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An axiomatic basis for the three objective social welfare function within a poverty context
89
Economxs Letters 20 (1986) 89-94 North-Holland
AN AXIOMATIC BASIS FOR THE THREE WITHIN A POVERTY CONTEXT Robert
1. BRENT
Fordhum
Un~rw.s/r~. New York, NY 10458. USA
Received
3 June 1985
OBJECTIVE
SOCIAL
This article argues that numbers of uncompencated losers. here equated with number5 accompany distribution and efficiency as a third element in the SWF. No constant weights. can satisfy two simple axioms.
WELFARE
FUNCTION
below the poverty line. should efficiency and distribution SWF
1. Introduction It is standard in applied welfare economics to consider social welfare as consisting of two elements, economic efficiency and income distribution. A recent empirical study by Brent (1976) aiming to reveal the implicit social welfare function (SWF) behind past railway closure decisions in the UK, began with such a presumption. However, what was observed in addition to the two presumed social objectives was a ‘numbers effect’ (N). That is, the number of uncompensated losers from the decisions was a third element in the SWF. One justification therefore for policy analysts to consider a three objective SWF is that, in the railway closure setting, it explained government behavior better than the standard version - see Brent (1984a). In this paper an alternative. theoretical justification for N will be sought. It will be shown that the three objective SWF satisfies some simple axioms from the poverty/inequality literature, while the two objective version does not. The concept that links the numbers effect to the existing theoretical literature is the ‘head count’ (H), the number of persons who are below the poverty level. A person in poverty is not a loser in the sense that their income necessarily has fallen, as is the case with uncompensated losers of public project decisions. Nonetheless, the two are closely related in that an increase in either would have an adverse effect on social welfare, independently of the size of aggregate net benefits (weighted or unweighted). To date the literature on poverty and inequality by Sen (1974a,1976) and Kakwani (1980) has been inconsistent with regard to the head count variable. When discussing welfare of a subset of society (the poor) H plays a role; but when concern is extended to society as a whole to assess inequality, H is omitted. One of the objectives of this paper is to integrate poverty and social welfare indicators when the domain consists of the total population. 2. Remarks and definitions The basic
notation
for this section
is due to Kakwani
’ The main difference
(1980). ’ Let individual
income
x be
between Kakwani’s and Sen’s notation is that Kakwani uses continuous distributions while Sen uses discrete summations. In his book Kakwani converts all of Sen’s theorems to a continuous basis. In this paper we also use the existential (3) and universal (V) quantifiers. and the conditional ( - ) and biconditional ( * ) connectives of mathematical logic.
0165.1765/86/$3.50
‘v’ 1986, Elsevier Science Publishers
B.V. (North-Holland)
considered a random variable with a density of f(x). Mean income p is given by ,/,$_xf(x)dx. 1-1can be interpreted as measuring the efficiency objective in a per capita form. A concern for income distribution can be reflected by a set of weights o(x. x) defined on the vector of incomes x. A two objective (per capita) SWF W is then the weighted average.
With Q-y*) as the share of the total population p that has an income and F( _x*) the share of total income by those below x*, we have ’
below the poverty
level x*.
p* is the mean income of those below the poverty line. F(x*) is called by Sen (1976) the headcount ratio. It equals H(.x*)/p where H(x*) is the number of people below .Y* and H(.u*) is given by i E { 1, 2..
H( s*) := #i.
,p }
and
x, I s*.
(4)
For any two projects (X, Y) which lead to the two respective number of uncompensated losers can be defined as ;E {I, 2 . . . . . p}
N( x, y) := #i,
and
income
distributions
(x. J>), the
.x,
(5)
H and N can be linked as follows. ’ Define the reference project Y* which has an equal distribution of income, i.e., (Vl) .r, =.I,*. If this happens to be set at the poverty level, .I-* = x* and we have i E { 1, 2,. . . p }
N( x. _Y*)= #i, If we disregard
the fact that (6) contains
and
s, < x*.
(6)
a strict inequality
while (4) does not,
H(.Y*)=N(x, x*).
