Proof of Theorem 2: If. We verify only that the leximin ordering satis…es the IDP. Suppose p LM q and consider p0; q0 with qi0 < p0i, p0i < pi, q0i < qi, some i 2 N, and pj = p0j and qj = qj0, j 6= i: We prove that p0 LM q0.
Note …rst that if it was qj0 p0j, then p0 would Pareto dominate q0 and the conclusion would be reached immediately. So in what follows assume qj0 > p0j. Therefore we have q0j = qj > p0j = pj. Furthermore, if qi > p0j = pj, then min fq1; q2g > min fp1; p2g contradicting p LM q, so we must have
qj0 > p0j qi
Given that qi > qi0, it follows from these inequalities that qj0 > p0j > qi0. Since by assumption p0i > q0i we can conclude that min fp01; p02g > min fq10; q20g
Only if. Let a social ordering< satisfy the axioms and let p; q 2 R2 with p6= q. If p LM q then pi = qj, i 6= j, and it follows immediately by A that
p q. So we show that p LM q implies that p q. If p Pareto dominates q we are done by SPO. If p Pareto dominates the permutation of q then we are done by A and SPO. So let pi 6= qj, i 6= j, and suppose by contradiction that p LM q but q p. Without loss of generality, let
p1 > q2 q1 > p2
Let " 2 R+. De…ne p0 = (p01; p02) = (q2+ "; p2) and q0 = (q10; q20) = (p2; q2).
Consider three cases:
a) q2 q1. In view of the displayed inequalities it follows that p1 > q2 q1 > p2, contradicting the assumption p LM q.
b) p1 p2 and q1 > q2. As by assumption p1 > q2 it follows that p2 p1 > q2. For a su¢ ciently small ", by the IDP q p implies q0 < p0, whereas by A and SPO it must be p0 q0, contradiction.
c) p1 > p2 and q1 > q2. Then p LM q implies p2 q2. If it was p2 = q2 then p LM q would imply p1 > q1. This yields p1 > q1 > p2 q2 and p would Pareto dominate q, contradicting q p. So let p2 > q2: if p1 q1 a contradiction would immediately obtain. Therefore let q1 > p1 and, by A, it is possible to repeat the reasoning of the previous case.
This shows that it must be p < q whenever p LM q. If p q, it is possible to slightly reduce the welfare of one of two agents to obtain r close enough to p such that r LM qbut (by SPO) q r, and the argument above can be applied to r and q, which concludes the proof.
Proof of Proposition 3:20 1. Consider two vectors p; q 2 R2. Without loss of generality, assume q2 > q1:
Case (i). Suppose p > q. By SPO p q and it is easy to check that for any p0, q0 2 R2 such that p0i < pi, qi0 < qi, and pj = p0j, qj = qj0 for j 6= i;if p0i > q0i then p0 q0 by SPO.
Case (ii). Suppose q1 < p1 p2 < q2. By HE and SPO, it is immediate to prove that it must be p q. Consider p0, q0 2 R2 such that p02 < p2, q20 < q2, and p1 = p01, q1 = q01: if p02 > q20 then p0 q0 by SPO. Consider p0, q0 2 R2 such that p01 < p1, q10 < q1, and p2 = p02, q2 = q02: if p01 > q10 then by HE p0 < q0.
20We thank Michele Lombardi for various suggestions that have greatly improved the proof.
Case (iii). Suppose q1 < p2 < p1 q2. Again, by HE and SPO, it must be p q and the rest of the proof follows as in the previous case except for the following combination: q10 < p2 p01 < q2. Yet, by HE and SPO it is immediate to prove that in the latter case, too, it must be p0 q0.
Case (iv). Suppose q1 < p2 < q2 < p1. By HE and SPO, it is immediate to prove that it must be p q. Consider p0, q0 2 R2 such that p02 < p2, q20 < q2, and p1 = p01, q1 = q10: if p02 > q02 then again p0 q0 by SPO. Consider p0, q0 2 R2 such that p01 < p1, q10 < q1, and p2 = p02, q2 = q02, and p01 > q10: If q10 < p2 < q2 < p01 the proof that p0 q0 follows as at the beginning of this case. If q01 < p2 < p01 q2 the proof that p0 q0 follows as at the beginning of case (iii). If q01 < p01 p2 < q2 then by SPO and HE, it must be p0 q0.
