# The Distribution Of Wealth

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Chapter 11

THE DISTRIBUTION OF WEALTH *

JAMES B. DAVIESa and ANTHONY E SHORROCKSb aDepartmentof Economics, University of Western Ontario, London N6A 5C2, Canada;bDepartmentof Economics, University of Essex, Colchester C04 3SQ, UK and Institutefor FiscalStudies, 7 Ridgmount Street, London WC1E 7AE, UK

Contents

1. Introduction

606

2. Theoretical approaches

608

2.1. Simple models of wealth distribution

609

2.2. Lifecycle accumulation

612

2.3. Intergenerational issues

621

2.4. Self-made fortunes

627

3. Empirical evidence on wealth inequality

628

3.1. Household surveys

629

3.2. Wealth tax data

635

3.3. Estate multiplier estimates

637

3.4. The investment income method

642

3.5. Direct wealth estimates for named persons

642

3.6. Portfolio composition

643

3.7. Explaining national differences and changes over time

645

4. Applied work on the determinants of wealth distribution

648

4.1. Simulation studies

648

4.2. Intergenerational wealth mobility

651

4.3. The aggregate importance of inherited versus lifecycle wealth

653

4.4. Empirical studies of transfer behaviour

657

5. Summary and conclusions

663

References

667

* We should like to thank Andrea Brandolini, John Flemming, John Piggott, Edward Wolff and, most especially, Tony Atkinson and Franqois Bourguignon, for their advice, comments and assistance. The first author also thanks the Social Sciences and Humanities Research Council of Canada for its support.

Handbook of Incozme Distribution:Volume 1. Edited by A. B. Atkinson and F Bourguignon (O1999 ElsevierScience B. V All rights reserved.

605

606 Abstract

JamesB. Davies andAnthony F Shorrocks

This chapter is concerned with the distribution of personal wealth, which usually refers to the material assets that can be sold in the marketpace, although on occasion pension rights are also included. We summarise the available evidence on wealth distribution for a number of countries. This confirms the well known fact that wealth is more unequally distributed than income, and points to a long term downward trend in wealth inequality over most of the twentieth century. We also review the various theories that help account for these feature. Lifecycle accumulation is one popular explanation of wealth differences, but inheritance is also widely recognised as playing a major role, especially at the upper end of the wealth range. A recurrent theme in work on wealth distribution is the relative importance of these two sources of wealth differences. We discuss the results of studies that assess the contributions of inheritance and lifecycle factors, and give attention also to a variety of related issues, such as the link between wealth status across generations, and the possible motives for leaving bequests.

Keywords: Wealth distribution, wealth inequality, portfolio composition, lifecycle saving, inheritance, bequests, intergenerational transfers

JEL codes. main: C81, D31, D91, E21, GlI; secondary: Dll, D12, D63, D64, J14)

1. Introduction

This chapter surveys what is known about the distribution of personal wealth and its evolution over time. We review the descriptive evidence as well as theoretical and applied research that attempts to explain the main features of wealth-holdings and wealth inequality observed in the real world.

There are many reasons for interest in personal wealth, and many ways in which the concept of wealth may be defined. If we were concerned with the overall distribution of economic well-being or resources, it would be appropriate to examine the distribution of "total wealth", that is, human plus nonhuman capital. But that is not our objective here. Instead, we exclude the human capital component and focus on material assets in the form of real property and financial claims. The term "wealth" will therefore usually refer to "net worth"-the value of nonhuman assets minus debts. Our aim is to examine the reasons for holding wealth, to document the observed differences in holdings across individuals and families, and to examine the causes of these observed differences.

One major reason for interest in wealth-holdings is that, unlike human capital, most real property and financial assets can be readily bought and sold. This allows nonhuman wealth to be used for consumption smoothing in periods when consumption is expected to be high (growing families) or income is expected to be low (retirement), and in periods of unanticipated shocks to either income or expenditure (the precautionary motive for

Ch. 11: The Distributionof Wealth

607

saving). This consumption smoothing role is particularly important when individuals face capital market imperfections or borrowing constraints. Wealth may also be accumulated, or retained, for the purpose of making bequests. Additional noneconomic reasons for studying wealth include the power or social status which may be associated with certain types of assets such as privately-owned businesses. The pattern of wealth-holdings across individuals, families, and subgroups of the population, is therefore capable of revealing a great deal about both the type of economy in which people operate, and the kind of society in which they live.

The concept of net worth may appear to be straightforward, but should we deal with intangible assets which cannot be readily bought and sold? This category covers pension rights, life insurance, and entitlement to future government transfers (including "social security wealth"). Any attempt to include the rights to uncertain future benefits has to confront a variety of difficult valuation problems. For example, it is not obvious what discount rates should be used for these assets. Should they be risk-adjusted? Should a special adjustment be made for people who are borrowing constrained? Satisfactory answers to these questions require a considerable amount of painstaking work. It is therefore not surprising to discover that most applied work on wealth-holdings and wealth distribution confines itself to marketable wealth. When reviewing the empirical evidence, we use the term "augmented wealth" to refer to the broader concept which includes entitlements to future pension streams.

There are certain important "stylized facts" about the distribution of wealth which it is useful to highlight at the outset. These are:

1. Wealth is distributed less equally than labour income, total money income or consumption expenditure. While Gini coefficients in developed countries typically range between about 0.3 and 0.4 for income, they vary from about 0.5 to 0.9 for wealth. Other indicators reveal a similar picture. The estimated share of wealth held by the top 1% of individuals or families varies from about 15-35%, for example, whereas their income share is usually less than 10%.

2. Financial assets are less equally distributed than nonfinancial assets, at least when owner-occupied housing is the major component of nonfinancial assets. However, in countries where land value is especially important, the reverse may be true.