(7)
A three objective SWF I@contains the weighted average term reflecting but includes also N as a separate third effect. That is,
efficiency
17(x. s) is used since this need not be the same as P( x, x) in the two objective ci/ will be assumed to be linear and (8) becomes Ci/=u,N+P
%F(x,
/0
and distribution,
SWF W. For simplicity
x)xf(.x)dx.
(9)
’ Eqs. (2) and (3) are (3.1) and (3.2) respectively of Kakwani
(1980).
The second equality
in (3) follows by definition
of the
total income of the poor. ’ A good way to think accumulated
about
the relation
between
H and
N
is this. The number
in poverty
can be considered
as the
number of losers of a whole series of past investment decisions. If new marginal public projects were to ‘spin
off‘ losers at the same rate in past investments, then the relation between
H and N would be proportionate.
where u,v and /3 are the proportional weights. a, plays two roles. Firstly. it denotes the weight on N relative to /3. Call this component (Y. Secondly, it records how individual elements of N are to be aggregated. Let y signify this consideration. The combined effect will be represented by uN
=
(10)
aY.
An axiomatic justification for (9) if it is to be consistent with the welfare economic tradition of the West, must begin with the democratic ideal. Each person in society must be valued equally. This applies to uncompensated losers as well as any other section of society. Democratic
uxiom (Al).
in a society of p individuals
Each individual
will be given a weight equal to
l/P.
From (Al ). y = l/p
and (9) becomes
=6(x,x)xf(
With H( x*) = pF( x*), and because
Lb’=aF(x*) Since ranking
(11)
x)dx.
of (7), we have
+/3L=C(x. x)xf(x)dx. by I# is the same as ranking
(12) by I@/p, (12) can be transformed
into
$ =$F(**) +kmu(x. x)xf(x)dx. So one can assume terms of
p = 1 with no loss of generality.
bi’=nF(~*)+~~iY~(x, x)xF(x)dx.
3. An axiomatic justification
(13) The three objective
SWF will be discussed
in
(14)
for F#’
The essential difference between (1) and (14) is that F(x*) in ri/ is separate with its own weight (Y. Our justification for # will proceed as follows. If W is the primitive concept then W should be general enough to admit a concern for poverty as a special case. However, when emphasis is given to a most basic weighting procedure, a concern for equity (Axiom A2) would imply a perverse sign for LX,i.e., a violation of a poverty axiom. The weights will be discussed first, followed by a statement of the axioms. Finally, a theorem highlighting the perverse effect will be presented. A recent survey of schools on the determination of distributional weights by Brent (1984b) argued that weights make most sense when (a) they are attached to characteristics of social need (e.g., age or race) rather than just income, and (b) they are discrete rather than continuous for any group that is considered in need. Clearly, being
below x* is an index of social need. Thus, the weights we shall be using in this paper will distinguish those in poverty from those who are not; and the simplest case of discrete weights is to consider constant weights for each group. What is required is some axiomatic basis for integrating poverty into the construction of a SWF where income weighting plays a role. Since social welfare relates to the total population and poverty to a proper subset, poverty can be considered a component of social welfare. Moreover. it is clearly an adverse component. Porjertl, uxiom (A?).
Increases
in any index of poverty
p reduce social welfare:
aW/ap
< 0.
In Sen’s (1974a) construction, the head count ratio was one of three elements in his p index. Two of these. the mean income of the poor and their internal inequality, are notions that need to be extended to the population as a whole. So our P index will depend only on F(s*). which was shown earlier to be equal to H(s*)/p: P = F( X*).
(15)
The second ingredient of our axiomatic construction of a SWF. the income requirement that those with the lower income receive the higher weight.
V(i.
X,>X,-)I’,(X,
j).
.X)>C’,(X.
weighting,
will be the
X).
Axioms A2 and A3 are reasonable to apply to any SWF. It turns out that the two objective SWF cannot accommodate both of them, if for the two groups. 1 those in poverty and 2 those who are not, the weights rl(x, s) are constants I’, and P,. 3 constmt \tseights di’ferentiuting
Theorem. Proo/‘.
