Far from equilibrium: Wealth reallocation in the United States
aa r X i v : . [ q -f i n . E C ] M a y Far from equilibrium: Wealth reallocation in the United States
Yonatan Berman ∗ School of Physics and Astronomy, Tel-Aviv University, Tel-Aviv, Israel
Ole Peters † London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, UK andSanta Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA
Alexander Adamou ‡ London Mathematical Laboratory, 14 Buckingham Street, London WC2N 6DF, UK (Dated: May 19, 2016)Studies of wealth inequality often assume that an observed wealth distribution reflects a systemin equilibrium. This constraint is rarely tested empirically. We introduce a simple model that allowsequilibrium but does not assume it. To geometric Brownian motion (GBM) we add reallocation:all individuals contribute in proportion to their wealth and receive equal shares of the amountcollected. We fit the reallocation rate parameter required for the model to reproduce observedwealth inequality in the United States from 1917 to 2012. We find that this rate was positive untilthe 1980s, after which it became negative and of increasing magnitude. With negative reallocation,the system cannot equilibrate. Even with the positive reallocation rates observed, equilibration istoo slow to be practically relevant. Therefore, studies which assume equilibrium must be treatedskeptically. By design they are unable to detect the dramatic conditions found here when data areanalysed without this constraint.
PACS numbers: 89.65.Gh, 05.40.Jc, 02.50.Ey
THE MODEL – REALLOCATING GBM
Our model of personal wealth is geometric Brown-ian motion (GBM) enhanced with a simple reallocationmechanism. The wealth of the i th individual as a functionof time, x i ( t ), in a population of N individuals followsthe stochastic differential equation, dx i = x i ( µdt + σdW i ) | {z } growth − x i τ dt + h x i N τ dt | {z } reallocation . (1) dx i is the change in wealth over the time period, dt . dW i is the increment in a Wiener process, whichis normally distributed with mean zero and variance dt . We refer to the parameters µ as the drift, σ asthe volatility, and τ as the reallocation rate. Angledbrackets with subscript N denote the sample mean, i.e. h x i N = N P Ni =1 x i .The term labelled ‘growth’ in Eq. (1) is the wealth in-crement in a GBM. It is a random proportion of x i , con-sisting of a certain part, x i µdt , and an uncertain part, x i σdW i . On its own this term generates noisy exponen-tial growth.The ‘reallocation’ term works as follows: each indi-vidual contributes x i τ dt , an amount proportional to itswealth, and receives h x i N τ dt , an equal share of the sumof all contributions. For τ >
0, low-wealth individualswith x i < h x i N are net recipients under this mechanism,while high-wealth individuals with x i > h x i N are netcontributors. For τ <
0, the roles are reversed. We call this model reallocating geometric Brownianmotion (RGBM) to distinguish it from GBM, to whichit reduces when τ = 0. Equation (1) may also be writtenas dx i = x i ( µdt + σdW i ) − τ ( x i − h x i N ) dt, (2)which shows that RGBM can be viewed as GBM withreversion to the sample mean at rate | τ | if τ is positive,and repulsion from the sample mean at rate | τ | if τ isnegative. Figure 1 depicts typical individual trajectoriesproduced by the RGBM model with positive, zero, andnegative reallocation (respectively τ > , τ = 0 , τ < Regimes of RGBM
The RGBM model produces qualitatively different be-haviour for different model parameters, population sizes,and timescales. There are three important regimes char-acterised by the value of τ . In the following discussion,as in Fig. 1, we let all individuals start with wealth ofone dollar, x i (0) = $1. τ = 0 Without reallocation the model is GBM, whoseproperties are well known (see, for example, [1]).Wealth follows a lognormal distribution whichbroadens indefinitely over time. There is no sta-tionary non-zero distribution to which it converges.In relative terms ( i.e. on logarithmic wealth scales) Time (year) W ea l t h ( $ ) -4-20246810 Time (year) W ea l t h ( $ ) -2 -1 Time (year) W ea l t h ( $ ) -4-20246810 Time (year) W ea l t h ( $ ) -2 -1 Time (year) W ea l t h ( $ ) -4-20246810 FIG. 1: Simulations of RGBM with N = 1000, µ = 0 .
021 year − , σ = 0 .
