On income and utilty Transfer Principles when households differ in needs

On Income and Utility Transfers Principles
when Households differ in Needs
Marc Dubois and Stéphane Mussard
LAMETA
Université de Montpellier
July 14th, 2015
Marc Dubois and Stéphane Mussard LAMETA Université de Montpellier
On Income and Utility Transfers Principles when Households differ in Needs

The aim of this paper
• Question: What the necessary conditions are to fulﬁll
whether the Pigou-Dalton Income Transfers Principle or the
Pigou-Dalton Income Transfers Principle in an additively
separable framework?
• First step: The additively separable SWF with homogeneous
individuals.
• Second step: The additively separable SWF with
heterogeneous households.
• Finish line: Whether in the homogeneous case or in the
heterogeneous case, inequality aversion is not a necessary
condition for the SWF to fulﬁll the Pigou-Dalton Income
Transfers Principle.

The starting point
• Adler and Treich (2014), Kaplow (2010) present an
extended-form of the SWF:
Deﬁnition
SW(Y) :=
ymax
0
g (u (y)) f(y)dy. (1)
Let g be a weighting function of utilities.
u(y) represents the individuals’ (identical) utility function
with income y; utility is positive.
Let g and u be strictly increasing but g is not supposed
concave.

Utility Transfers Principle of order 2
Following the Fishburn and Willig’s (1984) deﬁinition of transfer:
Deﬁnition
Utility Transfer of order 2. For all f ∈ F and for all
y ∈ [0, ymax], an utility mean-preserving transfer of order 2 is
given by:
T2
(α, u(y), δ) := T1
(α, u(y), δ)−T1
u (α, u(y)+δ, δ), s ∈ N+, (2)
such that,
0 < α f(y) at point u(y) + 2δ, and δ > 0.

Attitude towards inequality
Deﬁnition
Utility Transfers Principle of order 2. For all f ∈ F, for all
y ∈ [0, ymax] and s ∈ N; we have the following implication for the
utility transfer of order 2:
h = f + T2
(α, u(y), δ) =⇒ SW(h) SW(f) (3)
• The Pigou-Dalton Transfers Principle of utility characterizes
inequality aversion. The Prioritarian SWF respects such a
principle if and only if g is concave.

Income Transfers Principle of order 2
Deﬁnition
Income Transfer of order 2. For all f ∈ F and for all
y ∈ [0, ymax], an income mean-preserving transfer of order 2 is
given by:
T2
(α, y, δ) := T1
(α, y, δ) − T1
(α, y + δ, δ), s ∈ N+, (4)
such that,
0 < α f(y + 2δ), δ > 0.

Deﬁnition
Income Transfers Principle of order 2. For all f ∈ F, for all
y ∈ [0, ymax] and s ∈ N; we have the following implication for the
income transfer of order 2:
h = f + T2
(α, u(y), δ) =⇒ SW(h) SW(f) (5)
• The SWF respects such a principle if and only if g ◦ u is
concave.

The normative content of the Pigou-Dalton Income
Transfers Principle
Inequality aversion is not necessary for SW to fulﬁll the
Pigou-Dalton Income Transfers Principle.
Theorem
Let u be increasing and concave over incomes; and g be
increasing over utilities. Then, the following statements are
equivalent:
(i) g(2) ≤ g
(2)
+ with g
(2)
+ := −g(1)u(2)
[u(1)]2 > 0,
(ii) SW fulﬁlls the Pigou-Dalton Income Transfers Principle.

The new starting point
• Let consider now housholds that differ in needs rather than
identical individuals, the SWF becomes:
Deﬁnition
W(f) =
H
h=1
ymax
0
g(uh(y)) f(y, h) dy. (H1)
Let g be a weighting function of utilities.
uh(y) represents the utility function of household of type h
with income y; utility is positive.
Let g and u be strictly increasing but g is not supposed
concave.

From Blackorby, Bossert and Donaldson (2002, Theorem 8),
the following statement holds.
Theorem
The condition (H1) with g being continuous and increasing yield
information invariance with respect to Φ = (φ1, . . . , φH) if, and
only if, for each φh ∈ Φ there exists real bh and a ∈ R++, for all
ξ ∈ R and h = 1, . . . , H:
φh(ξ) = g−1
(ag(ξ) + bh) .

Informational Basis
D’Aspremont and Gevers (1977, Theorem 3) and Blackorby et
al. (2002, Theorems 9 and 10) demonstrate that at least as
strong axioms as the well-known Cardinal Full Comparability
(CFC) of utility allow for inter-households comparability of
transformed utility only if utility is not transformed!

