Monotonic redistribution of non-negative allocations: a case for proportional taxation revisited

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1 Mootoi redistributio of o-egative alloatios: a ase for proportioal taxatio revisited Adré Casajus a a Eoomis ad Iformatio Systems, HHL Leipzig Graduate Shool of Maagemet Jahallee 59, 0409 Leipzig, Germay Abstrat We reosider Casajus (204, Theoretial Eoomis, forthomig) haraterizatio of proportioal taxatio by three properties of redistributio: e iey, symmetry, ad mootoiity. Whe restrited to o-egative alloatios, these properties oly imply proportioal taxatio i a weaker sese the tax rate may vary with overall performae but ot i a arbitrary fashio. O the restrited domai, proportioal taxatio is haraterized by the afore-metioed properties together with positive homogeeity, i.e., upsalig performaes implies upsaled rewards after redistributio. Keywords: Redistributio, proportioal taxatio, e iey, symmetry, mootoiity, positive homogeeity 200 MSC: 9A2, 9B5 JEL: C7, D63, H20 Ad all the tithe of the lad, whether of the seed of the lad, or of the fruit of the tree, is the Lord s: it is holy uto the Lord. Levitius 27:30. Itrodutio The above quotatio from the Kig James Bible provides a early example of the idea of proportioal taxatio. I moder times, proportioal taxatio has bee advoated by otable people like MCulloh (845), Mill (848), Hayek (960), ad Friedma (962), or later by Hall ad Rabushka (985) ad Hall (996). Oly reetly, Casajus (204) makes a axiomati ase for proportioal taxatio via mootoi redistributio of performae-based alloatios. 2 We are grateful to Frak Huetter ad Hervé Mouli for helpful ommets o this paper. Fiaial support by the Deutshe Forshugsgemeishaft, grat CA 266/4- is gratefully akowledged. address: mail@asajus.de (Adré Casajus) URL: (Adré Casajus) Fleubaey ad Maiquet (20, Chapters 0 ad ), for example, provide a survey of axiomati foudatios of taxatio. 2 Lambert (2002) provides a overview o the formal treatmet of redistributio. Preprit submitted to PET5 April 20, 205

2 I partiular, Casajus osiders redistributio i a very simple model of a soiety where its members are haraterized by their idividual otributios to the soiety s wealth. A redistributio rule assigs to ay list of idividual performae-based alloatios (for short, performaes) a list of alloatios after redistributio. Cum grao salis 3, three properties of redistributio rules e iey, symmetry, ad mootoiity 4 etail proportioal taxatio i the followig sese. First, idividual performaes are taxed proportioally at a ertai rate. Ad seod, overall tax reveue is distributed equally withi the soiety. A peuliarity of this model is that idividual performaes may be egative. A fortiori, Casajus proof ritially makes use of egative idividual performaes. This a be justi ed whe oe osiders redistributio withi a ertai period of time where idividuals may iur losses. O the oe had, it is possible to iterpret the taxatio of losses. I may tax systems, losses i oe tax period may redue the tax burde i future periods. Taxatio of losses the meas that these redutios are aouted i the period where the losses arued. O the other had, oe would like to have a justi atio for proportioal taxatio of o-egative performaes i the same vei as for the full domai. Bad tidigs is that the restritio of Casajus (204, Theorem ) to o-egative performaes does ot hold true. However, thigs are t too bad. Withi the restrited domai, the three properties still imply a weaker versio of proportioal taxatio (Theorem 2). For ay xed overall performae, redistributio takes plae by proportioal taxatio. Yet, the tax rate may vary with overall performae but ot i a arbitrary fashio. I partiular, overall tax reveue must ot derease whe overall performae ireases, i.e., tax rate is ot allowed to drop too fast. Moreover, overall tax reveue o a xed overall performae aot irease more tha per-apita tax reveue with ireasig overall performae, i.e., tax rate aot rise too fast. While the former requiremet protets members of the soiety with a zero performae, the latter oe protets a member who is the sole performer i the soiety. Tehially, these oditios a be expressed by a lower ad upper boud o the elastiity of the tax rate futio that essetially has to be absolutely otiuous (Theorem 4). Our outerexample o the restrited domai fails homogeeity for positive salars, i.e., saled up performaes may be taxed at a di eret rate. It turs out that addig positive homogeeity to the list of properties yields a haraterizatio of proportioal taxatio (Theorem 5). The ext setio gives a formal aout ad disussio of these results. Some remarks olude the paper. Two appedies otai the legthier proofs of our results. 2. Mootoi redistributio rules ad proportioal taxatio Casajus (204) osiders redistributio i a partiularly simple model of a soiety. For 3 Casajus (204, Theorem ) does ot hold true for two-perso soieties. 4 E iey: redistributio has o ost. Symmetry: members of the soiety with the same performae obtai the same reward after redistributio. Mootoiity: wheever both the performae of a ertai member of the soiety as well as the overall performae of the soiety do ot derease, the this member s reward after redistributio should ot derease. 2

