On income and utilty Transfer Principles when households differ in needs

On Income and Utility Transfers Principleswhen Households differ in Needs

Marc Dubois and Stéphane MussardLAMETA

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 fulfillwhether the Pigou-Dalton Income Transfers Principle or thePigou-Dalton Income Transfers Principle in an additivelyseparable framework?

• First step: The additively separable SWF with homogeneousindividuals.

• Second step: The additively separable SWF withheterogeneous households.

• Finish line: Whether in the homogeneous case or in theheterogeneous case, inequality aversion is not a necessarycondition for the SWF to fulfill the Pigou-Dalton IncomeTransfers Principle.



The starting point

• Adler and Treich (2014), Kaplow (2010) present anextended-form of the SWF:

Definition

SW (Y ) :=

∫ ymax

0g (u (y)) f (y)dy . (1)

Let g be a weighting function of utilities.

u(y) represents the individuals’ (identical) utility functionwith income y ; utility is positive.

Let g and u be strictly increasing but g is not supposedconcave.



Utility Transfers Principle of order 2

Following the Fishburn and Willig’s (1984) defiinition of transfer:

DefinitionUtility Transfer of order 2. For all f ∈ F and for ally ∈ [0, ymax], an utility mean-preserving transfer of order 2 isgiven by:

T 2(α,u(y), δ) := T 1(α,u(y), δ)−T 1u (α,u(y)+δ, δ), s ∈ N+, (2)

such that,

0 < α 6 f (y) at point u(y) + 2δ, and δ > 0.



Attitude towards inequality

DefinitionUtility Transfers Principle of order 2. For all f ∈ F , for all

y ∈ [0, ymax] and s ∈ N; we have the following implication for theutility transfer of order 2:

h = f + T 2(α,u(y), δ) =⇒ SW (h) > SW (f ) (3)

• The Pigou-Dalton Transfers Principle of utility characterizesinequality aversion. The Prioritarian SWF respects such aprinciple if and only if g is concave.



Income Transfers Principle of order 2

DefinitionIncome Transfer of order 2. For all f ∈ F and for ally ∈ [0, ymax], an income mean-preserving transfer of order 2 isgiven by:

T 2(α, y , δ) := T 1(α, y , δ)− T 1(α, y + δ, δ), s ∈ N+, (4)

such that,0 < α 6 f (y + 2δ), δ > 0.




DefinitionIncome Transfers Principle of order 2. For all f ∈ F , for all

y ∈ [0, ymax] and s ∈ N; we have the following implication for theincome transfer of order 2:

h = f + T 2(α,u(y), δ) =⇒ SW (h) > SW (f ) (5)

• The SWF respects such a principle if and only if g ◦ u isconcave.



The normative content of the Pigou-Dalton IncomeTransfers Principle

Inequality aversion is not necessary for SW to fulfill thePigou-Dalton Income Transfers Principle.

Theorem

Let u be increasing and concave over incomes; and g beincreasing over utilities. Then, the following statements areequivalent:

(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 thanidentical individuals, the SWF becomes:

Definition

W (f ) =H∑

h=1

∫ ymax

0g(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 hwith income y ; utility is positive.

Let g and u be strictly increasing but g is not supposedconcave.



From Blackorby, Bossert and Donaldson (2002, Theorem 8),the following statement holds.

Theorem

The condition (H1) with g being continuous and increasing yieldinformation invariance with respect to Φ = (φ1, . . . , φH) if, andonly if, for each φh ∈ Φ there exists real bh and a ∈ R++, for allξ ∈ R and h = 1, . . . ,H:

φh(ξ) = g−1 (ag(ξ) + bh) .



Informational Basis

D’Aspremont and Gevers (1977, Theorem 3) and Blackorby etal. (2002, Theorems 9 and 10) demonstrate that at least asstrong axioms as the well-known Cardinal Full Comparability(CFC) of utility allow for inter-households comparability oftransformed 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).

DefinitionCardinal Ratio Comparability For all utility profilesU,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 allh = 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 toΦ = (φ1, . . . , φH) is:

W ρA(f ) =

H∑h=1

∫ ymax

0

[uh(y)]1−ρ

1− ρf (y ,h) dy with ρ 6= 1; (6)

when ρ = 1, W 1A(f ) =

∑Hh=1

∫ ymax0 log uh(y) f (y ,h) dy

Clearly the moral value g(uh(y)) = [uh(y)]1−ρ

1−ρ whenever ρ 6= 1.



Heterogeneity conditions

Households are supposed to be ranked increasingly withrespect to a non-income welfare attribute, the so-called needs.

Definition– Households needs – For all ` ∈ {1,2} and y ∈ [0, ymax]:

(−1)ù(`)1 (y) 6 (−1)ù(`)

2 (y) 6 · · · 6 (−1)ù(`)K (y) < 0 ; (H2a)

u1(y) 6 u2(y) 6 · · · 6 uK (ymax). (H2b)

Those conditions follow the idea of Moyes (2012) but they arenot similar to his C1− C5 conditions.



Heterogeneity conditions

If one translates the Moyes’ framework into ours, then theC1− 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) 6 (−1)`g ◦u(`)

2 (y) 6 · · · 6 (−1)`g ◦u(`)K (y) 6 0 ;

(H’2a)g ◦ u1(y) 6 g ◦ u2(y) 6 · · · 6 g ◦ uK (ymax). (H’2b)



Utility Transfer of order 2

DefinitionBetween-Type Utility Transfer of order 2. For all f ∈ F andfor all y ∈ [0, ymax], given two types of households q and r suchthat q < r ; a utility mean-preserving transfer of order 2 is givenby:

T 2(α,u{q,r}(y), δ) := T 1(α,uq(y), δ)− T 1(α,ur (y) + δ, δ), (7)

such that,

0 < α 6 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 asfollows.

DefinitionBetween-Type Utility Transfers Principle of order 2. For allf ∈ F , for all y ∈ [0, ymax] and s ∈ N, given two types ofhouseholds q and r such that q < r ; we have the followingimplication for the utility transfer of order 2:

h = f + T 2(α,u{q,r}(y), δ) =⇒ W (h) > W (f ) (8)



Income Transfer of order 2

DefinitionBetween-Type Income Transfer of order 2. For all f ∈ F andfor all y ∈ [0, ymax], given two types of households q and r suchthat q < r ; an income mean-preserving transfer of order 2 isgiven by:

T 2(α, y , δ,q, r) := T 1(α, y , δ,q)−T 1(α, y + δ, δ, r), s ∈ N+, (9)

such that,0 < α 6 f (y + 2δ, r)], δ > 0.




DefinitionBetween-Type Income Transfers Principle of order 2. For allf ∈ F , for all y ∈ [0, ymax] and s ∈ N, given two types ofhouseholds q and r such that q < r ; we have the followingimplication for the income transfer of order 2:

h = f + T 2(α, y , δ,q, r) =⇒ W (h) > W (f ) (10)



The normative content of the Pigou-Dalton UtilityTransfers Principle

Proposition

Given that (H2a) and (H2b) hold. If W ρA fulfills the Utility

Transfers Principle of order 2, then ρ > 0.

This statement characterizes Prioritarianism. In this case, W ρA

are the Atkinson SWFs.



The normative content of the Pigou-Dalton IncomeTransfers Principle

Proposition

Given that (H2a) and (H2b) hold. If W ρA 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 W ρ

A to fulfillthe Transfers Principle of order 2.



Thanks for your attention.



On income and utilty Transfer Principles when households differ in needs

Economy & Finance

Transcript of On income and utilty Transfer Principles when households differ in needs