Disparity in Wealth Accumulation — AFinancial Market Approach

October 26, 2014

(Preliminary and Incomplete Version)

Raphaele Chappe, The New School For Social Research

Willi Semmler, The New School For Social Research

Abstract

Disparity of wealth seems to be more severe than the disparity in in-come. In this paper we study to what extent the financial market hascontributed to wealth disparities. We assess current empirical work onthis issue and explore, in a dynamic asset accumulation model with het-erogeneous investors, the role of informational differences, risk aversion,saving rates and leveraging in their contribution to wealth disparities. Asshown, though those results are obtained in a stochastic approach, theoutcomes are less related to stochastic shocks but rather to some feed-back and scale effects operating in favor of some investors in the financialmarket.

1 IntroductionDisparities and inequality are rising in most developed economies — with respectto income, but more so with respect to wealth. In a recent opinion publishedwith the Financial Times, Robert Reich, the former Secretary of State for Laborfor the Clinton administration, points out that: “Since the start of the recession,the share of total U.S. national income going to profits has risen even as theshare going to labour has plunged. Profits in the U.S. corporate sector are nowat a 45-year high.” (Reich, 2012). Corporate earnings now represent the largestshare of the gross domestic product (and wages the smallest share) than at anytime in recorded history, leading Reich to conclude that 2013 has been the yearof “the great redistribution” (Reich, 2014). In other words, from a distributionalperspective the fraction of a smaller group of citizens accumulate a larger shareof income and wealth. Roughly speaking, we may ask whether this a result ofrising disparity in labor income or capital income. This paper studies the roleof the financial market for the secularly rising disparity in wealth holdings.

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Of course disparity and inequality are multi-faceted, with different compo-nents, and cannot be summarized by one single cause, measure or determinant.There is income inequality and wealth inequality. Income can be decomposedbetween, roughly speaking, labor income and capital income (and possibly anin-between category of entrepreneurial income for self-employed entrepreneurs).Labor income and capital income each have, of course, their own distribution.We can understand total income inequality in terms of the distribution of laborincome on the one hand, the distribution of capital income on the other (itselfdependent on capital ownership), and the share of each source of income interms of accounting for total income.

Top income shares in the U.S. are now higher than before World War II(Piketty and Saez, 2003). After the war, the share of the top decile income(excluding capital gains) fluctuated between 31 and 33 percent until the 1970s.By 1998 it had risen to 44 percent in the U.S. Today it is as much as 50 percent oftotal income, with the top 1 percent alone getting 20 percent (Piketty and Saez,2003, Wolff, 2010). Income inequality is worse in the U.S. than in other Westerncountries, as this compares with 35 (10) percent and 25 (7) percent going to thetop decile (centile) in Europe and Scandinavia respectively – see Table 7.3 fromPiketty (2013, p. 392) which summarizes the current (as of 2010) distributionof income for the U.S. as well as countries with lower inequality levels.

Studying the evolution of the composition of income over the reference period1913-1998, Piketty and Saez (2003) find a steady decline in top capital incomes(dividend, interest, rents, royalties, capital gains) since the 1960s, and a signif-icant increase of the share of wage income since 1929, for all income groups.This leads Piketty and Saez (2003) to conclude that there has been a changein how high income earners derive their income. They argue that “the workingrich have now replaced the coupon-clipping rentiers.” (Piketty & Saez, 2003, p.3). This could suggest that inequality today is mainly driven by factors thataffect top wage income. One hypothesis is that the wage distribution itself mayhave worsened through factors that shape the supply and demand of skilled andunskilled labor, such as technological and institutional change, globalization andoutsourcing, labor laws and job protection, union membership, etc. Regardingtop labor income specifically, Piketty expresses some reservations as to whetherthe standard theory of the marginal productivity of labor can properly accountfor the explosion of very high salaries for the top one percent (and especially0.1 percent) earners (Piketty 2013, p.529).

Yet, this is not to say, of course, that capital income does not play animportant role in explaining rising levels of inequality. The financial marketseems to have significantly contributed to wealth disparity, and capital incometends to play an increasingly important role relative to labor income as oneclimbs up the social ladder to top income earners (specifically within the topone percent). The top one percent is characterized by a combination of both topcapital income earners and top labor income earners, rather than the completereplacement of one by the other (Piketty, 2013, p. 475; Wolff & Zacharias,2009). Capital income exceeds labor income as a share of total income for thetop 0.1 percent of top income earners. On average, since the 1980s, the wealth of

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high net worth individuals has grown faster than that of average investors, andfaster than world GDP (see Table 12.1 in Piketty, 2013). There are now $13.7million of high net worth individuals (having investable assets of $1 million ormore), holding collectively $52.62 trillion as of 2013 (Capgemini & RBC WealthManagement, 2013).

It is well known that the wealth disparity is even greater than income in-equality. There are long swings in wealth inequality, declining with the riseof income tax in the prewar until the post War II period and then again sec-ularly rising again in the last few decades, in particular in the US. The fallin top capital incomes was for most countries concentrated around key macro-economic and fiscal shocks (World War I, the Great Depression, and World WarII). Piketty and Saez (2003) explain the decline in top capital incomes in termsof the inability of large fortunes to recover from such shocks, leading to a de-creased concentration of capital income (rather than a decline in the share ofaggregate capital income in the economy, which is relatively steady in the longrun at around 25-30 percent). At the beginning of the century capital/incomeratios were about 6-7 in Europe, and 4-5 in the U.S. They dropped significantlydue to the financial and physical destruction of capital during the Great De-pression and the two world wars (though Europe was more affected than theU.S.). These shocks to the capital/income ratio (and capital income) are largelyresponsible for the decline of top income shares in the first half of the century(Piketty & Saez, 2003).