3 2 N, the members of a -perso soiety are represeted by the rst atural umbers; N := f; : : : ; g; idividual performaes are give by a vetor x 2 R : A redistributio rule for a -perso soiety is a mappig f : R! R : For x 2 R ad i 2 N ; f i (x) deotes the reward of member i of the soiety after redistributio. 5 The properties of redistributio rules metioed i the itrodutio are formally de ed as follows. E iey, E. For all x 2 R ; we have P`2N f` (x) = P`2N x`: Symmetry, S. For all x 2 R ad i; j 2 N suh that x i = x j, we have f i (x) = f j (x) : Mootoiity, M. For all x; y 2 R ad i 2 N suh that P`2N x` P`2N y` ad x i y i ; we have f i (x) f i (y) : Cum grao salis, these properties already imply proportioal taxatio for soieties omprisig more tha two members 6 i the followig sese. Idividual performaes are taxed at a ertai rate ad overall tax reveue is distributed equally amog the soiety s members. 7 Theorem (Casajus, 204). Let > 2: A redistributio rule f : R! R satis es e iey (E), symmetry (S), ad mootoiity (M) if ad oly if there exists some 2 [0; ] suh that f i (x) = ( ) x i + X`2N x` for all x 2 R ad i 2 N : () The proof of this result, i partiular, the proof that the tax rate does ot deped o the overall performae of the soiety makes use of a ubouded-domai assumptio, i.e., the fat that we osider arbitrary great or small (egative) idividual performaes. Sie the iterpretatio of the taxatio of egative performaes is less oviig tha for oegative performaes, oe might woder whether Theorem remais true if oe restrits attetio to o-egative performaes. That is, the domai of redistributio rules is R +. 8 Moreover, the properties are required to hold oly for x; y 2 R +: We idiate this by a subsript + at the abbreviatios of the properties. Cosider the redistributio rule f ~ : R +! R give by ( 0; = 0; f ~ i (x) := ~ () x i + ~ for all x 2 R + ad i 2 N ; (2) () ; > 0 where = P`2N x`; ad the tax rate futio ~ : R ++! [0; ] ; ~ () := ; > ; ; for all 2 R ++ : (3) 5 This setup is related to bakrupty problems ad divisio rules. Thomso (2003, 205) provides surveys o these. 6 Throughout, we disregard the trivial ase = : 7 Casajus ad Huetter (204) obtai a similar result for oe-poit solutios of ooperative games with trasferable utility. For bakrupty problems ad divisio rules, Mouli (987, Theorem 2) establishes a related result. 8 We set R + := [0; +) ad R ++ = (0; +) : 3