3. The distribution of inherited wealth is much more unequal than that of wealth in general.

4. In all age groups there is typically a group of individuals and families with very low net worth, and in a number of countries, including the US, the majority have surprisingly low financial assets at all ages.

5. Wealth inequality has, on the whole, trended downwards in the twentieth century, although there have been interruptions and reversals, for example in the US where wealth inequality has increased since the mid 1970s. Possible explanations for these, and other, stylized facts will be investigated in this

chapter. Section 2 begins with a review of simple models that try to account for the overall shape of the distribution of wealth. We also show how the accounting identity relating

608

James B. Davies andAnthony F Shorrocks

changes in wealth to earnings, rates of return, consumption and capital transfers provides a useful framework for investigating the proximate determinants of wealth inequality and its trend over time. Attention then turns to more detailed models of wealth-holdings, including numerous versions of the lifecycle saving model, and a variety of models concerned with bequest behaviour and the distributional impact of inheritance.

Section 3 reviews the descriptive evidence on wealth distribution in a number of countries, and discusses possible explanations for national differences and trends over time. The roles of asset prices, inheritance taxation, and other factors are examined. Then, in Section 4 we look at applied research which attempts to assess the contribution of different factors to the distribution and evolution of wealth-holding. These studies address a variety of questions, and use a variety of approaches, including decomposition procedures, simulation exercises, and conventional econometrics. But, in broad terms, all the research is linked by a common objective: to cast light on the reasons for wealth accumulation, and on the importance of inheritance vis-A-vis lifecycle saving as determinants of the level and distribution of wealth.

2. Theoretical approaches

Many different types of theories have been used to model aspects of wealth-holdings and wealth distribution. To a large extent they reflect the empirical evidence available at the time the theories were proposed. Prior to the 1960s, data on wealth were obtained primarily from estate tax and wealth tax records, with other evidence pieced together from small unrepresentative surveys and a variety of other sources. These tended to confirm the widely held view that wealth was distributed very unequally, and that material inheritance was both a major cause of wealth differences and an important vehicle for the transmission of wealth status between generations. There were also grounds for believing that wealth inequality was declining over time, and that the shape of the distributions exhibited certain statistical regularities which could not have arisen by coincidence. Early theoretical work on wealth distribution sought to explain these statistical regularities, and to understand the interplay of basic forces that could account for high wealth concentration and a declining trend over time.

More recently, research has shifted away from a concern with the overall distributional characteristics, focusing instead on the causes of individual differences in wealthholdings. The change of emphasis was prompted in part by a recognition of the increasing importance of saving for retirement, and is reflected in the central role now assigned to the lifecycle saving model formulated by Modigliani and Brumberg (1954) and Ando and Modigliani (1963). The second major development has been the growth in the availability and sophistication of micro-data sets that offer not only estimates of the savings and asset holdings of individuals but also a range of other personal and household characteristics than can help account for differences in wealth.

Ch. 11: The Distributionof Wealth

609

This section reviews various models of wealth distribution and wealth differences, beginning with some of the early attempts to explain the overall distributional features and the evolution of wealth inequality. Section 2.2 discusses the simple lifecycle model and some of its many extensions, drawing out the implications for wealth accumulation over the course of the lifetime. Attention here is restricted to pure intra-generational models in which inheritances play no role. For the most part, Section 2.3 takes the opposite tack, suppressing interest in lifecycle variations and focusing instead on intergenerational links, especially those concerned with the motivation for bequests and the impact of inheritance.

The distinction between models of wealth accumulated for lifecycle purposes and models concentrating on the intergenerational connections reflects the theoretical literature on wealth distribution: very few contributions have attempted to deal simultaneously with both the lifecycle and inherited components of wealth. This is a major weakness of past theoretical work on wealth. While there may be some value in modelling lifecycle wealth in the absence of inheritance, it should be recognised that this exercise does not reveal the true pattern of lifecycle accumulation in the real world-since the real world also has inheritances. This rather obvious point is one which is often forgotten, particularly when attempts are made to assess the relative importance of accumulation and inheritance.

Current models of accumulation and bequests do not appear to capture the circumstances and motives of those who amass large fortunes in the course of their lifetimes. We look briefly at this issue in Section 2.4. Throughout, reference is made to relevant empirical evidence that has informed and influenced the development of the theories. Later, in Section 4, we consider in more detail some of the issues that have received most attention.

2.1. Simple models of wealth distribution

Early empirical work on personal wealth-holdings established two enduring features of the shape of the distribution of wealth. First, it is positively skewed, unlike the normal distribution, but roughly resembling a normal distribution when wealth is replaced by the logarithm of wealth. Second, the top tail is well approximated by a Pareto distribution, which yields a straight line graph when the logarithm of the number of persons with wealth above w is plotted against log w. These two statistical regularities were observed not only for wealth-holdings, but also for many other skewed distributions such as those for incomes, the turnover of firms, and the size of cities (see, for example, Steindl, 1965, Appendix B).

The fact that these size distributions are approximated by the lognormal suggests that, by appealing to the Central Limit Theorem, they can be generated by a random walk of the form:

In Wt == nn WWt + ut,

(2.1)

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James B. Davies andAnthony F Shorrocks

where Wt is wealth at time t, and u is a stochastic component (Aitchison and Brown, 1957). Equation (2.1) contains the implicit assumption that the factors influencing changes in wealth over time operate in a multiplicative fashion, rather than additively. This became known as Gibrat's "law of proportionate effect", following Gibrat (1931).

