Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions

Transfers Principles Revisited with Choquet’sLemma on Successive Differences

Marc DuboisLAMETA

Université Montpellier

Universidad del Rosario.August 13th, 2015.

Marc Dubois LAMETA Université Montpellier

Transfers Principles Revisited with Choquet’s Lemma on Successive Differences


Introduction

The social welfare function is the sum of utility functionsstrictly increasing over income:

W (F ) =

∫ ∞

0u(y)f (y)dy . (SWF)

In this framework, concavity of the u function is simplydefinitional of what inequality aversion is.




The most popular Transfers Principles

How do we formalize inequality aversion? Let us introducea Transfers Principle.

Definition (A weak Pigou-Dalton condition:)

Transfers 6= Transfers Principle.




The most popular Transfers Principles

A higher-order Transfers Principle is due to Kolm (1976).

Definition (A weak Kolm condition:)




The Generalized Transfers function

Definition

Generalized Transfers function of order s + 1. Amean-preserving transfer of order s + 1 is given by:

T s+1(α, x , δ) := T s(α, x , δ)− T s(α, x + δ, δ), s ∈ N+,

such that δ > 0, and for k being even:(s + 1

k

)α ∈ (0, f (x + kδ)].




The Transfers Principle of order s + 1

DefinitionThe Transfers Principle of order s + 1. For all f ,h ∈ Ω;[

h = f + T s+1(α, x , δ)]

=⇒ [W (H) > W (F )] . (TPs+1)

Notation: W (H)−W (F ) =: ∆W (T s+1(α, x , δ)).




Theorem on Transfers Principles

Fishburn and Willig (1984) show that:

Theorem

For any positive integer s, the two following statements areequivalent:

(i) the SWF satisfies all Transfers Principles up to the sth order.

(ii) the s first successive derivatives of u alternate in sign.




Successive Forward Differences

Let u(x) ∈ C s+1, s ∈ N+. Consider its first order forwarddifference:

∆1(x ,a1) = u(x + a1)− u(x) ∀a1 > 0.

Recursively, the second order forward difference is:

∆2(x ,a1,a2) = ∆1(x + a2,a1)−∆1(x ,a1) ∀a1,a2 > 0.




Successive Forward Differences

More generally, for i ∈ 1, . . . , s + 1, the s + 1-th order forwarddifference is:

∆s+1(x ,a1, . . . ,as+1) = ∆s(x+as+1,a1, . . . ,as)−∆s(x ,a1, . . . ,as)

∀ai > 0.




Choquet’s (1954) Lemma

Choquet (1954) states that if the s-th order derivative of the ufunction is non-negative [non-positive], then the s-th orderforward difference that involves u is non-negative [non-positive].

Lemma

Choquet (1954, p. 149): for any given s ∈ N andi ∈ 1, . . . , s + 1,[(−1)s+1u(s+1)(x) 6 0

]=⇒

[(−1)s+1∆s+1(x ,a1, . . . ,as+1) 6 0,

].

∀ai > 0.

Let us consider a1 = . . . = as+1 = δ.




Transfers and Successive differences

The scaled s + 1-th order forward difference α∆s+1characterizes the change in social welfare due to a transferfunction T s+1(α, x , δ) for s being odd, and −T s+1(α, x , δ) for sbeing even.

Lemma

The following equation is true for all non-negative integer s:[Hs+1] : ∆W (T s+1(α, x , δ)) = (−1)sα∆s+1(x , δ, . . . , δ).




New theorem on Transfers Principles

Theorem

For any positive integer s, the two following statements areequivalent:

(i) the SWF satisfies the Transfers Principle of order s + 1.

(ii) (−1)s+1u(s+1)(x) 6 0.




General interpretations

A higher-order principle is not stronger than a lower-orderone.

it is necessary to take into consideration attitudes relyingon lower-order Transfers Principles to determine theattitude towards inequality relying on the respect or strongdisrespect of a given order Transfers Principle.




Interpretation at order 3

Definition (Transfers Principle of order 3:)

Inequality aversion is not embodied by the Transfers Principleof order 3.




On higher orders

Definition (Transfers Principle of order 4:)

Transfers Principles of order higher than 3 take a position onwhether consideration about number people wins against theemphasize placed on few people at the lowest or highestincome levels in the distribution.




On higher orders

DefinitionNumbers do not win at order s: Numbers do not win at orders > 3 if

sgn (∆W (T s(α, x , δ))) = sgn(

∆W(

T 2(α, x + kδ, δ)))

with k = 0 whenever W displays a downside sensitivity, andk = s − 2 whenever W displays an upside sensitivity.




On higher orders

PropositionAll the even-order derivatives of u have the same sign up toorder s + 1 =⇒ numbers do not win at an even-order up tos + 1 for all s ∈ N and s > 3.

PropositionAll the odd-order derivatives of u have the same sign up toorder s + 1 =⇒ numbers do not win at an odd-order up to s + 1for all s ∈ N and s > 3.




On higher orders

CorollaryAll the even-order derivatives of u up to order s + 1 have thesame sign and all the odd-order derivatives of u up to orders + 1 have the same sign=⇒ numbers do not win up to order s + 1 for all s ∈ N ands > 3.




Extreme attitudes

4 cases of extreme attitudes to inequality (1/2):

Respect of all odd-order Transfers Principles and strongdisrespect of all even-order ones up to∞ =⇒ one aims atincreasing extreme richness (leximax).

Respect of all Transfers Principles up to∞ =⇒ one aims atreducing extreme poverty (leximin).




Higher orders

4 cases of extreme attitudes to inequality (2/2):

Strong disrespect of all Transfers Principles up to∞ =⇒one aims at increasing extreme poverty.

Respect of all even-order Transfers Principles and strongdisrespect of all odd-order ones up to∞ =⇒ One aims atreducing extreme richness.




References

Chateauneuf, A., Gajdos, T. and P.-H. Wilthien 2002, ThePrinciple of Strong Diminishing Transfer. Journal of EconomicTheory, 103(2), 311-333.

Choquet, G. 1954, Théorie des capacités, Ann. Inst. Fourier.5, 131-295.

Fishburn, P., R. Willig 1984, Transfer Principles in IncomeRedistribution. Journal of Public Economics 25: 323-328.

Kolm, S.-C. 1976, Unequal Inequalities II. Journal of EconomicTheory, 13, 82-111.



Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

Economy & Finance

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