transfers principles revisited with choquet’s lemma on successive differences
TRANSCRIPT
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 Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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)].
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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)).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 areequivalent:
(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
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 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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 forwarddifference 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
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 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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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+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).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 areequivalent:
(i) the SWF satisfies the Transfers Principle of order s + 1.
(ii) (−1)s+1u(s+1)(x) 6 0.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 Principleof order 3.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 onwhether consideration about number people wins against theemphasize placed on few people at the lowest or highestincome levels in the distribution.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 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).
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
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 strongdisrespect of all odd-order ones up to∞ =⇒ One aims atreducing extreme richness.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences
Introduction Transfers and Principles Transfers and Successive Differences The Theorem Discussions
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.
Marc Dubois LAMETA Université Montpellier
Transfers Principles Revisited with Choquet’s Lemma on Successive Differences