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Differentially monotonic redistribution of income
Economics Letters 141 (2016) 112–115
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Differentially monotonic redistribution of income✩ André Casajus Economics and Information Systems, HHL Leipzig Graduate School of Management, Jahnallee 59, 04109 Leipzig, Germany
highlights • We suggest a differential version of monotonicity for redistribution rules. • Increasing income differentials entail increasing post-redistribution differentials. • Differential monotonicity implies a flat tax combined with a basic income.
article
info
Article history: Received 25 December 2015 Received in revised form 30 January 2016 Accepted 12 February 2016 Available online 19 February 2016
abstract We suggest a differential version of monotonicity for redistribution rules: whenever the differential of two persons’ income weakly increases, then the differential of their post-redistribution rewards also weakly increases. Together with efficiency and non-negativity, differential monotonicity characterizes redistribution via taxation at a fixed rate and equal distribution of the total tax revenue. © 2016 Elsevier B.V. All rights reserved.
JEL classification: C71 D63 H20 MSC: 91A12 91B15 Keywords: Redistribution Flat tax Basic income Differential monotonicity
1. Introduction In modern societies, their members’ income is redistributed via various channels. A simple model to study redistribution of income in an n-member society is the following: Its members are numbered from 1 to n; Nn := {1, . . . , n}. Redistribution is modeled by mappings (redistribution rules) f : Rn+ → Rn .1 For x ∈ Rn+ and i ∈ Nn , fi (x) denotes member i’s income after redistribution.
✩ We are grateful to Hervé Moulin for helpful comments on this paper. Financial support by the Deutsche Forschungsgemeinschaft (grant CA 266/4-1) is gratefully acknowledged. E-mail address: [email protected]. URL: http://www.casajus.de. 1 Throughout, we disregard the trivial case n = 1. Further, we set R :=
[0, +∞).
http://dx.doi.org/10.1016/j.econlet.2016.02.013 0165-1765/© 2016 Elsevier B.V. All rights reserved.
+
Monotonicity principles or properties have a long tradition and are ubiquitous in economics and game theory (see e.g. Sprumont, 2008). In this note, we suggest and advocate a new monotonicity property for income redistribution rules—differential monotonicity: whenever the differential of two members’ income weakly increases, then the differential of their post-redistribution rewards does not decrease. Other than its non-differential cousin strong monotonicity, it leaves some room for real redistribution (Theorems 1 and 4). Strong monotonicity: whenever a member’s income weakly increases, then her post-redistribution reward also weakly increases. We first show that differential monotonicity is a tuned up version of the order preservation property (Theorem 2). Order preservation: a member with a weakly higher income than another one obtains a weakly higher post-redistribution reward than this other member. Second, we decompose differential monotonicity into a much weaker differential monotonicity property and a strong differential invariance property (Theorem 3).
A. Casajus / Economics Letters 141 (2016) 112–115
In our main result, we make use of differential monotonicity in order to characterize uniform proportional taxation, i.e., taxation by a tax rate that neither depends on individual income nor on the total income of the society, which is combined with an equal distribution of the total tax revenue among the members of the society (Theorem 4). Besides differential monotonicity, we employ two standard axioms, efficiency and non-negativity. Efficiency: total income before redistribution equals total reward after redistribution. Non-negativity: individual rewards after redistribution are non-negative. Note that our main result provides some support the idea of a flat tax combined with an unconditional basic income that depends on the total productivity of the society as suggested by Milner (1920), for example.2 The next section gives a formal account and discussion of these results. Some remarks conclude the paper. Appendix A contains the lengthier proof of our main result. 2. Differentially monotonic redistribution rules and uniform proportional taxation We first show that a natural, but strong monotonicity property for redistribution rules essentially prevents real redistribution. In order to avoid this peculiarity, we suggest a differential version of strong monotonicity and explore its basic properties and its relation to some standard properties. Finally, we show that differential monotonicity together with efficiency and nonnegativity characterizes redistribution by uniform proportional taxation combined with equal distribution of the total tax revenue. 2.1. Strong monotonicity and incentives The strong monotonicity property for redistribution rules below precludes adverse incentives to earn income, which may be viewed as desirable. When combined with rather innocuous standard properties as efficiency and non-negativity, however, strong monotonicity implies that the members of the society just keep their income. That is, strong monotonicity essentially rules out any kind of real redistribution or solidarity within the society. Strong monotonicity, M+ . For all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi , we have fi (x) ≤ fi (y) .3 Strong monotonicity requires a non-decreasing individual income to translate into a non-decreasing individual postredistribution reward. This property is strong because its implication holds irrespective of how the other members’ income changes. This implies that increasing ones income at the expense of other members’ income is not discouraged. Let 0 ∈ Rn+ be given by 0ℓ = 0 for all ℓ ∈ Nn . For i ∈ Nn , ei ∈ Rn+ is given by eii = 1 and eiℓ = 0 for all ℓ ∈ Nn \ {i} .