Using
the additivity
Let the differential
l’(X, x)=
when l’, IT? when
property
the poor ( (1,c2 ) in W sutisJvit?g A2 und A_<.
of integration,
(1) can be split into
weights on group 1 and 2 be x I x*. x > s*.
Based on (16). the two objective
(17) SWF using these weights becomes
(18) where use has been made of (3).
As 1_1*is always positive,
aW/aF(,x*) -=c0 -
1)2 >
the proof is completed
by recognizing
that, in (18). (19) Q.E.D.
c’,.
An increase in poverty, as specified by (15) lowers welfare if, and only if, the income of the non-poor receive a higher weight than that of the poor. To summarize: a two objective SWF is such that the number of uncompensated losers (in the form of the number of poor) could appear as an independent constituent via an income weighting exercise. But. the weights would violate one basic axiom that should be applied to any SWF. Under these circumstances it is not legitimate therefore to argue that a three objective SWF is merely the two objective version in disguise.
4. Final remarks and interpretations The impossibility theorem developed in section 2 tells us that the inclusion of N is more than just a concern with the distribution of income. What exactly is it that the numbers effect contributes to the SWF? Elsewhere, in Brent (1984a), it has been argued that the three objective SWF is consistent with Bentham’s maxim of ‘the greatest good for the greatest number’. Central to Bentham’s ideas is utilitarianism which requires that utility differences be comparable - see for example Sen (1974b). Sen (1974a,1976), on the other hand. has constructed his poverty and inequality measures on the precept of ordinal level comparability of utilities. f&’ seems to provide a vehicle for both kinds of interpersonal comparability: N shows an awareness of welfare levels and the distributional weights a( X, X) can be based on utility differences. A recent paper by Sen (1982, p. 341) appears to have the three objective SWF in mind when he suggests that W take the form W=A
+(l/q)C”
with
77< 1,
where C is per capita consumption used so widely in applied welfare income in (14), I@ would become
w= aF(x*)
and 7 is the isoelastic income weighting parameter that has been economics. 4 If such a weighting system is employed for mean
(21)
+P(l/q)pV.
Equivalence between (20) and (21) would be complete if A = aF(x*) (and C = 71). Sen regards A as a shift parameter dependent on time. It registers the impact of a rising subsistence level on the SWF. He clearly had poverty in mind with his formulation. f@ could then be legitimately regarded as an extension of his SWF, i.e.. F(x*) could be substituted for x*(t).
References Brent. R.J., 1976. The Minister of Transport’s social welfare (1963-1970) (University of Manchester, Manchester).
function:
A study of the factors
behind
railway
closure decisions
4 For a critique of the isoelastic weighting procedure. and a discussion of alternative frameworks. see Brent (1984b). Sen used (20) in the context of determining the Social Discount Rate. To see how the numbers effect can play a part in the SDR refer to Brent (1984~).
Brent, R.J., 1984a. A three objective social welfare function for cost-benefjt analysis. Applied Economics 16, 369-37X. Brent. R.J.. 1984b. Use of distributional weights in cost-benefit analysis: A survey of schools. Public Finance Quarterly 12, 213-230. Brent, R.J.. 1984~. The form of the social welfare function. the numbers effect and the social discount rate. Discussion paper (Fordham University, New York). Kakwani. N.C.. 19X0, Income inequality and poverty (Oxford University Press for the World Bank. New York). Sen. A.K., 1974a. Informational bases of alternative welfare approaches. Journal of Public Economics 3. 3X7-403. Sen. A.K.. 197413, Rawla versus Bentham: An axiomatic examination of the pure distribution problem, Theory and Decision 4, 301-309. Sen. A.K.. 1976. Poverty: An ordinal approach to measurement, Econometrica 44, 219-231. Sen. A.K.. 1982. Approaches to the choice of discount rates for social benefit-cost analysis. in: R.C. Lind. ed.. Discounting for time and risk in energy policy (Resources for the Future, Baltimore. MD).