As concerning recent decades, studies show that historically wealth distribu-tion has again become very concentrated in the U.S., since the end of the 1970s(See Wolff 1996 and 2010), this in part due to the gradual rise in capital/incomeratios in recent decades (Piketty, 2013). Wealth inequality has increased morethan income inequality since the late 1980s. In the U.S., the wealthiest 5 per-cent of American households held 54 percent of all wealth reported in the 1989Survey of Consumer Finance; this share has now reached 63 percent as of 2013(Yellen, 2014). The rise in top wealth share in the U.S. is well documented(Wolff, 2006), as well as fat tails for the wealth distribution (Nirei & Souma,2007). The top of the wealth distribution (people in the Forbes 400 list) exhibitsa power law distribution (Klass, Biham, Levy, Malcai, & Solomon, 2007). InEurope, the top 10 percent of wealthiest households own 50.4 percent of totalnet wealth (European Central Bank, 2013). The distribution of financial assetsis even more concentrated than the distribution of total wealth. In the U.S. thewealthiest 5 percent of households held nearly two-thirds of all financial assetsin 2013, and the bottom half of households hold as little as 2 percent (Yellen,2014).

Since the 2007/2008 financial crisis, there has been a drop in the wealth(networth) of households. However, the median household was affected morethan the top 1 percent of households (36.1 percent drop in wealth compared with11.1 percent) so that the wealth distribution is even more unequal now thanbefore the crisis (see Wolff, 2010).1 At the peak of the housing bubble in 2007,

1The findings in Wolff (2004) and Wolff (2010) are primarily based on the Survey of Con-

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the richest 1 percent held 34.6 percent of wealth – now they own 37.1 percent(Wolff, 2010). This result is somewhat surprising given that stock prices wereaffected by the financial crisis even more than housing prices. Yet the middleclass took a bigger hit from the decline in home prices than wealthy investorstook from the decline in stock prices because houses were a larger share of thegross assets of the middle class than stocks were for the wealthy (Wolff, 2010).The major asset of the middle class in the U.S. is their home. In the euro area,60.1 percent of households own their main residence (European Central Bank,2013).

In this paper we explore the forces causing disparities and inequalities thatare greatly coming from financial market. Running away of some “top laborincome” and “super star” incomes, may end up there, but important effectsfor rising wealth disparities seem to stem from forces in the financial markets.Though there is no clear evidence regarding the link between standard mone-tary policy and economic inequality, in a recent keynote address, Yves Mersch(member of the Executive Board of the European Central Bank) highlightedthat monetary policy could have an impact on wealth inequality precisely be-cause of households’ connection to financial markets. Expansionary monetaryshocks could increase inequality by redistributing wealth to those householdsmore connected to markets, as well as not benefit low income households whohold more cash and risk-free assets than high income households, see Mersch(2014).

For the rise of wealth in the last few decades – real and financial wealthas Piketty defines it – there might have been several factors at work: 1) risingsuperstar income enlarged through the financial market, 2) higher saving ratesof certain groups of wealth holders, (consumption may have upper constraints,so that saving rates can rise), 3) higher returns of certain groups of wealthholders on the higher end of the distribution relative to average investors andlow income earners, and 4) the possibility of leveraging up investments usinglarger assets and net worth as collaterals. These are the forces we want toexplore in our paper.

The history of economic theory provides us with some deeper explanations ofthe distribution of income and wealth. Those explanations start with the clas-sical economists (Smith, Ricardo, Marx) on the laws governing the allocationof income between wages, rent and profit. The modern theory of income distri-bution beginning after World War II (the works of Kaldor, Kuznets, Kalecki)mainly explored the issue of whether a country’s level of economic prosperityand stage of development has implications for income inequality. Kaldor (1956,1961) proposed a Keynesian model of economic growth, distribution and inequal-ity arising from different saving rates of workers and owners of capital. Kuznets(1955) studied the relation between income distribution and a country’s state ofdevelopment. Kuznet’s hypothesis was that the distribution of income is moreunequal in early stages of development (economic inequality first increases whilea country is developing), but that this trend is reversed after a certain income

sumer Finances (SCF) conducted by the Federal Reserve Board.

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threshold is reached – the famous inverted U-shape function (the Kuznet curve)relating inequality to the level of GDP. As the human-capital-led growth theorymade progress, the role played by education and human capital formation oninequality was explored. The endogenous growth literature further explored theissue of the exact causal relation between the factors of economic growth andincome distribution, in particular which ways it goes: inequality as an obstacleto growth or a factor of growth2.

Based on a unique data collection covering 20 countries and spanning (forsome) three centuries, Thomas Piketty’s recent book Capital in the Twenty-FirstCentury defies traditional economic thinking by suggesting that the trend ofincreasing inequality is the natural result of free-market dynamics. His positionis arguably closer to the gloomy views of Karl Marx than to the standard growthmodel where income differentials just arise from the marginal contributions ofthe factors of production3, or the optimistic hypothesis advanced by Kuznetsin the 1950s and 1960s (market forces will ultimately reduce inequality as acountry experiences industrialization and economic growth).

The success of Piketty’s book has placed distributional issues and inequalityat the forefront of the public economic debate. Piketty’s hypothesis is that themain driver of inequality is the tendency of returns on capital to exceed the rateof economic growth. This is a reasonable consideration if the returns on assetsfor certain groups of wealth holders are greater than the growth rate of incomeof the rest of the population. It of course also assumes that the consumptionfraction of income of the first group does not erode the asset accumulation ofthat group. As such, as he then argues, the prosperous decades that followed theGreat Depression and World War II (two major external shocks that reducedinequality) were more the exception than the rule in terms of making wealthinequality declining, a period somewhat unique and unlikely to be repeated. Therecent development, as many argue, is not without significance for the politicalsystem.

While most of the literature is focused on understanding disparities and in-equality in terms of income from labor, we here propose to study wealth inequal-ity and the process of wealth accumulation from a microeconomic perspectivefocusing on the financial market. A particular emphasis here is the fact thathigh net worth individuals are growing their wealth largely on the strength ofthe strong performance of global financial markets and with the help of largescale wealth management firms. Through their individual retirement accounts(401k, pension funds, etc.), average investors are likely to also be exposed tofinancial markets either directly or indirectly (e.g. via mutual funds) – the shareof households with indirect ownership of stocks increased from 23.5 percent in1989 to 47.7 percent in 2001 (Wolff, 2010). Yet their performance in terms ofwealth accumulation may show different results than for high net worth indi-viduals. As to personal investment and savings, low income investors usuallyinvest in low return assets – in risk free deposits in thrift organizations and

2See Greiner et al. (2005, ch 8)3Although one must admit that the Solow growth model would in fact also permit disparity

of income per capita in the long run, if one allows for differential saving rates.