As a model of wealth distribution, the simple random walk Eq. (2.1) has one major technical disadvantage: it predicts that wealth dispersion, as measured by variance of log wealth, increases over time without bound. It cannot therefore apply when wealth inequality is stable or decreasing. To overcome this problem it is necessary to introduce a mechanism which offsets the inequality increasing impact of the stochastic component. One simple solution is the formulation proposed by Galton (1889) in his study of inherited genetic characteristics, which yields

InWt = IlnWtl +ut,

O < < 1,

(2.2)

with /i indicating the degree of "regression towards the mean". Given appropriate constraints on the stochastic component ut, this process can generate a stable lognormal distribution of wealth in which changes in wealth at the level of individuals exactly balance out to maintain equilibrium in the aggregate (see, for example, Creedy, 1985).

A number of other authors, including Champernowne (1953), Wold and Whittle (1957), Steindl (1972), Shorrocks (1973, 1975b), and Vaughan (1975, 1979), proposed alternative types of stochastic models capable of generating distributions of wealth with upper tails that are asymptotically Pareto. In broad terms, these models all assume that a variant of the law of proportionate effect applies at high wealth levels, where the expected change in wealth must be negative in order to prevent the top tail from drifting upwards over time. Equilibrium is maintained by some mechanism lower down the distribution-such as a pool of low wealth-holders in the case of Wold and Whittle (1957)-which stops wealth from converging to zero in the long run. As with the Galtonian specification Eq. (2.2), these models typically view the observed wealth distribution as the outcome of a stochastic process in which individual wealth movements net out to produce a stable aggregate configuration.

The simplest types of stochastic models lack an explicit behavioural foundation for the parameter values and are perhaps best viewed as reduced forms in which terms like the regression coefficient p and the random component u in Eq. (2.2) capture in some unspecified way the influence of factors such as the impact of wealth effects on consumption and the randomness of investment returns. Attempts to incorporate explicitly the relevant explanatory variables rapidly produced complex models that were difficult to solve analytically (Sargan, 1957; Vaughan, 1975, 1979; Laitner, 1979), and these technical obstacles have hampered further development of this line of research.

Meade (1964, 1975) offers another simple framework for analysing wealth distribution based on the accounting identity

Wt = Wt,_ + Et + rtWt-I - Ct + It,

(2.3)

Ch. 11: The Distributionof Wealth

611

where Ct and Et are, respectively, consumption and earned income in period t, net of taxes and transfers; rt is the average (net) rate of return on investments; and It represents net "inheritances" (gifts and bequests) received in period t. In the Meade formulation, inheritances are suppressed (or, more accurately, absorbed into the initial wealth level Wo), and consumption is assumed to depend on both income and wealth. This yields

Wt = Wt_l + st(Et + rtWt_) - ctWt_1 = (1 + strt - ct)Wt-I + stEt = (1 -Pt)Wt- +stE t,

(2.4)

where st is the average rate of saving from current income, ct indicates the wealth effect on consumption, and Pt = ct - start represents the "internal rate of decumulation out of wealth".

Equation (2.4) provides a convenient framework for investigating the forces leading to greater or lesser wealth inequality. Assuming for the moment that Et, st and Pt are constant over time and across individuals, Eq. (2.4) has the solution

Wt = W* + (1 - )t(Wo - W*),

(2.5)

where W* = sE//. If is negative, the initial wealth differences expand over time and wealth inequality grows without bound. Conversely, if lies between 0 and 1, then wealth converges to the steady state level W*, which depends only on savings from earned income and the internal rate of decumulation. Under the above assumptions, this implies that the distribution of wealth will be completely equalized in the long run. But once allowance is made for variations in earnings across individuals, the model suggests that long run differences in wealth will mirror the differences in earnings or income. As already noted, the tendency for the level of wealth inequality in the twentieth century to decline towards the lower level observed for incomes is one of the best documented findings of empirical studies of wealth distribution.l

The simple relationship between income inequality and the equilibrium level of wealth inequality is modified by variability in the values of Et, st and /B across individuals or over time. For example, as Meade (1964) points out, individual differences in the rates of return received on investments are likely to be a disequalizing influence, particularly so if the average rate of return increases with the level of wealth. Another possible consideration is the general equilibrium connections between the components contained in Eq. (2.4). Stiglitz (1969) bases his analysis on a similar model, but also links earnings and rates of return to average wealth via a simple neoclassical production

1 It is interesting to note that the recent rise in wealth inequality in the US and, to a lesser extent, Sweden (see Fig. 1 below) have occurred during periods in which income inequality has grown. Wolff (1992) reports a regression of wealth inequality against income inequality in which the coefficient is not significantly different from one.

612

James B. Davies andAnthony F Shorrocks

function. His analysis shows that if the savings rate st is constant, is positive, and earnings are the same for all workers, then Et and Wt converge to their steady state values, and wealth distribution is again completely equalized in the long run. Assuming a stable balanced growth path for the economy, Stiglitz demonstrates that a variety of saving functions (linear, concave, depending on income, or depending on wealth) produce the same result, but other factors, such as wage differences and class saving behaviour, have disequalizing effects over time.

In principle, the framework proposed by Meade, and captured in Eq. (2.4), could absorb the insights gained from subsequent work on lifecycle saving behaviour, which suggests how savings depend on current income and wealth, and how the parameters may adjust to, say, increases in life expectancy. The rich potential of Eq. (2.4) is also evident in the fact that it can be interpreted equally well in an intergenerational context, as is done by Atkinson and Harrison (1978), with Wt referring to the "lifetime wealth" of generation t, and the coefficient f capturing the impact of bequest splitting and estate taxation.

