Monotonic redistribution of performance-based allocations: A case for proportional taxation

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1 Theoretical Ecoomics 10 (2015), / Mootoic redistributio of performace-based allocatios: A case for proportioal taxatio Adré Casajus Ecoomics ad Iformatio Systems, HHL Leipzig Graduate School of Maagemet Withi a simple setup, we show that proportioal taxatio is implied by three properties: efficiecy, symmetry, ad mootoicity. Efficiecy: redistributio has o cost. Symmetry: members of the society with the same performace obtai the same reward after redistributio. Mootoicity: wheever both the performace of a certai member of the society as well as the overall performace of the society do ot decrease, the this member s reward after redistributio does ot decrease. Keywords. Redistributio, proportioal taxatio, efficiecy, symmetry, mootoicity. JEL classificatio. D63, H20. The momet you abado the cardial priciple of exactig from all idividuals the same proportio of their icome or of their property, you are at sea without rudder or compass, ad there is o amout of ijustice ad folly you may ot commit. McCulloch (1975, p. 174) 1. Itroductio I 1845, McCulloch, author of the most extesive ad systematic treatmet of public fiace i the classical literature, made the above case for proportioal taxatio. Later o, otable others joied him, for example, Mill (1848, Chapter 2), Hayek (1960, Chapter 20), Friedma (1962, Chapter X), ad more recetly Hall ad Rabushka (1985) ad Hall (1996). 1 I this paper, we make a axiomatic case for proportioal taxatio 2 via mootoic redistributio of performace-based allocatios. 3 Particularly, we cosider a society i which its members first are rewarded based o their idividual cotributios to the society s wealth (idividual performace). Moder societies, however, base the allocatio Adré Casajus: mail@casajus.de We are grateful to Reé va de Brik, Faruk Gul, Adreas Hoffma, Frak Huetter, Erico Schöbel, Harald Uhlig, Thomas Steger, ad Harald Wiese for helpful commets o this paper. 1 For a geeral treatmet of taxatio, see Musgrave ad Musgrave (1976)orStiglitz (2000), for example. 2 Fleubaey ad Maiquet (2011, Chapters 10 ad 11), for example, provide a survey of axiomatic foudatios of taxatio. 3 Lambert (2002) provides a overview o the formal treatmet of redistributio. Copyright 2015 Adré Casajus. Licesed uder the Creative Commos Attributio-NoCommercial Licese 3.0. Available at DOI: /TE1897

2 888 Adré Casajus Theoretical Ecoomics 10 (2015) of wealth amog their members ot oly o idividual performace but also o egalitaria or solidarity priciples. This leads to the questio of how idividual cotributios should be redistributed withi a society. We cosider three properties of redistributio rules: efficiecy, symmetry, ad mootoicity. For the sake of simplicity, we assume that redistributio has o cost, i.e., redistributio is efficiet. Moreover, members of the society with the same performace obtai the same reward after redistributio, i.e., redistributio is symmetric. Our third property mootoicity requires that wheever both the performace of a certai member of the society as well as the overall performace of the society do ot decrease, the this member s reward after redistributio should ot decrease. For societies comprisig more tha two members, it turs out that the redistributio rules satisfyig these properties are of a particularly simple form. First, the idividual performace-based allocatios are taxed proportioally at a certai rate, ad secod, the overall tax reveue is distributed equally withi the society. The ext sectio gives a formal accout of this result. Some remarks coclude the paper. The Appedix cotais the proof of our result. 2. Mootoic redistributio rules ad proportioal taxatio I this paper, we cosider a particularly simple model of a society. Its members are distiguished oly by their idividual cotributios to the society s wealth i a certai period of time (for short, performace). I reality, the idividual gross icome may be viewed as a idicator for these idividual cotributios. Techically, we cosider the -member society N := {1 }, N, i.e., the society s members are represeted by atural umbers. The idividual performaces are give by a vector x R, i.e., we allow for egative performaces. This ca be justified by cosiderig a period of time i which a member of the society or the etire society does ot fare that well. I real-life societies, members are ot just rewarded accordig to their performaces. A cosiderable amout of the society s wealth is redistributed via taxatio ad public spedig. I our simple model, this is reflected by redistributio rules. A redistributio rule for a -perso society is a mappig f : R R.Forx R ad i N, f i (x) deotes the reward of member i of the society after redistributio. I this framework, the properties of redistributio rules advaced i the Itroductio ca be formalized as follows. Efficiecy (E). For all x R,wehave l N f l (x) = l N x l. The very idea of redistributio suggests that the sum of idividual rewards after redistributio should ot be greater tha before. I additio, efficiecy requires that redistributio comes at o cost. While i real life, redistributio is costly, efficiecy might be acceptable i our simple ad abstract framework. Symmetry (S). For all x R ad i j N such that x i = x j,wehavef i (x) = f j (x).