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commercial banks. They also typically cannot easily leverage up when higherreturns are expected.

We may then study the problem of wealth inequality by asking what charac-teristics of the wealth distribution can be obtained from the dynamic behaviorof heterogeneous investors over time. In order to study the process of wealthaccumulation and distribution, we turn to a recurring and classic problem in fi-nancial economics, the accumulation and allocation of funds into different typesof assets, following different investment conditions and features. This leads usto a dynamic portfolio theory using a stochastic dynamic model of wealth ac-cumulation with preferences, saving and consumption, difference risk behavior,and heterogeneity between investors. The rest of the paper is organized as fol-lows. In Section 2, we review the main body of related literature. Section 3outlines the stylized facts we wish to study. Section 4 introduces a stochasticdynamic model of wealth accumulation. Section 5 presents our main results andsimulations. Section 6 concludes the paper.

2 Related Literature

2.1 Standard Approach to Income InequalityOne research trend is how do we explain the recent trend of high inequalityof labor income in the U.S. specifically? The traditional theory is that wagesare equal to the marginal productivity of labor, which in turn depends on aworker’s skills and qualifications, as well as the supply and demand for suchqualifications in a given society. The demand for labor is dependent upon thetechnology used to produce goods and services (technically captured by the con-cept of a production function), and its supply is dependent on the educationalsystem (Piketty, 2013, pp. 482-483). From the 1960s to 1980s, the developmentof human-capital-led growth theory further studied the role played by educationand human capital formation in inequality. See Becker and Tomes (1979) foran approach taking into account the characteristics of the community individ-uals find themselves in. See also Becker (1962), Stiglitz (1975), Riley (1976).Bowles and Gintis (2002) emphasize the role of inherited wealth in the persis-tence of inequalities. See also Brock and Durlauf (2006) for the role played bysocial environment for socio-economic outcomes, focusing on social-interactiondynamics.4

Consistent with the standard theory that wages are equal to the marginalproductivity of labor, which in turn depends on supply and demand for skills,one theory is that the increase of labor income inequality in the U.S. in the pastthree decades can be explained in terms of insufficient investments in highereducation (Goldin & Katz, 2008). Another theory is that the main cause ofthe declining demand for unskilled workers and the corresponding deteriorationof their relative wage is international trade (rather than technological improve-ment). As production shifts to high-skill production, we can expect the wage

4See also Barsington, Kato and Semmler (2010).

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differential between skilled and unskilled workers to increase. Richardson (1995)proposes a basic model in which trade and technology are exogenous, and showsthat international trade might result in an increase in wage inequality due todislocations and the difficulties of adjustment in the labor market.5 A third lineof this research explores the relation between technical progress and inequality– how technology influences relative wages.6 We will note here in passing thatthe standard model encounters difficulties to properly account for the dispro-portional increase in the top salaries for the 1 percent and 0.1 percent. AsPiketty observes, there is a very sharp discontinuity of salaries between the top1 percent of wage earners and the remaining top 9 percent. This cannot be eas-ily explained in terms of qualifications or professional experience, as one wouldhave expected a gradual increase in salaries (Piketty, 2013, p. 498).

Piketty’s recent hypothesis is that the main driver of inequality is the ten-dency of returns on capital (note that Piketty uses “capital” interchangeablywith wealth) to exceed the rate of economic growth. This is Piketty’s key in-equality relationship r > g which in principle is obvious, following classicalgrowth theory (since part of those returns on assets is consumed and cannotmake the economy growing). If, as above mentioned we give this the interpreta-tion that the returns on assets for certain groups of wealth holders are greaterthan the growth rate of income of the rest of the population, this is plausible.This naturally holds, if the consumption rate of the former group is smaller – apattern we will explore further in our model variants below.

Piketty derives this rising share share of total national income flowing tocapital income (α) as equal to the rate of return on capital (r) multiplied bythe capital/income ratios (β). In the long run, β is determined by the savingrate and the growth rate, with β = s/g (which Piketty labels the “secondfundamental law of capitalism”). Piketty argues that recent low growth rates

5Yet, we should also expect, in the long run, the ratio of skilled to unskilled employmentto decline, as employment shifts to skilled labor. Krugman (1994) argues that internationaltrade cannot be the main driver of wage inequality since this shift of employment to skilled-intensive industries has not been observed empirically. Krugman finds that some commonfactor affecting all sectors must be the cause of the wage differential.

6Building on the Romer (1990) growth model, Murphy, Riddell, and Romer (1998) usethe movement in relative wages as indicators of changes in the demand for different types oflabor, which in turn is related to technological change. The concept of skill-biased techno-logical change can help explain why wages for skilled workers have grown significantly morethan wages for unskilled workers in spite of the fact that the number of high skilled workershas sharply increased in many countries. Under a supply / demand framework, we wouldexpect relative wages for skilled labor to decrease. Yet if technology is complementary toskills, the demand for skilled labor increases as technology grows. Further, if there is a largesupply of skilled-workers, new technologies will be skill-based and skill-complementary to theextent the increased supply of skilled workers will make it profitable. Based on these ideas,Rubart, Greiner and Semmler (2004) develop a growth model of the Romer type to explore theforces generating skill-based wage inequalities. In the model, innovation is based on technicalchange, there are positive externalities, the structure of the productive sector is similar toRomer (1990), labor is divided into two groups (high skilled and low skilled), and there aresubstitution effects among the groups. See also Greiner, Semmler and Gong (2005) ch. 7 fora similar model. Aghion (2002) attempts to model endogenous technical change with qualityimproving innovation with wage dynamics.

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(in part due to small population growth) have led to a rise in β in advancedeconomies (though less so in the U.S. than in Europe and Japan). Pikettyobserves the historical data that shows that the return on capital is generallypretty stable at around 4-5 percent in the long run and concludes that, short ofpolicy changes that might perturb the continued stability of this rate, or highereconomic growth than anticipated (due potentially to technological progress),the share of income flowing to capital will increase.

