aa r X i v : . [ ec on . GN ] J a n Dynamics of contentment
Alexey A. BurlukaFaculty of Engineering & Environment, Northumbria University,Newcastle-upon-Tyne, NE1 8ST, The United KingdomE-mail: [email protected] 15, 2021
Abstract
A continuous variable changing between 0 and 1 is introduced to charac-terise contentment, or satisfaction with life, of an individual and an equationgoverning its evolution is postulated from analysis of several factors likelyto affect the contentment. As contentment is strongly affected by materialwell-being, a similar equation is formulated for wealth of an individual andfrom these two equations derived an evolution equation for the joint distribu-tion of individuals’ wealth and contentment within a society. The equationso obtained is used to compute evolution of this joint distribution in a soci-ety with initially low variation of wealth and contentment over a long periodtime. As illustration of this model capabilities, effects of the wealth tax rateare simulated and it is shown that a higher taxation in the longer run maylead to a wealthier and more content society. It is also shown that lowerrates of the wealth tax lead to pronounced stratification of the society interms of both wealth and contentment and that there is no direct relationshipbetween the average values of these two variables.
Introduction
Taken at a level of an individual, the satisfaction with life, i.e. oneself, own envi-ronment, current moment and many other factors some of which are fleeting andminute may be considered as a strongly fluctuating quantity. To quantify it, sev-eral variables were suggested, such as the TSWLS ( Temporal Satisfaction With1ife Scale) Pavot et al. [1998], an integer variable taking values between 15 and105, or numerous other criteria often lacking precise mathematical definition, seee.g. Durayappah [2011] and references therein. Most of these criteria are averagequantities with an averaging period which may vary between several hours andmany years. It is hardly surprising that attempts to establish a statistical corre-lation of such parameters with objective factors affecting an individual met onlypartial success, Kahneman et al. [2006]. Typically, a model aimed at predictingsuch a “happiness” factor would be a more or less sophisticated algebraic regres-sion, Kopsov [2019]. At the same time, it is obvious that, regardless of how it maybe quantified, an individual’s happiness may come from external agents, e.g. thecurrent state of economy or level of cultural activity, and what is more, intrinsicfluctuations in one’s personal satisfaction with life strongly suggest an investiga-tion at the level of a “society” that is an ensemble of individuals sharing the samedaily activities, an ensemble sufficiently large so as to admit a statistical descrip-tion. Ultimately, it is the ensemble statistical properties of the suitably quantifiedhappiness which matter for a society: as an example, the important societal issueof growth of inequality cannot be described in terms of individual or even averagequantities and requires at least knowledge of the second-order moments, somesort of “root-mean-square happiness”.Rather than concentrating on one of the existing definitions for individual hap-piness departing from subjective factors, it seems convenient to introduce a newvariable and rather than trying to define it in terms of subjective perceptions ofan individual, this new variable is defined through an evolution equation takinginto account the major factors affecting the well-being. The term “contentment”for this new variable is used to distinguish it from the already existing measuresof the well-being. The evolution equation for the contentment of an individual ispostulated as an ordinary differential equation (ode) including a number of factors.Among other factors, to a large extent, the objective quality of life and its sub-jective perception by an individual both depend on the relative standing of thisindividual within the society and the rate at which this standing improves; onemay assume that the latter is correlated with self-fulfilment. To characterise thisstanding the variable representing wealth is introduced; while it is most readilyassociated with monetary wealth and such an association is used here for mathe-matical expression of specific factors affecting the dynamics of contentment, thenotion of wealth may be generalised to include other factors affecting the indi-vidual’s standing in the society, e.g. level of skills or some other non-monetaryforms of societal recognition. Similarly to the contentment, the temporal evolu-tion of the wealth of an individual is described in terms of an ordinary differential2quation. The ode’s for the wealth and contentment of an individual are coupledand from them is derived a partial differential equation (pde) for the joint proba-bility density function of the contentment and wealth within the society. In thisapproach, it turns out that the effects of marriage cannot be easily included in theode’s for an individual’s wealth and contentment and these effects are representedas an integral term in the joint distribution pde.Owing to the lack of the fundamental conservation laws for the wealth andcontentment, the mathematical model for these variables may only be postulatedbut not derived; the approach adopted here is to approximate the rate of a par-ticular effect with a simplest mathematical expression compatible with everydayexperience; reasoning is done in terms of characteristic times of each factor af-fecting wealth or contentment. Values of adjustable parameters entering such anexpression are chosen so that, taken individually, the effect in hand produces theexpected rate of change of wealth or contentment. This approach is entirely empir-ical and ad-hoc, however, it allows, firstly, a simple comparison of the magnitudeof individual effects, and secondly, a straightforward means of improving everyexpression when additional data becomes available.