4 Note that this is quite a reasoable rule. The soiety wishes to guaratee its members a miimum reward of but ot more. If this is ot possible, the soiety would like to be as lose as possible to this aim i the followig sese. If the soiety is ot so wealthy, P`2N x` ; redistributio is egalitaria, i.e., the tax rate is ad all members of the soiety ed up with the same reward. I a more a uet soiety, P`2N x` > ; redistributio is restrited to the level for P`2N x` = : That is, the tax rate is set to P`2N x` ; whih etails a overall tax reveue amoutig to : It is straightforward to show that the redistributio rule f ~ satis es the restrited versios of e iey, symmetry, ad mootoiity. Hee, the restritio of Casajus (204, Theorem ) to o-egative performaes does ot hold true. This triggers two questios. What are the impliatios of the restrited properties? How a the origial result be restored i the restrited setup? The rst questio is aswered by the ext theorem. Its proof is referred to Appedix A. The restritio of Casajus outerexample for = 2 also works o the domai of oegative alloatios. Theorem 2. Let > 2: A redistributio rule f : R +! R satis es e iey (E + ), symmetry (S + ), ad mootoiity (M + ) if ad oly if there is a mappig : R ++! [0; ] with the followig properties. (i) For all x 2 R + ad i 2 N ; we have f i (x) = ( 0; = 0; ( ()) x i + () ; > 0; where = X`2N x`: (4) (ii) For all ; d 2 R ++ suh that d ; we have (d) d () : (iii) For all ; d 2 R ++ suh that d ; we have (d) d () ( (d) ()) : The theorem says that, i the restrited setup, e iey, symmetry, ad mootoiity still imply proportioal taxatio but i a less strit way tha for the full domai. I partiular, for a give overall performae (), idividual performaes (x i ) are taxed a ertai rate ( ()) ad overall tax reveue ( () ) is distributed equally amog the members of the soiety. I otrast to Casajus (204, Theorem ), however, the tax rate may vary with the overall performae but ot i a arbitrary fashio. By oditio (ii) of the theorem, overall tax reveue aot derease with ireasig overall performae, i.e., the tax rate is ot allowed to drop too fast with ireasig total performae. Sie members of the soiety with a zero idividual performae obtai a fratio of overall tax reveue, this property protets the weakest members of the soiety. Give that oditio (ii) holds true, oditio (iii) is always satis ed whe the tax rate dereases. Hee, oditio (iii) requires the tax rate ot to irease too muh with ireasig overall performae. I partiular, overall tax paid (o a give overall performae) is 4

5 ot allowed to irease more tha the per-apita tax with ireasig total performae. This property protets the strogest member of a soiety, i.e., a member whose performae equals overall performae, beause this sole performer pays all the tax. As a (almost) immediate osequee of Theorem 2, zero taxatio has to be global, i.e., the tax rate is either zero for all positive overall performaes or positive for all positive overall performaes. Corollary 3. Let > 2: If a tax rate futio : R ++! [0; ] meets properties (ii) ad (iii) of Theorem 2 ad (d) = 0 for some d 2 R ++ ; the () = 0 for all 2 R ++ : Proof. Let : R ++! [0; ] meet properties (ii) ad (iii) of Theorem 2. Moreover, let d 2 R ++ be suh that (d) = 0: By (ii), 0 () ; i.e., () = 0 for all 2 (0; d] : By (iii), () d 0; i.e., () = 0 for all 2 [d; d) : Hee, () = 0 for all 2 [d; ) : I the ext theorem, we haraterize the tax rate futios from Theorem 2 by loal oditios. Its proof is referred to Appedix B. Theorem 4. Let > 2: A mappig : R ++! [0; ] satis es oditios (ii) ad (iii) of Theorem 2 if ad oly if it satis es the followig oditios. (a) The mappig is absolutely otiuous o ay ompat subiterval of R ++. (b) If is di eretiable at 2 R ++ ; the () 0 () : () If is di eretiable at 2 R ++ ; the 0 () () : I view of Corollary 3, we may fous o tax rate futios that ever assig a zero tax rate. For suh futios, oditios (a) ad (b) a be rewritte as 0 () () ; (5) i.e., oditios (b) ad () essetially represet a lower ad a upper boud o the elastiity of the tax rate futio. This idiates that there is rih family of tax rate futios that are ompatible with the three properties of redistributio. Below, we provide a kid of pathologial example that jumps bak ad forth betwee both boudaries. Some tehial remarks o Theorem 4 seem to be i order: Sie otiuity is a loal property ad sie absolute otiuity implies otiuity, the tax rate futio is otiuous o its whole domai. Moreover, i the proof of the theorem, we atually show that the tax rate futio eve is Lipshitz otiuous o ay subiterval that does ot otai a ope eighborhood of 0: However, the tax rate futio either eeds to be otiuously extedable to 0 or to be absolutely otiuous o R ++ : Let the tax rate futio z : R ++! [0; ] be give by 8 >< z () := 2 (k+) ; 2 < (k+) 2 ; (k+) >: 2 k ; 2 < for all 2 R ++ ad k 2 Z: (6) (k+) 2 k 5