2.2 Dynamic Portfolio Approach to Wealth Accumulationand Distribution

Most of this literature has focused on understanding inequality in terms of in-come. However, there are many reasons as to why wealth would be a morerelevant measure of inequality. Arguably, wealth is more representative of one’sability to consume (assets can be converte into cash) and overall economic well-being (the availability of financial assets can provide liquidity, a sense of security,and affect household behavior beyond mere income).7 Further, wealth distribu-tion is also relevant to the distribution of power in a representative democracy.The principal concept of networth used in most studies is that of the marketablevalue of assets (such as real estate, stocks, bonds, 401(k) plans), not includingconsumer goods not held for resale (such as cars), less liabilities (mortgage debt,consumer debt, etc.).8 In addition, economists also use the concept of financialwealth (i.e. wealth that is not tied to the value of one’s home), which can bedefined as networth minus net equity in owner-occupied housing.

The process of wealth accumulation can be approached from a microeco-nomic perspective, but it has considerable externalities. We may study theproblem of wealth inequality by asking what characteristics of the wealth distri-bution can be obtained from the dynamic behavior of individual households overtime. This approach differs from the human capital model (the marginal produc-tivity of labor) in that it explores the possibility that inequality may result frombehavior in financial markets, in connection with certain structural parametersof the economy (market risk, taxation, cost of asset management, availability ofleverage, etc.), investor characteristics (saving rates, discount rates, preferencesand risk aversion), as well as luck and stochastic forces (income shocks, or evensimple features of capital markets that inherently produce winners and losers),and not merely from the distribution of abilities, investment in human capital,or demand for specific skills.

Power-law dynamics can emerge from simple features of capital markets,such as the idiosyncratic risk component associated with the realization of cap-ital income, independently of any portfolio optimization problem and optimaldecisions about how much to save or consume. This idiosyncratic componentis not a standard assumption in most macroeconomic models, but see for in-stance Angeletos (2007), who introduces it in order to study aggregate savings

7See Conley (1999), Spilerman (2000).8See for example Woolf (2004).

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and growth (see also Benhabib and Zhu, 2008). This analytical developmentsuggests that inequality can result from luck and stochastic forces, rather thanthe distribution of abilities and investment in human capital.

For example, Klass et al. (2007) find that a simple stochastic multiplica-tive model whereby variations in the market value of an investment portfolioare characterized by random noise and no investor can consistently “beat themarket” (markets are efficient) can reproduce the Pareto wealth distributionfor the 400 wealthiest individuals based on the Forbes 400 data.9 Note how-ever that empirically, lower wealth percentiles follow an exponential rather thanpower-law distribution. The authors suggest that this may be explained lowerpercentiles of the population being less affected by financial market fluctua-tions to the extent they make fewer transactions in the stock market. Similardynamic models might account for both the power-law and exponential distri-butions at higher and lower percentiles respectively. A model which combinesa multiplicative asset accumulation process with an additive wage process cansuccessfully reproduce both exponential and power-law distributions – see forinstance Nirei and Souma (2007), finding the Pareto exponent for the power-lawtail distribution a function of the ratio of savings from labor income to assetincome.

Recent research reaches similar conclusions regarding the wealth distribu-tion by modeling households’ optimal decisions about how much to consume orsave from their labor and investment income. For instance, Fernholz and Fern-holz (2014) consider a set-up in which identical and infinitely-lived householdshave equal investment opportunities (equal expected returns) but face idiosyn-cratic investment risk (with luck alone affecting the evolution of wealth in theform of higher returns due the realizations of independent Brownian motions).They find that in the absence of any redistributive mechanisms (broadly de-fined as any process that proportionately affects wealthy households and poorhouseholds differently, including both direct and indirect transfers), the equilib-rium distribution of wealth is not stationary and over time becomes increasinglyright-skewed (at the limit wealth tends be entirely concentrated at the top).The major factor affecting the distribution of wealth is the idiosyncratic risk(the standard deviation of the independent Brownian motions). Accumulationwith the wealthiest households increases with individual households’ exposureto idiosyncratic risk. Importantly, the set up disregards differences in investorcharacteristics: households are identical in terms of their abilities to earn labor

9At each period, a randomly chosen wealth for one investor is multiplied by a factor λ drawnfrom a given distribution p (λ). Starting from a homogeneous distribution of wealth for 400investors, Klass et al. (2007) find that this process evolves towards a power-law distributionwith an exponent varying between 1 and 2 (a simulation done for 10,000 investors yields thesame result). A crucial assumption in Klass et al. (2007) is that the same distribution p (λ) beused for all the investors, as an implementation of the efficient market hypothesis (no investorcan consistently obtain a return distribution that is better than that of other investors).Though the short time gain/loss distribution is similar for all investors, differences betweenmore successful and less successful investors are magnified by the multiplicative dynamics inthe system (the impact of random stock price fluctuations on each investor being proportionalto the investor’s wealth).

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incomes, expected returns of risky asset, and preferences for consumption overtime (discount rates and risk aversion).

In addition to idiosyncratic risk, a stochastic labor endowment process canalso conceivably generate skewness in the wealth distribution. There is a largeliterature exploring the so-called “Bewley economies”, in which agents solvean infinite horizon consumption problem with incomplete markets, investingin a risk-free bond, and facing a stochastic process for labor earnings. Stan-dard Bewley models cannot generate heavy tails in wealth (Aiyagari, 1994;Huggett, 1993). To generate heavy tails some authors introduce features suchas preferences for bequest and entrepreneurial talent (Cagetti & De Nardi, 2008;Quadrini, 1999, 2000) and heterogeneous discount rates (Krusell & Smith Jr,1998). It can be shown that the Pareto distribution of the right tail is driven bycapital income risk rather than labor earnings (Benhabib, Bisin, & Zhu, 2011,2014). Wealth inequality increases with the capital income risk that house-holds face. Cagetti and De Nardi (2006) also give a comprehensive overview ofthe standard intertemporal economic models and the standard causes for theincrease in wealth. Their intergenerational model shows that more restrictiveborrowing constraints generate less inequality in wealth holdings, but also re-duce average firm size, the amount of entrepreneurial activities, and aggregatecapital accumulation.