Informational Basis
Hence we have to propose Cardinal Ratio Comparability (CRC)
that is a weaker axiom than (CFC).
Deﬁnition
Cardinal Ratio Comparability For all utility proﬁles
U, V ∈ U H, U is informationally equivalent to V with respect to
(CRC) if, and only if there exist vh(y) = ag(uh(y)) for all
h = 1, · · · , H such that a ∈ R++, for all y ∈ [0, ymax].

Informational Basis
Following Roberts (1980, p. 432), the following statement holds.
Theorem
If, (CRC) holds, then condition (H1) with respect to
Φ = (φ1, . . . , φH) is:
Wρ
A(f) =
H
h=1
ymax
0
[uh(y)]1−ρ
1 − ρ
f(y, h) dy with ρ = 1; (6)
when ρ = 1, W1
A(f) = H
h=1
ymax
0 log uh(y) f(y, h) dy
Clearly the moral value g(uh(y)) = [uh(y)]1−ρ
1−ρ whenever ρ = 1.

Heterogeneity conditions
Households are supposed to be ranked increasingly with
respect to a non-income welfare attribute, the so-called needs.
Deﬁnition
– Households needs – For all ∈ {1, 2} and y ∈ [0, ymax]:
(−1) u
( )
1 (y) (−1) u
( )
2 (y) · · · (−1) u
( )
K (y) < 0 ; (H2a)
u1(y) u2(y) · · · uK (ymax). (H2b)
Those conditions follow the idea of Moyes (2012) but they are
not similar to his C1 − C5 conditions.

Heterogeneity conditions
If one translates the Moyes’ framework into ours, then the
C1 − C5 conditions would be as follows.
Deﬁnition
– Households ranking in Moyes (2012) – For all ∈ {1, 2}
and y ∈ [0, ymax]:
(−1) g ◦u
( )
1 (y) (−1) g ◦u
( )
2 (y) · · · (−1) g ◦u
( )
K (y) 0 ;
(H’2a)
g ◦ u1(y) g ◦ u2(y) · · · g ◦ uK (ymax). (H’2b)

Utility Transfer of order 2
Deﬁnition
Between-Type Utility Transfer of order 2. For all f ∈ F and
for all y ∈ [0, ymax], given two types of households q and r such
that q < r; a utility mean-preserving transfer of order 2 is given
by:
T2
(α, u{q,r}(y), δ) := T1
(α, uq(y), δ) − T1
(α, ur (y) + δ, δ), (7)
such that,
0 < α f(y, r), at point ur (y) + 2δ, δ > 0.

Between-Type Utility Transfers Principle of order 2
From this deﬁnition, the Transfers Principle can be stated as
follows.
Deﬁnition
Between-Type Utility Transfers Principle of order 2. For all
f ∈ F, for all y ∈ [0, ymax] and s ∈ N, given two types of
households q and r such that q < r; we have the following
implication for the utility transfer of order 2:
h = f + T2
(α, u{q,r}(y), δ) =⇒ W(h) W(f) (8)

Income Transfer of order 2
Deﬁnition
Between-Type Income Transfer of order 2. For all f ∈ F and
for all y ∈ [0, ymax], given two types of households q and r such
that q < r; an income mean-preserving transfer of order 2 is
given by:
T2
(α, y, δ, q, r) := T1
(α, y, δ, q) − T1
(α, y + δ, δ, r), s ∈ N+, (9)
such that,
0 < α f(y + 2δ, r)], δ > 0.

Deﬁnition
Between-Type Income Transfers Principle of order 2. For all
f ∈ F, for all y ∈ [0, ymax] and s ∈ N, given two types of
households q and r such that q < r; we have the following
implication for the income transfer of order 2:
h = f + T2
(α, y, δ, q, r) =⇒ W(h) W(f) (10)

The normative content of the Pigou-Dalton Utility
Transfers Principle
Proposition
Given that (H2a) and (H2b) hold. If Wρ
A fulﬁlls the Utility
Transfers Principle of order 2, then ρ 0.
This statement characterizes Prioritarianism. In this case, Wρ
A
are the Atkinson SWFs.

The normative content of the Pigou-Dalton Income
Transfers Principle
Proposition
Given that (H2a) and (H2b) hold. If Wρ
A fulﬁlls the Income
Transfers Principle of order 2, then ρ
log
u
(1)
r (y+δ)
u
(1)
q (y)
log
ur (y+δ)
uq(y)
=: ρ 2.
Remark: a negative numerator and a positive denominator :
inequality aversion is not a necessary condition for Wρ
A to fulﬁll
the Transfers Principle of order 2.

Thanks for your attention.

On income and utilty Transfer Principles when households differ in needs

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