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\begin{document}

\title{Wealth and Inheritance in the Long Run}
\author{Thomas \textsc{Piketty} \\
%EndAName
Paris School of Economics \and Gabriel \textsc{Zucman} \\
%EndAName
London School of Economics and UC Berkeley}
\date{\today \thanks{%
Thomas Piketty: piketty@pse.ens.fr; Gabriel Zucman: zucman@berkeley.edu.
This article has been prepared for the \textit{Handbook of Income
Distribution} (North-Holland, volume 2). We are grateful to the editors and
to Daniel Waldenstrom for helpful comments. All series and \ figures
presented in this chapter are available in an on-line data appendix.} }
\maketitle

\begin{abstract}
This article offers an overview of the empirical and theoretical research on
the long run evolution of wealth and inheritance. Wealth-income ratios,
inherited wealth, and wealth inequalities were high in the 18$^{th}$-19$^{th}
$ centuries up until World War 1, then sharply dropped during the 20$^{th}$
century following World War shocks, and have been rising again in the late 20%
$^{th}$ and early 21$^{st}$ centuries. We discuss the models that can
account for these facts. We show that over a wide range of models, the long
run magnitude and concentration of wealth and inheritance are an increasing
function of $\overline{r}-g$, where $\overline{r}$ is the net-of-tax rate of
return on wealth and $g$ is the economy's growth rate. This suggests that
current trends toward rising wealth-income ratios and wealth inequality
might continue during the 21$^{st}$ century, both because of the slowdown of
population and productivity growth, and because of rising international
competition to attract capital.
\end{abstract}

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\section{Introduction}

Economists have long recognized that the magnitude and distribution of
wealth play an important role in the distribution of income -- both across
factors of production (labor and capital) and across individuals. In this
chapter, we ask three simple questions: What do we know about historical
patterns in the magnitude of wealth and inheritance relative to income? How
does the distribution of wealth vary in the long run and across countries?
And what are the models that can account for these facts?

%Kuznets (1955), for instance, saw the accumulation of large fortunes by upper income groups -- due to relatively high saving rates -- as the number one force that could make for rising income inequality in rich countries.  

In surveying the literature on these issues, we will focus the analysis on
three inter-related ratios. The first is the aggregate wealth to income
ratio, that is the ratio between marketable -- non human -- wealth and
national income. The second is the share of aggregate wealth held by the
richest individuals, say the top 10\% or top 1\%. The last is the ratio
between the stock of inherited wealth and aggregate wealth (or between the
annual flow of bequests and national income). As we shall see, to properly
analyze the concentration of wealth and its implications, it is critical to
study top wealth shares jointly with the macroeconomic wealth/income and
inheritance/wealth ratios. In so doing, this chapter attempts to build
bridges between income distribution and macroeconomics.

The wealth to income ratio, top wealth shares, and the share of inheritance
in the economy have all been the subject of considerable interest and
controversy -- but usually on the basis of limited data. For a long time,
economics textbooks have presented the wealth-income ratio as stable over
time -- one of the Kaldor facts.\footnote{%
See, e.g., Kaldor (1961) and Jones and Romer (2010).} There is, however, no
strong theoretical reason why it should be so: with a flexible production
function, any ratio can be a steady state. And until recently we lacked
comprehensive national balance sheets with harmonized definitions for wealth
that could be used to vindicate the constant-ratio thesis. Recent research
shows that wealth-income ratios, as well as the share of capital in national
income, are actually much less stable in the long run than what is commonly
assumed.

Following the Kuznets curve hypothesis -- first formulated in the 1950s --
another common view among economists has been that income inequality -- and
possibly wealth inequality as well -- should first rise and then decline
with economic development, as a growing fraction of the population joins
high-productivity sectors and benefits from industrial growth.\footnote{%
See Kuznets (1953, 1955).} However, following the rise in inequality that
has occurred in most developed countries since the 1970s-80s, this
optimistic view has become less popular.\footnote{%
See Atkinson, Piketty and Saez (2011). See also the survey chapter in this
Handbook by Roine and Walndestrom (2014).} As a consequence, most economists
are now fairly skeptical about universal laws regarding the long-run
evolution of inequality.

Last, regarding the inheritance share in total wealth accumulation, there
seems to exist a general presumption that it should tend to decline over
time. Although this is rarely formulated explicitly, one possible mechanism
could be the rise of human capital (leading maybe to a rise of the labor
share in income and saving), or the rise in lifecycle wealth accumulation
(itself maybe due to the rise of life expectancy). Until recently, however,
there was limited empirical evidence on the share of inherited wealth
available to test these hypotheses. The 1980s saw a famous controversy
between Modigliani (a life-cyle advocate, who argued that the share of
inherited wealth was as little as 20-30\% of US aggregate wealth) and
Kotlikoff-Summers (who instead argued that the inheritance share was as
large as 80\%, if not larger). Particularly confusing was the fact that both
sides claimed to look at the same data, namely US data from the 1960s-70s.%
\footnote{%
See Kotlikoff and Summers (1981, 1988) and Modigliani (1986, 1988).
Modigliani's theory of lifecycle saving was first formulated in the
1950s-60s, see the references given in Modigliani (1986).}

Since many of the key predictions about wealth and inheritance were
formulated a long time ago -- often in the 1950s-60s, sometime in the
1970s-80s -- and usually on the basis of relatively little long-run
evidence, it is high time to take a fresh look at them again on the basis of
the more reliable evidence now available.

We begin by reviewing in section 2 what we know about the historical
evolution of the wealth-income ratio $\beta $. In most countries, this ratio
has been following a U-shaped pattern over the 1910-2010 period -- with a
large decline between the 1910s and the 1950s, and a gradual recovery since
the 1950s. The pattern is particularly spectacular in Europe, where the
aggregate wealth-income ratio was as large as 600-700\% during the 18$^{th}$%
, 19$^{th}$ and early 20$^{th}$ centuries, then dropped to as little as
200\%-300\% in the mid-20$^{th}$ century. It is now back to about 500-600\%
in the early 21$^{st}$ century. These same orders of magnitude also seem to
apply to Japan (though the historical data is less complete than for
Europe). The U-shaped pattern also exists -- but is less marked -- in the US.

In section 3, we turn to the long run changes in wealth concentration. We
also find a U-shaped pattern over the past century, but the dynamics have
been quite different in Europe and the U.S. In Europe, the recent increase
in wealth inequality appears to be more limited than the rise of the
aggregate wealth-income ratio, so that European wealth seems to be
significantly less concentrated in the early 21$^{st}$ century than a
century ago. The top 10\% wealth share used to be as large as 90\%, while it
is around 60-70\% today (which is already quite large -- and in particular a
lot larger than the concentration of labor income). In the U.S., by
contrast, wealth concentration appears to have almost returned to its early
20$^{th}$ century level. While Europe was substantially more unequal than
the U.S. until World War I, the situation has reversed over the course of
the 20$^{th}$ century. Whether the gap between both economies will keep
widening in the 21$^{st}$ century is an open issue.

In section 4, we describe the existing evidence regarding the evolution of
the share $\varphi $ of inherited wealth in aggregate wealth. This is an
area in which available historical series are scarce and a lot of data has
yet to be collected. However existing evidence -- coming mostly from France,
Germany, the UK, and Sweden -- suggests that the inheritance share has also
followed a U shaped pattern over the past century. Modigliani's estimates --
with a large majority of wealth coming from life-cycle savings -- might have
been right for the immediate postwar period (though somewhat exaggerated).
But Kotlikoff-Summers' estimates -- with inheritance accounting for a
significant majority of wealth -- appear to be closer to what we generally
observe in the long-run, both in the 19$^{th}$ and early 20$^{th}$ centuries
and in the late 20$^{th}$ and early 21$^{st}$ centuries. Here again, there
could be some interesting difference between Europe and the US (possibly
running in the opposite direction than for wealth concentration).
Unfortunately the fragility of available US data makes it difficult to
conclude at this stage.

We then discuss in section 5 the theoretical mechanisms that can be used to
account for the historical evidence and to analyze future prospects. Some of
the evolutions documented in sections 2-4 are due to shocks. In particular,
the large U-shaped pattern of wealth-income and inheritance-income ratios
observed over the 1910-2010 period is largely due to the wars (which hit
Europe and Japan much more than the US). Here the main theoretical lesson is
simply that capital accumulation takes time, and that the world wars of the
20$^{th}$ century have had a long lasting impact on basic economic ratios.
This, in a way, is not too surprising and follows from simple arithmetic.
With a 10\% saving rate and a fixed income, it takes 50 years to accumulate
the equivalent of 5 years of income in capital stock. With income growth,
the recovery process takes even more time.

The more interesting and difficult part of the story is to understand the
forces that determine the new steady-state levels towards which each economy
tends to converge once it has recovered from shocks. In section 5, we show
that over a wide range of models, the long run magnitude and concentration
of wealth and inheritance are a decreasing function of $g$ and an increasing
function of $\overline{r}$ , where $g$ is the economy's growth rate and $%
\overline{r}$ is the net-of-tax rate of return to wealth. That is, under
plausible assumptions, our three inter-related sets of ratios -- the
wealth-income ratio, the concentration of wealth, and the share of inherited
wealth -- all tend to take higher steady-state values when the long-run
growth rate is lower or when the net-of-tax rate of return is higher. In
particular, a higher $\overline{r}-g$ tends to magnify steady-state wealth
inequalities. We argue that these theoretical predictions are broadly
consistent with both the time-series and the cross-country evidence. This
also suggests that the current trends toward rising wealth-income ratios and
wealth inequality might continue during the 21$^{st}$ century, both because
of population and productivity growth slowdown, and because of rising
international competition to attract capital.

Due to data availability constraints, the historical evolutions analyzed in
this chapter relate for the most part to today's rich countries (Europe,
North America, and Japan). However, to the extent that the theoretical
mechanisms unveiled by the experience of rich countries also apply
elsewhere, the findings presented here are also of interest for today's
emerging economies. In section 5, we discuss the prospects for the
global evolution of wealth-income ratios, wealth concentration and the share
of inherited wealth in the coming decades. Finally, section 6 offers
concluding comments and stresses the need for more research in this area.

\section{The long-run evolution of wealth-income ratios}

\subsection{Concepts, data sources and methods}

\subsubsection{Country balance sheets}

Prior to World War I, there was a vibrant tradition of national wealth
accounting: economists, statisticians and social arithmeticians were much
more interested in computing the stock of national wealth than the flows of
national income and output. The first national balance sheets were
established in the late seventeenth and early eighteenth centuries by Petty
(1664) and King (1696) in the U.K., Boisguillebert (1695) and Vauban (1707)
in France. National wealth estimates then became plentiful in the nineteenth
and early twentieth century, with the work of Colquhoun (1815), Giffen
(1889) and Bowley (1920) in the U.K., Foville (1893) and Colson (1903) in
France, Helfferich (1913) in Germany, King (1915) in the U.S., and dozens of
other economists.

The focus on wealth, however, largely disappeared in the interwar. The shock
of World War I, the Great Depression, and the coming of Keynesian economics
led to attention being switched from stocks to flows, with balance sheets
being neglected. The first systematic attempt to collect historical balance
sheets is due to Goldsmith (1985, 1991). Building upon recent progress made
in the measurement of wealth, and pushing forward Goldsmith's pioneering
attempt, Piketty and Zucman (2014) construct aggregate wealth and income
series for the top 8 rich economies. Other recent papers looking at specific
countries include Atkinson (2013) for the UK and Ohlsson, Roine and
Waldenstrom (2013) for Sweden. In this section, we rely upon the data
collected by Piketty and Zucman (2014) -- and closely follow the discussion
therein -- to present the long run evolution of wealth-income ratios in the
main developed economies.