3 Stylized Facts and our Set-upOverall, why might the assets of some asset holders grow faster than the assets ofothers? In general we could imagine three types of scale effects: First, investorshave more or better information about expected returns than others. Thoughit is commonly assumed that markets are efficient and no money is to be madeby forecasting (all information is already built into asset prices and returns),industry, firm and product knowledge, as well as knowledge on innovations,product development, and future market share are likely to give rise to bet-ter information and higher expected returns.10 Information flux could includeinsider information obtained through informal networks. There are economiesof scale associated with the size of the portfolio, meaning that wealthier in-vestors can afford more sophisticated asset managers, thereby securing higherreturns. Studying university endowments as a case study, Piketty finds thatthe rate of return depends on the size of the endowment (see Table 12.2 inPiketty, 2013). Further, for average investors there can also be restrictions onthe types of investments made, and on the corresponding associated expectedreturns. For example, though 401k funds invest in equities, most pension fundsof unions, firms and the public sector have constraints such that they shouldinvest a large portion in risk free assets (Treasury bills and Treasury bonds).

10This has often been the case in industry developments such as the oil boom, before WorldWar I, the boom in the auto industry after World War I, the boom in the steel industry, duringand after World War I, the high tech boom in the 1990s, the commodity price boom after1995, the real estate boom after 2001, and the recent banking and finance industry boom.

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In Europe the most prevalent financial assets for assets are deposits (sight orsaving accounts), which are owned by 96.4 percent of households, and voluntaryprivate pensions/whole life insurance (held by 33 percent of households) (Euro-pean Central Bank, 2013). All other financial assets are owned by less than 15percent of households.

Second, for large investors there are scale advantages, not only with respectto information but also with respect to leveraging. Investment opportunitiescan be explored more extensively with greater access to and lower cost of credit.Larger assets and larger net worth means larger collaterals, and availability offunding may rise and the cost of funding falls. Levering, and over-leveraginghas become one of the most common and well known strategies to harvest largegains from traditional as well as new financial instruments in the recent past.11

Third, one might reasonably assume that larger income will imply lower con-sumption rates and higher saving rates. This will result in a higher proportionof funds being reinvested. Numerous studies in the literature have used thisassumption showing that the wealth will be build up faster with higher savingratios. The faster build up of wealth is an expected outcome of this strategy.There is lots of evidence that higher income leads to higher saving rates.12Among households with heads aged 40 to 49, median saving rates on currentincome range from -23 percent in the lowest income quintile to 46 percent in thehighest on the basis of Consumer Expenditure Survey (CEX), the best availabledata on household consumption (Dynan et al., 2004, p. 416).13 Saving ratesestimated on the basis of the Survey of Consumer Finances (SCF) range from 1percent in the lowest income quintile to 24 percent in the highest, with estimatesfor households in the 95th and 99th percentile of the income distribution of 37and 51 percent respectively (Dynan et al., 2004, p. 416).14 Saving rates varyfor different age groups, and are lower for ages 30-39 and 50-59.

In order to study the process of wealth accumulation and distribution, weintroduce a stochastic dynamic portfolio model of wealth accumulation withpreferences and consumption. The choice of a portfolio is a recurring and clas-sic problem in financial economics. We wish to study the evolution of wealth

11There is this well-known Banker’s Paradox pointing to those scale effects: “The peoplewho most need the money are worst credits risks and thus cannot get a loan, whereas peoplewho least need the money are best credit risks and thus once again the rich get richer” (Toobyand Cosmides, 1996)

12Early contributions include Fisher (1930), Keynes (1936), Vickrey (1947), Duesenberry(1949), Hicks (1950), Pigou (1951), Friedman (1957), Friend and Kravis (1957), and Modiglianiand Ando (1960). For more recent work, see for example Saltz (1999), Dynan, Skinner andZeldes (2004), and Chakrabarty, Katayama and Moslen (2008).

13Dynan et al. (2004) define the saving rate for a CEX household as the difference be-tween consumption and after-tax income divided by after-tax income. Consumption equalstotal household expenditures plus imputed rent for home owners minus mortgage payments,expenditures on home capital improvements, life insurance payments, and spending on newand used vehicles.

14The saving rate variable used for the SCF calculations is equal to the change in real networth between 1983 and 1989 divided by six times the average of 1982 and 1989 divided bysix times the average of 1982 and 1988 total real household income. As such, it is computedover several years and is likely to be a less noisy measure of average savings.

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Figure 1:

simulations for three classes of investors, conservative, moderate and aggressive.We assume that such classes correspond, roughly, to investors belonging to thesocial classes identified by Piketty for thinking about distributional matters,mainly the working class (bottom 50 percent), the middle class (middle 40 per-cent), and the upper class (upper 10 percent) respectively. Unlike Fernholz andFernholz (2014) who use a relative risk aversion parameter for all investors, weuse different risk aversion parameters for three investor classes, as calibratedbelow.

Using an algorithm for nonlinear model predictive control developed by(Grüne & Pannek, 2011), made suitable for economic issues with discounting inGrune et al (2013), we can run two sets of simulations. This algorithm allowsto solve the model for a finite decision horizon, which we have set at twelveperiods. In the first set of simulations, all investors face identical market con-ditions and do not bear individual idiosyncratic risk (in short, all investors aretaken to face identical investment products and market conditions, such as theS&P500 or other well-diversified index), so as to isolate the impact of investorcharacteristics (risk aversion, leverage, saving rate) on the wealth distribution.In the second set of simulations (forthcoming), investors do bear idiosyncraticrisk with respect to the stock index, as in Klass, Biham, Levy, Malcai, andSolomon (2007), Nirei and Souma (2007), and Fernholz and Fernholz (2014).

In all scenarios we study the end wealth distribution and test for the robust-ness of the results by varying initial conditions slightly. We will test specificallyfor the isolated impact of investor level characteristics (risk aversion, leverage,saving rate) between the three classes of investors. All investors begin with thesame initial wealth ($50,000), and the same initial wage income of $30,000 (closeto the median wage, see Figure 1). We test for the impact on the end wealthdistribution of risk aversion, leverage, saving rate.

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4 The Model

4.1 A Stochastic ModelWe use a dynamic portfolio approach resembling Campbell and Viceira (2002),which includes a wealth equation as well as mean-reverting returns for the risk-free investment and for the risky-asset. Models with mean reversion in returnsare now frequently used in the portfolio modeling literature. There are stochas-tic shocks to the returns that take on the form of a Brownian motion. Thebasics of such model are analytically treated in Semmler (2011) Ch. 17 andCampbell and Viceira (2002). Similar models can be found, in different forms,in Chiarella, Hsiao and Semmler (2007), Wachter (2002), Munk, Sørensen, andNygaard Vinther (2004), Platen and Semmler (2009), and Brunnermeier andSannikov (2009).