Efficiency, E. For all x ∈ Rn+ , we have ℓ∈Nn fℓ (x) = ℓ∈Nn xℓ . The very idea of re-distribution suggests that the total rewards after redistribution should not be greater than total income before. In addition, efficiency requires that redistribution has no cost.
Non-negativity, NN. For all x ∈ Rn+ and i ∈ Nn , we have fi (x) ≥ 0. For non-negative income, non-negativity is a very natural requirement. No member of the society necessarily must end up with a negative post-redistribution reward.
2 The flat tax (or proportional tax) has been advocated in 1845 by McCulloch (1975) and later on by notable others as Mill (1848), Hayek (1960), and Friedman (1962), more recently by Hall and Rabushka (1985) and Hall (1996). Vanderborght and Van Parijs (2005) provide a survey on the idea of an unconditional basic income. 3 Moulin (1985) suggests a weak version of strong monotonicity called the nodisposal-of-utility property: For all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi and xℓ = yℓ for all ℓ ∈ Nn \ {i}, we have fi (x) ≤ fi (y) .
113
Theorem 1. A redistribution rule f : Rn+ → Rn satisfies efficiency (E), non-negativity (NN), and strong monotonicity (M + ) if and only if f (x) = x for all x ∈ Rn+ . Proof. It is immediate that the redistribution rule f : Rn+ → Rn given by f (x) = x for all x ∈ Rn meets E, NN, and M+ . Let now the redistribution rule f : Rn+ → Rn satisfy E, NN, and M+ . (i) E and NN imply f (0) = 0. (ii) By (i) and M+ , we have fi (x) = 0 for all x ∈ Rn+ and i ∈ Nn such that xi = 0. (iii) By M+ , we also have fi (x) = fi (y) for all x, y ∈ Rn+ and i ∈ Nn such that xi = yi and yℓ = 0 for all ℓ ∈ Nn \ {i} . (iv) By (ii) and E, we have fi (y) = xi . By (iii) and (iv), we finally have f (x) = x for all x ∈ Rn+ . 2.2. Differential monotonicity In order to allow for real redistribution without setting adverse incentives to earn income, we suggest and advocate a differential version of strong monotonicity. Differential monotonicity, DM. For all x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j such that xi − xj ≤ yi − yj , we have fi (x) − fj (x) ≤ fi (y) − fj (y) . This property demands non-decreasing income differentials of two members to translate into non-decreasing differentials of their post-redistribution rewards. As we will see in the next subsection, differential monotonicity still avoids adverse incentives with respect to increasing ones income but without preventing real redistribution. In the following, we explore the relation of differential monotonicity to some standard properties of redistribution rules. Order preservation, OP. For all x ∈ Rn+ and i, j ∈ Nn , i ̸= j such that xi ≤ xj , we have fi (x) ≤ fj (x) . This property guarantees that a member with a weakly higher income ends up with a weakly higher post-redistribution reward. Additivity, A. For all x, y ∈ Rn+ , we have f (x + y) = f (x) + f (y) . Nullity, NY. We have f (0) = 0. Since nullity is a rather weak requirement that is implied by additivity and cum grano salis, the theorem below says that differential monotonicity is a tuned up version of the orderpreservation property. In particular, differential monotonicity coincides with the order-preservation property in presence of additivity. Theorem 2. (i) If a redistribution rule f : Rn+ → R satisfies the order preservation property (OP) and additivity (A), then f obeys differential monotonicity (DM). (ii) If a redistribution rule f : Rn+ → R satisfies differential monotonicity (DM) and nullity (NY), then f obeys the order preservation property (OP). Proof. (i) Let the redistribution rule f : Rn+ → R satisfy OP and A. Further, let x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (*) xi − xj ≤ yi − yj . W.l.o.g., 0 ≤ xi − xj . Set (**) x∗ := x − xj − xi · ej and (***) y∗ := y − yj − yi · ej . Note that x∗ , y∗ ∈ Rn+ and x∗i = x∗j and y∗i = y∗j . We have
fj (x) − fi (x)
A
= (∗∗),OP
=
(∗),OP
≤
fj x∗ − fi x∗ + fj
xj − xi · e j
− fi xj − xi · ej fj xj − xi · ej − fi xj − xi · ej fj xj − xi · ej − fi xj − xi · ej + fj yj − yi − xj − xi · ej − fi yj − yi − xj − xi · ej
114
A. Casajus / Economics Letters 141 (2016) 112–115 A
=
fj
(∗∗∗),OP
=
=
j
yj − yi · e − fi
∗
fj y
− fi A
∗
− fi y + fj yj − yi · ej
yj − yi · ej
To see how Theorem 4 fails for n = 2, consider the redistribution rule f ♥ : R2+ → R2 given by
j
yj − yi · e
fi (x) ♥
fj (y) − fi (y) ,
=
which proves the first claim. (ii) Let the redistribution rule f : Rn+ → R satisfy DM and NY. Further, let x ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (****) xi ≤ xj . We have NY
0 = fj (0) − fi (0)
(∗∗∗∗),DM
≤
fj (x) − fi (x) ,
which proves the second claim.
