comparing their projections on a given simplex. In order to do so, we will make the following two definitions: DEFINITION 2. Let ei(r) = (0, 0 ,..., 0, r, 0 ,..., 0) be a vector in 9” whose components are all zero, except for the i-th component, ifs{I, 2,..., n> whose value is r, r > 0. Then the (n-dimensional) r simplex is
I
I: Z = i
DEFINITION LEMMA
clliei(r), f 01~= 1 and q 3 0 for all i = 1, 2 ,..., n
i=l
i-l
3. Q(r) = (r/n, r/n ,..., r/n)
r > 0.
4. Z and I’ are on the r-simplex for some r > 0 ly
i = i’.
The proof is immediate. In order to compare the Gini indices of vectors on the same r-simplex we shall use the fact that Z symmetric to I’ implies G(Z) = 41’). We shall,
380
KATS
therefore, restrict ourselves, without vectors that satisfy the following:
loss of generality,
z1 I>I 2 I>I 3 ,,‘.‘, > LEMMA
1>I,.
only to those (7)
5. Zf Isatisfies (7), then G(Z) = -$ i
(n - 2i + 1) Zi .
a=1
Proof.
Assuming that (7) is satisfied, we can write
-fl g 1zi - zj I = 2 ii Ii (4 - zJ i=l j=i Q.E.D.
and (8) follows.
The reader might notice that Eq. (8) gives the Gini index of an income distribution not in terms of absolute differences between the incomes of any two individuals but as a weighted sum of the income of each individual (when these are ordered from the greatest to the smallest), and the weights depend only on n, the number of individuals. Notice also that the weights sum up to zero, i.e., t (n - 2i + 1) = 0. i=l
It follows directly that G(Q(r)) = 0. We can now state 6. Zf H(a) satisfies condition (3) then for Z, I’ on an r-simplex b HV’) ifs
LEMMA
WI
i (n - 2i + l>(fi - Ii’> < 0, i=l
where 1, p are those permutations condition (7).
of Z and I’, respectively, that satisji
The proof follows from lemma 5. At this point we can specify another property of H. LEMMA
convex.
7.
On a given r-simplex the Gini-lower-contour
sets are strictly
381
THE SOCIAL WELFARE FUNCTION
Proof. Let Z, I’ be on the same r-simplex with G(Z) -=cG(Z). Let I” = hZ + (1 - X) I’ for 0 -C h < 1. Without loss of generality, assume that Z, I’ satisfy condition (7). It follows that I” also satisfies this condition. G(Z”) - G(Z’) = i
(n - 2i + l)(Zt” - Zi’)
i=l
= il (n - 2i + l)(Ui
+ (1 - A) Zi’ - li’)
= il (n - 2i + I) A(z~ - Ii’) = h(G(Z) - W’))
< 0. Q.E.D.
We can combine now the previous results into PROPOSITION 1. Let H(Z): Qn + R, then H(.) satisfies condition ifs the following hold simultaneously:
(3)
1. H is symmetric. 2. H is strictly increasing along rays from the origin. 3. Zf Z, I’ are on an r-simplex then H(Z) > H(Z’) QT tl (n - 2i + l)Cfi - Ii’) < 0 Proof: condition
.for all
r 3 0.
The necessity has been shown above. Sufficiency follows from (3) and the observations in Lemmas 3-6. Q.E.D.
At this point it might be of some interest to consider the following example: U(Zi) z Zi i = 1,2, (9) WW,), U(h)) = ZJs . Notice that this example satisfies all the conditions in Proposition 1 except for the fact that His not strictly increasing along rays that are not interior to Gn (see footnote 2). In other words, the social welfare does not change, and in fact remains in its lowest level, as long as there is at least one individual in the society that has no income whatsoever. To resolve this incompatibility between condition (3) and example (9) one can either restrict example (9) to all income distributions Z > 0 (i.e., a A ray j3 from the origin is interior to 52” iff for every I0 E 8, I0 # 0, there exists c > 0 such that {Z : 11Z - I0 11< C} C R”.
382
KATS
Zi > 0 for i = 1,2) or redefine condition (3) in such a way that it will take into account not only the Gini index per se but also will assume a minimum level of income to all members of society. If we restrict ourselves to only strictly increasing utility functions U (and as was indicated above there is enough reason, both economically and mathematically, to do so) then by reproducing the same analysis carried out previously one can state PROPOSITION 2. Let H(U(Z,), U(Z,),..., U(Z,)): EP -+ Iw, then H(.) satisfies condition (3) if the following hold simultaneously:
1. H is symmetric in U, i.e., Zsymmetric to I’ =>H( U(Z)) = H( U(Z’)). 2. H is strictly increasing along rays from the origin. 3. Zf U(Z), U(Z’) are on an r-simplex, then H( U(Z)) 3 H(U(Z’)) 23 i
(n - 2i + l)(U(Ii)
- U(Ii’))
< 0
for all
r 3 0.
i=l
Notice that the assumption that U is increasing implies that His defined on a domain which is bounded from below. Remark. Notice that for n = 1 we get Z-Z= H(U(Z,)), G(Z,) = 0 for all ZI >, 0 and H will satisfy condition (3) iff H is strictly increasing in Z, .
One can ask also if any of the conditions or the results will have to be modified if we restrict ourselves to the class of strictly concave utility functions U. By Lemma 1 this class is reduced to either monotone increasing or monotone decreasing functions. Proposition 2 gives the conditions for monotone increasing functions. Conditions 2 and 3 in that proposition will have to be reversed if U is monotone decreasing. ACKNOWLEDGMENT I wish to thank Henry Wan and Ehud Kalai for their valuable comments. REFERENCE 1. E. SHESHINSKI, Relation between a social welfare function and the Gini index of income inequality, J. Icon. Theory 4 (1972), 98-100.