Returning to the accounting identity Eq. (2.3), and interpreting it in terms of the lifetime experience of a family (or individual) which begins at age 0 with zero wealth, implies that wealth at the end of age t is given by

t

t

Wt = (E -C + Ik)

(1 + r).

(2.6)

k=1

j=k+l

From Eq. (2.6) it is clear that a family's wealth is determined by: (i) its age, and its history of; (ii) earnings; (iii) saving rates; (iv) rates of return; and (v) inheritances. A complete economic theory of wealth distribution needs both to determine the distributional impact of each of these five elements and to account for differences in the components across individuals and families. In principle, even the distribution of the population across age groups requires explanation, since fertility and mortality are in part economic phenomena. Although a complete theory of wealth distribution remains a distant objective, considerable progress has been made on many of the specific components. In the following subsections we review many of the relevant contributions, concentrating in particular on questions concerned with savings rates and inheritances. Theories of earnings distribution are considered in detail by Neal and Rosen in Chapter 7 of the Handbook, while fertility, mortality, and family formation are discussed in Rosenzweig and Stark (1997).

2.2. Lifecycle accumulation

The paradigm for intra-generational accumulation is the lifecycle saving model (LCM) pioneered by Modigliani and Brumberg (1954).2 This now comes in several different

2 The life-cycle model of Modigliani and Brumberg (1954) was developed contemporaneously with the closely related permanent income hypothesis of Friedman (1957). Insights from both these studies continue to have an important impact on current work.

Ch. 11: The Distributionof Wealth

613

variants, reflecting the degree of realism introduced into assumptions about institutions and about the uncertainties faced by savers. All versions of the model share the basic assumptions that: (i) consumers are forward-looking; (ii) their preferences are defined over present and future consumption, and possibly leisure time; and (iii) life is expected to end with a period of retirement. Extensions of the basic LCM allow individuals to be interested in the consumption of their offspring or in the size of their planned bequest. These are sometimes known as the "bequest-augmented LCM". The LCM can also be extended in many other ways-for example, by introducing capital market imperfections and/or borrowing constraints, and uncertainty in earnings, rates of return, or the length of life.

The simplest version of the LCM assumes there is no uncertainty; that everyone faces the same constant rate of return, r, and has the same length of life, T; and that there is no bequest motive. The consumer's problem is then:

Max U = U(Ci,.. ,CT),

(2.7a)

subject to:

CC =

( t <)t ( +E

E L,

t=l

t=l

((22..77bb)

where CL and EL respectively denote lifetime consumption and lifetime earnings, and R is the retirement date. If there is a nonworking period at the end of life, minimal restrictions on the functional form of U (-) will ensure that saving is undertaken for the purpose of financing consumption in retirement. This is the key explanation offered by the LCM for personal wealth accumulation.

To simplify the exposition, leisure has been neglected in the maximization problem Eq. (2.7). 3 In other respects, the specification is too general to produce precise conclusions about patterns of saving behaviour. The solution to the problem does include, however, what Browning and Lusardi (1996) call the "central tenet" of the modern view of the LCM: that the consumer attempts to equalize the discounted marginal utility of expenditure in all periods.4 In order to achieve this equalization, given diminishing marginal utility, consumers engage in "consumption smoothing". Retirement saving is one result. Another is that assets will fluctuate to keep consumption smooth. In addition, since earnings rise quickly in the initial working years, substantial net borrowing (i.e., negative net worth) is expected to be prevalent among young people. The fact that this is

3 We note below those occasions when endogenous labour-leisure choices have important implications for wealth-holding.

4 This goal can be achieved precisely in a world of certainty, but only in expectation when uncertainty is introduced.

614

James B. Davies andAnthony E Shorr-ocks

not observed 5 suggests either that individuals cannot readily borrow against future earnings (often referred to as a capital market imperfection), or that they have precautionary reasons for saving, as discussed below.

To make more concrete predictions about saving behaviour, intertemporal utility is typically assumed to be additively separable:

ut(Ct)

U = E (1 t ,

(2.8)

t=l ( +

where p is the rate of time preference. It is common to assume also that Ut = u, although intuition suggests that the instantaneous utility function is likely to change with age. Constant time preference is a requirement for consistent consumption planning over the lifetime (Strotz, 1956).

Various functional forms have been assumed for u. As discussed below, a quadratic function produces the certainty equivalent (CEQ) version of the LCM. Caballero (1991) and others use the constant absolute risk aversion (CARA) specification u(C) = -e-C, which, like the quadratic, yields closed form solutions under uncertain earnings. But the most popular specification by far is the constant relative risk aversion (CRRA) form given by:

C t (YCt U'(C)l

i~lYy y

1>>

0 O,

y

1

(2.9)

ut(C) =logCt,

=

where y is the coefficient of relative risk aversion.6 When Eqs. (2.8) and (2.9) are incorporated into Eq. (2.7), the optimal consumption path satisfies

Ct+l =

/ C) = (1 + g)Cr,

(2.10)

and planned consumption grows at a constant rate g, which is approximately (or, under continuous time, exactly) equal to (r - p)/y. 7 From Eq. (2.10) it is evident that the intertemporal elasticity of substitution is l/y.

5 While surveys find some individuals at all ages with negative net worth, and a higher incidence among the young, the majority have positive net worth even at low ages. See, for example, Hubbard et al. (1995).

6 The CRRA form is the only additively separable homothetic utility function. Many feel that it has intuitive plausibility, and not just analytical convenience, on its side. However, Attanasio and Browning (1995) find that it is decisively rejected in favour of a more general altemative.

7 More generally, the hyperbolic absolute risk aversion (HARA) family formulated by Merton (1971)which includes the CRRA, CARA and CEQ specifications as special cases-leads to a Euler equation similar to Eq. (2.10), but with an additional constant term.