xi , 1
|x1 − x2 | > 1,
Differential monotonicity can be decomposed into a strong invariance property and a considerably weaker monotonicity property. Differential invariance, DI. For all x, y ∈ R+ and i, j ∈ Nn , i ̸= j such that xi − xj = yi − yj , we have fi (x) − fj (x) = fi (y) − fj (y) . This property requires the differential of two members’ postredistribution reward to depend only on their income differential. This implies that this differential is neither affected by the other members’ income nor by the level of income in general. Note that this property is the differential cousin of the no-disposal-of-utility property (Moulin, 1985); see Footnote 3. n
Weak differential monotonicity, DM− . For all x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that xℓ = yℓ for all ℓ ∈ Nn \ {i} and yi ≥ xi , we have fi (x) − fj (x) ≤ fi (y) − fj (y) . In contrast to differential monotonicity, its weaker cousin only considers situations in which a single member’s income weakly increases. In such a case, the differential of post-redistribution rewards of this member and any other member is required not to decrease. Theorem 3. Let n > 2. A redistribution rule f : Rn+ → Rn satisfies differential monotonicity (DM) if and only if it satisfies differential invariance (DI) and weak differential monotonicity (DM − ). Proof. It is immediate that DM implies both DI and DM− . To see that DI and DM− entail DM, let x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (*) xi − xj ≤ yi − yj . Choose z ∈ Rn+ such that (**) zℓ = yℓ for all ℓ ∈ Nn \ {i} and zi = yj + xi − xj . This way, we have zi − zj = xi − xj . Hence, DI implies (***) fi (x) − fj (x) = fi (z ) − fj (z ). By (*) and (**), we have yi = yj + yi − yj ≥ yj + xi − xj = zi . Now, DM− entails fi (z ) − fj (z ) ≤ fi (y) − fj (y). By (***), we are done.
2
· (x1 + x2 ) ,
|x1 − x2 | ≤ 1
One easily checks that f ♥ satisfies all three properties but is not as in (1). Remark 5. Careful inspection of the proof of Theorem 4 shows that its extension to the domain Rn , i.e., the redistribution of gains and losses, holds true. Of course, efficiency and differential monotonicity have to be extended to Rn in the obvious way. 3. Concluding remarks In this note, we provide an axiomatic characterization of the redistribution of income by a flat tax combined with an unconditional basic income depending on the total income of the society. In particular, we invoke the differential monotonicity property of redistribution, which can be seen as a modification of strong monotonicity that leaves some room for real redistribution. Casajus (2015b) pursues another approach. In particular, he uses a relaxation of strong monotonicity called monotonicity but on the domain Rn , i.e., he considers the redistribution of gains and losses: for all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi and 4 ℓ∈Nn xℓ ≤ ℓ∈Nn yℓ , we have fi (x) ≤ fi (y) . Besides efficiency and monotonicity, he makes use of the symmetry property: for all x ∈ Rn+ and i, j ∈ Nn such that xi = xi , we have fi (x) = fj (x) in order to characterize the redistribution rules as in Theorem 4 on Rn . Unfortunately, this characterization does not work on Rn+ , i.e., for non-negative income (see Casajus, 2015a). Yet, one may regard monotonicity somewhat more natural than differential monotonicity. Appendix A. Proof of Theorem 4 Let n > 2. By construction, the redistribution rules as in (1) obey E, NN, and DM. Let now f : Rn+ → Rn meet E, NN, and DM. It is immediate that E and NN entail f (0) = 0.
(A.1)
Moreover, it is straightforward to show that DM and (A.1) imply fi (x) = fj (x)
for all x ∈ Rn+ and i, j ∈ Nn such that xi = xj . (A.2)
By DM, there are mappings dij : R → R for all i, j ∈ Nn such that dij xi − xj = fi (x) − fj (x)
2.3. Uniform proportional taxation
for all x ∈ R2+ and i ∈ N2 .
for all x ∈ Rn+ .
(A.3)
Of course, we have
For societies comprising more than two members, differential monotonicity combined with efficiency and non-negativity already implies uniform proportional taxation: individual income is taxed at a fixed rate and overall tax revenue is distributed equally among the society’s members.
dji (a) = −dij (a)
for all i, j ∈ Nn , i ̸= j and a ∈ R
(A.4)
and dij (a) + djk (b) = dik (a + b) for all i, j, k ∈ Nn , i ̸= j ̸= k ̸= i and a, b ∈ R.