Economxs Letters 20 (1986) 89-94 North-Holland
AN AXIOMATIC BASIS FOR THE THREE WITHIN A POVERTY CONTEXT Robert
1. BRENT
Fordhum
Un~rw.s/r~. New York, NY 10458. USA
Received
3 June 1985
OBJECTIVE
SOCIAL
This article argues that numbers of uncompencated losers. here equated with number5 accompany distribution and efficiency as a third element in the SWF. No constant weights. can satisfy two simple axioms.
WELFARE
FUNCTION
below the poverty line. should efficiency and distribution SWF
1. Introduction It is standard in applied welfare economics to consider social welfare as consisting of two elements, economic efficiency and income distribution. A recent empirical study by Brent (1976) aiming to reveal the implicit social welfare function (SWF) behind past railway closure decisions in the UK, began with such a presumption. However, what was observed in addition to the two presumed social objectives was a ‘numbers effect’ (N). That is, the number of uncompensated losers from the decisions was a third element in the SWF. One justification therefore for policy analysts to consider a three objective SWF is that, in the railway closure setting, it explained government behavior better than the standard version - see Brent (1984a). In this paper an alternative. theoretical justification for N will be sought. It will be shown that the three objective SWF satisfies some simple axioms from the poverty/inequality literature, while the two objective version does not. The concept that links the numbers effect to the existing theoretical literature is the ‘head count’ (H), the number of persons who are below the poverty level. A person in poverty is not a loser in the sense that their income necessarily has fallen, as is the case with uncompensated losers of public project decisions. Nonetheless, the two are closely related in that an increase in either would have an adverse effect on social welfare, independently of the size of aggregate net benefits (weighted or unweighted). To date the literature on poverty and inequality by Sen (1974a,1976) and Kakwani (1980) has been inconsistent with regard to the head count variable. When discussing welfare of a subset of society (the poor) H plays a role; but when concern is extended to society as a whole to assess inequality, H is omitted. One of the objectives of this paper is to integrate poverty and social welfare indicators when the domain consists of the total population. 2. Remarks and definitions The basic
notation
for this section
is due to Kakwani
’ The main difference
(1980). ’ Let individual
income
x be
between Kakwani’s and Sen’s notation is that Kakwani uses continuous distributions while Sen uses discrete summations. In his book Kakwani converts all of Sen’s theorems to a continuous basis. In this paper we also use the existential (3) and universal (V) quantifiers. and the conditional ( - ) and biconditional ( * ) connectives of mathematical logic.
0165.1765/86/$3.50
‘v’ 1986, Elsevier Science Publishers
B.V. (North-Holland)
considered a random variable with a density of f(x). Mean income p is given by ,/,$_xf(x)dx. 1-1can be interpreted as measuring the efficiency objective in a per capita form. A concern for income distribution can be reflected by a set of weights o(x. x) defined on the vector of incomes x. A two objective (per capita) SWF W is then the weighted average.
With Q-y*) as the share of the total population p that has an income and F( _x*) the share of total income by those below x*, we have ’
below the poverty
level x*.
p* is the mean income of those below the poverty line. F(x*) is called by Sen (1976) the headcount ratio. It equals H(.x*)/p where H(x*) is the number of people below .Y* and H(.u*) is given by i E { 1, 2..
H( s*) := #i.
,p }
and
x, I s*.
(4)
For any two projects (X, Y) which lead to the two respective number of uncompensated losers can be defined as ;E {I, 2 . . . . . p}
N( x, y) := #i,
and
income
distributions
(x. J>), the
.x,
(5)
H and N can be linked as follows. ’ Define the reference project Y* which has an equal distribution of income, i.e., (Vl) .r, =.I,*. If this happens to be set at the poverty level, .I-* = x* and we have i E { 1, 2,. . . p }
N( x. _Y*)= #i, If we disregard
the fact that (6) contains
and
s, < x*.
(6)
a strict inequality
while (4) does not,
H(.Y*)=N(x, x*).