In determining what is to be counted as wealth, we follow the U.N. System of
National Accounts (SNA). For the 1970-2010 period, the data come from
official national accounts that comply with the latest international
guidelines (SNA, 1993, 2008). For the previous periods, Piketty and Zucman
(2014) draw on the vast national wealth accounting tradition to construct
homogenous income and wealth series that use the same concepts and
definitions as in the most recent official accounts. The historical data
themselves were established by a large number of scholars and statistical
administrations using a wide variety of sources, including land, housing and
wealth censuses, financial surveys, corporate book accounts, etc. Although
historical balance sheets are far from perfect, their methods are well
documented and they are usually internally consistent. It was also somewhat
easier to estimate national wealth around 1900-1910 than it is today: the
structure of property was simpler, with less financial intermediation and
cross-border positions.\footnote{%
A detailed analysis of conceptual and methodological issues regarding wealth
measurement, as well as extensive country-specific references on historical
balance sheets, are provided by Piketty and Zucman (2014).}

\subsubsection{Concepts and definitions: wealth vs. capital}

We define private wealth $W_{t}$ as the net wealth (assets minus
liabilities) of households.\footnote{%
Private wealth also includes the assets and liabilities held by non-profit
institutions serving households (NPISH). The main reason for doing so is
that the frontier between individuals and private foundations is not always
clear. In any case, the net wealth of NPISH is usually small, and always
less than 10\% of total net private wealth: currently it is about 1\% in
France, 3\%-4\% in Japan, and 6\%-7\% in the U.S., see Piketty and Zucman
(2014, Appendix Table A65). Note also that the household sector includes all
unincorporated businesses.} Following SNA guidelines, assets include all the
non-financial assets -- land, buildings, machines, etc. -- and financial
assets -- including life insurance and pensions funds -- over which
ownership rights can be enforced and that provide economic benefits to their
owners. Pay-as-you-go social security pension wealth is excluded, just like
all other claims on future government expenditures and transfers (like
education expenses for one's children or health benefits). Durable goods
owned by households, such as cars and furniture, are excluded as well.%
\footnote{%
The value of durable goods appears to be relatively stable over time (about
30\%-50\% of national income, i.e. 5\%-10\% of net private wealth). See for
instance Piketty and Zucman (2014, Appendix Table US.6f) for the long-run
evolution of durable goods in the U.S.} As a general rule, all assets and
liabilities are valued at their prevailing market prices. Corporations are
included in private wealth through the market value of equities and
corporate bonds. Unquoted shares are typically valued on the basis of
observed market prices for comparable, publicly traded companies.

Similarly, public (or government) wealth $W_{gt}$ is the net wealth of
public administrations and government agencies. In available balance sheets,
public non-financial assets like administrative buildings, schools and
hospitals are valued by cumulating past investment flows and upgrading them
using observed real estate prices.

Market-value national wealth $W_{nt}$ is the sum of private and public
wealth:

\begin{equation*}
W_{nt}=W_{t}+W_{gt}
\end{equation*}

And national wealth can also be decomposed into domestic capital and net
foreign assets: 
\begin{equation*}
W_{nt}=K_{t}+NFA_{t}
\end{equation*}

In turn, domestic capital $K_{t}$ can be written as the sum of agricultural
land, housing, and other domestic capital (including the market value of
corporations, and the value of other non-financial assets held by the
private and public sectors, net of their liabilities).

Regarding income, the definitions and notations are standard. Note that we
always use net-of-depreciation income and output concepts. National income $%
Y_{t}$ is the sum of net domestic output and net foreign income: $%
Y_{t}=Y_{dt}+r_{t}\cdot NFA_{t}$.\footnote{%
National income also includes net foreign labor income and net foreign
production taxes -- both of which are usually negligible.} Domestic output
can be thought of as coming from some aggregate production function that
uses domestic capital and labor as inputs: $Y_{dt}=F(K_{t},L_{t})$.

One might prefer to think about output as deriving from a two-sector
production process (housing and non-housing sectors), or more generally from 
$n$ sectors. In the real world, the capital stock $K_{t}$ comprises
thousands of various assets valued at different prices (just like output $%
Y_{dt}$ is defined as the sum of thousands of goods and services valued at
different prices). We find it more natural, however, to start with a
one-sector formulation. Since the same capital assets (i.e., buildings) are
often used for housing and office space, it would be quite artificial to
start by dividing capital and output into two parts. We will later on
discuss the pros and cons of the one-sector model and the need to appeal to
two-sector models and relative asset price movements in order to properly
account for observed changes in the aggregate wealth-income ratio.

Another choice that needs to be discussed is the focus on market values for
national wealth and capital. We see market values as a useful and
well-defined starting point. But one might prefer to look at book-values,
for example for short run growth accounting exercices. Book values exceed
market values when Tobin's $Q$ is less than 1, and conversely when Tobin's $Q
$ is larger than 1. In the long run, however, the choice of book vs. market
value does not affect much the analysis (see Piketty and Zucman, 2014, for a
detailed discussion).

We are interested in the evolution of the private wealth-national income
ratio $\beta _{t}=W_{t}/Y_{t}$ and of the national wealth-national income
ratio $\beta _{nt}=W_{nt}/Y_{t}$. In a closed economy -- and more generally
in an open economy with a zero net foreign position -- the national
wealth-national income ratio $\beta _{nt}$ is the same as the domestic
capital-output ratio $\beta _{kt}=K_{t}/Y_{dt}$.\footnote{%
In principle, one can imagine a country with a zero net foreign asset
position (so that $W_{nt}=K_{t}$) but non-zero net foreign income flows (so
that $Y_{t}\neq Y_{dt}$). In this case the national wealth-national income
ratio $\beta _{nt}$ will slightly differ from the domestic capital-output
ratio $\beta _{kt}$. In practice today, differences between $Y_{t}$ and $%
Y_{dt}$ are very small -- national income $Y_{t}$ is usually between 97\%
and 103\% of domestic output $Y_{dt}$ (see Piketty and Zucman (2014,
Appendix Figure A57)). Net foreign asset positions are usually small as
well, so that $\beta _{kt}$ turns out to be usually close to $\beta _{nt}$
in the 1970-2010 period (see Piketty and Zucman (2014, Appendix Figure A67)).%
} In case public wealth is equal to zero, then both ratios are also equal to
the private wealth-national income ratio: $\beta _{t}=\beta _{nt}=\beta _{kt}
$. At the global level, the world wealth-income ratio is always equal to the
world capital-output ratio.

\subsection{The very long-run: Britain and France, 1700-2010}

Figure 2.1. and 2.2. present the very long-run evidence available for
Britain and France regarding the national wealth-national income ratio $%
\beta _{nt}$. Net public wealth -- either positive or negative -- is usually
a relatively small fraction of national wealth, so that the evolution of $%
\beta _{nt}$ mostly reflects the evolution of the private wealth-national
income ratio $\beta _{t}$ (more on this below).\footnote{%
For an historical account of the changing decomposition of national wealth
into private and public wealth in Britain and France since the 18th century,
see Piketty (2014, Chapter 3).}

The evolutions are remarkably similar in the two countries. First, the
wealth-income ratio has followed a spectacular U-shaped pattern. Aggregate
wealth was worth about 6-7 years of national income during the 18$^{th}$-19$%
^{th}$ centuries on both sides of the Channel, up until the eve of World War
1. Raw data sources available for these two centuries are not sufficiently
precise to make fine comparisons between the two countries or over time, but
the orders of magnitude appear to be reliable and roughly stable (they come
from a large number of independent estimates). Aggregate wealth then
collapsed to as little as 2-3 years of national income in the aftermath of
the two World Wars. Since the 1950s, there has been a gradual recovery in
both countries. Aggregate wealth is back to about 5-6 years of national
income in the 2000s-2010s, just a little bit below the pre-World War 1 level.

The other important finding that emerges from figures 2.1-2.2 is that the
composition of national wealth has changed in similar ways in both
countries. Agricultural land -- which made the majority of national capital
in the 18$^{th}$ century -- has been gradually replaced by real estate and
other domestic capital (which is for the most part business capital, i.e.,
structures and equipment used by private firms). The nature of wealth has changed entirely -- reflecting a
dramatic change in the structure of economic activity -- and yet the total
value of wealth is more or less the same as what it used to be before the
industrial revolution.

Net foreign assets also made a large part of national capital in the late 19$%
^{th}$ century and on the eve of World War 1: as much as two years of
national capital in the case of Britain and over a year in the case of
France. Net foreign asset positions were brought back to zero in both
countries following World Wars 1 and 2 shocks (including the loss of the
colonial empires). In the late 20$^{th}$ and early 21$^{st}$ centuries, net
foreign positions are close to zero in both countries, just like in the 18$%
^{th}$ century. In the very long run, net foreign assets do not matter too
much for the dynamics of the capital/income ratio in Britain or France. The
main structural change is the replacement of agricultural land by housing
and business capital.\footnote{%
It is worth stressing that should we divide aggregate wealth by disposable
household income (rather than national income), then today's ratios would be
around 700-800\% in Britain or France and would slightly surpass 18$^{th}$-19%
$^{th}$ century level. This mechanically follows from the fact that
disposable income was above 90\% in the 18$^{th}$-19$^{th}$ centuries and is
about 70-80\% of disposable income in the late 20$^{th}$-early 21$^{st}$
century. The rising gap between disposable and household income reflects the
rise of government provided services, in particular in health and education.
To the extent that these services are mostly useful (in their absence
households would have to purchase them on the market), it is more justified
for the purpose of historical and international comparisons to focus upon
ratios using national income as denominator. For wealth-income ratios using
disposable income as denominator, see Piketty and Zucman (2014, Appendix
Figure A9).}

\subsection{Old Europe vs. the New World}

It is interesting to contrast the case of Old Europe -- as illustrated by
Britain and France -- with that of the US.

As figure 2.3. shows, the aggregate value of wealth in the 18$^{th} $-19$%
^{th}$ centuries was markedly smaller in the New World than in Europe. At
the time of the Declaration of Independence and in the early 19$^{th}$
century, national wealth in the US was barely equal to 3-4 years of national
income, about half as in Britain or France. Although available
estimates are fragile, the order of magnitude again appears to be robust. In
section 5, we will attempt to account for this interesting contrast. At this
stage, we simply note that there are two obvious -- and potentially
complementary -- factors that can play a role: first, there had been less
time to save and accumulate wealth in the New World than in the Old World;
second, there was so much land in the New World that it was almost worthless
(its market value per acre was much less than in Europe).

The gap between the US and Europe gradually reduces over the course of the
19th century, but still remains substantial. Around 1900-1910, national
wealth is about 5 years of national income in the US (see figure 2.3), vs.
about 7 years in Britain and France. During the 20$^{th} $ century, the U.S.
wealth-income ratio also follows a U-shaped pattern, but less marked than in
Europe. National wealth falls less sharply in the US than in Europe
following World War shocks, which seems rather intuitive. Interestingly,
European wealth-income ratios have again surpassed US ratios in the late 20$%
^{th}$ and early 21$^{st}$ centuries.

This brief overview of wealth in the New World vs. Europe would be rather
incomplete if we did not mention the issue of slavery. As one can see from
figure 2.4, the aggregate market value of slaves was fairly substantial in
the US until 1865: about 1 to 1.5 year of national income according to the
best available historical sources. There were few slaves in Northern states,
but in the South the value of the slave stock was so large that it
approximately compensated -- from the viewpoint of slave owners -- the lower
value of land as compared to the Old World (see figure 2.5).

It is rather dubious, however, to include the market value of slaves into
national capital. Slavery can be viewed as the most extreme form of debt: it
should be counted as an asset for the owners and a liability for the slaves,
so that net national wealth should be unaffected. In the extreme case where
a tiny elite owns the rest of the population, the total value of slaves --
the total value of ``human capital" -- could be a lot larger than that of
non-human capital (since the share of human labor in income is typically
larger than 50\%). If the rate of return $r$ is equalized across all assets,
then the aggregate value of human capital -- expressed in proportion to
national income -- will be equal to $\beta _{h}=(1-\alpha )/r$, while the
value of (non-human) capital will be given by $\beta _{n}=\alpha /r$, where $%
\alpha $ is the capital share and $1-\alpha $ the labor share implied by the
production technology.\footnote{%
That is, $1-\alpha $ is the marginal product of labor times the labor
(slave) stock. The formula $\beta _{h}=(1-\alpha )/r$ implicitly assumes
that the fraction of output that is needed to feed and maintain the slave
stock is negligible (otherwise it would just need to be deducted from $%
1-\alpha $), and that labor productivity is unaffected by the slavery
condition (this is a controversial issue).} So for instance with $r=5\%$, $%
\alpha =30\%$, $1-\alpha =70\%$, the value of the human capital stock will
be as large as $\beta _{h}=(1-\alpha )/r=1400\%$ (14 years of national
income), and the value of the (non-human) capital stock will be $\beta
_{n}=\alpha /r=600\%$ (6 years of national income). Outside of slave
societies, however, it is unclear whether it makes much sense to compute the
market value of human capital and to add it to non-human capital.