(A.5)
Next, we show that Theorem 4. Let n > 2. A redistribution rule f : Rn+ → Rn satisfies efficiency (E), non-negativity (NN), and differential monotonicity (DM) if and only if there exists some τ ∈ [0, 1] such that fi (x) = (1 − τ ) · xi +
τ n
·
ℓ∈Nn
xℓ
for all x ∈ Rn and i ∈ Nn .
(1)
d := dij = dkℓ
for all i, j, k, ℓ ∈ Nn such that i ̸= j, k ̸= ℓ.
(A.6)
4 This property is related to re-allocation proofness (Moulin, 1985) used by Ju et al. (2007, Theorem 7) in order to characterize a class of more general proportional rules.
A. Casajus / Economics Letters 141 (2016) 112–115
Fix a ∈ R, a ≥ 0. Let i, j, k ∈ Nn and x ∈ Rn+ be such that i ̸= j ̸= k ̸= i, xi = a, and xj = xk = 0. We have dij (a) = dij xi − xj
(A.3)
(A.2)
= fi (x) − fj (x) = fi (x) − fk (x)
= dik (xi − xk ) = dik (a) .
Analogously, one shows dij (a) = dik (a) for a < 0 and dji (a) = dki (a) for a ∈ R. This already establishes (A.6). By (A.5) and (A.6), it is clear that d is additive. We now use this fact in order to show that f also is additive. For i ∈ Nn and x, y ∈ Rn+ , we have
=
fi (x + y) − fj (x + y)
d xi + yi − xj − yj
d xi − xj + d yi − yj
=
fi (x) − fj (x) + fi (y) − fj (y)
E
= n · (fi (x) + fi (y)) −
xℓ −
ℓ∈Nn
yℓ ,
ℓ∈Nn
i.e., fi (x + y) = fi (x) + fi (y). Hence, f is additive. By Aczél (1966, Theorem 1) and Aczél and Dhombres (1989, Proposition 1), the additivity of f and NN entail that f is linear. Let i, j ∈ Nn , i ̸= j. We have fi ei + fi ej = fi ei + ej
(A.2)
= fj ei + ej = fj ei + fj ej . (A.7)
Further, E and (A.2) entail fi ei + (n − 1) · fj ei = 1
and fj ej + (n − 1) · fi ej = 1.
(A.8) Solving (A.7) and (A.8) gives fi ei = fj ej
=
DM
(A.1)
f1 e1 − f2 e1 ≥ f1 (0) − f2 (0) = 0,
i.e., τ ∈ [0, 1] . Finally, we have fi (x) =
(A.10), (A.11) τ = xi · 1 − (n − 1) ·
xℓ · fi eℓ
τ · xℓ n
n
= (1 − τ ) · xi +
τ n
·
xℓ
ℓ∈Nn
j∈Nn
(A.10), (A.11)
References
j∈Nn \{i} (A.3) and (A.6)
(A.11)
for all i ∈ Nn and x ∈ Rn+ , where the first quality drops from the linearity of f . Hence, f is as in (1).
j∈Nn \{i}
=
1−τ
ℓ∈Nn \{i}
j∈Nn
=
n
+
(A.3) and (A.6)
E, (A.9) (A.10) τ = 1 − (n − 1) · fj ei = 1 − (n − 1) · .
fi ei
ℓ∈Nn
(xℓ + yℓ )
ℓ∈Nn E
This entails
Thus, NN and (A.10) imply 0 ≤ τ . Further, we have
(A.3)
n · fi ( x + y ) −
115
and
fi ej = fj ei .
(A.9)
Set
(A.9) τ := n · f2 e1 = n · fi ej
for all i, j ∈ Nn , i ̸= j.
(A.10)
Aczél, J., 1966. Lectures on Functional Equations and their Applications. Academic Press, New York. Aczél, J., Dhombres, J., 1989. Functional Equations in Several Variables: with Applications to Mathematics, Information Theory and to Natural and Social Sciences. Cambridge Universitiy Press, Cambridge. Casajus, A., 2015a. Monotonic redistribution of non-negative allocations: a case for proportional taxation revisited. Econom. Lett. 136, 95–98. Casajus, A., 2015b. Monotonic redistribution of performance-based allocations: a case for proportional taxation. Theor. Econ. 10 (3), 887–892. Friedman, M., 1962. Capitalism and Freedom. University of Chicago Press, (Chapter x) The Distribution of Income. Hall, R. E., 1996. Fairness and Efficiency in the Flat Tax. AEI Press, Washington, D.C. Hall, R.E., Rabushka, A., 1985. The Flat Tax. Hoover Institution Press, Stanford, CA. Hayek, F.A., 1960. The Constitution of Liberty. In: Taxation and Redistribution, University of Chicago Press, pp. 306–323 (Chapter 20). Ju, B.-G., Miyagawa, E., Sakai, T., 2007. Non-manipulable division rules in claim problems and generalizations. J. Econom. Theory 132, 1–26. McCulloch, J. R., 1975. A Treatise on the Principles and Practical Influence of Taxation and the Funding System. Longman, London. Mill, J. S., 1848. On the Influence of Government. In: Principles of Political Economy with some of their Applications to Social Philosophy, vol. V. Longmans, Green and Co, (Chapter II) On the General Principles of Taxation. Milner, D., 1920. Higher Production by a Bonus on National Output: A Proposal for a Minimum Income for all Varying with National Productivity. G. Allen & Unwin, London. Moulin, H., 1985. Egalitarianism and utilitarianism in quasi-linear bargaining. Econometrica 53 (1), 49–68. Sprumont, Y., 2008. Monotonicity and solidarity axioms in economics and game theory. In: Rational Choice and Social Welfare: Theory and Applications. Springer, Berlin Heidelberg, pp. 71–94. Vanderborght, Y., Van Parijs, P., 2005. L’Allocation Universelle. Editions La Découverte, Paris.
