An equilibrium-conserving taxation scheme for income from capital
aa r X i v : . [ q -f i n . E C ] A ug An equilibrium-conserving taxation scheme for income from capital
Jacques Tempere
Theory of Quantum and Complex Systems, Universiteit Antwerpen, Universiteitsplein 1, B-2610 Antwerpen, Belgium (Dated: October 9, 2018)Under conditions of market equilibrium, the distribution of capital income follows a Pareto power law, withan exponent that characterizes the given equilibrium. Here, a simple taxation scheme is proposed such that thepost-tax capital income distribution remains an equilibrium distribution, albeit with a di ff erent exponent. Thistaxation scheme is shown to be progressive, and its parameters can be simply derived from (i) the total amountof tax that will be levied, (ii) the threshold selected above which capital income will be taxed and (iii) the totalamount of capital income. The latter can be obtained either by using Piketty’s estimates of the capital / laborincome ratio or by fitting the initial Pareto exponent. Both ways moreover provide a check on the amount ofdeclared income from capital. PACS numbers: 89.65.Gh, 89.75.Da
I. INTRODUCTION
The distribution of income has been studied for a longtime in the economic literature, and has more recently be-come a topic of investigation for statistical physicists turningto econophysics [1–4]. The income distribution is character-ized by a density function f ( x ) such that f ( x ) dx is the numberof individuals earning income between x and x + dx . Fromthe empirical data obtained from tax records, two di ff erentregimes are readily distinguished. For income levels belowa certain threshold x c , the distribution follows and exponen-tial (Boltzmann) law, f ( x ) ∝ exp( − x / ¯ x ), whereas for incomelevels above x c the distribution is better fitted by a power law, f ( x ) ∝ x − γ . Note that for the bottom incomes, a deviationfrom the Boltzmann law is visible. This is due to redistri-bution (such as social security benefits) which lifts a certainamount of people above a poverty threshold x pov .For income above the poverty level but below x c the distri-bution is very well fitted by a Gibbs distribution [5, 6]. Taxrecords that keep track of the source of income indicate thatincome in this regime is dominated by labor income (salariesand wages) [7]. In this regime, economic transactions canbe modelled by additive processes [4, 6]: money exchangeshands between agents but the total amount of money is con-served over the transaction. For example, each month an em-ployee gets a certain sum of money added to his account,and this sum is subtracted from the account of the employer’scompany. Using this principle of local money conservation,Dragulescu and Yakovenko [5] have shown that the equilib-rium distribution of money over the agents involved in addi-tive transactions follows a Boltzmann-Gibbs exponential dis-tribution. Note that this is a strongly simplified model of eco-nomic activity: it is clear that in reality global money conser-vation is violated. Indeed, banks can issue (or recall) loans,thereby increasing (or decreasing) the total money supply.This brings us to the higher incomes, x > x c . As notedalready by Pareto in 1897, these follow a power law [8].Records that keep track of the source of income reveal that in-come in the power law regime is mainly capital income (rent,profits, interests, dividends,...) [9, 10]. For these types ofincome, economic transactions are better modeled by multi-plicative processes [6, 11, 12]. In contrast to a wage worker, a rentier expects that at each time step the money in his invest-ment is multiplied by an interest factor. For such processes,it is log( x ) rather than x which is conserved locally, and thisleads naturally to a power law rather than an exponential equi-librium distribution [13].The change from exponential regime to power-law regimeoccurs in a narrow interval [14, 15] around x c , allowing toseparate not only the income in two sources ( M lab from laborand M cap from capital), but also the population in two groups( N lab and N cap , respectively). Of course, these are not clearlydelineated, and also people filing tax forms for income below x c have a portion of their income coming from returnon cap-ital, but the problem can be greatly simplified by taking themain source of income to be the entire income. The value x c separating the exponential from the power-law regimes liesbetween three and four times the average income in the expo-nential part of the distribution [14–16].The question that I wish to address in this paper is how tolevy taxes such that the immediate after-tax income distribu-tion remains in equilibrium (i.e. of Boltzmann type for x < x c ,and of Pareto type for x > x c ). The parameters ( ¯ x and γ ) of thepre-tax and after-tax distributions will be di ff erent, reflectingthe change from one equilibrium to another rather than a shiftto a non-equilibrium distribution. Free-market advocates canargue that this type of equilibrium-to-equilibrium taxation isthe least disruptive choice. The conjecture behind this is thatwhen the income distribution is pushed strongly out of equi-librium the market is far from optimized as transactions thatare wished for may not take place. Hence a taxation schemethat results in an out-of-equilibrium after-tax income distri-bution would be more detrimental to the market. Of course,whether any scheme is just or desirable is well beyond thescope of this paper. Nevertheless, given the current debatetaking place (in the USA, the UK and the EU) of how to taxthe rich, I believe the results presented here can contribute toan informed discussion. II. CAPITAL AND LABOR INCOME FOR BELGIUM
To illustrate the results with numbers, I use my home coun-try of Belgium as an example, taking x c ≈
100 k e . There were x pov = € x = € × × × x ( k €) f ( x ) d x FIG. 1. (color online) The (annual) income distribution in Belgiumin 2014 as obtained from personal tax records [17], in bins of d x = e . The distribution follows a Boltzmann law (dashed line) exceptfor the lowest income. Redistributive taxation lifts people with an in-come below x pov (area shaded in dark red) above this poverty thresh-old (area shaded in light blue). N tot = . × people filing a non-zero income tax recordin Belgium in 2014, the latest year with complete information[17]. Of these, N cap = . × (or 2.8% of the total) in-dicate an income above x c . The remaining N lab = . × are in the exponential regime and represent a cumulative in-come of M lab = . e . As seen in Fig. 1, there is a strongdeviation of the exponential regime for income levels below x pov = .
25 k e : this is the poverty threshold. People belowthis threshold receive social security benefits lifting them toroughly the threshhold level and slightly above.Estimating the amount of capital income M cap for Belgiumis di ffi cult to do based on tax forms. The reason is that not allsources of capital income need to be declared in Belgium: Forinstance, income from renting out apartments or o ffi ces is notdeclared. Piketty [18] provides an alternative way to estimate M cap from M lab . His study of the historical ratio between cap-ital income and labor income for developed economies showsthat this income has its own slow dynamics over time. Cur-rently, the capital share of income in rich countries stands at25-30% of national income. The dynamics of the capital / laborincome ratio is slow enough such that the income distribu-tion is close to equilibrium at any time. Taking the above-mentioned capital share of the national income into accountresults in an estimate[ ? ] of M cap ≈
60 G e .The additive class, x pov < x < x c is subject to the (normal-ized) Boltzmann-Gibbs distrubution f lab ( x ) = N lab ¯ x exp( − x / ¯ x ) , (1)with ¯ x = M lab / N lab the average income in this class.The multiplicative class is subject to the power law distri-bution f cap ( x > x c ) = ( γ − N cap x c ( x / x c ) − γ . (2)The Pareto parameter γ is fixed by M cap through the normal- ization M cap = R ∞ x c x f cap ( x ) dx by γ = M cap − N cap x c M cap − N cap x c . (3)This restricts γ >
2, since M cap > N cap x c . The estimate M cap =
60 G e based on Piketty’s observations corresponds to γ = .
4. This value is in agreement with the estimate of γ ≈ . γ (in combination with x c and N cap ) may be used to estimatethe total amount of income from capital. Detailed data forthe high-income distribution is not publicly available in Bel-gium, and as mentioned above, exemption of some capital in-come sources in Belgium complicates data finding. However,for countries with an obligation to declare all capital income,Eq. 3 could be used to estimate the amount of undeclared in-come and hence the level of tax evasion by rentiers. A simi-lar proposal, based on deviations from the Pareto distribution,was introduced to estimate the size of shadow banking [19]. III. TAXING CAPITAL INCOME
Suppose one wants to levy taxes to raise a given amount ∆ M of money. For example, to bring all the poor to a mini-mum wage of x pov , one would need ∆ M = Z x c ( x c − x ) f lab ( x ) dx (4) = M lab h ( x pov / ¯ x ) − (1 − e − x pov / ¯ x ) i . (5)Using the numbers listed above for Belgium, the topping upof all lower incomes to x pov would require 16.4 G e .Suppose moreover that one wants to obtain the amount of ∆ M by taxing capital income in such a way that the after-taxcapital income distribution follows again a Pareto law. Thepost-tax power law distribution necessarily has a di ff erent ex-ponent η > γ , given by η = M cap − ∆ M ) − N cap x c M cap − ∆ M − N cap x c . (6)In our Belgian example, this would mean a change from γ = .
41 to η = .
66. To describe the taxation scheme, weintroduce a function X ( x ) that gives the post-tax net income X as a function of the pre-tax income x . The distribution ofpost-tax income is denoted by f post-taxcap ( X ) = ( η − N cap x c ( X / x c ) − η . (7)This distribution has to obey f post-taxcap ( X ) dX = f cap [ x ( X )] dx . (8)Substituting the Pareto distributions in the above equationyields a di ff erential equation for X(x),( η − x − η c X − η ( x ) dXdx = ( γ − x − γ c x − γ . (9) τ = τ = τ = τ = x / x c t a x r a t e ( % ) FIG. 2. Taxation levels as a function of capital income, for di ff erentvalues of the parameter τ , which is fixed by the amount of tax ∆ M that will be levied from the total capital income M cap . The tax rateis progressive and taxation starts at the income level x c separatingthe Boltzmann-type labor income distribution from the Pareto-typecapital income distribution. It solution depends on the parameter τ = γ − η − , (10)and is given by X ( x ) = x − τ c x τ . (11)As τ <
1, this solution corresponds to a weighted geo-metric averaging between the capital income and the thresh-old x c where main income switches from additive to multi-plicative. For a pre-tax income x the corresponding tax rate T ( x ) = ( x − X ) / x is given by T ( x ) = − ( x c / x ) − τ . (12)This represents a progressive tax rate. Taking our example τ = .
85, and the resulting tax rate is shown as the full curve in Fig. 2. Whereas for a capital income of 120 k e (slightlyabove the threshold when one can be called a rentier or “rich”)the tax rate is about 3%, at 200 k e it has risen to 10%, dou-bling again to 20% for 500 k e . IV. DISCUSSION AND CONCLUSIONS
Eqs. (6),(10) and (12) represent a simple taxation schemethat preserves the power law nature of capital income. In orderto implement it, policy makers need to select a threshold x c ofincome from capital above which the tax is levied, and choosea value of τ , or equivalently, an amount ∆ M of money thatthe tax should raise. The basic idea behind preserving thepower law is that this law represents a distribution in whichthe market is in equilibrium.Free market proponents claim, in the first fundamental the-orem of welfare economics, that a competetive market pro-duces a (non-unique) Pareto e ffi cient equilibrium outcome.Moreover, in this train of thought, a social planner could se-lect the most suitable e ffi cient outcome by lump sum transfers,according to the second fundamental theorem. In this context,the current proposal for taxation is precisely a way to organ-ise a transfer that links one equilibrium for capital income toanother.What if one would use the same logic to labor income?Changing one Boltzmann distribution into another only re-quires a scale change x → α x where α = − ∆ M / M lab . Thiscorresponds to a proportional tax system (a “flat tax”, such asa fixed sales tax). This is not always seen as the socially mostdesirable outcome as it penalizes the low-income segment ofthe population, who have less disposable income. In essence,the current proposal represents a flat tax on log( x ), modifiedby the presence of a the threshold x c . Regardless of the desir-ability debate, it is clear that proponents of a flat tax for laborincome who base their arguments on market e ffi ciency, shouldthen logically advocate the current progressive tax on capitalincome. Another commonly encountered argument for a flattax is its simplicity. In this respect, the proposal for a progres-sive capital income taxation put forward in this paper o ff ers ascheme which, at least to a physicist, is of similar simplicity. [1] Econophysics of Wealth Distributions , ed. A. Chatterjee, S.Yarlagadda, B.K. Chakrabarti (Springer Verlag, Milan, 2005).[2] A. Chatterjee and B.K. Chakrabarti, Eur. Phys. J. B , 135(2007).[3] V.M. Yakovenko and J. Barkley Rosser, Rev. Mod. Phys. ,1703 (2009).[4] B.K. Chakrabarti, A. Chakraborti, S.R. Chakravarti, and A.Chatterjee, Econophysics of Income and Wealth Disitributions (Cambridge University Press, Cambridge, 2013).[5] A.A. Dragulescu and V.M. Yakovenko, Eur. Phys. J. B , 723(2000).[6] S. Ispolatov, P.L. Krapivsky, S. Redner, Eur. Phys. J. B , 267(1998).[7] G. Willis and J. Mimkes, e-print arXiv:cond-mat / Cours d’Economie Politique (ed. F. Rouge, Librairie de l’Universit´e, Lausanne, 1897).[9] F. Clementi and M. Gallegati,
Income Inequality Dynamics:Evidence from a Pool of Major Industrialized Countries , Talkat the International Workshop of Econophysics of WealthDistributions, Kolkata, March 15-19, 2005. Retrieved from .[10] T.L. Hungerford,
Changes in the Distribution of In-come Among Tax Filers Between 1996 and 2006: TheRole of Labor Income, Capital Income, and Tax Pol-icy , Congressional Research Service, 2011. Retrieved from http://taxprof.typepad.com/files/crs-1.pdf [11] M. Levy and H. Levy, Rev. Econ. Stat. , 709 (2003).[12] Y. Fujiwara, W. Souma, H. Aoyama, T. Kaizoji, and M. Aoki,Physica A , 598 (2003).[13] M. Levy and S. Solomon, Int. J. Mod. Phys. C , 595 (1996). [14] A.A. Dragulescu and V.M. Yakovenko, Physica A , 213(2001).[15] A.A. Dragulescu and V.M. Yakovenko, Statistical mechanicsof money, income, and wealth: a short survey . In Garrido,P.L., and Marro, J. (eds.),
Modeling of Complex Systems: Sev-enth Granada Lectures , vol. 661. American Institute of Physics(AIP) Conference Proceedings. Melville, NY, AIP (2003).[16] A.C. Silva and V.M. Yakovenko, Europhys. Lett. , 304(2005).[17] Data source: Algemene Directie Statistiek - Statistics Belgium, Fiscale inkomens, http://statbel.fgov.be/nl/modules/publications/statistiques/arbeidsmarkt_levensomstandigheden/Statistique_fiscale_des_revenus.jsp ,retrieved on July 1st, 2017.[18] T. Piketty, Capital in the Twenty-First Century (translationA. Goldhammer, Harvard University Press, Cambridge, USA,2014).[19] D. Fiaschi, I. Kondor, M. Marsili, V. Volpati, PLoS ONE9