3 Theoretical Ecoomics 10 (2015) Redistributio of performace-based allocatios 889 I our simple framework, the society s members are fully described by their performace. Therefore, rewards after redistributio should be the same for members with thesameproductivity. Mootoicity (M). For all x y R ad i N such that l N x l l N y l ad x i y i,wehavef i (x) f i (y). This property relates rewards of a member of the society whe idividual performaces chage i a mootoic way. Wheever both the society s overall performace ad the performace of a particular member of the society do ot decrease, the this member s reward after redistributio does ot decrease. The ituitio behid this property is as follows. Nodecreasig overall performace guaratees that o member of the society ecessarily has to be rewarded less tha before the chage. For a member whose performace does ot decrease at the same time, mootoicity esures that she actually is ot awarded less, which seems to be desirable. The followig theorem shows that redistributio rules that obey the above three properties etail proportioal taxatio for societies comprisig more tha two members, i a sese. Its proof is deferred to the Appedix. Of course, the oe-member case is trivial. Theorem 1. Let 2. A redistributio rule f : R R satisfies efficiecy (E), symmetry (S), ad mootoicity (M) if ad oly if there exists some τ [0 1] such that f i (x) = (1 τ) x i + τ for all x R ad i N (1) l N x l It is immediate that the redistributio rules (1) satisfy all the properties, eve for = 2. Moreover, oe might have the strog feelig that there caot be other redistributio rules that do so. Yet, the theorem fails for = 2. It is straightforward to show that the redistributio rule f : R 2 R 2 give by f (x) = (x 1 x 2 ) if x 1 >x 2 ad f (x) = ( x 1+x 2 2 x 1+x 2 2 ) if x 1 x 2 for all x R 2 satisfies the three properties. To see how we ca iterpret formula (1) i terms of proportioal taxatio, ote that performace is taxed at a rate of τ. This way, member i N keeps a amout of (1 τ) x i from his performace x i. I additio, member i obtais oe th of the overall tax reveue of τ l N x l. The latter ca be iterpreted as that all members of the society beefit equally from public spedig, which fits our assumptio that the society s members are equal up to performace. Note that Theorem 1 leaves the tax rate idetermiate. Mootoic redistributio oly requires proportioal taxatio at a certai rate. Hece, the tax rate has to be determied by other criteria, for example, optimality cosideratios. 3. Cocludig remarks I this paper, we cosider the possibly simplest meaigful framework to study redistributio withi a society. We show that three ituitive ad plausible properties of redistributio rules efficiecy, symmetry, ad mootoicity joitly etail proportioal taxatio of performace-based allocatios.