Contents lists available at ScienceDirect
Economics Letters journal homepage: www.elsevier.com/locate/ecolet
Differentially monotonic redistribution of income✩ André Casajus Economics and Information Systems, HHL Leipzig Graduate School of Management, Jahnallee 59, 04109 Leipzig, Germany
highlights • We suggest a differential version of monotonicity for redistribution rules. • Increasing income differentials entail increasing post-redistribution differentials. • Differential monotonicity implies a flat tax combined with a basic income.
article
info
Article history: Received 25 December 2015 Received in revised form 30 January 2016 Accepted 12 February 2016 Available online 19 February 2016
abstract We suggest a differential version of monotonicity for redistribution rules: whenever the differential of two persons’ income weakly increases, then the differential of their post-redistribution rewards also weakly increases. Together with efficiency and non-negativity, differential monotonicity characterizes redistribution via taxation at a fixed rate and equal distribution of the total tax revenue. © 2016 Elsevier B.V. All rights reserved.
JEL classification: C71 D63 H20 MSC: 91A12 91B15 Keywords: Redistribution Flat tax Basic income Differential monotonicity
1. Introduction In modern societies, their members’ income is redistributed via various channels. A simple model to study redistribution of income in an n-member society is the following: Its members are numbered from 1 to n; Nn := {1, . . . , n}. Redistribution is modeled by mappings (redistribution rules) f : Rn+ → Rn .1 For x ∈ Rn+ and i ∈ Nn , fi (x) denotes member i’s income after redistribution.
✩ We are grateful to Hervé Moulin for helpful comments on this paper. Financial support by the Deutsche Forschungsgemeinschaft (grant CA 266/4-1) is gratefully acknowledged. E-mail address: [email protected]. URL: http://www.casajus.de. 1 Throughout, we disregard the trivial case n = 1. Further, we set R :=
[0, +∞).
http://dx.doi.org/10.1016/j.econlet.2016.02.013 0165-1765/© 2016 Elsevier B.V. All rights reserved.
+
Monotonicity principles or properties have a long tradition and are ubiquitous in economics and game theory (see e.g. Sprumont, 2008). In this note, we suggest and advocate a new monotonicity property for income redistribution rules—differential monotonicity: whenever the differential of two members’ income weakly increases, then the differential of their post-redistribution rewards does not decrease. Other than its non-differential cousin strong monotonicity, it leaves some room for real redistribution (Theorems 1 and 4). Strong monotonicity: whenever a member’s income weakly increases, then her post-redistribution reward also weakly increases. We first show that differential monotonicity is a tuned up version of the order preservation property (Theorem 2). Order preservation: a member with a weakly higher income than another one obtains a weakly higher post-redistribution reward than this other member. Second, we decompose differential monotonicity into a much weaker differential monotonicity property and a strong differential invariance property (Theorem 3).