14 year − / , x i (0) = $1. Effect of reallocationparameter τ . Top left: τ = 0 year − , inequality grows indefinitely. Magenta lines denote the largest and smallest wealths. Thedistribution of individual wealth, on logarithmic scales, is symmetric about the median exp (cid:2)(cid:0) µ − σ / (cid:1) t (cid:3) (black) rather thanthe sample mean (green). Blue lines illustrate five randomly chosen wealth trajectories. Top right: τ = 0 . − , inequalitygrows initially but is limited by reallocation. The distribution of individual wealth is confined around the sample mean (green), h x i N ≈ exp ( µt ). Bottom left: same as top left, zoomed in, and on a linear wealth scale. Bottom middle: same as top right,zoomed in, and on a linear wealth scale. Bottom right: τ = − . − , the system is in a qualitatively different regime inwhich wealth can become negative. the wealth distribution is symmetric about the me-dian, exp (cid:2)(cid:0) µ − σ / (cid:1) t (cid:3) .GBM exhibits an extreme form of a phenomenonknown as wealth condensation [2]: over time ameasure-zero proportion of the population ends upwith a measure-one proportion of wealth. In prac-tical terms, this corresponds to one person holdingall the wealth, i.e. perfect inequality. τ > τ → ∞ . In this limit, all individualshave equal wealth which is well approximated byexp (cid:2)(cid:0) µ − σ / N (cid:1) t (cid:3) . For N and σ appropriate tohuman populations, this is very close to exp ( µt ).For finite τ , wealths disperse. However, their distri-bution remains confined around the sample mean, h x i N , to which they are connected by the reallo-cation mechanism. For realistic model parameters,populations and timescales, h x i N ≈ exp ( µt ). Thewealth distribution broadens as τ → + to resemblethat of GBM. The wealth distribution for RGBM with τ > τ increases, successively higher mo-ments become finite as the distribution becomesmore closely confined around the sample mean. τ < τ ≥ τ < x i < h x i N into negative wealth. The creation oflarge wealths is fuelled not only by multiplicativegrowth but also by direct transfers from the poor,whose wealths become increasingly negative. Nostationary wealth distribution exists. In practi-cal terms this corresponds to creditors and debtorswhose wealths diverge exponentially away from thesample mean. Interpretation of τ In the GBM model ( i.e. τ = 0), some individuals in apopulation would see their wealth grow over a short timeperiod, while others would see it shrink. It might looklike wealth is being transferred between individuals but,mechanistically in the model, these apparent transfersare the results of random fluctuations.We wish to model a population of individuals whosewealths are connected through social and economic struc-tures such as governments and markets. Such a popu-lation will likely exhibit resource allocation that is notcaptured by GBM alone. The mechanism we introduceto model these interactions creates systematic transfersof wealth between individuals. We stress that these areover and above the apparent transfers observed in a pop-ulation whose wealths follow GBM.Our reallocation mechanism has many possible corre-spondences to aspects of real economies. With τ > τ < τ > τ < i.e. fixed-rate disbursements to many people.The argument who benefits more cannot be settled bylong lists of examples. It should be addressed empirically.We find a single time-varying value of τ – an effective re-allocation rate summarising the net action of all parts ofthe economy, both public and private – which best de-scribes observed wealth inequalities in the United States. EMPIRICAL STUDYEstimating µ We estimate µ from historical private wealth data forthe US [4] by dividing the total private wealth by the to-tal population size. In the analysed data “private wealth [...] is the net wealth (assets minus liabilities) of house-holds and non-profit institutions serving households. [...]Assets include all the non-financial assets – land, build-ings, machines, and so on – and financial assets, includinglife insurance and pensions funds, over which ownershiprights can be enforced and that provide economic benefitsto their owners” [4, p. 1268].Starting at t = 1917 years AD, we assume the averagehistorical private wealth follows exp [ µ ( t − t )] and find µ = 0 . ± .
001 year − using a least-squares regression(Fig. 2). For the timescale of interest, about 100 years,the growth rate of the population average is about thegrowth rate of the expectation value, since the populationis large enough (see Model section and also [1] and [5]). Year < x > N ( $ ) · · · · · · · Data
021 year − (red). Datafrom [4]. Estimating σ For each year we estimate σ ( t ) as the standard devi-ation of daily logarithmic changes, annualised by mul-tiplying by (250 / year) / , of the Dow Jones IndustrialAverage (data from Quandl database [6]). The valuesusually lie within the range 0 . − . − , with an av-erage of 0 .
16. Using a fixed σ within the range 0 . − . Inferring τ Our key interest is to fit a time series, τ ( t ), that re-produces the annually observed wealth shares in [7]. Thewealth share S q is defined as the proportion of totalwealth, P Ni x i , owned by the richest fraction q of thepopulation, e.g. S = 80% means that the richest 10%of the population own 80% of the total wealth.For an empirical wealth share time series, S data q ( t ), weproceed as follows.Step 1 Initialize N individual wealths, { x i ( t ) } , as randomvariates of a lognormal distribution whose param-eters are chosen to match S data q ( t ).Step 2 Propagate { x i ( t ) } according to Eq. (1) over ∆ t = 1year, using the value of τ that minimizes the dif-ference between the wealth share in the modelledpopulation, S model q ( t + ∆ t, τ ), and S data q ( t + ∆ t ).We use the Nelder-Mead algorithm [9].Step 3 Repeat Step 2 until the end of the time series in2012.We consider historical wealth shares of the richest q =10%, 5%, 1%, 0 . .