THE DISTRIBUTION OF WEALTH *

JAMES B. DAVIESa and ANTHONY E SHORROCKSb aDepartmentof Economics, University of Western Ontario, London N6A 5C2, Canada;bDepartmentof Economics, University of Essex, Colchester C04 3SQ, UK and Institutefor FiscalStudies, 7 Ridgmount Street, London WC1E 7AE, UK

Contents

1. Introduction

606

2. Theoretical approaches

608

2.1. Simple models of wealth distribution

609

2.2. Lifecycle accumulation

612

2.3. Intergenerational issues

621

2.4. Self-made fortunes

627

3. Empirical evidence on wealth inequality

628

3.1. Household surveys

629

3.2. Wealth tax data

635

3.3. Estate multiplier estimates

637

3.4. The investment income method

642

3.5. Direct wealth estimates for named persons

642

3.6. Portfolio composition

643

3.7. Explaining national differences and changes over time

645

4. Applied work on the determinants of wealth distribution

648

4.1. Simulation studies

648

4.2. Intergenerational wealth mobility

651

4.3. The aggregate importance of inherited versus lifecycle wealth

653

4.4. Empirical studies of transfer behaviour

657

5. Summary and conclusions

663

References

667

* We should like to thank Andrea Brandolini, John Flemming, John Piggott, Edward Wolff and, most especially, Tony Atkinson and Franqois Bourguignon, for their advice, comments and assistance. The first author also thanks the Social Sciences and Humanities Research Council of Canada for its support.

Handbook of Incozme Distribution:Volume 1. Edited by A. B. Atkinson and F Bourguignon (O1999 ElsevierScience B. V All rights reserved.

605

606 Abstract

JamesB. Davies andAnthony F Shorrocks

This chapter is concerned with the distribution of personal wealth, which usually refers to the material assets that can be sold in the marketpace, although on occasion pension rights are also included. We summarise the available evidence on wealth distribution for a number of countries. This confirms the well known fact that wealth is more unequally distributed than income, and points to a long term downward trend in wealth inequality over most of the twentieth century. We also review the various theories that help account for these feature. Lifecycle accumulation is one popular explanation of wealth differences, but inheritance is also widely recognised as playing a major role, especially at the upper end of the wealth range. A recurrent theme in work on wealth distribution is the relative importance of these two sources of wealth differences. We discuss the results of studies that assess the contributions of inheritance and lifecycle factors, and give attention also to a variety of related issues, such as the link between wealth status across generations, and the possible motives for leaving bequests.

Keywords: Wealth distribution, wealth inequality, portfolio composition, lifecycle saving, inheritance, bequests, intergenerational transfers

JEL codes. main: C81, D31, D91, E21, GlI; secondary: Dll, D12, D63, D64, J14)

1. Introduction

This chapter surveys what is known about the distribution of personal wealth and its evolution over time. We review the descriptive evidence as well as theoretical and applied research that attempts to explain the main features of wealth-holdings and wealth inequality observed in the real world.

There are many reasons for interest in personal wealth, and many ways in which the concept of wealth may be defined. If we were concerned with the overall distribution of economic well-being or resources, it would be appropriate to examine the distribution of "total wealth", that is, human plus nonhuman capital. But that is not our objective here. Instead, we exclude the human capital component and focus on material assets in the form of real property and financial claims. The term "wealth" will therefore usually refer to "net worth"-the value of nonhuman assets minus debts. Our aim is to examine the reasons for holding wealth, to document the observed differences in holdings across individuals and families, and to examine the causes of these observed differences.

One major reason for interest in wealth-holdings is that, unlike human capital, most real property and financial assets can be readily bought and sold. This allows nonhuman wealth to be used for consumption smoothing in periods when consumption is expected to be high (growing families) or income is expected to be low (retirement), and in periods of unanticipated shocks to either income or expenditure (the precautionary motive for

Ch. 11: The Distributionof Wealth

607

saving). This consumption smoothing role is particularly important when individuals face capital market imperfections or borrowing constraints. Wealth may also be accumulated, or retained, for the purpose of making bequests. Additional noneconomic reasons for studying wealth include the power or social status which may be associated with certain types of assets such as privately-owned businesses. The pattern of wealth-holdings across individuals, families, and subgroups of the population, is therefore capable of revealing a great deal about both the type of economy in which people operate, and the kind of society in which they live.

The concept of net worth may appear to be straightforward, but should we deal with intangible assets which cannot be readily bought and sold? This category covers pension rights, life insurance, and entitlement to future government transfers (including "social security wealth"). Any attempt to include the rights to uncertain future benefits has to confront a variety of difficult valuation problems. For example, it is not obvious what discount rates should be used for these assets. Should they be risk-adjusted? Should a special adjustment be made for people who are borrowing constrained? Satisfactory answers to these questions require a considerable amount of painstaking work. It is therefore not surprising to discover that most applied work on wealth-holdings and wealth distribution confines itself to marketable wealth. When reviewing the empirical evidence, we use the term "augmented wealth" to refer to the broader concept which includes entitlements to future pension streams.

There are certain important "stylized facts" about the distribution of wealth which it is useful to highlight at the outset. These are:

1. Wealth is distributed less equally than labour income, total money income or consumption expenditure. While Gini coefficients in developed countries typically range between about 0.3 and 0.4 for income, they vary from about 0.5 to 0.9 for wealth. Other indicators reveal a similar picture. The estimated share of wealth held by the top 1% of individuals or families varies from about 15-35%, for example, whereas their income share is usually less than 10%.

2. Financial assets are less equally distributed than nonfinancial assets, at least when owner-occupied housing is the major component of nonfinancial assets. However, in countries where land value is especially important, the reverse may be true.