A. Casajus / Economics Letters 141 (2016) 112–115
In our main result, we make use of differential monotonicity in order to characterize uniform proportional taxation, i.e., taxation by a tax rate that neither depends on individual income nor on the total income of the society, which is combined with an equal distribution of the total tax revenue among the members of the society (Theorem 4). Besides differential monotonicity, we employ two standard axioms, efficiency and non-negativity. Efficiency: total income before redistribution equals total reward after redistribution. Non-negativity: individual rewards after redistribution are non-negative. Note that our main result provides some support the idea of a flat tax combined with an unconditional basic income that depends on the total productivity of the society as suggested by Milner (1920), for example.2 The next section gives a formal account and discussion of these results. Some remarks conclude the paper. Appendix A contains the lengthier proof of our main result. 2. Differentially monotonic redistribution rules and uniform proportional taxation We first show that a natural, but strong monotonicity property for redistribution rules essentially prevents real redistribution. In order to avoid this peculiarity, we suggest a differential version of strong monotonicity and explore its basic properties and its relation to some standard properties. Finally, we show that differential monotonicity together with efficiency and nonnegativity characterizes redistribution by uniform proportional taxation combined with equal distribution of the total tax revenue. 2.1. Strong monotonicity and incentives The strong monotonicity property for redistribution rules below precludes adverse incentives to earn income, which may be viewed as desirable. When combined with rather innocuous standard properties as efficiency and non-negativity, however, strong monotonicity implies that the members of the society just keep their income. That is, strong monotonicity essentially rules out any kind of real redistribution or solidarity within the society. Strong monotonicity, M+ . For all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi , we have fi (x) ≤ fi (y) .3 Strong monotonicity requires a non-decreasing individual income to translate into a non-decreasing individual postredistribution reward. This property is strong because its implication holds irrespective of how the other members’ income changes. This implies that increasing ones income at the expense of other members’ income is not discouraged. Let 0 ∈ Rn+ be given by 0ℓ = 0 for all ℓ ∈ Nn . For i ∈ Nn , ei ∈ Rn+ is given by eii = 1 and eiℓ = 0 for all ℓ ∈ Nn \ {i} .
Efficiency, E. For all x ∈ Rn+ , we have ℓ∈Nn fℓ (x) = ℓ∈Nn xℓ . The very idea of re-distribution suggests that the total rewards after redistribution should not be greater than total income before. In addition, efficiency requires that redistribution has no cost.
Non-negativity, NN. For all x ∈ Rn+ and i ∈ Nn , we have fi (x) ≥ 0. For non-negative income, non-negativity is a very natural requirement. No member of the society necessarily must end up with a negative post-redistribution reward.
2 The flat tax (or proportional tax) has been advocated in 1845 by McCulloch (1975) and later on by notable others as Mill (1848), Hayek (1960), and Friedman (1962), more recently by Hall and Rabushka (1985) and Hall (1996). Vanderborght and Van Parijs (2005) provide a survey on the idea of an unconditional basic income. 3 Moulin (1985) suggests a weak version of strong monotonicity called the nodisposal-of-utility property: For all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi and xℓ = yℓ for all ℓ ∈ Nn \ {i}, we have fi (x) ≤ fi (y) .
113
Theorem 1. A redistribution rule f : Rn+ → Rn satisfies efficiency (E), non-negativity (NN), and strong monotonicity (M + ) if and only if f (x) = x for all x ∈ Rn+ . Proof. It is immediate that the redistribution rule f : Rn+ → Rn given by f (x) = x for all x ∈ Rn meets E, NN, and M+ . Let now the redistribution rule f : Rn+ → Rn satisfy E, NN, and M+ . (i) E and NN imply f (0) = 0. (ii) By (i) and M+ , we have fi (x) = 0 for all x ∈ Rn+ and i ∈ Nn such that xi = 0. (iii) By M+ , we also have fi (x) = fi (y) for all x, y ∈ Rn+ and i ∈ Nn such that xi = yi and yℓ = 0 for all ℓ ∈ Nn \ {i} . (iv) By (ii) and E, we have fi (y) = xi . By (iii) and (iv), we finally have f (x) = x for all x ∈ Rn+ . 2.2. Differential monotonicity In order to allow for real redistribution without setting adverse incentives to earn income, we suggest and advocate a differential version of strong monotonicity. Differential monotonicity, DM. For all x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j such that xi − xj ≤ yi − yj , we have fi (x) − fj (x) ≤ fi (y) − fj (y) . This property demands non-decreasing income differentials of two members to translate into non-decreasing differentials of their post-redistribution rewards. As we will see in the next subsection, differential monotonicity still avoids adverse incentives with respect to increasing ones income but without preventing real redistribution. In the following, we explore the relation of differential monotonicity to some standard properties of redistribution rules. Order preservation, OP. For all x ∈ Rn+ and i, j ∈ Nn , i ̸= j such that xi ≤ xj , we have fi (x) ≤ fj (x) . This property guarantees that a member with a weakly higher income ends up with a weakly higher post-redistribution reward. Additivity, A. For all x, y ∈ Rn+ , we have f (x + y) = f (x) + f (y) . Nullity, NY. We have f (0) = 0. Since nullity is a rather weak requirement that is implied by additivity and cum grano salis, the theorem below says that differential monotonicity is a tuned up version of the orderpreservation property. In particular, differential monotonicity coincides with the order-preservation property in presence of additivity. Theorem 2. (i) If a redistribution rule f : Rn+ → R satisfies the order preservation property (OP) and additivity (A), then f obeys differential monotonicity (DM). (ii) If a redistribution rule f : Rn+ → R satisfies differential monotonicity (DM) and nullity (NY), then f obeys the order preservation property (OP). Proof. (i) Let the redistribution rule f : Rn+ → R satisfy OP and A. Further, let x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (*) xi − xj ≤ yi − yj . W.l.o.g., 0 ≤ xi − xj . Set (**) x∗ := x − xj − xi · ej and (***) y∗ := y − yj − yi · ej . Note that x∗ , y∗ ∈ Rn+ and x∗i = x∗j and y∗i = y∗j . We have
fj (x) − fi (x)
A
= (∗∗),OP
=
(∗),OP
≤
fj x∗ − fi x∗ + fj
xj − xi · e j
− fi xj − xi · ej fj xj − xi · ej − fi xj − xi · ej fj xj − xi · ej − fi xj − xi · ej + fj yj − yi − xj − xi · ej − fi yj − yi − xj − xi · ej
114
A. Casajus / Economics Letters 141 (2016) 112–115 A
=
fj
(∗∗∗),OP
=
=
j
yj − yi · e − fi
∗
fj y
− fi A
∗
− fi y + fj yj − yi · ej
yj − yi · ej
To see how Theorem 4 fails for n = 2, consider the redistribution rule f ♥ : R2+ → R2 given by
j
yj − yi · e
fi (x) ♥
fj (y) − fi (y) ,
=
which proves the first claim. (ii) Let the redistribution rule f : Rn+ → R satisfy DM and NY. Further, let x ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (****) xi ≤ xj . We have NY
0 = fj (0) − fi (0)
(∗∗∗∗),DM
≤
fj (x) − fi (x) ,
which proves the second claim.