6 It is straightforward to show that domai ad rage of z are as i Theorem 4 as well as that z satis es oditios (a), (b), ad (). I partiular, the elastiities of z o the iterior of the subdomais i (6) are ad ; respetively. Yet, i ay ope eighborhood of 0; z assumes ay value betwee ad (ompare Royde, 988, Problem 2, p. 0): 2 We olude this setio by aswerig the seod questio. Obviously, the redistributio rule f ~ is ot ivariat uder upsalig performaes, i.e., f ~ fails the followig property for > : Homogeeity, H +. For all x 2 R ; i 2 N, ad 2 R +, we have f i ( x) = f i (x) : O the oe had, homogeeity ould be iterpreted as that redistributio is ot a eted by the urrey used. Give this iterpretatio homogeeity is a rather atural requiremet o redistributio rules. Iterpretig homogeeity as sale ivariae of redistributio, o the other had, it is ot that iouous. Ayway, addig this property to the list of Casajus (204, Theorem ) restores its impliatios whe restrited to o-egative performaes. Note that the restritio of Casajus outerexample for = 2 also meets homogeeity. Theorem 5. Let > 2: A redistributio rule f : R +! R satis es e iey (E + ), symmetry (S + ), mootoiity (M + ), ad homogeeity (H + ) if ad oly if there exists some 2 [0; ] suh that f i (x) = ( ) x i + X`2N x` for all x 2 R + ad i 2 N : (7) Proof. Let > 2: By Theorem 2, the redistributio rule i (7) satis es E +, S +, ad M + : Sie 2 [0; ] ad x 2 R +; it also meets H +. Let f : R +! R meet E +, S +, M +, ad H +. By Theorem 2, there is a mappig : R ++! R suh that f is as i (4). Let 2 R ++ ad x; y 2 R + be suh that x` = y` = 0 for all ` 2 f; : : : ; g ; x = ; ad y =, i.e., y = x: Hee, we have () (4) = f (x) H + = f (y) (4) = () ad therefore () = () for all 2 (0; ). 3. Coludig remarks The mai isight of this paper is that mootoi redistributio of o-egative alloatios breathes the spirit of Casajus (204) haraterizatio. I partiular, mootoi redistributio implies that idividual taxatio essetially does ot deped o idividual performae but o overall performae. Yet, the elastiity of the tax rate with respet to overall performae is bouded above ad below. Oe limitatio of our approah is that, for example, we do ot distiguish betwee a low idividual performae due to low potetial ad a low idividual performae due to low e ort. I the former ase, the soiety might wish support suh a member of the soiety via redistributio, while i the latter ase it would t. Hee, it seems to desirable to study redistributio i a framework where members of the soiety are desribed by both their atual performae ad their potetial. 6

7 Appedix A. Proof of Theorem 2 First, the oly-if part. Let > 2 ad let f : R +! R meet E +, S +, ad M +. Set R 2 + := x 2 R 2 + j x x 2 : By M +, there are mappigs F i : R 2 +! R; i 2 N suh that! f i (x) = F i x i ; X`2N x` for all x 2 R + ad i 2 N : (A.) Next, we show that F i = F j =: F for all i; j 2 N. Let (a; ) 2 R 2 + ad i; j; k 2 N ; i 6= j 6= k 6= i: Let y; z 2 R + be give by y i = a ad y` = a for all ` 2 N fig (A.2) ad We have z j = a ad z` = a for all ` 2 N fjg : (A.3) F i (a; ) (A.) = f i (y) (A.2),E +,S + = ( ) f k (y) (A.2),(A.3),M + = ( ) fk (z) (A.3),E +,S + = f j (z) (A.) = F j (a; ) : By E + ad M +, the mappig F has the followig properties. E iey, E*. For all a 2 R +; we have P`2N F a`; P k2n a k = P`2N a`: Mootoiity, M*. For all a; b; ; d 2 R + suh that b a ad d ; we have F (b; d) F (a; ) : For 2 R + ; > 0; let the mappig : [0; ]! R be give by For a; b 2 [0; ] ; a + b ; we have (a) := F (a; ) F (0; ) for all a 2 [0; ] : (A.4) F (a; ) + F (b; ) + ( By (A.4) ad (A.5), we further obtai a b 2) F 2 ; E* = F (a + b; ) + F (0; ) + ( 2) F a b 2 ; : (A.5) (a) + (b) = (a + b) : (A.6) This already etails ( a) = (a) for all a 2 [0; ] ad 2 Q + (A.7) 7