The computations reported on figures 2.4-2.5 illustrate the ambiguous
relationship of the New World with wealth, inequality and property. To some
extent, America is the land of opportunity, the place where wealth
accumulated in the past does not matter too much. But it is also the place
where a new form of wealth and class structure -- arguably more extreme and
violent than the class structure prevailing in Europe -- flourished, whereby
part of the population owned another part. 


Available historical series suggest that the sharp U-shaped pattern for the
wealth-income ratio in Britain and France is fairly representative of Europe
as a whole. For Germany, the wealth-income ratio was approximately the same
as for Britain and France in the late 19$^{th}$ and early 20$^{th}$
centuries, then fell to a very low level in the aftermath of the World Wars,
and finally has been rising regularly since the 1950s (see figure 2.6).
Although the German wealth-income ratio is still below that of the U.K. and
France, the speed of the recovery over the past few decades is similar.%
\footnote{%
The factors that can explain the lower German wealth-income ratio are the
following. Real estate prices have increased far less in Germany than in
Britain or France, which could be due in part to the lasting impact of
German reunification and to stronger rent regulations. This could also be
temporary. Next, the lower market value of German firms could be due to a
stakeholder effect. Finally, the return to German foreign portfolio (where a
large part of German savings were directed) was particularly low in the most
recent period. See Piketty and Zucman (2014, Section V.C) and Piketty (2014,
chapter 3).} On Figure 2.7, we compare the European wealth-income ratio (obtained as a simple average of Britain, France, Germany, and Italy, the latter being only available for the most recent decades) to the U.S. one. The European
wealth-income ratio was substantially above that of the US until World War
1, then fell significantly below in the aftermath of World War 2, and
surpassed it again in the late 20$^{th}$ and early 21$^{st}$ centuries (see
figure 2.7).

\subsection{The return of high wealth-income ratios in rich countries}


Turning now to the 1970-2010 period, for which we have annual series covering most
rich countries, the rise of wealth-income ratios -- particularly private
wealth-national income ratios -- appears to be a general phenomenon. In the
top 8 developed economies, private wealth is between 2 and 3.5 years of
national income around 1970, and between 4 and 7 years of national income
around 2010 (see figure 2.8). Although there are chaotic short-run
fluctuations (reflecting the short-run volatility of asset prices), the
long-run trend is clear. Take Japan. The huge asset price bubble of the late 1980s should not obscure the 1970-2010 rise of the wealth-income ratio, fairly comparable in magnitude to what we observe in
Europe. (For instance, the Japanese and Italian patterns are relatively close: both countries go from about 2-3 years of
national income in private wealth around 1970 to 6-7 years by 2010.)

Although we do not have national wealth estimates for Japan for the late 19$%
^{th}$ and early 20$^{th}$ centuries, there are reasons to believe that the
Japanese wealth-income ratio has also followed a U-shaped evolution in the
long run, fairly similar to that observed in Europe over the 20$^{th}$
century. That is, it seems likely that the wealth-income ratio was
relatively high in the early 20$^{th}$ century, fell to low levels in the
aftermath of World War 2, and then followed the recovery process that we see
on Figure 2.8.\footnote{%
The early 20$^{th}$ century Japanese inheritance tax data reported by
Morigushi and Saez (2010) are consistent with this interpretation.}

To some extent, the rise of private wealth-national income ratios in rich
countries since the 1970s is related to the decline of public wealth (see
figure 2.9). Public wealth has declined pretty much everywhere due both to the rise of public debt and the
privatization of public assets. In some countries, such as Italy, public
wealth has become strongly negative. The rise in private wealth, however, is
quantitatively much larger than the decline in public wealth. As a result,
national wealth -- the sum of private and public wealth -- has increased
substantially, from 250-400\% of national income in 1970 to 400-650\% in
2010 (see figure 2.10). In Italy, for instance, net government wealth fell
by the equivalent of about one year of national income, but net private
wealth rose by over four years of national income, so that national wealth
increased by the equivalent of over three years of national income.

Figure 2.10 also depicts the evolution of net foreign wealth. Net
foreign asset positions are generally small compared to national wealth. In
other words, the evolution of national wealth-national income ratios mostly
reflects the evolution of domestic capital-output ratios. There are two
caveats, however. First,  gross cross-border
positions have risen a lot in recent decades, which can generate large portfolio valuation effects at the country level. Second, Japan and Germany have
accumulated significant net foreign wealth (with net positions around 40\%
and 70\% of national income respectively in 2010). Although these are still
much smaller than the positions held by France and Britain on the eve of
World War 1 (around 100\% and 200\% of national income, respectively), they
are becoming relatively large (and were rising fast in the case of Germany
in the first half of the 2010s, due to the large German trade surpluses).

\section{The long-run evolution of wealth concentration}

\subsection{Concepts, data sources and methods}

We now turn to the evidence on the long-run evolution of wealth
concentration. This question can be studied with different data sources (see
Davies and Shorrocks, 1999, for a detailed discussion). Ideally, one would
want to use annual wealth tax declarations for the entire population. Annual
wealth taxes, however, often do to exist, and when they do the data
generally do not cover long periods of time.

The key source used to study the long-run evolution of wealth inequality has
traditionally been inheritance and estate tax declarations.\footnote{%
The difference between inheritance and estate taxes is that inheritance
taxes are computed at the level of each inheritor, whereas estate taxes are
computed at the level of the total estate (total wealth left by the
decedent). The raw data coming from these two forms of taxes on wealth
transfers are similar.} By definition, estates and inheritance returns only
provide information about wealth at death. The standard way to use
inheritance tax data in order to study wealth concentration was invented
over a century ago. Shortly before World War I, a number of British and
French economists developed what is known as the mortality multiplier
technique, whereby wealth-at-death is weighted by the inverse of the
mortality rate of the given age and wealth group in order to generate
estimates for the distribution of wealth among the living.\footnote{%
See Mallet (1908), Seailles (1910), Strutt (1910), Mallet and Strutt (1915),
and Stamp (1919).} This approach was later followed in the U.S. by Lampman
(1962) and Kopczuk and Saez (2004), who use estate tax data covering the
1916-1956 and 1916-2000 period respectively, and in the U.K. by Atkinson and
Harrison (1978), who exploit inheritance tax data covering the 1922-1976
period.

To measure historical trends in the distribution of wealth, one can also use
individual income tax returns and capitalize the dividends, interest, rents
and other forms of capital income declared on such returns. The
capitalization technique was pioneered by King (1927), Stewart (1939),
Atkinson and Harrison (1978) and Greenwood (1983), who used it to estimate
the distribution of wealth in the U.K. and U.S. for some years in isolation.
To obtain reliable results, it is critical to have detailed income data,
preferably at the micro level, and to carefully reconcile the tax data with
household balance sheets, so as to compute the correct capitalization
factors. Drawing on the very detailed U.S. income tax data and Flow of Funds
balance sheets, Saez and Zucman (2014) use the capitalization technique to
estimate the distribution of U.S. wealth annually since 1913.

For the recent period, one can also use wealth surveys. Surveys, however,
are never available on a long-run basis and raise serious difficulties
regarding self-reporting biases, especially at the top of the distribution.
Tax sources also raise difficulties at the top, especially for the recent
period, given the large rise of offshore wealth (Zucman, 2013). Generally
speaking it is certainly more difficult for the recent period to accurately
measure the concentration of wealth than the aggregate value of wealth, and
one should be aware of this limitation. One needs to be pragmatic and
combine the various available data sources (including the global wealth
rankings published by magazines such as Forbes, which we will refer to in
section 5).

The historical series that we analyze in this chapter combine works by many
different authors (more details below), who mostly relied on estate and
inheritance tax data. They all relate to the inequality of wealth among the
living.

We focus upon simple concentration indicators such as the share of aggregate
wealth going to the top 10\% individuals with the highest net wealth and the
share going to the top 1\%. In every country and historical period for which
we have data, the share of aggregate wealth going to the bottom 50\% is
extremely small (usually less than 5\%). So a decline in the top 10\% wealth
share can for the most part be interpreted as a rise in the share going to
the middle 40\%. Note also that wealth concentration is usually almost as
large within each age group as for the population taken as a whole.\footnote{%
See, e.g., Atkinson (1983) and Saez and Zucman (2014).}

\subsection{The European pattern: France, Britain and Sweden, 1810-2010}

\subsubsection{France}

We start with the case of France, the country for
which the longest time series are available. French inheritance tax data is
exceptionally good, for one simple reason. As early as 1791, shortly after
the abolition of the tax privileges of the aristocracy, the French National
Assembly introduced a universal inheritance tax, which has remained in force
since then. This inheritance tax was universal because it applied both to
bequests and to inter-vivos gifts, at any level of wealth, and for nearly
all types of property (both tangible and financial assets). The key
characteristic of the tax is that the successors of all decedents with
positive wealth, as well as all donees receiving a positive gift, have
always been required to file a return, no matter how small the estate was,
and no matter whether any tax was ultimately owed.

In other countries, available data is less long-run and/or less systematic.
In the UK, one has to wait until 1894 for the unification of inheritance
taxation (until this date the rules were different for personal and real
estates), and until the early 1920s for unified statistics to be established
by the UK tax administration. In the US, one has to wait until 1916 for the
creation of a federal estate tax and the publication of federal statistics
on inheritance.

In addition, individual-level inheritance tax declarations have been well
preserved in French national archives since the time of the revolution, so
that one can use tax registers to collect large representative micro
samples. Together with the tabulations by inheritance brackets published by
the French tax administration, this allows for a consistent study of wealth
inequality over a two-century-long period (see Piketty, Postel-Vinay and
Rosenthal, 2006, 2013).

The main results are summarized on figures 3.1-3.2.\footnote{%
The updated series used for figures 3.1-3.2 are based upon the historical
estimates presented by Piketty, Postel-Vinay and Rosenthal (2006) and more
recent fiscal data. See Piketty (2014, chapter 10, figures 10.1-10.2).}
First, wealth concentration was very high -- and rising -- in France during
the 19th and early 20th centuries. There was no decline in wealth
concentration prior to World War 1, quite the contrary: the trend towards
rising wealth concentration did accelerate during the 1870-1913 period. The
orders of magnitude are quite striking: in 1913, the top 10\% wealth share
is about 90\%, and the top 1\% share alone is around 60\%. In Paris, which
hosts about 5\% of the population but as much as 25\% of aggregate wealth,
wealth is even more concentrated: more than two thirds of the population
owns zero or negligible wealth, and 1\% of the population owns 70\% of the
wealth.

Looking at figures 3.1-3.2, one naturally wonders whether wealth
concentration would have kept increasing without the 1914-1945 shocks. 
It would maybe have stabilized at a very high level. It could also have
started to decline at some point. In any
case, it is clear that the war shocks induced a violent regime change.

The other interesting fact is that wealth concentration has started to
increase again in France since the 1970s-80s -- but it is still much
lower than on the eve of World War 1. According to the
most recent data, the top 10\% wealth share is slightly above 60\%. Given
the relatively low quality of today's wealth data,
especially regarding top global wealth holders, one should be 
cautious about this estimate. It could well be that we somewhat underestimate
the recent rise and the current level of wealth concentration.%
\footnote{%
In contrast, the 19$^{th}$ and early 20$^{th}$ centuries estimates are
probably more precise (the tax rates were so low at that time that there was
little incentive to hide wealth).} In any case, a share
of 60\% for the top decile is already high, especially compared to
the concentration of labor income: the top 10\% labor earners typically
receive less than 30\% of aggregate labor income.

\subsubsection{Britain}

Although the data sources for other countries are not as systematic and
comprehensive as the French sources, existing evidence suggests that the
French pattern extends to other European countries. For the UK, on Figure
3.3 we have combined historical estimates provided by various authors --
particularly Atkinson and Harrison (1978) and Lindert (1986) -- as well as
more recent estimates using inheritance tax data. These series are not fully
homogenous (in particular the 19$^{th}$ century computations are based on
samples of private probate records and are not entirely comparable to the 20$%
^{th}$ century inheritance tax data), but they deliver a consistent picture.
Wealth concentration was high and rising during the 19$^{th}$ century up
until World War 1, then fell abruptly following the 1914-1945 shocks, and
has been rising again since the 1980s. 