xi , 1
|x1 − x2 | > 1,
Differential monotonicity can be decomposed into a strong invariance property and a considerably weaker monotonicity property. Differential invariance, DI. For all x, y ∈ R+ and i, j ∈ Nn , i ̸= j such that xi − xj = yi − yj , we have fi (x) − fj (x) = fi (y) − fj (y) . This property requires the differential of two members’ postredistribution reward to depend only on their income differential. This implies that this differential is neither affected by the other members’ income nor by the level of income in general. Note that this property is the differential cousin of the no-disposal-of-utility property (Moulin, 1985); see Footnote 3. n
Weak differential monotonicity, DM− . For all x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that xℓ = yℓ for all ℓ ∈ Nn \ {i} and yi ≥ xi , we have fi (x) − fj (x) ≤ fi (y) − fj (y) . In contrast to differential monotonicity, its weaker cousin only considers situations in which a single member’s income weakly increases. In such a case, the differential of post-redistribution rewards of this member and any other member is required not to decrease. Theorem 3. Let n > 2. A redistribution rule f : Rn+ → Rn satisfies differential monotonicity (DM) if and only if it satisfies differential invariance (DI) and weak differential monotonicity (DM − ). Proof. It is immediate that DM implies both DI and DM− . To see that DI and DM− entail DM, let x, y ∈ Rn+ and i, j ∈ Nn , i ̸= j be such that (*) xi − xj ≤ yi − yj . Choose z ∈ Rn+ such that (**) zℓ = yℓ for all ℓ ∈ Nn \ {i} and zi = yj + xi − xj . This way, we have zi − zj = xi − xj . Hence, DI implies (***) fi (x) − fj (x) = fi (z ) − fj (z ). By (*) and (**), we have yi = yj + yi − yj ≥ yj + xi − xj = zi . Now, DM− entails fi (z ) − fj (z ) ≤ fi (y) − fj (y). By (***), we are done.
2
· (x1 + x2 ) ,
|x1 − x2 | ≤ 1
One easily checks that f ♥ satisfies all three properties but is not as in (1). Remark 5. Careful inspection of the proof of Theorem 4 shows that its extension to the domain Rn , i.e., the redistribution of gains and losses, holds true. Of course, efficiency and differential monotonicity have to be extended to Rn in the obvious way. 3. Concluding remarks In this note, we provide an axiomatic characterization of the redistribution of income by a flat tax combined with an unconditional basic income depending on the total income of the society. In particular, we invoke the differential monotonicity property of redistribution, which can be seen as a modification of strong monotonicity that leaves some room for real redistribution. Casajus (2015b) pursues another approach. In particular, he uses a relaxation of strong monotonicity called monotonicity but on the domain Rn , i.e., he considers the redistribution of gains and losses: for all x, y ∈ Rn+ and i ∈ Nn such that xi ≤ yi and 4 ℓ∈Nn xℓ ≤ ℓ∈Nn yℓ , we have fi (x) ≤ fi (y) . Besides efficiency and monotonicity, he makes use of the symmetry property: for all x ∈ Rn+ and i, j ∈ Nn such that xi = xi , we have fi (x) = fj (x) in order to characterize the redistribution rules as in Theorem 4 on Rn . Unfortunately, this characterization does not work on Rn+ , i.e., for non-negative income (see Casajus, 2015a). Yet, one may regard monotonicity somewhat more natural than differential monotonicity. Appendix A. Proof of Theorem 4 Let n > 2. By construction, the redistribution rules as in (1) obey E, NN, and DM. Let now f : Rn+ → Rn meet E, NN, and DM. It is immediate that E and NN entail f (0) = 0.