(7)
A three objective SWF I@contains the weighted average term reflecting but includes also N as a separate third effect. That is,
efficiency
17(x. s) is used since this need not be the same as P( x, x) in the two objective ci/ will be assumed to be linear and (8) becomes Ci/=u,N+P
%F(x,
/0
and distribution,
SWF W. For simplicity
x)xf(.x)dx.
(9)
’ Eqs. (2) and (3) are (3.1) and (3.2) respectively of Kakwani
(1980).
The second equality
in (3) follows by definition
of the
total income of the poor. ’ A good way to think accumulated
about
the relation
between
H and
N
is this. The number
in poverty
can be considered
as the
number of losers of a whole series of past investment decisions. If new marginal public projects were to ‘spin
off‘ losers at the same rate in past investments, then the relation between
H and N would be proportionate.
where u,v and /3 are the proportional weights. a, plays two roles. Firstly. it denotes the weight on N relative to /3. Call this component (Y. Secondly, it records how individual elements of N are to be aggregated. Let y signify this consideration. The combined effect will be represented by uN
=
(10)
aY.
An axiomatic justification for (9) if it is to be consistent with the welfare economic tradition of the West, must begin with the democratic ideal. Each person in society must be valued equally. This applies to uncompensated losers as well as any other section of society. Democratic
uxiom (Al).
in a society of p individuals
Each individual
will be given a weight equal to
l/P.
From (Al ). y = l/p
and (9) becomes
=6(x,x)xf(
With H( x*) = pF( x*), and because
Lb’=aF(x*) Since ranking
(11)
x)dx.
of (7), we have
+/3L=C(x. x)xf(x)dx. by I# is the same as ranking
(12) by I@/p, (12) can be transformed
into
$ =$F(**) +kmu(x. x)xf(x)dx. So one can assume terms of
p = 1 with no loss of generality.
bi’=nF(~*)+~~iY~(x, x)xF(x)dx.
3. An axiomatic justification
(13) The three objective
SWF will be discussed
in
(14)
for F#’
The essential difference between (1) and (14) is that F(x*) in ri/ is separate with its own weight (Y. Our justification for # will proceed as follows. If W is the primitive concept then W should be general enough to admit a concern for poverty as a special case. However, when emphasis is given to a most basic weighting procedure, a concern for equity (Axiom A2) would imply a perverse sign for LX,i.e., a violation of a poverty axiom. The weights will be discussed first, followed by a statement of the axioms. Finally, a theorem highlighting the perverse effect will be presented. A recent survey of schools on the determination of distributional weights by Brent (1984b) argued that weights make most sense when (a) they are attached to characteristics of social need (e.g., age or race) rather than just income, and (b) they are discrete rather than continuous for any group that is considered in need. Clearly, being
below x* is an index of social need. Thus, the weights we shall be using in this paper will distinguish those in poverty from those who are not; and the simplest case of discrete weights is to consider constant weights for each group. What is required is some axiomatic basis for integrating poverty into the construction of a SWF where income weighting plays a role. Since social welfare relates to the total population and poverty to a proper subset, poverty can be considered a component of social welfare. Moreover. it is clearly an adverse component. Porjertl, uxiom (A?).
Increases
in any index of poverty
p reduce social welfare:
aW/ap
< 0.
In Sen’s (1974a) construction, the head count ratio was one of three elements in his p index. Two of these. the mean income of the poor and their internal inequality, are notions that need to be extended to the population as a whole. So our P index will depend only on F(s*). which was shown earlier to be equal to H(s*)/p: P = F( X*).
(15)
The second ingredient of our axiomatic construction of a SWF. the income requirement that those with the lower income receive the higher weight.
V(i.
X,>X,-)I’,(X,
j).
.X)>C’,(X.
weighting,
will be the
X).
Axioms A2 and A3 are reasonable to apply to any SWF. It turns out that the two objective SWF cannot accommodate both of them, if for the two groups. 1 those in poverty and 2 those who are not, the weights rl(x, s) are constants I’, and P,. 3 constmt \tseights di’ferentiuting
Theorem. Proo/‘.