Our model consists of N investors whose wealth is given at any moment intime by Wnt, where the index n = 1, ..., N represents each investor (for the easeof notation we drop the subscript n going forward with the understanding that itis implied). Wealth can be consumed or reinvested, with only reinvested wealthearning the portfolio return in the next time period, so that consumption todayis achieved at the expense of reinvesting assets that may increase consumptionin the future. So, a decrease in consumption will lead to greater savings andwealth accumulation. Each investor is maximizing the expected power utility15

of consumption over a finite horizon (T ), and is assumed to have a power utilityfunction U(Ct) = C1−γ

t /(1 − γ) with the risk-aversion parameter γ. Powerutility implies that absolute risk aversion is declining in wealth, with relativerisk aversion a constant. Each investor is also choosing the allocation of wealthin the portfolio (the fraction of assets invested in risky and risk-free assets). Toexplain the evolution of income and wealth we use four stochastic processes:

First, a process for wealth, whereby the investor starts with an initial wealth,and is earning labor income in each period [equation (2)]. We add labor incometo the wealth equation in order to properly account for the share of labor in-come being continually saved-up and invested during a person’s life (for instancefunding an employee’s 401k during employment). From the perspective of finan-cial economics labor income can be thought of as a dividend on an individual’s“human wealth”, the expected present discounted value of future labor earn-ings. However, to the extent human wealth is not a tradable financial asset (itis difficult to monetize or sell claims against future labor income, due to im-perfect capital markets), investors cannot directly borrow against future laborincome. A fraction (1− αt) of assets is invested in a risk-free investment (risk-free here means that the return is known over the holding period, earning therisk free interest rate rt), and a fraction (αt) is invested in risky assets (earninga risk-premium xt above the risk free rate). If αt > 1, there is leverage and thefraction “invested” in the risk-free asset is interpreted as the cost of borrowing.Note that for the time being we have disregarded the possibility that an investor

15Note that we could take log utility here, in which case we would be back to a classicalgrowth model.

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own equity in his or her home and potentially finance current consumption byborrowing against this equity (with for instance a home-equity line of credit).We are only considering financial wealth.

Second, a process of a mean-reverting equity premium on risky assets (thestock index) described by an Ornstein-Uhlenbeck process, whereby the time-varying excess return is xt and the expected mean of the equity premium x[equation (3)]. λ is an adjustment coefficient, representing the adjustment speedof the equity premium towards the mean. All future trajectories of the equitypremium will evolve around x, and will do so (in the long run) with a long-termvariance of δ2x/2λ.

Third, a process of a mean-reverting risk-free interest rate, described by aCox-Ingersoll-Ross process, whereby the time-varying risk-free interest rate isrt and the mean interest rate θ [equation (4)]. The drift is exactly the same asin the Ornstein-Uhlenbeck process (ensuring mean reversion in the long run),with the adjustment coefficient κ. The level dependent volatility avoids thepossibility of negative interest rates. All future trajectories of the risk-free ratewill evolve around θ, and will do so (in the long run) with a long-term varianceof δ2r/2κ. The dynamics described in equation (4) are for nominal interest rates.Our model does not take into account inflation uncertainty, whereby the realprice of assets could be determined by deflating by the price index (the nominalprice of the real consumption good in the economy, the dynamics of whichcould be determined by another system of differential equations, as in Munk etal. (2004), for example).

Fourth, a process for the evolution of labor income. We model the processof labor income following Campbell and Viceira (2002) in Ch. 6. We assumethat expected labor income growth is constant, and allowing for the possibilityof transitory income shocks [equation (5)]. We also explore retirement horizoneffects (by introducing a positive probability that the investor retires).

For each investor, the continuous-time (with discount rate δ) optimal portfolioand consumption problem for agents is as follows:

maxα,C

ˆ T

t=0

e−δtCt

1−γ

1− γdt (1)

s.t.dW = {[αt(rt + xt) + (1− αt)rt]Wt + Lt − Ct}dt+ σwWtdz1t (2)

dxt = λ(x− xt)dt+ σxdz2t (3)

drt = κ(θ − rt)dt+√rtσrdz3t (4)

dLt = gLtdt+ σLdz4t (5)

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Table 1: Capital market parameters from Munk et al. (2004)

where Ct is the total amount consumed in the period, determined as a percent-age of wealth, and T the finite investment horizon. Note that in our simulationscapital income earned in one period is available for consumption in the next pe-riod, while labor income is immediately available for consumption in the currentperiod. A stochastic shock is imposed on each dynamic equation, where theterms are the increments in the Brownian motions (Wiener processes). Eachprocess has its own volatility parameter, with an excess return (equity premium)volatility parameter σx, an interest rate volatility σt, and a labor income volatil-ity σL. Though the Wiener processes are possibly different for all four Brownianmotions, we assume that the Wiener process in the wealth equation can be writ-ten as a function of the Wiener processes for the risky and risk-return, with thesame framework as in Munk et al. (2004):

σwWtdz1t = −{αtσSdz2t + (1− αt)σBdz3t} (6)

where σS is the stock return volatility, σB is the bond return volatility, withσB = σr when there is no interest rate premium and the duration of a bond is 1.The expected excess return and the stock return processes are instantaneouslyperfectly negatively correlated, as are the short interest rate and the returnon the bond. We assume, as in Munk et al. (2004) that the two basic Wienerprocesses that generate the dynamics of the excess stock return and the nominalinterest rate (dz2t and dz3t) are correlated with a constant correlation coefficientρxr. The correlation between the stock index and the nominal interest rate is−ρxr.

4.2 Calibration of the ModelWe assume T = 12. To calibrate the model to U.S. data, we assume capitalmarket parameters (Table 1) and investor level parameters (Table 2) empiricallyestimated in Munk et al. (2004):

For purposes of setting limits on the maximum consumption, we refer to thefollowing saving rates estimates (Table 3) from Dynan et al. (2004).