According to these estimates, wealth concentration was also somewhat larger
in the UK than in France in the 19th and early 20th centuries. Yet the gap is much smaller than what French contemporary observers
claimed. Around 1880-1910, it was very common among French republican elites
to describe France as a ``country of little property owners" (``un pays de
petits propri\'{e}taires"), in contrast to aristocratic Britain. Therefore,
the argument goes, there was no need to introduce progressive taxation in
France (this should be left to Britain). The data show
that on the eve of World War 1 the concentration of wealth was almost as
extreme on both sides of the Channel: the top 10\% owns about 90\% of wealth
in both countries, and the top 1\% owns 70\% of wealth in Britain, vs. 60\%
in France. True, aristocratic landed estates were more present in the UK
(and to some extent still are today). But given that the share of
agricultural land in national wealth dropped to low levels during the 19th
century (see figures 2.1-2.2), this does not matter much. At the end of the
day, whether the country is a republic or a monarchy seems to have little
impact on wealth concentration in the long-run.

\subsubsection{Sweden}

Although widely regarded  as an egalitarian
haven today, Sweden was just as unequal as France and Britain in the 19th and
early 20th centuries. This is illustrated by figure 3.3, where we plot some
of the estimates constructed by Roine and Waldenstrom (2009) and Waldenstrom
(2009).

The concentration of wealth is quite similar across European countries, both for the more ancient and
the more recent estimates. Beyond national specificities, a European pattern
emerges: the top 10\% wealth share went from about 90\%
around 1900-1910 to about 60-70\% in 2000-2010, with a recent rebound. In
other words, about 20-30\% of national wealth has been redistributed away
from the top 10\% to the bottom 90\%. Since most of
this redistribution benefited the middle 40\% (the bottom 50\% still
hardly owns any wealth), this evolution can
be described as the rise of a patrimonial middle class.

In the case of Sweden, Roine and Waldenstrom (2009) have also computed
corrected top 1\% wealth shares using estimates of offshore wealth held
abroad by rich Swedes. They find that under plausible assumptions the top
1\% share would shift from about 20\% of aggregate wealth to over 30\%
(i.e., approximately the levels observed in the UK, and not too far away
from the level observed in the U.S.). This illustrates the limitations of
our ability to measure recent trends and levels, given the rising importance
of tax havens.

\subsection{The great inequality reversal: Europe vs. the US, 1810-2010}

Comparing wealth concentration in Europe and the U.S., the main
finding is a fairly spectacular reversal. In the 19$^{th}$ century, the US
was to some extent the land of equality (at least for white men): the
concentration of wealth was much less extreme than in Europe (except in the
South). Over the course of the 20$^{th}$ century, this ordering was
reversed: wealth concentration has become significantly higher in the US.
This is illustrated by figure 3.5, where we combine the estimates due to
Lindert (2000) for the 19$^{th}$ century with those of Saez and Zucman
(2014) for the 20$^{th}$ and 21$^{st}$ centuries to form long-run U.S. series, and by figure 3.6., where we compare the U.S. to Europe (defined as the
arithmetic average of France, Britain and Sweden).

The reversal comes from the fact that Europe has become significantly less
unequal over the course of the 20$^{th}$ century, while the US has not. The
U.S. has almost returned to its early 20$^{th}$ century wealth concentration
level: at its peak in the late 1920s, the 10\% wealth share was about 80\%,
in 2012 it is about 75\%; similarly the top 1\% share peaked at about 45\%
and is back to around 40\% today. Note, however, that the US never reached
the extreme level of wealth concentration of 19$^{th}$ and early 20$^{th}$
centuries Europe (with a top decile of 90\% or more). The US has always had
a patrimonial middle class, although one of varying importance. The wealth
middle-class appears to be shrinking since the 1980s.

US economists of the early 20$^{th}$ century were
very concerned about the possibility that their country becomes as unequal
as Old Europe. Irving Fisher, then president of the American Economic Association, gave his presidential
address in 1919 on this topic. He argued that the concentration of income and
wealth was becoming as dangerously excessive in America as it had been for a
long time in Europe. He called for steep tax progressivity to counteract this tendency. Fisher was particularly concerned
about the fact that as much as half of US wealth was owned by just 2\% of US
population, a situation that he viewed as ``undemocratic" (see Fisher,
1920). One can indeed interpret the spectacular rise of tax progressivity
that occurred in the US during the first half of the 20th century as an
attempt to preserve the egalitarian, democratic American ethos (celebrated a
century before by Tocqueville and others). Attitudes towards inequality are dramatically different today. Many US observers now view Europe as excessively egalitarian
(and many European observers view the US as excessively inegalitarian). 

\section{The long-run evolution of the share of inherited wealth}

\subsection{Concepts, data sources and methods}

We now turn to our third ratio of interest, the share
of inherited wealth in aggregate wealth. We should make clear at the outset
that this is an area where available evidence is scarce and incomplete.
Measuring the share of inherited wealth requires a lot more data than the
measurement of aggregate wealth-income ratios or even wealth concentration.
It is also an area where it is important to be be particularly careful about
concepts and definitions. Purely definitional conflicts have caused substantial confusion in
the past. Therefore it it critical to start from there.

\subsubsection{Basic notions and definitions}

The most natural way to define the share of inherited wealth in aggregate
wealth is to cumulate past inheritance flows. That is, assume that we
observe the aggregate wealth stock $W_{t}$ at time $t$ in a given country,
and that we would like to define and estimate the aggregate
inherited wealth stock $W_{Bt}\leq W_{t}$ (and conversely aggregate
self-made wealth, which we simply define as $W_{St}=W_{t}-W_{Bt}$). Assume
that we observe the annual flow of inheritance $B_{s}$ that occured in any
year $s\leq t$. At first sight, it might seem natural to define the stock of
inherited wealth $W_{Bt}$ as the sum of past inheritance flows:

\  \  \  \ 
\begin{equation*}
\ W_{Bt}=\int \limits_{s\leq t}B_{s}\cdot ds
\end{equation*}

However, there are several practical and conceptual difficulties with this
ambiguous definition, which need to be addressed before the formula can be
applied to actual data. First, it is critical to include in this sum not
only past bequest flows $B_{s}$ (wealth transmissions at death) but also
inter vivos gift flows $V_{s}$ (wealth transmissions inter vivos). That is,
one should define $W_{Bt}$ as $\ W_{Bt}=\int \limits_{s\leq t}B_{s}^{\ast
}\cdot ds$., with $B_{s}^{\ast }=B_{s}+V_{s}$.

Alternatively, if one cannot observe directly the gift flow $V_{s}$, one
should replace the observed bequest flow $B_{s}$ by some grossed up level $%
B_{s}^{\ast }=(1+v_{s})\cdot B_{s}$, where $v_{s}=V_{s}/B_{s}$ is an
estimate of the gift/bequest flow ratio. In countries where adequate data is
available, the gift/bequest ratio is at least 10-20\%, and is often higher
than 50\%, especially in the recent period.\footnote{%
See below. Usually one only includes formal, monetary capital gifts, and one
ignores informal presents \ and in-kind gifts. In particular in-kind gifts
made to minors living with their parents (i.e. the fact that minor children
are usually catered by their parents) are generally left aside.} It is thus critical to include gifts in one way or another. In countries where
fiscal data on gifts is insufficient, one should at least try to estimate a
gross-up factor $1+v_{s}$ using surveys
(which often suffers from severe downward biases) and harder administrative
evidence from other countries.

Next, in order to properly apply this definition, one should only take into
account the fraction of the aggregate inheritance flow $B_{st}\leq B_{s}$
that was received at time $s$ by individuals who are still alive at time $t$%
. The problem is that doing so properly requires very detailed
individual-level information. At any time $t$, there are always individuals
who received inheritance a very long time ago (say, 60 years ago) but who
are still alive (because they inherited at a very young age and/or are
enjoying a very long life). Conversely, a fraction of the inheritance flow
received a short time ago (say, 10 years ago) should not be counted (because
the relevant inheritors are already dead, e.g., they inherited
at an old age or died young). In practice, however, such unusual events tend
to balance each other, so that a standard simplifying assumption is to
cumulate the full inheritance flows observed the previous $H$ years, where $%
H $ is the average generation length, i.e. the average age at which parents
have children (typically $H=$ 30 years). Therefore we obtain the
following simplified definition:

\begin{equation*}
W_{Bt}=\int \limits_{t-30\leq s\leq t}(1+v_{s})\cdot B_{s}\cdot ds
\end{equation*}

\subsubsection{The Kotlikoff-Summers-Modigliani controversy}

Assume now that these two difficulties can be addressed (i.e., that
we can properly estimate the gross up factor $1+v_{s}$ and the average
generation length $H$). There are more substantial difficulties ahead.
First, in order to properly compute $W_{Bt}$, one needs to be able to
observe inheritance flows $B_{s}^{\ast }$ over a relatively long time period
(typically, the previous 30 years). In the famous
Kotlikoff-Summers-Modigliani (KSM) controversy, both Kotlikoff-Summers
(1981, 1988) and Modigliani (1986, 1988) used estimates of the US
inheritance flow for only one year (and a relatively ancient year: 1962).
They simply assumed that this estimate could be used for other years.
Namely, they assumed that the inheritance flow-national income ratio (which
we note $b_{ys}=B_{s}^{\ast }/Y_{s}$) is stable over time. One problem with
this assumption is that it might not be verified. As we shall
see below, extensive historical data on inheritances recently collected in
France show that the $b_{ys}$ ratio has changed tremendously over the past
two centuries, from about 20-25\% of national income in the 19$^{th}$ and
early 20$^{th}$ centuries, down to less than 5\% at mid-20$^{th}$ century,
back to about 15\% in the early 21$^{st}$ century (Piketty, 2011). So one
cannot simply use one year of data and assume that we are in a steady-state:
one needs long-run time series on the inheritance flow in order to estimate
the aggregate stock of inherited wealth.

Next, one needs to decide the extent to which past inheritance flows need to
be upgraded or capitalized. This is the main source of disagreement and
confusion in the KSM controversy.

Modigliani (1986, 1988) chooses zero capitalization. That is, he simply
defines the stock of inherited wealth $W_{Bt}^{M}$ as the raw sum of past
inheritance flows with no adjustment whatsoever (except for the GDP price
index):

\begin{equation*}
W_{Bt}^{M}=\int \limits_{t-30\leq s\leq t}B_{s}^{\ast }\cdot ds
\end{equation*}

Assume a fixed inheritance flow-national income ratio $b_{y}=B_{s}^{\ast
}/Y_{s}$, growth rate $g$ (so that $Y_{t}=Y_{s}\cdot e^{g(t-s)}$),
generation length $H$, and aggregate private wealth-national income ratio $%
\beta =W_{t}/Y_{t}$. Then, according to the Modigliani definition, the
steady-state formulas for the stock of inherited wealth relative to national
income $W_{Bt}^{M}/Y_{t}$ and for the share of inherited wealth $\varphi
_{t}^{M}=W_{Bt}^{M}/W_{t}$ are given by:

\  \  \  \ 
\begin{equation*}
\ W_{Bt}^{M}/Y_{t}=\frac{1}{Y_{t}} \int \limits_{t-30\leq s\leq
t}B_{s}^{\ast }\cdot ds=\frac{1-e^{-gH}}{g}\cdot b_{y}
\end{equation*}

\begin{equation*}
\  \varphi _{t}^{M}=W_{Bt}^{M}/W_{t}=\frac{1-e^{-gH}}{g}\cdot \frac{b_{y}}{%
\beta }
\end{equation*}

In contrast, Kotlikoff and Summers (1981, 1988) choose to capitalize past
inheritance flows by using the economy's average rate of return to wealth
(assuming it is constant and equal to $r$). Following the Kotlikoff-Summers
definition, the steady-state formulas for the stock of inherited wealth
relative to national income $W_{Bt}^{KS}/Y_{t}$ and for the share of
inherited wealth $\varphi _{t}^{KS}=W_{Bt}^{KS}/W_{t}$ are given by:

\  \  \  \ 
\begin{equation*}
\ W_{Bt}^{KS}/Y_{t}=\frac{1}{Y_{t}} \int \limits_{t-30\leq s\leq
t}e^{r(t-s)}\cdot B_{s}^{\ast }\cdot ds=\frac{e^{(r-g)H}-1}{r-g}\cdot b_{y}
\end{equation*}

\begin{equation*}
\  \varphi _{t}^{KS}=W_{Bt}^{KS}/W_{t}=\frac{e^{(r-g)H}-1}{r-g}\cdot \frac{%
b_{y}}{\beta }
\end{equation*}

In the special case where growth rates and rates of return are negligible
(i.e., infinitely close to zero), then both definitions coincide. That is,
if $g=0$ and $r-g=0$, then $(1-e^{-gH})/g=(e^{(r-g)H}-1)/(r-g)=H$ , so that $%
W_{Bt}^{M}/Y_{t}=W_{Bt}^{KS}/Y_{t}=H b_{y}$ and $\varphi _{t}^{M}=\varphi
_{t}^{KS}=H b_{y}/\beta$.