Development of the model
Let C , a real variable varying from 0 to 1, denote the contentment so that a pperfectly unhappy person be attributed a zero value and someone at the state ofperfect bliss unity. Clearly, a unit of contentment may be established by attributinga specific value of C , say between 0 and 1, taken as degree of satisfaction withlife resulting from a particular objective and observable event, however, this is notpursued here and Use of continuous rather than discreet variable is convenient asit allows subsequent comparisons with diverse previous measures ranging from3-point scale of Easterlin [2003] to 105-point TSWLS Pavot et al. [1998]. Theunit of time, t , is taken here as one year.While the real variable quantifying “wealth” M is not, sensu stricto , not boundedfrom either above or below, as one may, at least in theory, have an infinitely largedebt or fortune, it is taken here as non-negative, i.e. debt is neglected. Its unitis taken here as half of the average household wealth as it makes it easier to for-mulate the model so that it yields the temporal evolution of wealth compatiblewith the relatively well documented trends, e.g data from the Bank of England,Hills et al. [2017], or Office for National Statistics in the UK, for National Statistics[2019]. For the numerical simulations illustrating the model, it is taken bounded3rom above by a positive value M max of the average wealth of the upper decile inthe UK in 2018. Evolution of contentment of an individual
It is assumed that the rate of changeof contentment for every individual comes from several additive factors, namely,satisfaction with own wealth W and income W , satisfaction with the infrastruc-ture of the society W , quality of the neighbourhood W , and dissatisfaction withhigh taxes W and inequality in the society W . Besides, there are two additionalterms, one, W describing the fact that everyone is seeking and finding satisfactionin things independent of the wealth and another W representing rapidly changingand statistically uncorrelated events affecting contentment. The rate of evolutionof the individual contentment is then simply a sum of these factors: dCdt = X i W i (1)Notice that the above ode does not include change of contentment resulting frommarriage, or formation a family, which is treated later. These factors cover a mixof subjective, objective, direct, and indirect aspects Benjamin et al. [2014].Contentment due to wealth increases or decreases when the wealth is greateror smaller than the average and, in absence of other factors, it usually takes aboutseveral years before this difference of wealth is considered as permanent and itleads to a lasting improvement of own satisfaction with life. Therefore, one mayuse a simple linear expression for W : W ( M ) = K ( M − h M i ) (2)where K is a reciprocal of the characteristic time for this factor and the angularbrackets denote average for the society. As postulated, it is obvious that, shouldboth the individual wealth M and the society average wealth h M i stay constant,this term alone would leave to a change of the individual contentment proportionalto the time as C ( t ) = C ( t = 0) + K ( M − h M i ) t . For an approximation of K ,one may think that material comfort of life for someone with twice the averagewealth would lead to a perfect contentment within 10 years, thus K ≈ . .That the contentment should increase with income may sound controversial,e.g. it has been argued that income has no influence as may be derived fromdata obtained when tracing satisfaction with life of differently educated groups ofpopulation, taking the education for proxy measure of income, see e.g. Fig. 4in Easterlin [2003]. However, these very data do show a significant increase in4ontentment for both the more and the less educated groups at ages where onemay expect a rapid career growth with stagnation afterwards corresponding againto an age where most careers tend to achieve a plateau. Besides, it seems naturalthat not all sources of individual’s income contribute equally to contentment: it isdifficult to imagine that the income from welfare contributes to satisfaction withlife even though it may cover the material necessities. While the analysis of in-fluence of income on happiness show that it changes with the age of an individualEasterlin [2003], the age distribution of the society is not considered in the presentmodel. Furthermore, what matters is not the magnitude of income as such but theexcess of the disposable income I d over the cost of living L , or what is generallyperceived to be a “good income” G . Thus, one may tentatively suggest: W ( M, C ) = K ( I d ( M, C ) − G ) (3)A specific expression for the disposable income I d ( M, C ) is formulated later; thereciprocal of the characteristic time for this effect K ≈ . − . .The term related to the satisfaction with the infrastructure W at this levelof generalisation should reflect the fact that it does not depend on the individualcircumstances but on the general level of the expenditure of the society on thecommon use facilities such as education, culture and sport institutions. Ratherimportantly, it is this term which includes here investment of the society into thehealth care, thus indirectly it includes the important contribution of good healthinto the contentment. All such expenses are assumed to come from the generaltaxation and be spread evenly among the individuals; this choice is an idealisa-tion reflecting the UK NHS model but more refined expressions may be easilyformulated for applications of the model to other circumstances. Thus W = K α I T t (4)where T t is the total tax intake in the society and α I is the fraction of it going intoinfrastructure. Reflecting the fact that the infrastructure projects have relativelylong time scales of order of 10-20 years, the estimation for the product α I K ≈ . may be made. As the taxation is always positive, this term leads to uniformincrease of C while the dissatisfaction with the high taxation W should decrease C in proportion to the tax paid by the individual T ( M, ˙ M ( M, C )) which maydepend on the individual’s wealth M and total gross income ˙ M ( M, C ) : W = − K T ( M, ˙ M ( M, C )) (5)The characteristic time of this term should be of order of the time between thegeneral elections, at least for the democratic societies as the pledge to decrease5axes often proves to be an election winner thus indicating that the dissatisfactionwith the tax level may overtake other considerations. As the overall tax burdenin the OECD countries is somewhere between 30-60%, the K value should besomewhere in the range 0.3 to 0.7; K = 0 . may thus be suggested.The “quality” of the neighbourhood is an integral characteristic of the society,and average contentment h C i is taken here as its proxy, making straightforward toapproximate: W = K (cid:18) h C i − (cid:19) (6)This expression also subsumes the stronger tendency of the more contented soci-ety to take better care of its members as compared with a society in strife charac-terised by low values of h C i .Along with the well-known Gini coefficient, the wealth inequality, i.e. thewidth of the spread of the marginal probability density P ( M ) in the society maybe characterised with the root-mean-square (rms) value D M ′ E / = Z M max ( M − h M i ) P ( M ) dM ! / However, it is not the rms magnitude per se but rather its ratio to the mean wealth,or “intensity of the inequality” which seems to cause the discontent. This leads to W = − K h M ′ i / h M i (7)where K ≈ . ÷ . as the time scale for this effect is 10 ÷
20 years as evidencedby the timescales for raise and fall of large-scale social movements.Regardless of wealth or income, everyone seeks solace finding it in variousthings, so that it may be assumed that there is a general tendency of increase ofcontentment and this increase is the larger the smaller is the contentment, decreas-ing to zero at C = 1 . The simplest form of W reflecting this would be: W = K (1 − C ) β (8)where β > to ensure the smoothness of W at C = 1 . The characteristictime scale for this term depends on the probability distribution of C but if toconsider that takes several, say between 2 and 10, years for someone initially atfull discontent to find solace, K ≈ . ÷ . ; the values of K = 0 . and β = 2 are adopted in what follows. 6
20 40 60 80Time0123456 < M > , [-] L =0.02L =0.04L =0.06L =0.08 < C > , [-] < M ’ > / , [-] < C ’ > / , [-] a) b)c) d) Figure 1: Temporal evolution of the first two central moments of the wealth: a)average and c) root mean square, and contentment: b) average and d) root meansquare. The values of the coefficient for the wealth tax in Eq. 12 are shown in thelegend.Finally, there are always fluctuations in anyone’s satisfaction of life the timescale of which is very short in comparison with any term introduced above. Thesemay be caused by multitude of events, big and small, ranging from burnt toast toa life-changing lottery win with no casual relationship between them. Some ofthose decrease and some increase contentment but their effect on the average con-tentment is zero; their action is to reduce the variations of contentment betweenthe individuals. The contribution of such events to the contentment may thereforebe described as a stochastic function of time ζ ( t ) : W = ζ ( t ) (9)such that h ζ ( t ) i = 0 at any time and h ζ ( t ) ζ ( t ′ ) i = γδ ( t − t ′ ) where δ ( t ) is Dirac’sdelta-function and γ is the magnitude of the fluctuations of the contentment in-7uced by these random events. Owing to the normalisation of C , the characteris-tic time scale for reduction of the contentment variations is γ − / . The levellingof the contentment variations caused by this random events happens at the timescale of the same order as an average life expectancy, say 75 years, thus γ ≈ . .Obviously, the numerical values of the rate factors in different terms in Eq. 1 areapproximate; moreover, one and the same factor would affect different individualsat different magnitude. It is possible to formulate more refined mathematical for-mulation for example assuming some statistical distribution of parameter valuesbut the resulting complexity seems unwarranted at this stage. Evolution of wealth of an individual
Clearly, both the distribution of wealthwithin the society and the individual wealth influence the distribution of content-ment, thus investigation of the dynamics of the latter should be conducted jointlywith the consideration of the temporal evolution of the wealth and its statisticaldistribution. Even though the “wealth” and “income” affecting the contentmentare not necessarily pecuniary, the model formulation is much easier to do asso-ciating these notions with the aggregated form of monetary wealth making nodistinction between various types of financial means, property, pensions etc. Themain assumption made here is is that the rate of change of an individual’s wealthequals the sum of his productivity U , some amount of the added value U pro-duced by others that he alienates from them, taxes U , and welfare assistance U .The only coupling between the contentment and the rate of change of wealth maycome from the fact that the productivity of an individual with very low content-ment is low but is increasing, more or less rapidly, as his contentment rises, as,obviously, neither taxes nor welfare payment do not depend on the contentment.The dependency of the productivity U ( M, C ) on the contentment C is ap-proximated with a power function reflecting fast drop in productivity as C tendsto zero and much slower rate of change as C tends to one: U ( M, C ) = L C α p M α p ( M max − M ) α p (10)The dependency of the productivity on the wealth M is approximated here witha polynomial such that the productivity has a broad maximum for M in between1 and 2 and decreases to zero when M tends to either 0 or its maximum value M max . For low values of M , this functional form reflects the statistical correla-tion between the destitution and unemployment, ill-health, lack of education andany other factor contributing to low productivity. It should be mentioned that theusual statistical data, see e.g. for National Statistics [2019], clearly demonstrate8his correlation between the income and wealth, however, they do not allow oneeasily to disentangle the individual’s own productivity from other contributions tothe rate of change of his wealth. The value of the factor L governs the overallproductivity of the society; in what follows it is fixed such that the productivityof an individual of average wealth M = 1 is about . M . The value of α p de-termines how steep is the decline of productivity with the discontent: with thevalue α p = 1 / retained here, with decrease of C , decrease of U is slow atfirst, down to approximately 60% of the maximum when C = 0 . , followed by avery rapid precipitation to zero for lower C . The choice of the power exponents α p and α p determine how flat is the dependency of U on M , for α p = 1 and α p = 3 the prodictivity stays within 20% of its maximum value in the interval . ≤ M ≤ . .The term U ( M ) describing redistribution of the income within the societytowards the elites must necessarily be negative for low M and positive for large M , thus there should be a value M ⋆ such that U ( M ⋆ ) = 0 as there is no reason toassume U discontinuous. This term must not change h M i . Its extremum valuesmay only be attained at the ends of the interval [0 , M max ] . While there is an infinitenumber of possible functions satisfying these requirements, a simple exponentialform is adopted here: U ( M ) = L (cid:18) exp (cid:18) M − M ⋆ M s (cid:19) − (cid:19) (11)where the parameter M s determines how fast is the growth of profit from capital.Equation 11 means that this growth is faster than linear reflecting the correla-tion between higher levels of investment, risk and return open to the wealthiestBlanchet et al. [2017]. For the developed countries, the ratio between the aver-age income and the nominal gross domestic product per capita is approximately0.7-0.8 suggesting that L ≈ . ÷ . For L = 0 . the value M s = 2 . yieldspre-tax rate of the wealth increase of the wealthiest individuals considered here, M = M max = 7 . , of approximately 20% in line with the average values reportedfor the developed countries, e.g. Blanchet et al. [2017]; these values are used inwhat follows.While here is a great variety of tax regimes, the simplified treatment usedhere assumes a flat tax on income supplemented with a wealth tax T w imposed onindividuals with the wealth above a threshold M w : U = L T ( U + U ) + T w ( M ) L T is the total rate of income tax including direct, and indirect, e.g. value-added, taxation. L t = 0 . is taken here as an approximation to the sum of the cur-rent basic rate of the UK income tax, National Insurance contribution and ValueAdded Tax for a person of average income making little savings. The wealth taxis approximated as: T w ( M ) = ( if M ≤ M w L exp (( M − M w )( M max − M )) if M ≤ M w (12)Even though most OECD countries currently have no direct wealth tax, they dohave inheritance taxes and Eq. 12 represents those. Its combination with flat in-come tax to some extent also describes progressive taxation, e.g. assuming de-layed capital gains tax etc. Equation 12 yields progressive decrease of T w for verylarge values of M ≈ M max representing effects of tax evasion and avoidance bythe wealthiest individuals. While it is straightforward to include in the descriptiona progressive income taxation, it is thought that the simplified representation ofthe taxation split into two parts, one depending on the wealth M and the other onits rate of change, is sufficient for illustration of tax on the contentment dynamics.At the same level of simplification, the welfare payments are assumed here tosupplement income of individuals with M < h M i tapering to zero at M = h M i : U ( M ) = ( L (cid:16) − M h M i (cid:17) α w if M ≤ h M i if M ≥ h M i (13)where variation of the power exponent α w achieves representation of the differentwelfare policies: values of α w >> means most of the welfare concentratingaround M ≈ with steep decrease of U for M ≈ h M i , i.e an alms-type welfare. / ≤ α w ≤ produce nearly linear decrease of U ; α w = 1 / is used in whatfollows. The value of the L factor is so determined that U compensates exactlyall other factors except marriages resulting in zero net change of wealth at M = 0 .Finally, the rate of change of an individual may be written as dMdt ( M, C ) = (1 − L T ) ( U ( M, C ) + U ( M )) − T w ( M ) + U ( M ) (14)This expression with the values of different parameters specified above producesoverall yearly increase of the average household wealth of approximately 8% inline with the average UK trends over the period 1998-2008. “Disposable” income I d affecting the contentment through Eq. 3 may now be written as I d = (1 − L T ) ( U ( M, C ) + U ( M )) − T w ( M ) and the “good” income G may simply be taken as the average D dMdt E .10igure 2: The joint pdf P ( M, C ) of the wealth M and contentment C for: left) lowwealth tax at L = 0 . , T ≈ . and right) high wealth tax at L = 0 . , T ≈ . .Figure 3: The flow in the wealth-contentment space induced by Eqs. 1 and 14for: left) low wealth tax at L = 0 . , T ≈ . and right) high wealth taxat L = 0 . , T ≈ . . Values of P ( M, C ) , see Fig. 2, are shown on thesuperimposed isolines. 11 epresentation of effects of formation of family/marriage From the view ofsociety as an ensemble of individuals and neglecting polygamy, the marriage, ormore generally, formation of a family is a pairwise interaction process. In terms ofevolution of wealth of individuals this process results in redistribution of wealth ofthe interacting pair into two equal parts; this leaves the average wealth unchangedand decreases its variance. Effects of marriage on wealth are remarkably similarto the phenomenon of scalar small-scale mixing in a turbulent flow, the subjectof numerous investigations, see e.g. Dopazo [1994]. Based on this similarly, aconvenient starting point for expressing how P ( M, C, t ) is affected by marriagesis provided by so-called integral models of mixing pioneered in Frost [1960] andCurl [1963].There is some evidence that, in average, married individuals show satisfactionwith life slightly higher than the celibate ones, however, this effect seems to betransitional and of the same magnitude as the spread of the “mean happiness” seee.g. Fig. 3 in Easterlin [2003], therefore, an assumption is made here the effect ofmarriage on the contentment is to equipartion it, producing equal probability ofany value C regardless of the initial contentment. Secondly, it is assumed that theprobability of marriage between two individuals depends only on their wealth butnot contentment. The dependency of the probability of marriage on the differenceof the wealth of the two individuals is described with a weight function f , unity forzero difference, and decreasing to zero when this difference becomes very large. Itis thought that introduction of such weight function reflects the general tendencyof marriages within the same social strata. Root-mean-square value of the wealthin the society is chosen as the measure for the disparity of the wealth resulting indecrease of the probability of marriage. For an individual, the marriage happenswith the characteristic time scale τ f which is simply the average life expectancydivided by an average number of marriages over the lifetime; the latter is currentlyabout 1 in the UK so τ f ≈ .With these assumptions, the expression for I f may be written as: I f = P ( M, C ) R dC ′ P ( M, C ′ ) · τ f " Z M dzf (cid:16) z, D M ′ E(cid:17) Z dC ′ P ( M + z, C ′ ) Z dC ′′ P ( M − z, C ′′ ) − P ( M, C )4 Z M max dzf ( z, D M ′ E ) Z dC ′ P ( z, C ′ ) (15)It may be shown that Eq. 15 preserves normalisation of P and h M i . In what12igure 4: The joint pdf P ( M, C ) of the wealth M and contentment C for: left) lowwealth tax at L = 0 . , T ≈ . and right) high wealth tax at L = 0 . , T ≈ . .follows, exponentially decaying weight function was used: f (cid:16) z, D M ′ E(cid:17) = (cid:16)D M ′ E(cid:17) / exp − z ( h M ′ i ) / ! (16)and it should be noticed that use of alternative forms had not led to qualitativelydifferent results. Evolution equation for P ( M, C ) The established evolution equations for theindividual’s contentment, Eq. 1, and wealth, Eq. 14, allow one to write an equationdescribing temporal evolution of the joint probability density P ( M, C, t ) of thesetwo variables for the entire society, Moyal [1949],Pope [1994], as: ∂P ( M, C, t ) ∂t + ∂∂M * dMdt ( M, C, t ) + P ( M, C, t ) + ∂∂C * dCdt ( M, C, t ) + P ( M, C, t )+ ∂ ∂C Z t dt ′ * dCdt ( M, C, t ) dCdt ( M, C, t ′ ) + P ( M, C, t )= I f ( M, C, t ) (17)where the term I f ( M, C, t ) is given by Eq. 15. Equation 17 is a truncated Kramers-Moyal equation where the dropped terms are identically zero owing to the struc-ture of Eqs. 1 and 14. The conditional averages in Eq. 17 are trivially found for13igure 5: The flow in the wealth-contentment space induced by Eqs. 1 and 14for: left) low wealth tax at L = 0 . , T ≈ . and right) high wealth taxat L = 0 . , T ≈ . . Values of P ( M, C ) , see Fig. 4, are shown on thesuperimposed isolines.all terms in Eq. 14 and all terms in Eq. 1 but W . The latter term upon averagingyields only the diffusion-like term with the second partial derivative with respectto C . The final equation for P ( M, C, t ) thus becomes: ∂P ( M, C, t ) ∂t + ∂∂M [(1 − L T ) ( U ( M, C ) + U ( M )) − T w ( M ) + U ( M )] P ( M, C, t )+ ∂∂C " X i =1 W i P ( M, C, t ) = γ ∂ P ( M, C, t ) ∂C + I f ( M, C, t ) (18)The boundary conditions imposed on P are zero normal derivatives at C =0 , and M = 0 , M max lines ensuring the preservation of normalisation of P .It should however be noticed that non-zero dMdt ( M, C, t ) and P at M = M max would require dynamic adjustment of M max ( t ) = max C dMdt ( M max , C, t ) but thisis not attempted here. The main reason for keeping constant M max is that, as willbe seen later, P ( M, C, t ) is small and concentrated in a fairly small vicinity ofthe point ( M max , affecting little the dynamics of contentment in the regions oflarge P representing the most of the society. The second reason is that numericalmethods for moving computational domain boundary are more complicated andthe extra complexity is not warranted at the stage of the formulation of the modelprinciples. 14igure 6: The joint pdf P ( M, C ) of the wealth M and contentment C for: left) lowwealth tax at L = 0 . , T ≈ . and right) high wealth tax at L = 0 . , T ≈ . . Illustration of the model: evolution of an initially ho-mogeneous society
Numerical implementation for P ( M, C ) The computational domain was cov-ered by uniform grid on which Eq.18 was discretised using control volume methodwith implicit upwind expressions for the fluxes on the boundaries of grid cells,the time derivative was calculated with simple first-order approximation. Aver-ages and root-mean-square values and the integrals in Eq. 15 were calculated ex-plicitly using method of overlapping parabolas implemented in DAVINT routineJones [1969]. The resulting sparse algebraic system of equations was solved usingGaussian elimination with partial pivoting and LU decomposition and precondi-tioning implemented in SUPERLU-MT set of routines Demmel et al. [1999]. Thesensitivity to the grid size and the time step selected so that equivalent to CFLcriterion in fluid mechanics does not exceed 0.8 were verified to ensure that theresults were not sensitive to their choice.
Evolution of P ( M, C ) for homogeneous society: effects of tax on wealth Thejoint probability density P ( M, C ) for society completely homogeneous in termsof contentment and wealth is δ ( M − h M i ) δ ( C − h C i ) . Joint action of marriagesand various random factors acting on a very short time-scale, rhs of Eq. 18, broad-15ns the initially infinitesimally narrow peak of contentment distribution into a nor-mal distribution which is then distorted and clipped at the boundaries C = 0 , until it attains a constant value of unity between these boundaries. The model pos-tulates that contentment increases the productivity, Eq. 10, thus non-zero h C ′ i / will produce broadening of the wealth distribution and non-zero h M ′ i / : thedomain where P ( M, C ) = 0 becomes a curve in M − C space. Under wealth-distributing action of the marriages, this P ( M, C ) carrier curve will spread alongconstant C lines becoming two-dimensional area. In simple words, this modelpredicts that in a society where everyone was initially equally rich or poor andequally satisfies with life, after a certain time there will be be a significant proba-bility of meeting an individual at any state of contentment or satisfaction with lifetogether with a certain variation of wealth. It is possible that a particular choice ofwelfare and tax rates and functional shapes may produce a steady-state P ( M, C ) but this aspect is beyond the scope of this work.In reality no society is completely homogeneous, thus the homogeneous soci-ety here is defined here as having a normal distribution for P ( M, C ) with non-zerorms values h M ′ i / and h C ′ i / . By construction, the initial value of h M i = 1 ,and thus the initial average wealth becomes a unity for scaling. However, the ini-tial value of h C i remains arbitrary because no scale is defined for C between itsextreme values of 0 and 1. The normal distribution is clipped at the boundariesand renormalised to unity. In what follows, the parameters of the initial P ( M, C ) are h M ′ i / = 0 . , h C i = 0 . h C ′ i / = 0 . ; use of different values yieldsqualitatively similar results. The simulations were run for the time period of 75units (years).In order to demonstrate the performance of the developed model the paramet-ric studies of tax on wealth given by Eq. 12, the magnitude of which is governedby the L parameter. The lower limit at which the wealth tax is levied is keptfixed at M w = 2 . . Figure 1 shows the temporal evolution of the average androot-mean-square wealth and contentment for a four-fold variation of the rate ofthe wealth tax. Initially, for the first approximately 25 years or one change of gen-erations, the wealth tax magnitude has very little influence: the the chosen initial P ( M, C ) corresponds to a very small fraction of individuals with M ≥ M w . Thisis corroborated with Figs. 2 and 3 showing the P ( M, C ) and its rate of change inthe midst of this period for the smallest and largest wealth tax. Figure 3 shows theiso-contours of the P ( M, C ) superimposed on the field of the its rate of change.One can see that the society undergoes stratification with appearance of noticeablefractions of individuals who are nearly entire destitute, see the rather sharp peak16igure 7: The flow in the wealth-contentment space induced by Eqs. 1 and 14for: left) low wealth tax at L = 0 . , T ≈ . and right) high wealth taxat L = 0 . , T ≈ . . Values of P ( M, C ) , see Fig. 4, are shown on thesuperimposed isolines.at M = 0 , C = 0 but also those with the more than twice the average wealth.Rather unsurprisingly, the model yields the destitute extremely unhappy and richvery content. What is less trivial is that, though both the average wealth, andthe wealth of the largest fraction of the population are steadily increasing, thisincrease is accompanied with decreasing average contentment and formation ofvery wide contentment distribution, see Fig. 3.The model postulates that contentment increases the productivity, Eq. 10, thusnon-zero h C ′ i / leads to a modest growth of h M ′ i / , see Fig. 1c). The initialgrowth of h C ′ i / is very fast owing to the combined effects of marriages, Eq. 15,random events, Eq. 9, and dependency of contentment on income, Eq. 3 andwealth Eq. 2. At this moment, the effects of the wealth tax rate L are small:larger L lead to small decrease of fraction of individuals with M ≈ , C ≈ .These effects, however, become more and more pronounceable in the longerrun. Even though the average