Cases (v) and (vi). If q1 = p2 < p1 q2 or q1 = p2 < q2 < p1, it is easily checked that IDP holds in these cases, too.
Finally, all other con…gurations of p; q 2 R2 are dealt with using sym-metrical arguments.
2. We want to show that there exists a SWO< on R2 that satis…es SPO and IDP but not HE. Consider the following lexicographically dictatorial SWO<LD by p LD q, p = q and p LD q, p1 > q1 or p1 = q1 & p2 > q2:
<LD satis…es IDP and SPO but not HE.
Proof of Theorem 5: 1. Consider p; q 2 R2, p 6= q, such that pi > qi. If q < p, then by WPO it is immediate to prove that there exist p0; q0 2 R2, p0 6= q0, such that p0i > qi0, q0j > p0j, and q0 p0.
2. By ND and step 1, there exist p; q 2 R2 such that p1 < q1, p2 > q2, and q p. By ND and step 1 above, there also exist x; y 2 R2 such that x1 < y1, x2 > y2, and x y. We show that there exist vectors p0; q0,x0; y0 such that q0 < p0, x0 < y0, p0 >> x0, and y0 >> q0 .
3. There are in principle a number of cases to consider. Case (i): suppose that x1 > p1, x2 < p2, y1 > q1, and y2 < q2. Then it is possible to construct two vectors x0; y0 such that x2 < x02 < p2, y2 < q2 < y20, x1 = x01, y1 = y10, and y02 < x02: by NI, it follows that x0 < y0. Then, construct two vectors p0; q0, such that p01 > x01 = x1 > p1, y10 = y1 > q10 > q1, p02 = p2, q20 = q2, and q10 > p01: by NI, it follows that q0 < p0. Furthermore, by WPO, it follows that p0 x0 and y0 q0, yielding the desired contradiction.
4. It is easily checked that starting from any initial con…guration of the vectors p; q,x; y , a contradiction is obtained with a similar reasoning in a
…nite number of steps using NI and WPO.
References
[1] Arneson, R. (2000) “Lockean Self-Ownership: Towards a Demolition”in (H. Steiner and P. Vallentyne, eds.) Left-Libertarianism and Its Critics, Palgrave, Basingstoke.
[2] Berger, F.R. (1984) Happiness, Justice, and Freedom, University of Cal-ifornia Press, Berkeley.
[3] Cohen, G.A. (2000) “Self-Ownership, World-Ownership and Equality”
in (H. Steiner and P. Vallentyne, eds.) Left-Libertarianism and Its Crit-ics, Palgrave, Basingstoke.
[4] Danley, J.R. (1979) “Robert Nozick and the Libertarian Paradox”, Mind 88: 419-423.
[5] d’Aspremont, C. (1985) “Axioms for Social Welfare Orderings”; in Hur-wicz, L., Schmeidler, D. and Sonnenschein, H. (Eds.) Social Goals and Social Organization: Essays in Memory of Elisha Pazner, Cambridge University Press, Cambridge.
[6] Dunn, J. (1979) Western political theory in the face of the future, Cam-bridge University Press, CamCam-bridge.
[7] Gibbard, A. (2000) “Natural Property Rights”in (H. Steiner and P. Val-lentyne, eds.) Left-Libertarianism and Its Critics, Palgrave, Basingstoke.
[8] Gray, J. and G.W. Smith (1991) “Introduction” in (J. Gray and G.W.
Smith, eds.) J.S. Mill ’On Liberty in focus, Routledge, London.
[9] Hammond, P. (1976) “Equity, Arrow’s Conditions and Rawls’Di¤erence Principle”, Econometrica 44: 793-804.
[10] Kymlicka, W. (2000) “Property Rights and the Self-Ownership Argu-ment” in (H. Steiner and P. Vallentyne, eds.) Left-Libertarianism and Its Critics, Palgrave, Basingstoke.