8 suh that a 2 [0; ] : By M*, is mootoi, i.e., (a) (b) for all a; b 2 [0; ] suh that a b: Sie Q + is a dese subset of R +, (A.7) etails ( a) = (a) for all a 2 [0; ] ad 2 R + (A.8) suh that a 2 [0; ] : By E*, F (0; 0) = 0: For all 2 R ++ ; set := By (A.4), (A.8), ad (A.9), we have F (; ) F (0; ) = () : (A.9) F (a; ) = a + F (0; ) for all (a; ) 2 R 2 +: (A.0) Moreover, we obtai E* = F ; (A.0) = + F (0; ) ; i.e., F (0; ) = ( ) for all 2 R ++ ad therefore F (a; ) = a + ( ) for all (a; ) 2 R 2 +; > 0: (A.) By (A.9) ad M*, we have 0 for all 2 R ++ : Moreover, we have 0 E* = F (0; 0) M* F (0; ) (A.) = ( ) ; i.e., for all 2 R ++ : Hee, the rage of the mappig : R ++! R give by () := for all 2 (0; ) (A.2) is as i the theorem. By (A.), (A.), ad (A.2), also meets part (i). For ; d 2 R ++ ; d > ; we have () (A.),(A.2) = F (0; ) M* F (0; d) (A.),(A.2) = (d) d ; i.e., satis es part (ii) of the theorem. Part (iii) drops from ( ()) + () (A.),(A.2) = F (; ) M* F (; d) (A.),(A.2) = ( (d)) + (d) d : Ad ow, the if part. Let the mappig : R ++! [0; ] obey (i) (iii) ad let the redistributio rule f : R +! R be give as i (4). It is immediate that f meets E ad S. 8

9 Let x; y 2 R + ad i 2 N be suh that y i x i ad d := P`2N y` P`2N x` =: : If = 0; the x i = 0 ad f i (x) = 0 f i (y) by (4) ad (d) 2 [0; ] : For > 0; we have f i (y) (4) = ( (d)) y i + (d) d (d)2[0;] ( (d)) x i + (d) d ( (d)) + (d) d + x i x i h( ()) + () i + x i = x i (ii),(iii) = ( ()) x i + () (4) = f i (x) : (d) d h () i Hee, f also meets M +. Appedix B. Proof of Theorem 4 First, the if part. Let > 2: Further, let the mappig : R ++! [0; ] satisfy (a), (b), ad (). Fix ; d 2 R ++ ; d >. By (a) ad Royde (988, p. 0), is di eretiable almost everywhere o [; d], its derivative 0 is Lebesgue itegrable, ad (a) = () + Z a 0 (t) dt for all a 2 [; d] : Cosider the mappig R : [; d]! R + give by R (a) = (a) a for all a 2 [; d] : By ostrutio, R is absolutely otiuous, i.e., R is di eretiable almost everywhere o [; d], its derivative R 0 is Lebesgue itegrable, ad R (d) = R () + Z d R 0 (t) dt: (B.) If is di eretiable at t 2 [; d] ; the R is also di eretiable at t ad R 0 (t) = 0 (t)t+ (t) : By (b), R 0 (t) 0: Hee, (B.) implies (d) d = R (d) R () = () : Sie ad d were hose arbitrarily, satis es oditio (ii) of Theorem 2. Cosider the mappig S : [; d]! R give by S (a) = (a) a (a) for all a 2 [; d] : If (d) () ; the (ii) implies S (d) S () : Suppose (d) > () : By (a), is otiuous. Hee, there is some b 2 (; d] suh that (b) = (d) ad (a) (d) for all a 2 [; b] : (B.2) Thus, S (d) = (d) d (d) (b) b (b) = S (b) : (B.3) 9