3. The distribution of inherited wealth is much more unequal than that of wealth in general.

4. In all age groups there is typically a group of individuals and families with very low net worth, and in a number of countries, including the US, the majority have surprisingly low financial assets at all ages.

5. Wealth inequality has, on the whole, trended downwards in the twentieth century, although there have been interruptions and reversals, for example in the US where wealth inequality has increased since the mid 1970s. Possible explanations for these, and other, stylized facts will be investigated in this

chapter. Section 2 begins with a review of simple models that try to account for the overall shape of the distribution of wealth. We also show how the accounting identity relating

608

James B. Davies andAnthony F Shorrocks

changes in wealth to earnings, rates of return, consumption and capital transfers provides a useful framework for investigating the proximate determinants of wealth inequality and its trend over time. Attention then turns to more detailed models of wealth-holdings, including numerous versions of the lifecycle saving model, and a variety of models concerned with bequest behaviour and the distributional impact of inheritance.

Section 3 reviews the descriptive evidence on wealth distribution in a number of countries, and discusses possible explanations for national differences and trends over time. The roles of asset prices, inheritance taxation, and other factors are examined. Then, in Section 4 we look at applied research which attempts to assess the contribution of different factors to the distribution and evolution of wealth-holding. These studies address a variety of questions, and use a variety of approaches, including decomposition procedures, simulation exercises, and conventional econometrics. But, in broad terms, all the research is linked by a common objective: to cast light on the reasons for wealth accumulation, and on the importance of inheritance vis-A-vis lifecycle saving as determinants of the level and distribution of wealth.

2. Theoretical approaches

Many different types of theories have been used to model aspects of wealth-holdings and wealth distribution. To a large extent they reflect the empirical evidence available at the time the theories were proposed. Prior to the 1960s, data on wealth were obtained primarily from estate tax and wealth tax records, with other evidence pieced together from small unrepresentative surveys and a variety of other sources. These tended to confirm the widely held view that wealth was distributed very unequally, and that material inheritance was both a major cause of wealth differences and an important vehicle for the transmission of wealth status between generations. There were also grounds for believing that wealth inequality was declining over time, and that the shape of the distributions exhibited certain statistical regularities which could not have arisen by coincidence. Early theoretical work on wealth distribution sought to explain these statistical regularities, and to understand the interplay of basic forces that could account for high wealth concentration and a declining trend over time.

More recently, research has shifted away from a concern with the overall distributional characteristics, focusing instead on the causes of individual differences in wealthholdings. The change of emphasis was prompted in part by a recognition of the increasing importance of saving for retirement, and is reflected in the central role now assigned to the lifecycle saving model formulated by Modigliani and Brumberg (1954) and Ando and Modigliani (1963). The second major development has been the growth in the availability and sophistication of micro-data sets that offer not only estimates of the savings and asset holdings of individuals but also a range of other personal and household characteristics than can help account for differences in wealth.

Ch. 11: The Distributionof Wealth

609

This section reviews various models of wealth distribution and wealth differences, beginning with some of the early attempts to explain the overall distributional features and the evolution of wealth inequality. Section 2.2 discusses the simple lifecycle model and some of its many extensions, drawing out the implications for wealth accumulation over the course of the lifetime. Attention here is restricted to pure intra-generational models in which inheritances play no role. For the most part, Section 2.3 takes the opposite tack, suppressing interest in lifecycle variations and focusing instead on intergenerational links, especially those concerned with the motivation for bequests and the impact of inheritance.

The distinction between models of wealth accumulated for lifecycle purposes and models concentrating on the intergenerational connections reflects the theoretical literature on wealth distribution: very few contributions have attempted to deal simultaneously with both the lifecycle and inherited components of wealth. This is a major weakness of past theoretical work on wealth. While there may be some value in modelling lifecycle wealth in the absence of inheritance, it should be recognised that this exercise does not reveal the true pattern of lifecycle accumulation in the real world-since the real world also has inheritances. This rather obvious point is one which is often forgotten, particularly when attempts are made to assess the relative importance of accumulation and inheritance.

Current models of accumulation and bequests do not appear to capture the circumstances and motives of those who amass large fortunes in the course of their lifetimes. We look briefly at this issue in Section 2.4. Throughout, reference is made to relevant empirical evidence that has informed and influenced the development of the theories. Later, in Section 4, we consider in more detail some of the issues that have received most attention.

2.1. Simple models of wealth distribution

Early empirical work on personal wealth-holdings established two enduring features of the shape of the distribution of wealth. First, it is positively skewed, unlike the normal distribution, but roughly resembling a normal distribution when wealth is replaced by the logarithm of wealth. Second, the top tail is well approximated by a Pareto distribution, which yields a straight line graph when the logarithm of the number of persons with wealth above w is plotted against log w. These two statistical regularities were observed not only for wealth-holdings, but also for many other skewed distributions such as those for incomes, the turnover of firms, and the size of cities (see, for example, Steindl, 1965, Appendix B).

The fact that these size distributions are approximated by the lognormal suggests that, by appealing to the Central Limit Theorem, they can be generated by a random walk of the form:

In Wt == nn WWt + ut,

(2.1)

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James B. Davies andAnthony F Shorrocks

where Wt is wealth at time t, and u is a stochastic component (Aitchison and Brown, 1957). Equation (2.1) contains the implicit assumption that the factors influencing changes in wealth over time operate in a multiplicative fashion, rather than additively. This became known as Gibrat's "law of proportionate effect", following Gibrat (1931).