[11] Lombardi, M. and R. Veneziani (2009a) “Liberal Egalitarianism and the Harm Principle”, Working Paper 649, Queen Mary University of London.
[12] Lombardi, M. and R. Veneziani (2009b) “Liberal Principles for Social Welfare Relations in In…nitely-Lived Economies”, Working Paper 650, Queen Mary University of London.
[13] Mariotti, M. and R. Veneziani (2009a) “Non-Interference Implies Equal-ity”, Social Choice and Welfare 32: 123-8.
[14] Mariotti, M. and R. Veneziani (2009b) “The Impossibility of Non-Interference”, Mimeo, Queen Mary University of London.
[15] Miller, G.P. (1987) “Economic E¢ ciency and the Lockean Proviso”, Harvard Journal of Law & Public Policy 10: 401-410.
[16] McCloskey, H.J. (1963) “Mill’s Liberalism”, The Philosophical Quarterly 13: 143-156.
[17] Moulin, H. (1988) Axioms of Cooperative Decision Making, Cambridge University Press, Cambridge.
[18] Nozick, R. (1974) Anarchy, State and Utopia, Basic Books, New York.
[19] Rees, J.C. (1991) “A Re-Reading of Mill On Liberty” in (J. Gray and G.W. Smith, eds.) J.S. Mill ’On Liberty in focus, Routledge, London.
[20] Riley, J. (1985) “On the Possibility of Liberal Democracy”, The Amer-ican Political Science Review 79: 1135-1151.
[21] Sanders, J.T. (1987) “Justice and the Initial Acquisition of Property”, Harvard Journal of Law & Public Policy 10: 367-399.
[22] Sen, A.K. (1970) “The Impossibility of a Paretian Liberal”, Journal of Political Economy 78: 152-57.
[23] Sen, A.K. (1983) “Liberty and Social Choice”, Journal of Philosophy 80: 5-28.
[24] Steiner, H. (2000) “Original Rights and Just Redistribution” in (H.
Steiner and P. Vallentyne, eds.) Left-Libertarianism and Its Critics, Pal-grave, Basingstoke.
[25] Steiner, H. and P. Vallentyne (Eds.) Left-Libertarianism and Its Critics, Palgrave, Basingstoke.
[26] Sugden, R. (1993) ”Rights: Why Do They Matter, and To Whom?”, Constitutional Political Economy 4: 127-152.
[27] Tungodden, B. (2000) “Hammond Equity: A Generalization”, Discus-sion paper 20/99, Norwegian School of Economics and Business Admin-istration.
[28] Vallentyne, P. (2000) “Introduction” in (H. Steiner and P. Vallentyne, eds.) Left-Libertarianism and Its Critics, Palgrave, Basingstoke.
[29] Wallack, M. (2006) “Justice between generations: the limits of proce-dural justice”in (J.C. Tremmel, ed.) Handbook of Intergenerational Jus-tice, Elgar, Aldershot.
[30] Warnock, M. (1962) (Ed.) Utilitarianism, Fontana Press, Glasgow.
[31] Wolf, C. (1995) “Contemporary Property Rights, Lockean Provisos, and the Interests of Future Generations”, Ethics 105: 791-818.
Figure 1: Leximin - Necessity
Agent 2 45˚ line
q2 = q'2 q' q
p2 = p'2 p' p
p''2 p''
q''2 q''
q'1 p'1 q1 = q''1 p1 = p''1 Agent 1
Figure 2: Leximin - Sufficiency
Agent 2 45˚ line
p2 = p'2 p' p
q2 = q'2 q' q
p'1 p1 q'1 q1 Agent 1
Figure 3: The impossibility of NonInterference
Agent 2
p p'
x'
x
y'
q q'
y
Agent 1
This working paper has been produced by the Department of Economics at
Queen Mary, University of London
Copyright © 2009 Marco Mariotti and Roberto Veneziani
Department of Economics
Queen Mary, University of London Mile End Road
London E1 4NS
Tel: +44 (0)20 7882 5096 All rights reserved