Thus, in case growth and capitalization effects can be neglected, one simply
needs to multiply the annual inheritance flow by generation length. If the
annual inheritance flow is equal to $b_{y}=10\%$ of national income, and
generation length is equal to $H=30$ years, then the stock of inherited
wealth is equal to $W_{Bt}^{M}=W_{Bt}^{KS}=300\%$ of national income
according to both definitions. In case aggregate wealth amounts to $\beta
=400\%$ of national income, then the inheritance share is equal to $\varphi
_{t}^{M}=\varphi _{t}^{KS}=75\%$ of aggregate wealth.

However, in the general case where $g$ and $r-g$ are significantly different
from zero, the two definitions can lead to widely different conclusions. For
instance, with $g=2\%$, $r=4\%$ and $H=30$, we have the following
capitalization factors: $(1-e^{-gH})/(g\cdot H)=0.75$ and $%
(e^{(r-g)H}-1)/((r-g)\cdot H)=1.37$. In this example, for a given
inheritance flow $b_{y}=10\%$ and aggregate wealth-income ratio $\beta
=400\% $, we obtain $\varphi _{t}^{M}=56\%$ and $\varphi _{t}^{KS}=103\%$.
About half of wealth comes from inheritance according to the Modigiani
definition, and all of it according to the Kotlikoff-Summers definition.

This is the main reason why Modigliani and Kotlikoff-Summers disagree
so much about the inheritance share. They both use the same (relatively fragile)
estimate for the US $b_{y}$ in 1962. But Modigliani does not capitalize past
inheritance flows and concludes that the inheritance share is as low as
20-30\%. Kotlikoff-Summers do capitalize the
same flows and conclude that the inheritance share is as large as 80-90\%
(or even larger than 100\%). Both sides also
disagree somewhat about the measurement of $b_{y}$, but the main source of
disagreement comes from this capitalization effect.\footnote{%
In effect, Modigliani favors a $b_{y}$ ratio around 5-6\%, while
Kotlikoff-Summers find it more realistic to use a $b_{y}$ ratio around
7-8\%. Given the data sources they use, it is likely that both sides tend to
somewhat underestimate the true ratio. See below the discussion for the case
of France and other European countries.}

\subsubsection{The limitations of KSM definitions}

Which of the two definitions is most justified? In our view, both are
problematic. It is wholly inappropriate not to capitalize at all past
inheritance flows. But full capitalization is also inadequate.

The key problem with the KSM representative-agent approach is that it fails
to recognize that the wealth accumulation process always involves two
different kinds of people and wealth trajectories. In every economy, there
are inheritors (people who typically consume part of the return to their
inherited wealth), and there are savers (people who do not inherit much but
do accumulate wealth through labor income savings). This is an important
feature of the real world that must be taken into account for a proper
understanding of the aggregate wealth accumulation process.

The Modigliani definition is particularly problematic, since it simply fails
to recognize that inherited wealth produces flow returns. This mechanically
leads to artificially low numbers for the inheritance share $\varphi
_{t}^{M} $ (as low as 20\%-40\%), and to artificially high numbers for the
lifecycle share in wealth accumulation, which Modigliani defines as $%
1-\varphi _{t}^{M}$ (up to 60\%-80\%). As Blinder (1988) argues:
\textquotedblleft a Rockefeller with zero lifetime labor income and
consuming only part of his inherited wealth income would appear to be a
lifecycle saver in Modigliani's definition, which seems weird to
me.\textquotedblright \ One can easily construct illustrative examples of
economies where all wealth comes from inheritance (with dynasties of the
sort described by Blinder), but where Modigliani would still find an
inheritance share well below 50\%, simply because of his definition. This
makes little sense.\footnote{%
It is worth stressing that the return to inherited wealth (and the
possibility to save and accumulate more wealth out of the return to
inherited wealth) is a highly relevant economic issue not only for
high-wealth dynasties of the sort referred to by Blinder, but also for
middle-wealth dynasties. For instance, it is easier to save if one has
inherited a house and has no rent to pay. An inheritor saving less than the
rental value of his inherited home would be described as a lifecycle saver
according to Modigliani's definition, which again seems odd.}

The Kotlikoff-Summers definition is conceptually more satisfactory than
Modigliani's. But it suffers from the opposite drawback, in the sense that
it mechanically leads to artificially high numbers for the inheritance share 
$\varphi _{t}^{KS}$. In particular, $\varphi _{t}^{KS}$ can easily be larger
than 100\%, even though there are lifecycle savers and self-made wealth
accumulators in the economy, and a significant fraction of aggregate wealth
accumulation comes from them. This will arise whenever the cumulated return
to inherited wealth consumed by inheritors exceeds the savers' wealth
accumulation from their labor savings. In the real world, this condition
seems to hold not only in prototype rentier societies such as Paris
1872-1937 (see Piketty, Postel-Vinay and Rosenthal, 2013), but also in
countries and time periods when aggregate inheritance flow are relatively
low. For instance, aggregate French series show that the capitalized bequest
share $\varphi _{t}^{KS}$ has been larger than 100\% throughout the 20th
century, including in the 1950s-1970s, a period where a very significant
amount of new self-made wealth was accumulated (Piketty, 2011).

In sum: the Modigliani definition leads to estimates of the inheritance
share that are artificially close to 0\%, while the Kotlikoff-Summers leads
to inheritance shares that tend to be structurally above 100\%. Neither of
them offers an adequate way to look at the data.

\subsubsection{The PPVR definition}

In an ideal world with perfect data, the conceptually consistent way to
define the share of inherited wealth in aggregate wealth is the
following. It has first been formalized and applied to Parisian wealth data
by Piketty, Postel-Vinay and Rosenthal (2013), so we refer to it as the PPVR
definition.

The basic idea is to split the population into two groups. First, there are
\textquotedblleft inheritors\textquotedblright \ (or \textquotedblleft
rentiers"), whose assets are worth less than the capitalized value of the
wealth they inherited (over time they consume more than their labor income).
The second group is composed of \textquotedblleft savers\textquotedblright \
(or \textquotedblleft self-made individuals"), whose assets are worth more
than the capitalized value of the wealth they inherited (they consume less
than their labor income). Aggregate inherited wealth can then be defined as
the sum of inheritors' wealth plus the inherited fraction of savers' wealth,
and self-made wealth as the non-inherited fraction of savers' wealth. By
construction, inherited and self-made wealth are less than 100\% and sum to
aggregate wealth, which is certainly a desirable property. Although the
definition is fairly straightforward, it differs considerably from the
standard KSM definitions based upon representative agent models. The PPVR
definition is conceptually more consistent, and provides a more meaningful
way to look at the data and to analyze the structure of wealth accumulation
processes. In effect, it amounts to defining inherited wealth at the
individual level as the minimum between current wealth and capitalized
inheritance.

More precisely, consider an economy with population $N_{t}$ at time $t$.
Take a given individual $i$ with wealth $w_{ti}$ at time $t$. Assume he or
she received bequest $b_{ti}^{0}$ at time $t_{i}<t$. Note $b_{ti}^{\ast
}=b_{ti}^{0}\cdot e^{r(t-t_{i})}$ the capitalized value of $b_{ti}^{0}$ at
time $t$ (where $e^{r(t-t_{i})}$ is the cumulated rate of return between
time $t_{i}$ and time $t$). Individual $i$ is said to be an ``inheritor" (or
a ``rentier") if $w_{ti}<$ $b_{ti}^{\ast }$ and a ``saver" (or a ``self-made
individual") if $w_{ti}\geq b_{ti}^{\ast }$. We define the set of inheritors
as $N_{t}^{r}=\{i$ s.t. $w_{ti}<$ $b_{ti}^{\ast }$ $\}$ and the set of
savers as $N_{t}^{s}=\{i$ s.t. $w_{ti}\geq $ $b_{ti}^{\ast }$ $\}$.

We note $\rho _{t}=N_{t}^{r}/N_{t}$ and $1-\rho _{t}=N_{t}^{s}/N_{t}$ the
corresponding population shares of inheritors and savers; $%
w_{t}^{r}=E(w_{ti}|w_{ti}<b_{ti}^{\ast })$ and $w_{t}^{s}=E(w_{ti}|w_{ti}%
\geq b_{ti}^{\ast })$ the average wealth levels of both groups; $%
b_{t}^{r\ast }=E(b_{ti}^{\ast }|w_{ti}<b_{ti}^{\ast })$ and $b_{t}^{s\ast
}=E(b_{ti}^{\ast }|w_{ti}\geq b_{ti}^{\ast })$ the levels of their average
capitalized bequest; and $\pi _{t}=\rho _{t}\cdot w_{t}^{r}/w_{t}$ and $%
1-\pi _{t}=(1-\rho _{t})\cdot w_{t}^{s}/w_{t}$ the share of inheritors and
savers in aggregate wealth.

We define the total share $\varphi _{t}$ of inherited wealth in aggregate
wealth as the sum of inheritors' wealth plus the inherited fraction of
savers' wealth, and the share $1-\varphi _{t}$ of self-made wealth as the
non-inherited fraction of savers' wealth: 
\begin{equation*}
\varphi _{t}=[\rho _{t}\cdot w_{t}^{r}+(1-\rho _{t})\cdot b_{t}^{s\ast
}]/w_{t}=\pi _{t}+(1-\rho _{t})\cdot b_{t}^{s\ast }/w_{t}
\end{equation*}

\begin{equation*}
1-\varphi _{t}=(1-\rho _{t})\cdot (w_{t}^{s}-b_{t}^{s\ast })/w_{t}=1-\pi
_{t}-(1-\rho _{t})\cdot b_{t}^{s\ast }/w_{t}
\end{equation*}

The downside of this definition is that it is more demanding in terms of
data availability. While Modigliani and Kotlikoff-Summers could compute
inheritance shares in aggregate wealth by using aggregate data only, the
PPVR definition requires micro data. Namely, we need data on the joint
distribution $G_{t}(w_{ti},b_{ti}^{\ast })$ of current wealth $%
w_{ti}$ and capitalized inherited wealth $b_{ti}^{\ast }$ in order to
compute $\rho _{t}$, $\pi _{t}$ and $\varphi _{t}$. This does require
high-quality, individual-level data on wealth and inheritance over two
generations, which is often difficult to obtain. It is worth stressing,
however, that we do not need to know anything about the individual labor
income or consumption paths $(y_{Lsi},c_{si},s<t)$ followed by individual $i$
up to the time of observation.\footnote{%
Of course more data are better. If we also have (or estimate) labor income
or consumption paths, then one can compute lifetime individual savings rate $%
s_{Bti}$, i.e. the share of lifetime resources that was not consumed up to
time $t$: $s_{Bti}=w_{ti}/(b_{ti}^{\ast }+y_{Lti}^{\ast })=1-c_{ti}^{\ast
}/(b_{ti}^{\ast }+y_{Lti}^{\ast })$, with $y_{Lti}^{\ast }=\int
\limits_{s<t}y_{Lsi}e^{r(t-s)}ds$ = capitalized value at time $t$ of past
labor income flows, and $c_{ti}^{\ast }=\int \limits_{s<t}c_{si}e^{r(t-s)}ds$
= capitalized value at time $t$ of past consumption flows.By definition,
inheritors are individuals who consumed more than their labor income (i.e. $%
w_{ti}<b_{ti}^{\ast }\leftrightarrow c_{ti}^{\ast }>y_{Lti}^{\ast }$), while
savers are individuals who consumed less than their labor income (i.e. $%
w_{ti}\geq b_{ti}^{\ast }\leftrightarrow c_{ti}^{\ast }\leq y_{Lti}^{\ast }$%
). But the point is that we only need to observe an individual's wealth ($%
w_{ti}$) and capitalized inheritance ($b_{ti}^{\ast }$) in order to
determine whether he or she is an inheritor or a saver, and in order to
compute the share of inherited wealth.}

For plausible joint distributions $G_{t}(w_{ti},b_{ti}^{\ast })$, the PPVR
inheritance share $\varphi _{t}$ will typically fall somewhere in the
interval $[\varphi _{t}^{M},\varphi _{t}^{KS}]$. There
is, however, no theoretical reason why it should be so in general. Imagine for
instance an economy where inheritors consume their bequests the very day
they receive it, and never save afterwards, so that wealth accumulation
entirely comes from the savers, who never received any bequest (or
negligible amounts), and who patiently accumulate savings from their labor
income. Then with our definition $\varphi _{t}=0\%$: in this economy, 100\%
of wealth accumulation comes from savings, and nothing at all comes from
inheritance. However with the Modigliani and Kotlikoff-Summers definitions,
the inheritance shares $\varphi _{t}^{M}$ and $\varphi _{t}^{KS}$ could be
arbitrarily large.