(A.1)
Moreover, it is straightforward to show that DM and (A.1) imply fi (x) = fj (x)
for all x ∈ Rn+ and i, j ∈ Nn such that xi = xj . (A.2)
By DM, there are mappings dij : R → R for all i, j ∈ Nn such that dij xi − xj = fi (x) − fj (x)
2.3. Uniform proportional taxation
for all x ∈ R2+ and i ∈ N2 .
for all x ∈ Rn+ .
(A.3)
Of course, we have
For societies comprising more than two members, differential monotonicity combined with efficiency and non-negativity already implies uniform proportional taxation: individual income is taxed at a fixed rate and overall tax revenue is distributed equally among the society’s members.
dji (a) = −dij (a)
for all i, j ∈ Nn , i ̸= j and a ∈ R
(A.4)
and dij (a) + djk (b) = dik (a + b) for all i, j, k ∈ Nn , i ̸= j ̸= k ̸= i and a, b ∈ R.
(A.5)
Next, we show that Theorem 4. Let n > 2. A redistribution rule f : Rn+ → Rn satisfies efficiency (E), non-negativity (NN), and differential monotonicity (DM) if and only if there exists some τ ∈ [0, 1] such that fi (x) = (1 − τ ) · xi +
τ n
·
ℓ∈Nn
xℓ
for all x ∈ Rn and i ∈ Nn .
(1)
d := dij = dkℓ
for all i, j, k, ℓ ∈ Nn such that i ̸= j, k ̸= ℓ.
(A.6)
4 This property is related to re-allocation proofness (Moulin, 1985) used by Ju et al. (2007, Theorem 7) in order to characterize a class of more general proportional rules.
A. Casajus / Economics Letters 141 (2016) 112–115
Fix a ∈ R, a ≥ 0. Let i, j, k ∈ Nn and x ∈ Rn+ be such that i ̸= j ̸= k ̸= i, xi = a, and xj = xk = 0. We have dij (a) = dij xi − xj
(A.3)
(A.2)
= fi (x) − fj (x) = fi (x) − fk (x)
= dik (xi − xk ) = dik (a) .
Analogously, one shows dij (a) = dik (a) for a < 0 and dji (a) = dki (a) for a ∈ R. This already establishes (A.6). By (A.5) and (A.6), it is clear that d is additive. We now use this fact in order to show that f also is additive. For i ∈ Nn and x, y ∈ Rn+ , we have
=
fi (x + y) − fj (x + y)
d xi + yi − xj − yj
d xi − xj + d yi − yj
=
fi (x) − fj (x) + fi (y) − fj (y)
E
= n · (fi (x) + fi (y)) −
xℓ −
ℓ∈Nn
yℓ ,
ℓ∈Nn
i.e., fi (x + y) = fi (x) + fi (y). Hence, f is additive. By Aczél (1966, Theorem 1) and Aczél and Dhombres (1989, Proposition 1), the additivity of f and NN entail that f is linear. Let i, j ∈ Nn , i ̸= j. We have fi ei + fi ej = fi ei + ej
(A.2)
= fj ei + ej = fj ei + fj ej . (A.7)
Further, E and (A.2) entail fi ei + (n − 1) · fj ei = 1
and fj ej + (n − 1) · fi ej = 1.
(A.8) Solving (A.7) and (A.8) gives fi ei = fj ej
=
DM
(A.1)
f1 e1 − f2 e1 ≥ f1 (0) − f2 (0) = 0,
i.e., τ ∈ [0, 1] . Finally, we have fi (x) =
(A.10), (A.11) τ = xi · 1 − (n − 1) ·
xℓ · fi eℓ
τ · xℓ n
n
= (1 − τ ) · xi +
τ n
·
xℓ
ℓ∈Nn
j∈Nn
(A.10), (A.11)
References
j∈Nn \{i} (A.3) and (A.6)
(A.11)
for all i ∈ Nn and x ∈ Rn+ , where the first quality drops from the linearity of f . Hence, f is as in (1).
j∈Nn \{i}
=
1−τ
ℓ∈Nn \{i}
j∈Nn
=
n
+
(A.3) and (A.6)
E, (A.9) (A.10) τ = 1 − (n − 1) · fj ei = 1 − (n − 1) · .
fi ei
ℓ∈Nn
(xℓ + yℓ )
ℓ∈Nn E
This entails
Thus, NN and (A.10) imply 0 ≤ τ . Further, we have
(A.3)
n · fi ( x + y ) −
115
and
fi ej = fj ei .
(A.9)
Set
(A.9) τ := n · f2 e1 = n · fi ej
for all i, j ∈ Nn , i ̸= j.
(A.10)
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