individual wealth increases slightly faster for smallerwealth tax for the first 40 years, it attains a plateau and decreases afterwards, whilegrowth of h M i with larger wealth tax is continuous for the entire period of simula-tions, cf. thin and thick solid lines in Fig. 1a). Rather curiously, lower wealth taxwith its faster growth of h M i does not yield proportional increase in average con-tentment: for larger L , after the initial decrease h C i stagnates and then decreaseswhile it grows steadily for larger L , so that the final values of average content-ment are about twice larger for larger wealth tax, cf. thin and thick solid lines in17ig. 1b). Larger rate of the wealth tax also decreases stratification of the societymeaning much slower growth and smaller values of root-mean-square both wealthand contentment, cf. thin and thick solid lines in Fig. 1c) and d).Figure 4 illustrates the difference in the stratification of the society for the in-stant when the smaller wealth tax leads to the stagnation in growth of the averagewealth. One may clearly see that, while unsurprisingly there is larger probabilityof larger, M > M w , individual wealth, there is also very much larger fraction ofthe society in the destitution and discontent M ≈ , C ≈ . Combination of twofactors explain this increased stratification: the first relates to the appropriationof the added value by the rich, Eq. 11, and the second is that the redistributiveaction of marriage is ineffectual for reducing the wealth extremes when their dif-ference exceeds two rms wealth values, cf. Eq. 16. Even though the society atthis instant is about three times wealthier in average, the stratification caused bysmaller wealth tax yields appreciably smaller average contentment. It may be in-ferred from Fig. 5 that the added value is alienated by progressively smaller fromprogressively larger fraction of the society.These trends continue and towards the end of simulation, T ≈ , the lowwealth tax case show a fully stratified society, see Figs. 6 and 7. The high wealthtax society is more homogeneous, with zero probability of very high individualwealth M ≥ and much smaller fraction of the destitute; Fig. 1 shows thatthis case has much higher average wealth and contentment. Comparison of thetemporal evolution of P ( M, C ) clearly shows that there is no simple relationshipbetween the wealth and contentment and the average values of these variables arenot sufficient to predict further evolution of the society. An interesting featureof the P ( M, C ) distribution in this case is the fairly wide spread of contentmentamong the well-off part of the population with noticeable probability of meetingsimultaneously wealth well above and contentment well below the average valuesfor the society as a whole. Conclusions
It is proposed to use a variable continuously changing between 0 and 1 for con-tentment, or satisfaction with life, of an individual and several factors affectingit are analysed in order to obtain an ordinary differential equation governing thecontentment. From it, and a similar equation for an individual’s wealth, it provesthen possible to formulate a mathematical model describing temporal evolution ofthe joint distribution of individuals’ wealth and contentment within a society re-18orting to fairly general arguments about time-scales of influence of different fac-tors. With very simple expressions for these factors an evolution equation, Eq. 18,for this distribution is derived; it yields results which are at least qualitativelyplausible. It should be stressed that alternative expressions for any of the factorsaffecting contentment (or wealth) may be easily incorporated into the model with-out affecting its underlying principles. It is hope that the proposed model couldbe of use in further quantitative works clarifying statistical characteristics of thesatisfaction with life.The first application of the evolution equation for the joint distribution ofwealth and contentment is made here to illustrate the effects of tax on the wealthon the society. It is shown that for a long periods of time this form of tax has asignificant impact on the degree of the stratification in the society in terms of bothcontentment and wealth and the less stratified society is, in longer run, wealthierand its members have generally greater contentment.
Acknowledgements
The impetus to this work was given by discussions with Prof. Roland Borghi.
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