Using
the additivity
Let the differential
l’(X, x)=
when l’, IT? when
property
the poor ( (1,c2 ) in W sutisJvit?g A2 und A_<.
of integration,
(1) can be split into
weights on group 1 and 2 be x I x*. x > s*.
Based on (16). the two objective
(17) SWF using these weights becomes
(18) where use has been made of (3).
As 1_1*is always positive,
aW/aF(,x*) -=c0 -
1)2 >
the proof is completed
by recognizing
that, in (18). (19) Q.E.D.
c’,.
An increase in poverty, as specified by (15) lowers welfare if, and only if, the income of the non-poor receive a higher weight than that of the poor. To summarize: a two objective SWF is such that the number of uncompensated losers (in the form of the number of poor) could appear as an independent constituent via an income weighting exercise. But. the weights would violate one basic axiom that should be applied to any SWF. Under these circumstances it is not legitimate therefore to argue that a three objective SWF is merely the two objective version in disguise.
4. Final remarks and interpretations The impossibility theorem developed in section 2 tells us that the inclusion of N is more than just a concern with the distribution of income. What exactly is it that the numbers effect contributes to the SWF? Elsewhere, in Brent (1984a), it has been argued that the three objective SWF is consistent with Bentham’s maxim of ‘the greatest good for the greatest number’. Central to Bentham’s ideas is utilitarianism which requires that utility differences be comparable - see for example Sen (1974b). Sen (1974a,1976), on the other hand. has constructed his poverty and inequality measures on the precept of ordinal level comparability of utilities. f&’ seems to provide a vehicle for both kinds of interpersonal comparability: N shows an awareness of welfare levels and the distributional weights a( X, X) can be based on utility differences. A recent paper by Sen (1982, p. 341) appears to have the three objective SWF in mind when he suggests that W take the form W=A
+(l/q)C”
with
77< 1,
where C is per capita consumption used so widely in applied welfare income in (14), I@ would become
w= aF(x*)
and 7 is the isoelastic income weighting parameter that has been economics. 4 If such a weighting system is employed for mean
(21)
+P(l/q)pV.
Equivalence between (20) and (21) would be complete if A = aF(x*) (and C = 71). Sen regards A as a shift parameter dependent on time. It registers the impact of a rising subsistence level on the SWF. He clearly had poverty in mind with his formulation. f@ could then be legitimately regarded as an extension of his SWF, i.e.. F(x*) could be substituted for x*(t).
References Brent. R.J., 1976. The Minister of Transport’s social welfare (1963-1970) (University of Manchester, Manchester).
function:
A study of the factors
behind
railway
closure decisions
4 For a critique of the isoelastic weighting procedure. and a discussion of alternative frameworks. see Brent (1984b). Sen used (20) in the context of determining the Social Discount Rate. To see how the numbers effect can play a part in the SDR refer to Brent (1984~).
Brent, R.J., 1984a. A three objective social welfare function for cost-benefjt analysis. Applied Economics 16, 369-37X. Brent. R.J.. 1984b. Use of distributional weights in cost-benefit analysis: A survey of schools. Public Finance Quarterly 12, 213-230. Brent, R.J.. 1984~. The form of the social welfare function. the numbers effect and the social discount rate. Discussion paper (Fordham University, New York). Kakwani. N.C.. 19X0, Income inequality and poverty (Oxford University Press for the World Bank. New York). Sen. A.K., 1974a. Informational bases of alternative welfare approaches. Journal of Public Economics 3. 3X7-403. Sen. A.K.. 197413, Rawla versus Bentham: An axiomatic examination of the pure distribution problem, Theory and Decision 4, 301-309. Sen. A.K.. 1976. Poverty: An ordinal approach to measurement, Econometrica 44, 219-231. Sen. A.K.. 1982. Approaches to the choice of discount rates for social benefit-cost analysis. in: R.C. Lind. ed.. Discounting for time and risk in energy policy (Resources for the Future, Baltimore. MD).