Directory UMM :Data Elmu:jurnal:E:Economics Letters:Vol70.Issue1.Jan2001:

Economics Letters 70 (2001) 95–98

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Pareto principles from Inch to Ell

*

Kotaro Suzumura

The Institute of Economic Research, Hitotsubashi University, Naka 2-1, Kunitachi, Tokyo 186-8603, Japan

Received 3 April 2000; accepted 29 June 2000

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.  2001 Elsevier Science B.V. All rights reserved.

Keywords: Pareto principles; Divisible and indivisible goods; Possibility of compensation

JEL classification: D63; D71

1. Introduction

Consider a society with n individuals, where 2 # _n, 1 `_{. Let R}5_{(R , R , . . . , R ) denote the}

1 2 n

profile of individual preference orderings on the set X of all social states such that, for all social states

x, y _[X, xR y means that x is at least as good as y according to i’s personal judgements. R denotes_i

the social welfare ordering corresponding to R.

In the literature of Paretian welfare economics and social choice theory, one or the other of the 1

following versions of the Pareto principle plays an important role.

Pareto Indifference Principle

If xI(R )y holds for all i_[N:5_{h1, 2, . . . , nj, then xI(R)y holds.}

i

Weak Pareto Principle

If xP(R )y holds for all i_{i [} N, then xP(R)y holds.

*Corresponding author. Tel.: 181-42-580-8353; fax:1 181-42-580-8353. E-mail address: suzumura@ier.hit-u.ac.jp (K. Suzumura).

1

For any binary relation Q on X, P(Q ) and I(Q ) denote, respectively, the asymmetric part of Q, and the symmetric part of Q, which are defined by P(Q )5_{h(x, y)[X}3_{Xu(x, y)[Q and ( y, x)[⁄ Qj and I(Q )}5_{h(x, y)[X}3_{Xu(x, y)[Q and ( y,} x)_[Qj.

96 K. Suzumura / Economics Letters 70 (2001) 95 –98

Strong Pareto Principle

If xR y holds for all i_{i [}N and xP(R )y holds for at least one j_{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.

2. An equivalence theorem

n

Suppose that the set X has the following decomposable structure: X:5_E 3V_{, where E is the set} of real numbers, whereasV _{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

n

individual, e:5_{(e , e , . . . , e )[ E , and the other features of the world which is captured by}

1 2 n

v_[V_.

Assumption (i). Each and every R is responsive only to i’s own assignment of the divisible good andi n2₁

Assumption (ii). Each and every Ri is continuous and monotonically increasing in i’s own assignment of the divisible good.

Assumption (iii). Each and every R admits the possibility of compensation in terms of i’s own_i

n

assignment of the divisible good, viz., for all e, e9_{[ E and,}v_,v 9_[V _{such that (e,}v_{)P(R )(ei} 9_,v9_),

99

9

99

99

9

there exists an e _[ E satisfying e ,_{e and ((e ; e ),}v9_{)P(R )(e,}v_).

i i i i 2i i

Assumption (iv). R is continuous and monotonically increasing in each and every individual’s own

n

₉

We are now ready to assert the folllowing:

Theorem (Pareto Principle from Inch to Ell). Under the assumptions (i)–(iv), the Pareto

Indifference, the Weak Pareto, the Strong Pareto, and the Full Pareto are all mutually equivalent.

Proof. Suppose that x, y [X are such that xI(R )y holds for all ii [N and yet xI(R)y does not hold, so

K. Suzumura / Economics Letters 70 (2001) 95 –98 97

define a new social state, say z, by starting from y and giving an additional amount ´._{0 of the} divisible good to each and every individual. By virtue of the responsiveness and monotonicity of each

R in this good, we have zP(R )yI(R )x, hence zP(R )x by transitivity, for all ii i i i [N, whereas the

continuity of R in this good ensures that xP(R)z holds as far as is small enough. If the Weak Pareto happens to be satisfied, we are led to conclude that zP(R)x in contradiction with xP(R)z. By reductio

ad absurdum, we have shown that the Weak Pareto cannot but imply the Pareto Indifference.

Suppose next that xR y holds for all ii [N and xP(R )y holds for at least one ji [N, and yet xP(R)y

does not hold, so that the Strong Pareto fails to be satisfied. R being complete, we must then have yRx. Start from y and increase the amount of the divisible good only for those individuals who said xP(R )yi by ´ ._{0 so that the resulting social state z satisfies xI(R )z for all i[N. Since R is responsive,}

i i i

continuous, monotonically increasing and admits the possibility of compensation in the divisible good, such an´ ._{0 can be found. Since R is also monotonically increasing and yRx, we must have zP(R)x.}

i

Suppose now that the Pareto Indifference is satisfied. Then we have xI(R)z in contradiction with

zP(R)x. Thus, the Pareto Indifference must imply the Strong Pareto, so that the Pareto Indifference

implies the Full Pareto. This completes the proof. h

3. 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 check

2

It is clear that all the conditions other than the possibility of compensation are satisfied by this example. Although the Pareto Indifference is satisfied by this example, however, the Weak Pareto is violated, as we have ((1, 1,v _{), (2, 2,}v _{))[ P(R )> P(R ) and yet ((1, 1,}v _{), (2, 2,}v _{))[⁄ R must}

1 2 1 2 1 2

hold.

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

2

98 K. Suzumura / Economics Letters 70 (2001) 95 –98

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

Alkan, A., Demange, G., Gale, D., 1991. Fair allocation of indivisible goods and criteria of justice. Econometrica (59), 1023–1039.