10 By ostrutio, S is absolutely otiuous, i.e., S is di eretiable almost everywhere o [; d], the derivative S 0 is Lebesgue itegrable, ad S (a) = S () + Z a S 0 (t) dt for all a 2 [; d] : (B.4) If is di eretiable at t 2 [; d] ; the S is also di eretiable at t ad S 0 (t) = 0 (t) t + (t) 0 (t) () 0 (t) t + ( ) 0 (t) d 0 (t) = 0 (t) (t ) : (B.5) Hee, we have Fially, we obtai S (b) (B.4),(B.5) S () + = S () + 0 (t) (t 0 (t) t + (t) dt = S () + (b) (b ) = S () + (B.2) S () : (b) (t) dt ) dt (t) dt (t) dt 0 (t) dt (B.6) (d) d (B.3),(B.6) (d) = S (d) S () = () () : Sie ad d were hose arbitrarily, satis es oditio (iii) of Theorem 2. Ad ow, the oly-if part. Let the mappig : R ++! [0; ] satisfy oditios (ii) ad (iii) of Theorem 2. Fix ; d 2 R ++ ; d >. For a; b 2 [; d] ; b > a; we have ad Hee, (b) (a) (ii) (b) a (b)2[0;];b>a (b a) (b)2[0;];b>a (b a) (b) a j (b) (a)j jb aj : (b a) (B.7) (b a) (iii) (b) (a) : (B.8) 0

11 Sie a ad b were hose arbitrarily, is Lipshitz otiuous o [; d] : By Royde (988, Problem 20 (a), p. 2), is absolutely otiuous o [; d] : Let be di eretiable at a 2 R ++ : By (B.7), we have (b) b (a) a (b) for all b 2 (a; ) : a Takig the limit as b approahes a gives 0 (a) a (a) ; where we make use of the fat that a absolutely otiuous mappig is otiuous ad that otiuity is loal property. Hee, satis es (b). By (B.8), we have (b) (b) (a) b a a for all b 2 (a; ) : Takig the limit as b approahes a gives (a) 0 (a) a: Hee, satis es (). Akowledgemets We are grateful to Frak Huetter ad Hervé Mouli for helpful ommets o this paper. Fiaial support by the Deutshe Forshugsgemeishaft (grat CA 266/4-) is gratefully akowledged. Referees Casajus, A., 204. Mootoi redistributio of performae-based alloatios: a ase for proportioal taxatio. Theoretial Eoomis (forthomig). Casajus, A., Huetter, F., 204. Weakly mootoi solutios of ooperative games. Joural of Eoomi Theory 54, Fleubaey, M., Maiquet, F., 20. A Theory of Fairess ad Soial Welfare. Cambridge Uiversity Press. Friedma, M., 962. Capitalism ad Freedom. Uiversity of Chiago Press, Ch. x. The Distributio of Iome. Hall, R. E., 996. Fairess ad E iey i the Flat Tax. AEI Press, Washigto, D.C. Hall, R. E., Rabushka, A., 985. The Flat Tax. Hoover Istitutio Press, Staford, CA. Hayek, F. A., 960. The Costitutio of Liberty. Uiversity of Chiago Press, Ch. 20, Taxatio ad Redistributio, pp Lambert, P., The Distributio ad Redistributio of Iome, 3rd Editio. Mahester Uiversity Press. MCulloh, J. R., 845. A Treatise o the Priiples ad Pratial I uee of Taxatio ad the Fudig System. Logma, Brow, Gree, ad Logmas, Lodo. Mill, J. S., 848. O the I uee of Govermet. Vol. V of Priiples of Politial Eoomy with some of their Appliatios to Soial Philosophy. Logmas, Gree ad Co., Ch. II. O the Geeral Priiples of Taxatio. Mouli, H., 987. Equal or proportioal divisio of a surplus, ad other methods. Iteratioal Joural of Game Theory 6 (3), Royde, H. L., 988. Real Aalysis, 3rd Editio. Mamilla, New York. Thomso, W., Axiomati ad game-theoreti aalysis of bakrupty ad taxatio problems: A survey. Mathematial Soial Siees 45, Thomso, W., 205. Axiomati ad game-theoreti aalysis of bakrupty ad taxatio problems: A update. Mathematial Soial Siees 74, 4 59.