Using the estimates prepared on the basis of the SCF, we cap consumptionto reflect the following saving rates for our different investor group (using the

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Table 2: Investor level parameters from Munk et al. (2004)

Table 3: Saving rates estimates from Dynan et al. (2004)

data of a specific quintile as a representative of the investor class, though thisis somewhat imperfect because the quintile categories do not perfectly matchout distributional categories): 0 percent for the bottom 50 percent (the secondquintile), 17 percent for the middle 40 percent (the fourth quintile), and 23percent for the top 10 percent (the fifth quintile). Regarding labor income, weassume a yearly growth rate of 2 percent, as consistent with the past 5 years,and a labor volatility σL = 0.1.

If αt > 1, there is leverage and the fraction “invested” in the risk-free asset isinterpreted as the cost of borrowing. We assume that only aggressive investors(the upper decile) have access to leverage. Average investors typically do notuse borrowed funds to invest in the stock market, and investment productsavailable to average investors (e.g. through a pension fund or a 401k) typicallydo not include leverage. In the mutual fund industry, for instance, the use ofleverage is restricted by Section 18(f) of the Investment Company Act of 1940.This is not the case for alternative investments and other products offered bysophisticated asset managers, which are typically less regulated. The hedge fundindustry is unrestricted in terms of leverage both in the U.S. and the EU.16 Inour simulations we set the maximum leverage to αt = 2.17

5 SimulationsWe consider the wealth distribution after 30 periods for a population of N =1, 000 and study the impact of risk aversion, leverage, and saving rate. Fornow we use a step size of one (so that 30 periods are equivalent to 30 years).As mentioned before, when we run simulations with leverage, we assume that

16Neither Dodd-Frank nor the Alternative Investment Fund Managers Directive seem toprovide for concrete limits on leverage, only disclosure.

17On average at the fund level, the hedge fund industry has a 1.5 to 1 debt to equity ratio.

16

leverage is capped at αt ≤ 2, but only made available to the top 10% of investors.In all simulations we assume as a starting point a risk-free rate of 1% anda risk premium of 5%. As mentioned above, in all simulations and resultsbelow, 50 percent of investors are taken to belong to the lower class and investconservatively (saving rate cannot be lower than 17%, and the risk aversionparameter γ = 4.177). The top 10 percent of investors are taken to invest asaggressive investors (saving rate cannot be lower than 23%, and the risk aversionparameter γ = 2.198).

In the first set of simulations, all investors face identical market conditionsand do not bear individual idiosyncratic market risk. Further, we retain thesame random shocks in all simulations, so as to test for the impact of othervariables. When investors do earn labor income, we assume that each investorbears individual idiosyncratic risk with respect to labor income, due to personalcircumstances and career path (hence in our figures below, the indication thatthere is no idiosyncratic risk refers to market risk only). We also explore re-tirement horizon effects (by introducing a positive probability that the investorretires) - we beging with a small probability that the investor retire in the nextperiod (2%). This probability increases by 2% in each forecasted period in theinvestor’s investment horizon (12 periods) T , as well as at the beginning of eachperiod in the overall simulation (30 periods). Figure 2a shows the evolution ofthe risk-free interest rate and the risk premium throughout the simulation (30periods) as driven by Brownian motions. Figure 2b shows the shocks to thestock index. These are taken to be identical for all investors, since we do notconsider idiosyncratic risk for this first set of simulations.

All investors begin with the same initial wealth (we test for the robustness ofthe results by varying initial conditions slightly and consider both initial wealthof $10,000 and $50,000 – we can assume any currency at this stage, though forconvenience sake we use US Dollars). We run simulations with and withoutlabor income (given the same initial wage income of $30,000 for all investors,close to the median wage in the U.S., see Figure 1). To show what the detailsof any given simulation might look like, Figure 3a shows for a given investor theevolution of wealth through the simulation (30 periods), and the consumptionratio (as a percentage of the sum of financial wealth at the start of period andincome earned in the current period – including both labor income and capitalgains/dividends earned with respect to investments made in the prior period).Figure 3b shows the path of labor income given the process in equation (5) forone specific investor (with his or her own idiosyncratic to labor income) in asimulation over 30 periods where investors begin with $10,000 in the first period.In forthcoming work, we ultimately intend to test the robustness of our resultsby running Monte-Carlo simulations for each investors (i.e. running simulationsfor many possible market processes, given different shock paths, with again thepossibility of investors bearing individual idiosyncratic risk).

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Figure 2: Risk free rate (%, in blue), risk premium (%, in red), and stock market(index) shocks, over 30 periods

(a) Risk free rate and risk premium (b) Shocks to stock index

Figure 3: Evolution of wealth (in ’ooo, in blue) and consumption ratio (as a %of both starting wealth and income for the period), over 30 periods

(a) Evolution of wealth and consumption ratio (b) Evolution of labor income

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Figure 4: Wealth distribution (without labor income) at the end of 30 periods,without leverage (αt ≤ 1), no idiosyncratic shocks

(a) Starting wealth 10k (b) Starting wealth 50k

Figures 4a and 4b show the wealth distribution (histogram) at the end ofsimulation period, where all investors start with $10,000 (Figure 3a) and $50,000(Figure 3b), and earn no labor income throughout the simulation. The only dif-ference between investors is the minimum saving rate and risk aversion, assignedon the basis on of the investor category (50 percent of investors are assumedto be conservative, 40 percent are assumed to be moderate, and 10 percent areassumed to be aggressive). Since all investors begin with the same wealth, atthis stage, these categories do not correspond to the bottom, middle and up-per distributional percentiles, but are still helpful to understand the impact ofinvestor characteristics on the wealth distribution. We find that investors be-longing to any given investor class (conservative, moderate or aggressive) endup with the same amount of wealth at the end of the simulation as all other in-vestors belong to the same class, since all investors in a given class are essentiallyidentical at that stage (and there is no idiosyncratic risk). Hence the end wealthdistribution has three single separate peaks. Interestingly, the characteristics ofaggressive investors in terms of risk aversion (γ = 2.198 as opposed to γ = 4.177and γ = 8.689 for moderate and conservative investors respectively), and sav-ing rate (a minimum of 23% of current year income for aggressive investors, asopposed to 17% and 0% for moderate and conservative investors respectively)already create a fatter distributional tail.