4 890 Adré Casajus Theoretical Ecoomics 10 (2015) I our simple framework, however, we caot distiguish betwee icome tax ad cosumptio tax. Hece, Theorem 1 implies proportioal overall taxatio. I view of the regressive effect usually attributed to cosumptio tax, our fidigs do ot rule out progressive taxatio of icome. Appedix: Proof of Theorem 1 It is straightforward to show that the redistributio rules i formula (1)obeyE,S,adM. Let f : R R meet E, S, ad M. For = 1, the secod part of the claim drops from E. Let ow >2. By M, there are mappigs F i : R 2 R, i N such that ( f i (x) = F i x i ) x l for all x R ad i N (A.1) l N Next, we show that F i = F j =: F for all i j N. Let (a c) R 2 ad i j k N, i j k i. Lety z R be give by ad We have y i = a ad y l = c a 1 z j = a ad z l = c a 1 for all l N \{i} for all l N \{j} (A.2) (A.3) F i (a c) (A.1) = f i (y) (A.2) E S = c ( 1) f k (y) (A.2),(A.3) M = c ( 1) f k (z) (A.3) E S = f j (z) (A.1) = F j (a c) By E ad M, the mappig F has the followig properties. Efficiecy (E*). For all a R,wehave l N F(a l k N a k ) = l N a l. Mootoicity (M*). For all a a c c R such that a a ad c c,wehavef(a c) F(a c ). For c R, let the mappig c : R R be give by c (a) := F(a c) F(0 c) for all a R (A.4) For a b c R,wehave ( ) c a b F(a c)+ F(b c)+ ( 2) F 2 c E = F(a+ b c) + F(0 c)+ ( 2) F ( c a b 2 ) c (A.5)

5 Theoretical Ecoomics 10 (2015) Redistributio of performace-based allocatios 891 By (A.4) ad(a.5), we obtai c (a) + c (b) = c (a + b) for all a b c R This already etails c (ρ a) = ρ c (a) for all a c R ad ρ Q (A.6) By M*, c is mootoic, i.e., c (a) c (b) for all a b R such that a b. Sice Q is a dese subset of R,(A.6) etails For all c R,set c (ρ a) = ρ c (a) for all a c ρ R (A.7) By (A.4), (A.7), ad (A.8), we have α c := F(1 c) F(0 c) (A.8) Moreover, we obtai F(a c)= α c a + F(0 c) for all a c R (A.9) ( ) c E = c F c (A.9) = α c c + F(0 c) i.e., F(0 c)= (1 α c ) c for all c R ad, therefore, F(a c)= α c a + (1 α c ) c for all a c R (A.10) Now, we show that α c does ot deped o c. Letc c R, c>c. Suppose α c α c. By (A.10), we have F(a c) F(a c ) = (α c α c ) a + (1 α c ) c (1 α c c ) for all a R i.e., oe ca fid some a R such that F(a c )>F(a c), cotradictig M*. Thus, α c = α c =: α for all c c R. By (A.8)adM*,wehaveα 0. Moreover, we have 0 (A.10) = F(0 0) M F(0 1) (A.10) = 1 α i.e., 1 α. Fially, by (A.1), (A.10), (1), ad our fidigs o α, f is as i formula (1). Refereces Fleubaey, Marc ad Fraçois Maiquet (2011), A Theory of Fairess ad Social Welfare. Cambridge Uiversity Press, Cambridge. [887] Friedma, Milto (1962), Capitalism ad Freedom. Uiversity of Chicago Press, Chicago. [887]

6 892 Adré Casajus Theoretical Ecoomics 10 (2015) Hall, RobertE. (1996), Fairess ad Efficiecy i the Flat Tax. AEI Press, Washigto, DC. [887] Hall, Robert E. ad Alvi Rabushka (1985), The Flat Tax. Hoover Istitutio Press, Staford. [887] Hayek, Friedrich A. (1960), The Costitutio of Liberty. Uiversity of Chicago Press, Chicago. [887] Lambert, Peter (2002), The Distributio ad Redistributio of Icome, third editio. Machester Uiversity Press, Machester. [887] McCulloch, Joh R. (1975), A Treatise o the Priciples ad Practical Ifluece of Taxatio ad the Fudig System. Logmas, Brow, Gree, ad Logmas, Lodo. [887] Mill, Joh S. (1848), Priciples of Political Ecoomy with Some of Their Applicatios to Social Philosophy, Vol. V, O the Ifluece of Govermet. Logmas, Gree ad Co., Lodo. [887] Musgrave, Richard A. ad Peggy B. Musgrave (1976), Public Fiace i Theory ad Practice, secod editio. McGraw-Hill, New York. [887] Stiglitz, Joseph E. (2000), Ecoomics of the Public Sector, third editio. W.W. Norto, New York. [887] Submitted Fial versio accepted Available olie