This document discusses transfers principles and their relationship to the derivatives of utility functions. It presents Choquet's lemma relating successive differences of a function to the signs of its derivatives. The author proves that a social welfare function satisfies the transfers principle of order s+1 if and only if the (s+1)th derivative of the utility function has a particular sign. This characterizes different attitudes toward inequality based on the signs of higher-order derivatives. The paper also discusses when "numbers don't win" at higher transfer orders.

1. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Transfers Principles Revisited with Choquet’s
Lemma on Successive Differences
Marc Dubois
LAMETA
Université Montpellier
Universidad del Rosario.
August 13th, 2015.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

2. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Introduction
The social welfare function is the sum of utility functions
strictly increasing over income:
W(F) =
∞
0
u(y)f(y)dy. (SWF)
In this framework, concavity of the u function is simply
definitional of what inequality aversion is.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

3. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
The most popular Transfers Principles
How do we formalize inequality aversion? Let us introduce
a Transfers Principle.
Definition (A weak Pigou-Dalton condition:)
Transfers = Transfers Principle.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

4. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
The most popular Transfers Principles
A higher-order Transfers Principle is due to Kolm (1976).
Definition (A weak Kolm condition:)
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

5. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
The Generalized Transfers function
Definition
Generalized Transfers function of order s + 1. A
mean-preserving transfer of order s + 1 is given by:
Ts+1
(α, x, δ) := Ts
(α, x, δ) − Ts
(α, x + δ, δ), s ∈ N+,
such that δ > 0, and for k being even:
s + 1
k
α ∈ (0, f(x + kδ)].
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

6. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
The Transfers Principle of order s + 1
Definition
The Transfers Principle of order s + 1. For all f, h ∈ Ω;
h = f + Ts+1
(α, x, δ) =⇒ [W(H) W(F)] . (TPs+1)
Notation: W(H) − W(F) =: ∆W(Ts+1(α, x, δ)).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

7. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Theorem on Transfers Principles
Fishburn and Willig (1984) show that:
Theorem
For any positive integer s, the two following statements are
equivalent:
(i) the SWF satisfies all Transfers Principles up to the sth order.
(ii) the s first successive derivatives of u alternate in sign.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

8. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Successive Forward Differences
Let u(x) ∈ C s+1, s ∈ N+. Consider its first order forward
difference:
∆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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

9. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Successive Forward Differences
More generally, for i ∈ {1, . . . , s + 1}, the s + 1-th order forward
difference is:
∆s+1(x, a1, . . . , as+1) = ∆s(x+as+1, a1, . . . , as)−∆s(x, a1, . . . , as)
∀ai > 0.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

10. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Choquet’s (1954) Lemma
Choquet (1954) states that if the s-th order derivative of the u
function is non-negative [non-positive], then the s-th order
forward difference that involves u is non-negative [non-positive].
Lemma
Choquet (1954, p. 149): for any given s ∈ N and
i ∈ {1, . . . , s + 1},
(−1)s+1
u(s+1)
(x) 0 =⇒ (−1)s+1
∆s+1(x, a1, . . . , as+1) 0, .
∀ai > 0.
Let us consider a1 = . . . = as+1 = δ.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

11. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Transfers and Successive differences
The scaled s + 1-th order forward difference α∆s+1
characterizes the change in social welfare due to a transfer
function Ts+1(α, x, δ) for s being odd, and −Ts+1(α, x, δ) for s
being even.
Lemma
The following equation is true for all non-negative integer s:
[Hs+1] : ∆W(Ts+1(α, x, δ)) = (−1)sα∆s+1(x, δ, . . . , δ).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

12. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
New theorem on Transfers Principles
Theorem
For any positive integer s, the two following statements are
equivalent:
(i) the SWF satisfies the Transfers Principle of order s + 1.
(ii) (−1)s+1u(s+1)(x) 0.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

13. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
General interpretations
A higher-order principle is not stronger than a lower-order
one.
it is necessary to take into consideration attitudes relying
on lower-order Transfers Principles to determine the
attitude towards inequality relying on the respect or strong
disrespect of a given order Transfers Principle.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

14. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Interpretation at order 3
Definition (Transfers Principle of order 3:)
Inequality aversion is not embodied by the Transfers Principle
of order 3.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

15. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
On higher orders
Definition (Transfers Principle of order 4:)
Transfers Principles of order higher than 3 take a position on
whether consideration about number people wins against the
emphasize placed on few people at the lowest or highest
income levels in the distribution.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

16. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
On higher orders
Definition
Numbers do not win at order s: Numbers do not win at order
s > 3 if
sgn (∆W (Ts
(α, x, δ))) = sgn ∆W T2
(α, x + kδ, δ)
with k = 0 whenever W displays a downside sensitivity, and
k = s − 2 whenever W displays an upside sensitivity.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

17. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
On higher orders
Proposition
All the even-order derivatives of u have the same sign up to
order s + 1 =⇒ numbers do not win at an even-order up to
s + 1 for all s ∈ N and s 3.
Proposition
All the odd-order derivatives of u have the same sign up to
order s + 1 =⇒ numbers do not win at an odd-order up to s + 1
for all s ∈ N and s 3.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

18. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
On higher orders
Corollary
All the even-order derivatives of u up to order s + 1 have the
same sign and all the odd-order derivatives of u up to order
s + 1 have the same sign
=⇒ numbers do not win up to order s + 1 for all s ∈ N and
s 3.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

19. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
Extreme attitudes
4 cases of extreme attitudes to inequality (1/2):
Respect of all odd-order Transfers Principles and strong
disrespect of all even-order ones up to ∞ =⇒ one aims at
increasing extreme richness (leximax).
Respect of all Transfers Principles up to ∞ =⇒ one aims at
reducing extreme poverty (leximin).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

20. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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 strong
disrespect of all odd-order ones up to ∞ =⇒ One aims at
reducing extreme richness.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences

21. Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
References
Chateauneuf, A., Gajdos, T. and P.-H. Wilthien 2002, The
Principle of Strong Diminishing Transfer. Journal of Economic
Theory, 103(2), 311-333.
Choquet, G. 1954, Théorie des capacités, Ann. Inst. Fourier.
5, 131-295.
Fishburn, P., R. Willig 1984, Transfer Principles in Income
Redistribution. Journal of Public Economics 25: 323-328.
Kolm, S.-C. 1976, Unequal Inequalities II. Journal of Economic
Theory, 13, 82-111.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences