Pareto principles from Inch to Ell

doi:10.1016/S0165-1765(00)00339-6

Economics Letters

Volume 70, Issue 1, January 2001, Pages 95-98

https://doi.org/10.1016/S0165-1765(00)00339-6 Get rights and content

Abstract

Under mild conditions including the possibility of compensation by monetary transfer, this paper shows that all versions of the Pareto Principle, viz., the Pareto Indifference, the Weak Pareto, the Strong Pareto, and the Full Pareto, are in fact mutually equivalent.

Introduction

Consider a society with n individuals, where 2 ≤ n<+∞. Let R=(R₁, R₂,…, R_n) denote the profile of individual preference orderings on the set X of all social states such that, for all social states x, y ∈X, xR_iy means that x is at least as good as y according to i’s personal judgements. R denotes the social welfare ordering corresponding to R.

In the literature of Paretian welfare economics and social choice theory, one or the other of the following versions of the Pareto principle plays an important role.¹

Pareto Indifference Principle

If xI(R_i)y holds for all i∈N:={1, 2,…, n}, then xI(R)y holds.

Weak Pareto Principle

If xP(R_i)y holds for all i∈ N, then xP(R)y holds.

Strong Pareto Principle

If xR_iy holds for all i∈N and xP(R_j)y holds for at least one j∈N, then xP(R)y holds.

Full Pareto Principle

The Strong Pareto as well as the Pareto Indifference holds.

By their very definitions, it is obvious that the Full Pareto implies the Strong Pareto and the Pareto Indifference, whereas the Strong Pareto implies the Weak Pareto. In general, this is all we can assert about the logical relations which hold among these Pareto principles. It is shown in this note, however, that all these Pareto principles are in fact mutually equivalent under several mild conditions including the possibility of compensation, which are standardly assumed in the literature on fair allocations with indivisible commodities.

Section snippets

An equivalence theorem

Suppose that the set X has the following decomposable structure: $X:=E^{n} ×Ω,$ where E is the set of real numbers, whereas Ω is the set whose structure need not be specified any further. The intended interpretation is that there exists an infinitesimally divisible good, and each and every social state x∈X is described by specifying the amount of this divisible good accruing to each and every individual, e:=(e₁, e₂,…, e_n)∈ Eⁿ, and the other features of the world which is captured by $ω∈Ω.$

Assumption (i)

Each and

A remark on the possibility of compensation

Among the assumptions which support the validity of the equivalence theorem, the only one which may seem strong is that of the possibility of compensation, viz., the Assumption (iii). Note, however, that this assumption is in fact standardly invoked in the literature on fair allocations with indivisible goods. See Alkan et al. (1991) and Tadenuma and Thomson (1991). It is also worth pointing out that this assumption is indispensable for the validity of the equivalence theorem. We have only to

Acknowledgements

This research was completed while I was visiting the Centre for Philosophy of Natural and Social Sciences, the London School of Economics, UK. I am grateful to the warm hospitality and research facilities provided by the Centre. Thanks are also due to Professors Claude d’Aspremont, Louis Gevers, Michel Le Breton, Amartya K. Sen, Koichi Tadenuma, and Yongsheng Xu for their helpful comments and discussions. Needless to say, they should not be held responsible for any defect that may remain.

References (2)

A. Alkan et al.
Fair allocation of indivisible goods and criteria of justice
Econometrica
(1991)
K. Tadenuma et al.
No-envy and consistency in economies with indivisible goods
Econometrica
(1991)

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