1% and 0 .
01% and obtain timeseries of fitted effective reallocation rates, τ q ( t ), shownin Fig. 3. For each value of q we perform 10 independentruns of the simulation for N = 10 and average overthe obtained results. Since in practice dW is randomlychosen, each run of the simulation will result in slightlydifferent τ q ( t ) values. However, the differences betweensuch calculations are negligible. We observe a short spin-up period of approximately 3 years, after which τ q ( t ) isno longer sensitive to the initial distribution.Figure 3 shows large annual changes in τ q ( t ). We areinterested in longer-term changes in reallocation drivenby structural economic and political changes. To eluci-date these we take a central moving average, e τ q ( t ), witha 10-year window truncated at the ends of the time se-ries. This smoothing reveals a recognizable history ofwealth reallocation, e.g. the stock-market boom of the1920s which benefited wealthy shareholders ( e τ q ( t ) < e τ q ( t ) > S data q ( t ). This makes little difference: the samebehaviour of e τ ( t ), including the substantial negative val-ues from the 1980s onward, is obtained for all inequalitymeasures we analysed.To ensure that the smoothing does not introduce ar-tificial biases, we reversed the procedure and used e τ q ( t )to propagate the initial lognormally distributed { x i ( t ) } The lognormal is a reasonable representation of observed distri-butions [8], and is generated by Eq. (1) for τ = 0. Our resultsdo not depend strongly on the initial distribution. and determine the wealth shares S model q ( t ). The goodagreement with S data q ( t ) suggests that the smoothed e τ q ( t ) is meaningful, see Fig. 3. DISCUSSION
GBM, without reallocation, is a model of an economywithout social structure. Wealth inequality in this modelincreases indefinitely. We considered this unrealistic, andexpected the net effect of real social institutions to bereflected in positive τ . Therefore, the negative realloca-tion rates we found came as a shock. We didn’t imaginethat explicitly reallocative structures such as taxationand welfare spending had been overridden to the pointof reversal.Having said that, with realistic parameters σ and µ ,and non-negative τ everyone gets wealthier over time un-der the RGBM model. Therefore the recent observationthat “the wealth owned by the bottom half of humanityhas fallen by a trillion dollars in the past five years” [10,p. 2] is consistent with negative τ , not only in the US butalso globally.Studies of inequality often make the following assump-tions that go under the headings of equilibrium, ergodic-ity, stationarity, or stability.A. The system can equilibrate, i.e. a stationary dis-tribution exists to which the observed distributionconverges in the long-time limit. For example, arecent study states: “We impose assumptions [...]that guarantee the existence and uniqueness of alimit stationary distribution” [11, p. 130].B. The system equilibrates quickly, i.e. the observeddistribution gets close to the stationary distributionafter a time shorter than the timescale of observa-tion. This assumption is often left unstated, but itis necessary for the stationary distribution to havepractical relevance.Such studies fit parameters of the stationary distribu-tion which reproduce observed inequality. For example,a researcher would observe a wealth share and then findthe reallocation rate, τ eqm q , for which the model producesthe same wealth share in the long-time limit.We do not make the equilibrium assumptions. Fitting τ in RGBM allows the data to speak without constraintas to whether the assumptions are valid. We find themto be invalid because:A. The system cannot equilibrate for τ ≤
0. Themodel generates a non-stationary distribution andrunning it for longer produces greater inequality.B. The system does not equilibrate quickly for realisticparameters. The shortest equilibration time we in-ferred from e τ was approximately 50 years when Year R ea ll o c a t i on r a t e ( y ea r - ) -0.05-0.04-0.03-0.02-0.0100.010.020.030.04 τ ( t ) e τ ( t ) Year R ea ll o c a t i on r a t e ( y ea r - ) -0.25-0.2-0.15-0.1-0.0500.050.10.150.20.25 τ . ( t ) e τ . ( t ) Year R ea ll o c a t i on r a t e ( y ea r - ) -0.1-0.08-0.06-0.04-0.0200.020.040.060.080.1 e τ . ( t ) e τ . ( t ) e τ . ( t ) e τ ( t ) e τ ( t ) e τ ( t ) Year S % ( t ) ( % ) S data ( t ) S model ( t, τ ( t )) S model ( t, e τ ( t )) FIG. 3: Fitted effective reallocation rates using µ = 0 .