As a model of wealth distribution, the simple random walk Eq. (2.1) has one major technical disadvantage: it predicts that wealth dispersion, as measured by variance of log wealth, increases over time without bound. It cannot therefore apply when wealth inequality is stable or decreasing. To overcome this problem it is necessary to introduce a mechanism which offsets the inequality increasing impact of the stochastic component. One simple solution is the formulation proposed by Galton (1889) in his study of inherited genetic characteristics, which yields

InWt = IlnWtl +ut,

O < < 1,

(2.2)

with /i indicating the degree of "regression towards the mean". Given appropriate constraints on the stochastic component ut, this process can generate a stable lognormal distribution of wealth in which changes in wealth at the level of individuals exactly balance out to maintain equilibrium in the aggregate (see, for example, Creedy, 1985).

A number of other authors, including Champernowne (1953), Wold and Whittle (1957), Steindl (1972), Shorrocks (1973, 1975b), and Vaughan (1975, 1979), proposed alternative types of stochastic models capable of generating distributions of wealth with upper tails that are asymptotically Pareto. In broad terms, these models all assume that a variant of the law of proportionate effect applies at high wealth levels, where the expected change in wealth must be negative in order to prevent the top tail from drifting upwards over time. Equilibrium is maintained by some mechanism lower down the distribution-such as a pool of low wealth-holders in the case of Wold and Whittle (1957)-which stops wealth from converging to zero in the long run. As with the Galtonian specification Eq. (2.2), these models typically view the observed wealth distribution as the outcome of a stochastic process in which individual wealth movements net out to produce a stable aggregate configuration.

The simplest types of stochastic models lack an explicit behavioural foundation for the parameter values and are perhaps best viewed as reduced forms in which terms like the regression coefficient p and the random component u in Eq. (2.2) capture in some unspecified way the influence of factors such as the impact of wealth effects on consumption and the randomness of investment returns. Attempts to incorporate explicitly the relevant explanatory variables rapidly produced complex models that were difficult to solve analytically (Sargan, 1957; Vaughan, 1975, 1979; Laitner, 1979), and these technical obstacles have hampered further development of this line of research.

Meade (1964, 1975) offers another simple framework for analysing wealth distribution based on the accounting identity

Wt = Wt,_ + Et + rtWt-I - Ct + It,

(2.3)

Ch. 11: The Distributionof Wealth

611

where Ct and Et are, respectively, consumption and earned income in period t, net of taxes and transfers; rt is the average (net) rate of return on investments; and It represents net "inheritances" (gifts and bequests) received in period t. In the Meade formulation, inheritances are suppressed (or, more accurately, absorbed into the initial wealth level Wo), and consumption is assumed to depend on both income and wealth. This yields

Wt = Wt_l + st(Et + rtWt_) - ctWt_1 = (1 + strt - ct)Wt-I + stEt = (1 -Pt)Wt- +stE t,

(2.4)

where st is the average rate of saving from current income, ct indicates the wealth effect on consumption, and Pt = ct - start represents the "internal rate of decumulation out of wealth".

Equation (2.4) provides a convenient framework for investigating the forces leading to greater or lesser wealth inequality. Assuming for the moment that Et, st and Pt are constant over time and across individuals, Eq. (2.4) has the solution

Wt = W* + (1 - )t(Wo - W*),

(2.5)

where W* = sE//. If is negative, the initial wealth differences expand over time and wealth inequality grows without bound. Conversely, if lies between 0 and 1, then wealth converges to the steady state level W*, which depends only on savings from earned income and the internal rate of decumulation. Under the above assumptions, this implies that the distribution of wealth will be completely equalized in the long run. But once allowance is made for variations in earnings across individuals, the model suggests that long run differences in wealth will mirror the differences in earnings or income. As already noted, the tendency for the level of wealth inequality in the twentieth century to decline towards the lower level observed for incomes is one of the best documented findings of empirical studies of wealth distribution.l

The simple relationship between income inequality and the equilibrium level of wealth inequality is modified by variability in the values of Et, st and /B across individuals or over time. For example, as Meade (1964) points out, individual differences in the rates of return received on investments are likely to be a disequalizing influence, particularly so if the average rate of return increases with the level of wealth. Another possible consideration is the general equilibrium connections between the components contained in Eq. (2.4). Stiglitz (1969) bases his analysis on a similar model, but also links earnings and rates of return to average wealth via a simple neoclassical production

1 It is interesting to note that the recent rise in wealth inequality in the US and, to a lesser extent, Sweden (see Fig. 1 below) have occurred during periods in which income inequality has grown. Wolff (1992) reports a regression of wealth inequality against income inequality in which the coefficient is not significantly different from one.

612

James B. Davies andAnthony F Shorrocks

function. His analysis shows that if the savings rate st is constant, is positive, and earnings are the same for all workers, then Et and Wt converge to their steady state values, and wealth distribution is again completely equalized in the long run. Assuming a stable balanced growth path for the economy, Stiglitz demonstrates that a variety of saving functions (linear, concave, depending on income, or depending on wealth) produce the same result, but other factors, such as wage differences and class saving behaviour, have disequalizing effects over time.

In principle, the framework proposed by Meade, and captured in Eq. (2.4), could absorb the insights gained from subsequent work on lifecycle saving behaviour, which suggests how savings depend on current income and wealth, and how the parameters may adjust to, say, increases in life expectancy. The rich potential of Eq. (2.4) is also evident in the fact that it can be interpreted equally well in an intergenerational context, as is done by Atkinson and Harrison (1978), with Wt referring to the "lifetime wealth" of generation t, and the coefficient f capturing the impact of bequest splitting and estate taxation.

Returning to the accounting identity Eq. (2.3), and interpreting it in terms of the lifetime experience of a family (or individual) which begins at age 0 with zero wealth, implies that wealth at the end of age t is given by

t

t

Wt = (E -C + Ik)

(1 + r).