\subsubsection{A simplified definition: inheritance flows vs. saving flows}

When available micro data is not sufficient to apply the PPVR definition,
one can also use a simplified, approximate definition based upon the
comparison between inheritance flows and saving flows. 

Assume that
all we have is macro data on inheritance flows $b_{yt}=B_{t}/Y_{t}$ and
savings flows $s_{t}=S_{t}/Y_{t}$. Suppose for simplicity that both flows
are constant over time: $b_{yt}=$ $b_{y}$ and $s_{t}=s$. We want to estimate
the share $\varphi =W_{B}/W$ of inherited wealth in aggregate wealth. The
difficulty is that we typically do not know which part of the aggregate
saving rate $s$ comes the return to inherited wealth, and which part comes
from labor income (or from the return to past savings). Ideally, one would
like to distinguish between the savings of inheritors and savers (defined
along the lines defined above), but this requires micro data over two
generations. In the absence of such data, a natural starting point would be
to assume that the propensity to save is on average the same whatever the
income sources. That is, a fraction $\varphi \cdot \alpha $ of the saving
rate $s$ should be attributed to the return to inherited wealth, and a
fraction $1-\alpha +(1-\varphi )\cdot \alpha $ should be attributed to labor
income (and to the return to past savings), where $\alpha =Y_{K}/Y$ is the
capital share in national income and $1-\alpha =Y_{L}/Y$ is the labor share.
Assuming again that we are in steady-state, we obtain the following
simplified formula for the share of inherited wealth in aggregate wealth:

\begin{equation*}
\  \varphi =\frac{b_{y}+\varphi \cdot \alpha \cdot s}{b_{y}+s}
\end{equation*}

\begin{equation*}
\  \text{I.e., }\varphi =\frac{b_{y}}{b_{y}+(1-\alpha )\cdot s}
\end{equation*}%
Intuitively, this formula simply compares the size of the inheritance and
saving flows. Since all wealth must originate from one of the two flows, it
is the most natural way to estimate the share of inherited wealth in total
wealth.\footnote{%
Similar formulas based upon the comparison of inheritance and saving flows
have been used by De Long (2003) and Davies et al (2012, p.123-124). One
important difference is that these authors do not take into account the fact
that the saving flow partly comes from the return to inherited wealth. We
return to this point in section 5.4 below.}

There are a number of caveats with this simplified formula. First,
real-world economies are generally out of steady-state, so it is important
to compute average values of $b_{y}$, $s$ and $\alpha $ over relatively long
periods of time (typically over the past $H$ years, with $H=30$ years). If
one has time-series estimates of the inheritance flow $b_{ys}$, capital
share $\alpha_{s}$ and saving rate $s_{s}$ then one can use the following
full formula, which capitalizes past inheritance and savings flows at rate $%
r-g$:

\begin{equation*}
\varphi =\dfrac{\int \limits_{t-H\leq s\leq t}e^{(r-g)(t-s)}\cdot
b_{ys}\cdot ds}{\int \limits_{t-H\leq s\leq t}e^{(r-g)(t-s)}\cdot
(b_{ys}+(1-\alpha _{s})\cdot s_{s})\cdot ds}
\end{equation*}

With constant flows, the full formula boils down to $\varphi =\dfrac{b_{y}}{%
b_{y}+(1-\alpha )\cdot s}$.

Second, one should bear in mind that the simplified formula $\varphi =b_{y}/(b_{y}+(1-\alpha )\cdot s)
$ is an approximate formula. In general, as we show below, it tends to
under-estimate the true share of inheritance, as computed from micro data
using the PPVR definition. The reason is that individuals who only have
labor income tend to save less (in proportion to their total income) than
those who have large inherited wealth and capital income, which in turns
seems to be related to the fact that wealth (and particularly inherited
wealth) is more concentrated than labor income.

On the positive side, simplified estimates of $\varphi $ seem to follow 
micro-based estimates relatively closely (much more closely than KSM
estimates, which are either far too small or far too large), and they are
much less demanding in terms of data. One only needs to estimate macro
flows. Another key advantage of the simplified definition over KSM
definitions is that it does not depend upon the sensitive choice of the rate
of return or the rate of capital gains or losses. Whatever these rates might
be, they should apply equally to inherited and self-made wealth (at least as
a first approximation), so one can simply compare inheritance and saving
flows.

\subsection{The long-run evolution of inheritance in France 1820-2010}

\subsubsection{The inheritance flow-national income ratio $b_{yt}$}

What do we empirically know about the historical evolution of inheritance?
We start by presenting the evidence on the dynamics of the inheritance to
national income ratio $b_{yt}$ in France, a country for which, as we have
seen in section 3, historical data sources are exceptionally good
(Piketty, 2011). The main conclusion is that $b_{yt}$ has followed a
spectacular U-shaped pattern over the 20th century. The inheritance flow was
relatively stable around 20--25\% of national income throughout the
1820--1910 period (with a slight upward trend), before being divided by a
factor of about 5--6 between 1910 and the 1950s, and then multiplied by a
factor of about 3--4 between the 1950s and the 2000s (see figure 4.1).

These are enormous historical variations, but they appear to be well founded
empirically. In particular, the patterns for $b_{yt}$ are similar with two
independent measures of the inheritance flow. The first, what we call the
fiscal flow, uses bequest and gift tax data and makes allowances for
tax-exempt assets such as life insurance. The second measure, what we call
the economic flow, combines estimates of private wealth $W_{t}$, mortality
tables and observed age-wealth profile, using the following accounting
equation:\  \  \ 

\begin{equation*}
\ B_{t}^{\ast }=(1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}\cdot m_{t}\cdot W_{t}
\end{equation*}

Where: $m_{t}=$ mortality rate (number of adult decedents divided by total
adult population)

$%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}=$ ratio between average adult wealth at death and average adult wealth
for the entire population

$v_{t}=V_{t}/B_{t}=$ estimate of the gift/bequest flow ratio

The gap between the fiscal and economic flows can be interpreted as
capturing tax evasion and other measurement errors. It is approximately
constant over time and relatively small, so that the two series deliver
consistent long-run patterns (see figure 4.1).

The economic flow series allow -- by construction -- for a straightforward
decomposition of the various effects at play in the evolution of $b_{yt}$.
In the above equation, dividing both terms by $Y_{t}$ we get:$\ $%
\begin{equation*}
b_{yt}=\ B_{t}^{\ast }/Y_{t}=(1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}\cdot m_{t}\cdot \beta _{t}
\end{equation*}

Similarly, dividing by $W_{t}$ we can define the rate of wealth transmission 
$b_{wt}$:

\begin{eqnarray*}
b_{wt} &=&\ B_{t}^{\ast }/W_{t}=(1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}\cdot m_{t}=%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}^{\ast }\cdot m_{t} \\
\text{with }%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}^{\ast } &=&(1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}\text{ = gift-corrected ratio}
\end{eqnarray*}

If $\mu_{t}=1$ (i.e., decedents have the same average wealth as the living)
and $v_{t}=0$ (no gift), then the rate of wealth transmission is simply
equal to the mortality rate: $b_{wt}=m_{t}$ (and $b_{yt}=\ m_{t}\cdot \beta
_{t}$). If $\mu_{t}=0$ (i.e., decedents die with zero wealth, like in
Modigliani's pure life-cycle theory of wealth accumulation) and $v_{t}=0$
(no gift), then there is no inheritance at all: $b_{wt}=$ $b_{yt}=0$.

Using these accounting equations, we can see that the U-shaped pattern
followed by the French inheritance-income ratio $b_{yt}$ is the product of
two U-shaped evolutions. First, it partly comes from the U-shaped evolution
of the private wealth-income ratio $\beta_{t}$. The U-shaped evolution of $%
b_{yt}$, however, is almost twice as marked at that of $\beta_{t}$. The
wealth-income ratio was divided by a factor of about 2-3 between 1910 and
1950 (from 600-700\% to 200-300\%, see figure 2.2), while the inheritance
flow was divided by a factor around 5-6 (from 20-25\% to about 4\%, see
figure 4.1). The explanation is that the rate of wealth transmission $%
b_{wt}=\mu_{t}^{\ast }\cdot m_{t}$ has also been following a U-shaped
pattern: it was almost divided by two between 1910 and 1950 (from over 3.5\%
to just 2\%), and has been rising again to about 2.5\% in 2010 (see figure
4.2).

The U-shaped pattern followed by $b_{wt}$, in turn, entirely comes from $%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}^{\ast }$. The relative wealth of decedents was at its lowest historical
level in the aftermath of World War 2 (which, as we shall see below, is
largely due to the fact that it was too late for older cohorts to recover
from the shocks and re-accumulate wealth after the war). Given that
aggregate wealth was also at its lowest historical level, the combination of
these two factors explain the exceptionally low level of the inheritance
flow in the 1950s-1960s. By contrast, the mortality rate $m_{t}$ has been
constantly diminishing: this long run downward trend is the mechanical
consequence of the rise in life expectancy (for a given cohort size).%
\footnote{%
The mortality rate, however, is about to rise somewhat in coming decades in
France due to baby boomers (see Piketty, 2011). This effect will be even
stronger in countries where cohort size has declined in recent decades (like
Germany or Japan) and will tend to push inheritance flows toward even higher
levels.}

In the recent decades, a very large part of the rise in $%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}^{\ast }=(1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}$ comes from the rise in the gift-bequest ratio $v_{t}$, which used to
be about 20\% during most of the 19th-20th centuries, and has gradually
risen to as much as 80\% in recent decades (see Figure 4.3). That is, the
gift flow is currently almost as large as the bequest flow.

Although there is still much uncertainty about the reasons behind the rise
in gifts, the evidence suggests that it started before the introduction of
new tax incentives for gifts in the 1990s-2000s, and has more to do with the
growing awareness by wealthy parents that they will die old and
that they ought to transmit part of their wealth inter-vivos if they want
their children to fully benefit from it.

In any case, one should not underestimate the importance of gifts. In
particular, one should not infer from a declining age-wealth profile at old
ages or a relatively low relative wealth of decedents that inheritance is
unimportant: this could simply reflect the fact that decedents have already
given away a large part of their wealth.

\subsubsection{The inheritance stock-aggregate wealth ratio $\protect \varphi %
_{t}$}

How do the annual inheritance flows transmit into cumulated inheritance
stocks? Given the data limitations we face, we report on figure 4.4 two
alternative estimates for the share $\varphi _{t}$ of total inherited wealth
in aggregate French wealth between 1850 and 2010. According to both
measures, there is again a clear U-shaped pattern. The share of inherited
wealth $\varphi _{t}$ was as large as 80-90\% of aggregate wealth in
1850-1910, down to as little as 35-45\% around 1970, and back up to 65-75\%
by 2010.

The higher series, which we see as the most reliable, was obtained by
applying the micro-based PPVR definition (see section 4.1.4 above). The
limitation here is that the set of micro data on wealth over two generations
that has been collected in French historical archives is more complete for
Paris than for the rest of France (see Piketty, Postel-Vinay and Rosenthal,
2006, 2013). For years with missing data for the rest of France, the
estimates reported on Figure 4.4 were extrapolated on the basis of the
Parisian data. On-going data collection suggests that the final estimates
will not be too different from the approximate estimates reported here.

The lower series, which we see as a lower bound, comes from the simplified
definition based upon the comparison of inheritance and saving flows (see
section 4.1.5 above).\footnote{%
The series was computed as $\varphi =b_{y}/(b_{y}+(1-\alpha )\cdot s)$ using
30-year averages for saving rates, capital shares and inheritance flows.}
The key advantage of this simplified definition is that it requires much
less data: it can readily be computed from the inheritance flow series $%
b_{yt}$ that were reported above. It delivers estimates of the inheritance
share $\varphi _{t}$ that are always somewhat below the micro-based
estimates, with a gap that appears to be approximately constant. The gap
seems to be due to the fact that the simplified definition attributes too
much saving to pure labor earners with little inheritance.

In both series, the share $\varphi _{t}$ of total inherited wealth in
aggregate wealth reaches its lowest historical point in the 1970s, while the
inheritance flow $b_{yt}$ reaches its lowest point in the immediate
aftermath of World War 2. The reason is that the stock of inherited wealth
comes from cumulating the inheritance flows of the previous decades -- hence
the time lag.

\subsection{Evidence from other countries}

What do we know about the importance of inheritance in countries other than
France? A recent wave of research attempts to construct estimates of the
inheritance flow-national income ratio $b_{yt}$ in a number of European
countries. The series constructed by Atkinson (2013) for Britain and Schinke
(2013) for Germany show that $b_{yt}$ has also followed a U-shaped pattern
in these two countries over the past century (see figure 4.5). Data
limitations, however, make it difficult at this stage to make precise
comparisons between countries.

For Britain, the inheritance flow $b_{yt}$ of the late 19th-early
20th centuries seems to be similar to that of France, namely about 20-25\%
of national income. The flow then falls  following the 1914-1945 shocks,
albeit less spectacularly than in France, and recovers in recent decades.
Karagiannaki (2011), in a study of inheritance in the UK from 1984 to 2005,
also finds a marked increase in that period. The rebound, however, seems to
be less strong in Britain than in France, so that the inheritance flow
appears smaller than in France today. We do not know yet whether this
finding is robust. Available British series are pure ``fiscal flow" series
(as opposed to French series, for which we have both an ``economic" and a
``fiscal'' estimate). As pointed out by Atkinson (2013), the main reason for
the weaker British rebound in recent decades is that the gift/bequest ratio $%
v_{t}$ has not increased at all according to fiscal data ($v_{t}$ has
remained relatively flat at a low level, around 10-20\%). According to
Atkinson, this could  be due to substantial under-reporting of gifts to tax
authorities.

Germany also exhibits a U-shaped pattern of inheritance flow $b_{yt}$ that
seems to be broadly as sharp as in France. In particular, just like
in France, the strong German rebound in recent decades comes with a large
rise in the gift/bequest ratio $v_{t}$ during the 1990s-2000s ($v_{t}$ is
above 50-60\% in the 2000s). The overall levels of $b_{yt} $ are generally
lower in Germany than in France, which given the lower aggregate
wealth-income ratio $\beta _{t}$ is not surprising. Should we
compare the rates of wealth transmission (i.e., $b_{wt}=b_{yt}/\beta _{t}$),
then the levels would be roughly the same in both countries in 2000-2010.

We report on figure 4.6 the corresponding estimates for the share $\varphi
_{t}$ of total inherited wealth in aggregate wealth, using the simplified
definition $\varphi =b_{y}/(b_{y}+(1-\alpha)s)$. For Germany, the inheritance share $\varphi _{t}$ appears to be
generally smaller than in France. In particular, it reaches very low levels
in the 1960s-1970s, due to the extremely low inheritance flows in Germany in
the immediate postwar period, and to large saving rates. In recent decades,
the German $\varphi _{t}$ has been rising fast and seems to catch up with
France's. In the UK, the inheritance share $\varphi _{t}$ apparently never
fell to the low levels observed in France and Germany in the 1950s, and
seems to be always higher than on the Continent. The reason, for the recent
period, is that the UK has had relatively low saving rates since the 1970s.%
\footnote{%
In effect, British saving rates in recent decades are insufficient to
explain the large rise in the aggregate wealth-income ratio, which can only
be accounted for by large capital gain (Piketty and Zucman, 2014).
The simplified definition of $\varphi _{t}$ based upon the comparison
between inheritance and saving flows assumes the same capital
gains for inherited and self-made wealth.}

Recent historical research suggests that inheritance flows have also
followed U-shaped patterns in Sweden (see Ohlsson, Roine and Waldenstrom,
2013). Here $b_{yt}$ appears to be smaller than in France, but this again
seems largely due to lower $\beta _{t}$ ratios. When we look at the
implied $b_{wt}$ and $\varphi _{t}$ ratios, which in a way are the most
meaningful ratios to study, then both the levels and shape are
relatively similar across European countries. As shown by Figure 4.7, the share of inherited wealth followed the same evolution in Sweden and France in the twentieth century (the main difference is that it seems to have increased a bit less in Sweden than in France in recent decades, due to a rise in the private saving rate).  We stress again, however, that
a lot more data needs to be collected -- and to some extent is currently
being collected -- on the historical evolution of inheritance before we can
make proper international comparisons.

Prior to the recent inheritance flow estimates surveyed above, a first wave
of research, surveyed by Davies and Shorrocks (1999), mostly focused on the
U.S., with conflicting results -- the famous Modigliani-Kotlikoff-Summers
controversy. More recently, Edlund and Kopczuk (2009) observe that in estate
tax data, the share of women among the very wealthy in the U.S. peaked in
the late 1960s (at nearly one-half) and then declined to about one-third.
They argue that this pattern reflects changes in the importance of
inheritance, as women are less likely to be entrepreneurs. Wolff and
Gittleman (2013) analyze SCF data and find little evidence of a rise in
inheritances since the late 1980s. Looking at Forbes data, Kaplan and Rauh
(2013) find that Americans in the Forbes 400 are less likely to have
inherited their wealth today than in the 1980s. It is unclear, however,
whether this result reflects a true economic phenomenon or illustrates the
limits of Forbes and other wealth rankings. Inherited wealth holdings are
probably harder to spot than self-made wealth, first because inheritors'
portfolios tend to be more diversified, and also because inheritors may not
like to be in the press, while entrepreneurs usually enjoy it and do not
attempt to dissimulate their wealth nearly as much. The conclusions about
the relative importance of inherited vs self-made wealth obtained by
analyzing Forbes list data may thus be relatively fragile.

In the end, there remain important uncertainties about the historical
evolution of inheritance in the US. There are reasons to believe that
inheritance has historically been less important in the US than in Europe,
because population growth has been much larger (more on this below). It is
unclear whether this still applies today, however. Given the relatively low
US saving rates in recent decades, it is possible that even moderate
inheritance flows imply a relatively large share $\varphi _{t}$ of total
inherited wealth in aggregate wealth (at least according to the simplified
definition of $\varphi $ based upon the comparison between $b_{y}$ and $s$).

One difficulty is that US fiscal data on bequests and gifts are relatively
low quality (in particular because the federal estate tax only covers few
decedents; in 2012 only about 1 decedent out of
1,000 pays the estate tax). One can use survey data (e.g., from the Survey
of Consumer Finances) in order to estimate the relative wealth of decedents $%
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}$ and compute the economic inheritance flow $b_{yt}=\ (1+v_{t})\cdot 
%TCIMACRO{\U{b5}}%
%BeginExpansion
{\mu}%
%EndExpansion
_{t}\cdot m_{t}\cdot \beta _{t}$. One key problem is that one needs to find
ways to estimate the gift/bequest ratio $v_{t}$, which is not easy to do in
the absence of high-quality fiscal data. Self-reported, retrospective data
on bequest and gift receipts usually suffer from large downward biases and
should be treated with caution. In countries where there exists exhaustive
administrative data on bequests and gifts (such as France, and to some
extent Germany), survey-based self-reported flows appear to be less than
50\% of fiscal flows. This may contribute to explain the low level of
inheritance receipts found by Wolff and Gittleman (2013).\footnote{%
One additional challenge in this study is that inherited assets are
generally valued using asset prices at the time the assets were transmitted:
no capital gain is included -- which probably contributes to a relatively
low estimated inheritance share in total US wealth (about 20\%, just like in
Modigiani's estimates). A comparison between inheritance flows and saving
flows (using the simplified formula) would likely lead to more balanced
results.}

\section{Accounting for the evidence: models and predictions}

\subsection{Shocks versus steady-states}

How can we account for the historical evidence on the evolution of
the aggregate wealth-income ratio, the concentration of wealth, and the
share of inherited wealth? In this section, we describe the theoretical
models that have been developed to address this question. While we still
lack a comprehensive model able to rigorously and quantitatively asses the
various effects at play, the literature makes it possible to highlight some
of the key forces.

We are primarily concerned here about the determinants of long run
steady-states. In practice, as should be clear from the historical series
presented above, real-world economies often face major shocks and changes in
fundamental parameters, so that we observe large deviations from
steady-states. In particular, the large decline in the aggregate
wealth-income ratios $\beta _{t}$ between 1910 and 1950 is due to the shocks
induced by the two World Wars. By using detailed series on saving flows and
war destructions, one can estimate the relative importance of the various
factors at play (Piketty and Zucman, 2014). In the case of France and
Germany, three factors of comparable magnitude each account for
approximately one third of the total 1910-1950 fall of $\beta _{t}$:
insufficient national savings (a large part of private saving was absorbed
by public deficits); war destructions; and the fall of relative assets
prices (real estate and equity prices were historically low in 1950-1960,
partly due to policies such as rent control and nationalization). In the
case of Britain, war destructions were relatively minor, and the other two
factors each account for about half of the fall in the ratio of wealth to
income (war-induced public deficits were particularly large).\footnote{%
For detailed decompositions of private and national wealth accumulation over
the various sub-periods, see Piketty and Zucman (2014).}

In thinking about the future, is the concept of a steady state a relevant
point of reference? Historical evidence suggests that is is. While the
dynamics of wealth and inequality has been chaotic in the 20th century, 18th
and 19th century UK and France can certainly be analyzed as being in a
steady state characterized by low growth, high wealth-income ratios, high
levels of wealth concentration and inheritance flows. This is true despite
the fact that there were huge changes in the nature of wealth and of
economic activity (from agriculture to industry).\footnote{%
In particular, private wealth/income ratios and inheritance flows seemed
quite stable in 19th century France (with maybe a slight upward trend at the
end of the century), despite major structural economic changes. This
suggests that although the importance of inheritance and wealth may rise and
fall in response to the waves of innovation, a steady state analysis is a
fruitful perspective.} The shocks of the 20th century put an end to this
steady state, and it seems justified to ask: if countries are to converge to
a new steady state in the 21st century (that is, if the shocks of the 20th
century do not happen again), which long-term ratios will they reach?

We show that over a wide range of models, the long run magnitude and
concentration of wealth and inheritance are a decreasing function of $g$ and
an increasing function of $\overline{r}$ , where $g$ is the economy's growth
rate and $\overline{r}$ is the net-of-tax rate of return to wealth. That is,
under plausible assumptions, both the wealth-income ratio and the
concentration of wealth tend to take higher steady-state values when the
long-run growth rate is lower and when the net-of-tax rate of return is
higher. In particular, a higher $\overline{r}-g$ tends to magnify
steady-state wealth inequalities. Although there does not exist yet any
rigorous calibrations of these theoretical models, we argue that these
predictions are broadly consistent with both the time-series and
cross-country evidence. These findings also suggests that the current trends
toward rising wealth-income ratios and wealth inequality might continue
during the 21$^{st}$ century, both because of population and productivity
growth slowdown, and because of rising international competition to attract
capital.

\subsection{The steady-state wealth-income ratio: $\protect \beta =s/g$}

The most useful steady-state formula to analyze the long-run evolution of
wealth-income and capital-output ratios is the Harrod-Domar-Solow
steady-state formula:%
\begin{equation*}
\beta _{t}\rightarrow \beta =s/g
\end{equation*}

With: $s=$ long-run (net-of-depreciation) saving rate

$g=$ long-run growth rate.\footnote{%
When one uses gross-of-depreciation saving rates rather than net rates, the
steady-state formula writes $\beta =s/(g+\delta )$ with $s$ the gross saving
rate, and $\delta $ the depreciation rate expressed as a proportion of the
wealth stock.}

The steady-state formula $\beta =s/g$ is a pure accounting equation. By
definition, it holds in the steady-state of any micro-founded, one-good
model of capital accumulation, independently of the exact nature of saving
motives. It simply comes from the wealth accumulation equation $%
W_{t+1}=W_{t}+S_{t}$, which can be rewritten in terms of wealth-income ratio 
$\beta _{t}=W_{t}/Y_{t}$ :

\begin{equation*}
\beta _{t+1}=\frac{\beta _{t}+s_{t}}{1+g_{t}}
\end{equation*}

With: $1+g_{t}=Y_{t+1}/Y_{t}$ = growth rate of national income

$s_{t}=S_{t}/Y_{t}$ = net saving rate

It follows immediately that if $s_{t}\rightarrow s$ and $g_{t}\rightarrow g$%
, then $\beta _{t}\rightarrow \beta =s/g$.

The Harrod-Domar-Solow says something trivial but important: in a low-growth
economy, the sum of capital accumulated in the past can become very large,
as long the saving rate remains sizable.