021 year − and Dow-Jones-derived σ ( t ). Top left: τ ( t ) (black) and e τ ( t ) (red). Top right: τ . ( t ) (black) and e τ . ( t ) (magenta). Translucent envelopes indicate one standard error in themoving averages. Bottom left: e τ ( t ) q reproducing the wealth shares listed in the legend. Bottom right: S data10% (blue), S model10% based on the annual τ ( t ) (dashed black), based on the 10-year moving average e τ ( t ) (red). e τ was at its maximum of 0 .
02 year − . This ismuch longer than the observation time of one year. Figure 4 contrasts τ eqm10% ( t ) as would be found in anequilibrium study with e τ ( t ) as found in this study. Ifequilibration were always possible and fast, then the twovalues would be identical within statistical uncertainties.They are not.The equilibrium assumptions preclude what we find The equilibration time was estimated as the exponential rate ofconvergence of the variance of the distribution of relative wealth x i / h x i N to its asymptotic value for prevailing levels of τ and σ . in this study, namely reallocation rates that correspondto wealth distributions which are either non-stationary( e τ q ( t ) ≤
0) or which equilibrate slowly (all observed e τ q ( t ) > i.e. in 2012, the latest year ofavailable data) the system is in a state best describedin RGBM by τ <
0. Each time we observe it, we see asnapshot of a distribution in the process of diverging. Itis much like taking a photo of an explosion in space: itwill show a fireball whose finite extent tells us nothing ofthe eventual distance between pieces of debris. Studies– of both fireballs and inequality – that assume equilib-rium must be treated skeptically, as they are incapable ofdetecting the dramatic conditions one finds without thisassumption.
Year R ea ll o c a t i on r a t e ( y ea r - ) -0.012-0.01-0.008-0.006-0.004-0.00200.0020.0040.0060.0080.010.0120.0140.0160.0180.02 e τ ( t ) τ eqm ( t ) Year S % ( t ) ( % ) S data ( t ) S model ( t, e τ ( t )) S model t, τ eqm ( t ) FIG. 4: Comparison of dynamic and equilibrium reallocation rates. Left: e τ ( t ) (red, same as in the top left of Fig. 3). τ eqm10% ( t )(green), defined such that lim t ′ →∞ S model10% (cid:0) t ′ , τ eqm10% ( t ) (cid:1) = S data10% ( t ). It is impossible by design for this value to be negative. Thesignificant difference between the red and green lines demonstrates that the equilibrium assumption is invalid for the problemunder consideration. Right: S data10% (blue), S model10% based on the 10-year moving average e τ ( t ) (red), based on τ eqm10% ( t ) (green).The reallocation rates found under equilibrium assumptions generate model wealth shares which bear little relation to reality. YB acknowledges travel support and hospitality fromLondon Mathematical Laboratory (LML). This researchwas partly funded by the LML Summer School 2015. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] O. Peters, W. Klein, Ergodicity breaking in geomet-ric Brownian motion, Phys. Rev. Lett. 110 (10) (2013)100603. doi:10.1103/PhysRevLett.110.100603 .[2] J.-P. Bouchaud, M. M´ezard, Wealth condensation in asimple model of economy, Physica A: Statistical Me-chanics and its Applications 282 (4) (2000) 536–545. doi:10.1016/S0378-4371(00)00205-3 .[3] O. Peters, A. Adamou,The evolutionary advantage of cooperation,arXiv:1506.03414.URL http://arxiv.org/abs/1506.03414 [4] T. Piketty, G. Zucman, Capital is back: Wealth-income ratios in rich countries, 1700-2010, The Quar-terly Journal of Economics 129 (3) (2014) 1255–1310. doi:10.1093/qje/qju018 . [5] J.-P. Bouchaud, On growth-optimal tax rates and the issue of wealtharXiv:1508.00275.URL http://arXiv.org/abs/1508.00275 [6] Dow Jones Industrial Average, , accessed:19/04/2016.[7] E. Saez, G. Zucman, Wealth inequality in the UnitedStates since 1913: Evidence from capitalized income taxdata, Tech. rep., National Bureau of Economic Research(2014). doi:10.3386/w20625 .[8] J. D. Sargan, The distribution of wealth, Economet-rica, Journal of the Econometric Society (1957) 568–590 doi:10.2307/1905384 .[9] J. A. Nelder, R. Mead, A simplex method for functionminimization, The Computer Journal 7 (4) (1965) 308–313. doi:10.1093/comjnl/7.4.308 .[10] D. Hardoon, R. Fuentes-Nieva, S. Ayele, An economy forthe 1%: How privilege and power in the economy driveextreme inequality and how this can be stopped, Tech.rep., Oxfam International (2016).[11] J. Benhabib, A. Bisin, S. Zhu, The distribution ofwealth and fiscal policy in economies with finitelylived agents, Econometrica 79 (1) (2011) 123–157. doi:10.3982/ECTA8416doi:10.3982/ECTA8416