Figures 5a and 5b show the wealth distribution for the exact same simula-tions, but for the fact that leverage is allowed for the agressive investors. For astarting wealth of $10,000, aggressive investors end up with $25,000 as opposedto $6,500 (Figure 5a vs. Figure 4a). For a starting wealth of $50,000, they endwith $120,000 as opposed to $33,000 (Figure 5b vs. Figure 4b). We can seethat the availability of leverage (αt ≤ 2 ) for 10 percent of investors (aggressive)

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Figure 5: Wealth distribution (without labor income) at the end of 30 periods,with leverage (αt ≤ 2), no idiosyncratic shocks

(a) Starting wealth 10k (b) Starting wealth 50k

creates a fat tail – investors with leverage opportunities end up almost six timeswealthier at the end of the simulation (compare $25,000 for aggressive investorsrelative to $5,000 for the others in Figure 5a, and $120,000 relative to $25,000in Figure 5b).

Figures 6a and 6b show the wealth distribution when investors earn laborincome (initial wage of $30,000), without any leverage, while Figures 7a and 7bshow the wealth distribution when investors earn labor income with leverage.We find that the presence of idiosyncratic labor income risk means that thereis more dispersion in the end wealth distribution, which looks like a normaldistribution skewed to the right. When investors start with $10,000, availabilityof leverage for aggressive investors results in an added $500,000 in wealth at theend of the simulation. When investors begin with $50,000, leverage results inas much as an added $1,000,000 for the fat tail of the distribution. Hence, wealso find that leverage creates fatter tails.

In a second set of simulations, we intend to model the results when investorsdo bear idiosyncratic risk with respect to the stock index, as in Klass, Biham,Levy, Malcai, and Solomon (2007), Nirei and Souma (2007), and Fernholz andFernholz (2014) [forthcoming].

We also run a set of simulations where we impose the constraint that αt ≤ 0.1to reflect the fact that the lower wealth group mainly invests in low risk assets:deposits of banks, Treasury bills and Treasury bonds. Pension fund rules alsoimpose similar conservative requirements. We find that this restriction resultsin a severe double peak distribution, with the lower class on the lower end of thewealth distribution (Figure 8a and 9a show the end wealth distribution without

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Figure 6: Wealth distribution (with labor income) at the end of 30 periods,without leverage (αt ≤ 1), no idiosyncratic shocks

(a) Starting wealth 10k (b) Starting wealth 50k

leverage, both without and with labor income). As before, the presence of id-iosyncratic risk for labor income results in a more dispersed wealth distribution.Leverage further intensifies the problem, both without and with labor income(Figure 8b and 9b respectively). In Figure 8b, we see quite well the impactof leverage as allowing the wealth of the top 10 percent to really take off ascompared with the rest of investors.

6 ConclusionIn order to study the process of wealth accumulation and distribution, we intro-duce a stochastic dynamic portfolio model of wealth accumulation with prefer-ences and consumption. In our simulations, investors decide on how to allocatefinancial assets (between a risk free asset and a risky asset) as well as how muchto consume (as a percentage of both current period income and wealth availableat the beginning of each period). We study the problem of wealth inequality byasking what characteristics of the wealth distribution can be obtained from thedynamic behavior of heterogeneous investors over time. For this purpose, we dis-tinguish between three groups of investors that we assume roughly correspondto investors belonging to the social classes identified by Piketty for thinkingabout distributional matters, mainly the bottom 50 percent (conservative), themiddle 40 percent (moderate), and the upper 10 percent (aggressive). Eachinvestor category has different saving rates and risk aversion parameters. Forthe time being we assume away the existence of idiosyncratic risk for investors,

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Figure 7: Wealth distribution (with labor income) at the end of 30 periods, withleverage (αt ≤ 2), no idiosyncratic shocks

(a) Starting wealth 10k (b) Starting wealth 50k

Figure 8: Wealth distribution (without labor income) at the end of 30 periods,lower investors constrained to αt ≤ 0.1, no idiosyncratic shocks

(a) Starting wealth 10k, without leverage(αt ≤ 1) for other investors

(b) Starting wealth 10k, with leverage (αt ≤2) for top 10% investors

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Figure 9: Wealth distribution (with labor income) at the end of 30 periods,lower investors constrained to αt ≤ 0.1, no idiosyncratic shocks

(a) Starting wealth 10k, without leverage(αt ≤ 1) for other investors

(b) Starting wealth 10k, with leverage (αt ≤2) for top 10% investors

considering that there are no winners or losers in the stock market, but ratherthat all investors have access to identical investment opportunities. In a sec-ond set of simulations, we will specifically consider the impact of idiosyncraticcapital market risk on the wealth distribution.

We find that the differences in saving rates (capping consumption) and riskaversion alone are sufficient to impact the wealth distribution over time, andthat the availability of leverage for some investors (the upper 10 percent) createsfatter tails. The presence of labor income does not change these results, thoughit does create a greater degree of dispersion with respect to the end wealthdistribution (due to the presence of idiosyncratic shocks to labor income).

We can also use this basic framework to test for the impact of higher ex-pected returns available to some investors. For the time being, we have limitedour simulations to the constraint that some investors on the lower end of thedistribution are limited to investing in low risk assets with lower expected re-turn. The simulations show the increase in wealth inequality resulting from thisconstraint. We can prolong this work by studying the impact of informationaladvantages that would allow investors on the higher end of the distribution tobenefit from investment opportunities that may allow them to earn expectedreturns that are slightly higher, in the long run.

Overall, using a dynamic model of asset accumulation with heterogeneousinvestors, we highlight the important role of informational differences, risk aver-sion, saving rates and leveraging in terms of their contribution to wealth dispar-ities. As shown, though those results are obtained in stochastic approach theoutcomes are less related to stochastic shocks but rather to some feedback andscale effects operating in favor of some investors in the financial market.

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[68] Yellen, J. (2014). Perspectives on Inequality and Opportunity fromthe Survey of Consumer Finances. Speech given at the Confer-ence on Economic Opportunity and Inequality, Federal ReserveBank of Boston, Massachusetts (October 17, 2014) available athttp://www.federalreserve.gov/newsevents/speech/yellen20141017a.htm#f15

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