(2.6)

k=1

j=k+l

From Eq. (2.6) it is clear that a family's wealth is determined by: (i) its age, and its history of; (ii) earnings; (iii) saving rates; (iv) rates of return; and (v) inheritances. A complete economic theory of wealth distribution needs both to determine the distributional impact of each of these five elements and to account for differences in the components across individuals and families. In principle, even the distribution of the population across age groups requires explanation, since fertility and mortality are in part economic phenomena. Although a complete theory of wealth distribution remains a distant objective, considerable progress has been made on many of the specific components. In the following subsections we review many of the relevant contributions, concentrating in particular on questions concerned with savings rates and inheritances. Theories of earnings distribution are considered in detail by Neal and Rosen in Chapter 7 of the Handbook, while fertility, mortality, and family formation are discussed in Rosenzweig and Stark (1997).

2.2. Lifecycle accumulation

The paradigm for intra-generational accumulation is the lifecycle saving model (LCM) pioneered by Modigliani and Brumberg (1954).2 This now comes in several different

2 The life-cycle model of Modigliani and Brumberg (1954) was developed contemporaneously with the closely related permanent income hypothesis of Friedman (1957). Insights from both these studies continue to have an important impact on current work.

Ch. 11: The Distributionof Wealth

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variants, reflecting the degree of realism introduced into assumptions about institutions and about the uncertainties faced by savers. All versions of the model share the basic assumptions that: (i) consumers are forward-looking; (ii) their preferences are defined over present and future consumption, and possibly leisure time; and (iii) life is expected to end with a period of retirement. Extensions of the basic LCM allow individuals to be interested in the consumption of their offspring or in the size of their planned bequest. These are sometimes known as the "bequest-augmented LCM". The LCM can also be extended in many other ways-for example, by introducing capital market imperfections and/or borrowing constraints, and uncertainty in earnings, rates of return, or the length of life.

The simplest version of the LCM assumes there is no uncertainty; that everyone faces the same constant rate of return, r, and has the same length of life, T; and that there is no bequest motive. The consumer's problem is then:

Max U = U(Ci,.. ,CT),

(2.7a)

subject to:

CC =

( t <)t ( +E

E L,

t=l

t=l

((22..77bb)

where CL and EL respectively denote lifetime consumption and lifetime earnings, and R is the retirement date. If there is a nonworking period at the end of life, minimal restrictions on the functional form of U (-) will ensure that saving is undertaken for the purpose of financing consumption in retirement. This is the key explanation offered by the LCM for personal wealth accumulation.

To simplify the exposition, leisure has been neglected in the maximization problem Eq. (2.7). 3 In other respects, the specification is too general to produce precise conclusions about patterns of saving behaviour. The solution to the problem does include, however, what Browning and Lusardi (1996) call the "central tenet" of the modern view of the LCM: that the consumer attempts to equalize the discounted marginal utility of expenditure in all periods.4 In order to achieve this equalization, given diminishing marginal utility, consumers engage in "consumption smoothing". Retirement saving is one result. Another is that assets will fluctuate to keep consumption smooth. In addition, since earnings rise quickly in the initial working years, substantial net borrowing (i.e., negative net worth) is expected to be prevalent among young people. The fact that this is

3 We note below those occasions when endogenous labour-leisure choices have important implications for wealth-holding.

4 This goal can be achieved precisely in a world of certainty, but only in expectation when uncertainty is introduced.

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James B. Davies andAnthony E Shorr-ocks

not observed 5 suggests either that individuals cannot readily borrow against future earnings (often referred to as a capital market imperfection), or that they have precautionary reasons for saving, as discussed below.

To make more concrete predictions about saving behaviour, intertemporal utility is typically assumed to be additively separable:

ut(Ct)

U = E (1 t ,

(2.8)

t=l ( +

where p is the rate of time preference. It is common to assume also that Ut = u, although intuition suggests that the instantaneous utility function is likely to change with age. Constant time preference is a requirement for consistent consumption planning over the lifetime (Strotz, 1956).

Various functional forms have been assumed for u. As discussed below, a quadratic function produces the certainty equivalent (CEQ) version of the LCM. Caballero (1991) and others use the constant absolute risk aversion (CARA) specification u(C) = -e-C, which, like the quadratic, yields closed form solutions under uncertain earnings. But the most popular specification by far is the constant relative risk aversion (CRRA) form given by:

C t (YCt U'(C)l

i~lYy y

1>>

0 O,

y

1

(2.9)

ut(C) =logCt,

=

where y is the coefficient of relative risk aversion.6 When Eqs. (2.8) and (2.9) are incorporated into Eq. (2.7), the optimal consumption path satisfies

Ct+l =

/ C) = (1 + g)Cr,

(2.10)

and planned consumption grows at a constant rate g, which is approximately (or, under continuous time, exactly) equal to (r - p)/y. 7 From Eq. (2.10) it is evident that the intertemporal elasticity of substitution is l/y.

5 While surveys find some individuals at all ages with negative net worth, and a higher incidence among the young, the majority have positive net worth even at low ages. See, for example, Hubbard et al. (1995).

6 The CRRA form is the only additively separable homothetic utility function. Many feel that it has intuitive plausibility, and not just analytical convenience, on its side. However, Attanasio and Browning (1995) find that it is decisively rejected in favour of a more general altemative.

7 More generally, the hyperbolic absolute risk aversion (HARA) family formulated by Merton (1971)which includes the CRRA, CARA and CEQ specifications as special cases-leads to a Euler equation similar to Eq. (2.10), but with an additional constant term.

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