For instance, if the long run saving rate is $s=10\%$, and if the economy
permanently grows at rate $g=2\%$, then in the long run the wealth income
ratio has to be equal to $\beta =500\%$, because it is the only ratio such
that wealth rises at the same rate as income: $s/\beta =2\%=g$. If the long
run growth rate declines to $g=1\%$, and the economy keeps saving at rate $%
s=10\%$, then the long run wealth income ratio will be equal to $\beta
=1000\% $.

In the long run, output growth $g$ is the sum of productivity and population
growth. In the standard one-good growth model, output is given by $%
Y_{t}=F(K_{t},L_{t})$, where $K_{t}$ is non-human capital input and $L_{t}$
is human labor input (i.e., efficient labor supply). $L_{t}$ can be written
as the product of raw labor supply $N_{t}$ and labor productivity parameter $%
h_{t}$. That is, $L_{t}=N_{t}\cdot h_{t}$, with $N_{t}=N_{0}\cdot (1+n)^{t}$
($n$ is the population growth rate) and $h_{t}=h_{0}\cdot (1+h)^{t}$ ($h$ is
the productivity growth rate). The economy's long-run growth rate $g$ is
given by the growth rate of $L_{t}$. Therefore it is equal to $%
1+g=(1+n)\cdot (1+h)$, i.e. $g\approx n+h$.\footnote{%
In order to obtain the exact equality $g=n+h$, one needs to use
instantaneous (continuous time) growth rates rather than annual (discrete
time) growth rates. That is, with $N_{t}=N_{0}\cdot e^{nt}$ (with $n=$
population growth rate) and $h_{t}=h_{0}\cdot e^{ht}$, we have $%
L_{t}=N_{t}\cdot h_{t}=L_{0}\cdot e^{gt}$, with $g=n+h$.} The long run $g$
depends both on demographic parameters (in particular fertility rates) and
on productivity-enhancing activities (in particular the pace of innovation).

The long-run saving rate $s$ also depends on many forces: $s$ captures the
strength of the various psychological and economic motives for saving and
wealth accumulation (dynastic, lifecycle, precautionary, prestige, taste for
bequests, etc.). The motives and tastes for saving vary a lot across
individuals and potentially across countries. Whether savings come primarily
from a lifecycle or a bequest motive, the $\beta =s/g$ formula will hold in
steady-state. In case saving is exogenous (as in the Solow model), the
long-run wealth-income ratio will obviously be a decreasing function of the
income growth rate $g$. This conclusion, however, is also true in a broad
class of micro-founded, general equilibrium models of capital accumulation
in which $s$ can be endogenous and can depend on $g$. That is the case, in
particular, in the infinite-horizon, dynastic model (in which $s$ is
determined by the rate of time preference and the concavity of the utility
function), in \textquotedblleft bequest-in-the-utility-function" models (in
which the long run saving rate $s$ is determined by the strength of the
bequest or wealth taste), and in most endogenous growth models (see box
below). In all cases, for given preference parameters, the long-run $\beta
=s/g$ tends to be higher when the growth rate is lower. A growth slowdown --
coming from a decrease in population or productivity growth -- tends to lead
to higher capital-output and wealth-income ratios.

\begin{framed}
\textbf{Box: The steady-state wealth-income ratio in macro models}\\

\textit{Dynastic model}

Assume that output is given by $
Y_{t}=F(K_{t},L_{t})$, where $K_{t}$ is the capital stock and $L_{t}$
is efficient labor and grows exogenously at rate $g$. Output is either consumed or added to the capital stock. We assume a closed economy, so the wealth-income ratio is the same as the capital-output ratio. In the infinite-horizon,
dynastic model, each dynasty maximizes: 

$$V=\int_{t\geq s} e^{-\theta t} U(c_{t})$$ 

where $\theta$ is the rate of time preference and $U(c_{t})=c^{1-\gamma}/(1-\gamma )$ is a standard utility function with a constant intertemporal elasticity of substitution equal to $1/\gamma$. This elasticity of substitution is often found to be small, typically between 0.2 and 0.5, and is in any case smaller than one. Therefore $\gamma$ is typically bigger than one.

The first-order condition describing the optimal consumption path of each dynasty is: $dc_{t}/dt=(r-\theta)\cdot c_{t}/\gamma$, i.e. utility-maximizing agents want their consumption path to grow at rate $g_{c}=(r-\theta)/\gamma$. This is a steady-state if and only if $g_{c}=g$, i.e. $r=\theta+\gamma g$,  what is known as the modified Golden rule of capital accumulation. The long run rate of return $r=\theta+\gamma g$ is entirely determined by
preference parameters and the growth rate and is larger than $g$.

The steady-state saving rate is equal to $s=\alpha \cdot
g/r=\alpha \cdot g/(\theta +\gamma g)$, where $\alpha =r\cdot \beta $ is
the capital share. Intuitively, a fraction $g/r$ of capital income is saved
in the long-run, so that dynastic wealth grows at the same rate $g$ as
national income. The saving rate $s=s(g)$ is an increasing function of the
growth rate, but rises less fast than $g$, so that the steady-state
wealth-income ratio $\beta =s/g$ is a decreasing function of the
growth rate.

For instance, with a Cobb-Douglas production function (in which case the capital share is entirely set by technology and is constantly equal to $\alpha$), the
wealth-income ratio is given by $\beta =\alpha /r=\alpha /(\theta
+\gamma \cdot g)$ and takes its maximum value $\overline{\beta }=\alpha
/\theta $ for $g=0$.

One unrealistic feature of the dynastic model is that it assumes an infinite
long-run elasticity of capital supply with respect to the net-of-tax rate of
return, which mechanically entails extreme consequences for optimal capital tax
policy (namely, zero tax).  The \textquotedblleft
bequest-in-the-utility-function" model provides a less extreme and more
flexible conceptual framework in order to analyze the wealth accumulation
process.\\


\textit{Wealth-in-the utility function model}

Consider a dynamic economy with a discrete set of generations $0,1,..,t,...$%
, zero population growth, and exogenous labor productivity growth at rate $%
g>0$. Each generation has measure $N_{t}=N$, lives one period, and is
replaced by the next generation. Each individual living in generation $t$
receives bequest $b_{t}=w_{t}\geq 0$ from generation $t-1$ at the beginning
of period $t$, inelastically supplies one unit of labor during his lifetime
(so that labor supply $L_{t}=N_{t}=N$), and earns labor income $y_{Lt}$. At
the end of period, he then splits lifetime resources (the sum of labor
income and capitalized bequests received) into consumption $c_{t}$ and
bequests left $b_{t+1}=w_{t+1}\geq 0$, according to the following budget
constraint:%
\begin{equation*}
c_{t}+b_{t+1}\leq y_{t}=y_{Lt}+(1+r_{t})b_{t}
\end{equation*}


The simplest case is when the utility function is defined directly over
consumption $c_{t}$ and the increase in wealth $\Delta w_{t}=w_{t+1}-w_{t}$
and takes a simple Cobb-Douglas form: $V(c,\Delta w)=c^{1-s}\Delta w^{s}$.
(Intuitively, this corresponds to a form of \textquotedblleft
moral\textquotedblright \ preferences where individuals feel that they
cannot possibly leave less wealth to their children than what they have
received from their parents, and derive utility from the increase in wealth, maybe because this is a signal of their ability or virtue). Utility maximization then leads to a fixed saving rate: $%
w_{t+1}=w_{t}+sy_{t} $. By multiplying per capita
values by population $N_{t}=N$ we have the same linear transition equation
at the aggregate level: $W_{t+1}=W_{t}+sY_{t}$. The long-run wealth-income ratio is given by $\beta _{t}\rightarrow \beta =s/g$. It depends on the strength of
the bequest motive and on the rate of productivity growth. 

With other functional forms for the utility
function, e.g., with $V=V(c,w)$, or with heterogenous labor productivities
or saving tastes across individuals, one simply needs to replace the
parameter $s$ by the properly defined average wealth or bequest taste parameter. For
instance, with $V(c,w)=c^{1-s}w^{s}$, utility maximization leads to $w_{t+1}=s\cdot (w_{t}+y_{t})$
and $\beta _{t}\rightarrow \beta =s/(g+1-s)=\widetilde{s}/g$, with $%
\widetilde{s}=s(1+\beta )-\beta $ the conventional saving rate (i.e., defined relative to income). See section 5.4.1 below for a simple application of this model to the analysis of the steady-state distribution of wealth.\\


\textit{Endogenous growth models}

In endogenous growth models with imperfect international capital flows, the
growth rate might rise with the saving rate, but it will usually rise less
than proportionally. It is only in what is known as the $AK$ closed-economy
model that the growth rate rises proportionally with the saving rate. To see this,
assume zero population growth ($n=0$) and a Cobb-Douglas production function 
$Y=K^{\alpha }\cdot (A_{L}\cdot L)^{1-\alpha }$. Further assume that the
productivity parameter is endogenously determined by an economy-wide capital
accumulation externality, such that $A_{L}=A_{0}\cdot K$. Then we have $%
Y=A\cdot K$, with $A=(A_{0}\cdot L_{0})^{1-\alpha }$. For a given saving
rate $s>0$, the growth rate is given by $g=g(s)=s\cdot A$. The growth
rate rises proportionally with the saving rate, so that the wealth-income
ratio is entirely set by technology: $\beta =s/g=1/A$ is a constant. 

In more general endogenous growth models, the rate of productivity growth 
depends not only on the pace of capital accumulation, but also -- and probably
more importantly -- on the intensity of innovation activities, the importance
of eduction spendings, the position on the international technological frontier, and a
myriad of other policies and institutions, so that the resulting growth rate
 rises less than proportionally with the saving rate.\\ \

\end{framed}

The slowdown of income growth is the central force explaining the rise of
wealth-income ratios in rich countries over the 1970-2010 period,
particularly in Europe and Japan, where population growth has slowed
markedly (and where saving rates are stil high relative to the US). As
Piketty and Zucman (2014) show, the cumulation of saving flows explains the
1970-2010 evolution of $\beta $ in the main rich countries relatively well.
An additional explanatory factor over this time period is the gradual
recovery of relative asset prices. In the very long run, however, relative
asset price movements tend to compensate each other, and the one-good
capital accumulation model seems to do a good job at explaining the
evolution of wealth-income ratios.

It is worth stressing that the $\beta =s/g$ formula works both in
closed-economy and open-economy settings. The only difference is that
wealth-income and capital-output ratios are the same in closed-economy
settings but can differ in open-economy environments.

In the closed economy case, private wealth is equal to domestic capital: $%
W_{t}=K_{t}$.\footnote{%
For simplicity we assume away government wealth and saving.} National income 
$Y_{t}$ is equal to domestic output $Y_{dt}=F(K_{t},L_{t})$. Saving is equal
to domestic investment, and the private wealth-national income ratio $\beta
_{t}=W_{t}/Y_{t}$ is the same as the domestic capital-output ratio $\beta
_{kt}=K_{t}/Y_{dt}$

In the open economy case, countries with higher saving rates $s_{a}>s_{b}$
accumulate higher wealth-ratios $\beta _{a}=s_{a}/g>\beta _{b}=s_{b}/g$ and
invest some their wealth in countries with lower saving rates, so that the
capital-output ratio is the same everywhere (assuming perfect capital
mobility). Noting $N_{a}$ and $N_{b}$ the population of countries $a$ and $b$%
, $N=N_{a}+$ $N_{b}$ the world population, $Y=Y_{a}+$ $Y_{b}$ the world
output, and $s=(s_{a}\cdot Y_{a}+$ $s_{b}\cdot Y_{b})/Y$ the world saving
rate, and assuming that each country's effective labor supply is
proportional to population and grows at rate $g$, then the long-run
wealth-income and capital-output ratio at the world level will be equal to $%
\beta =s/g$. With perfect capital mobility, each country will operate with
the same capital-output ratio $\beta =s/g$. Country $a$ with wealth $\beta
_{a}>\beta $ will invest its extra wealth $\beta _{a}-\beta $ in country $b$
with wealth $\beta _{b}<\beta $. Both countries have the same per capita
output $y=Y/N$, but country $a$ has a permanently higher per capita national
income $y_{a}=y+r\cdot (\beta _{a}-\beta )>y$, while country $b$ has a
permanently lower per capita national income $y_{b}=y-r\cdot (\beta -\beta
_{b})<y$. In the case of Britain and France at the eve of World War 1, the
net foreign wealth position $\beta _{a}-\beta $ was of the order of
100-200\%, the return on net foreign assets was about r=5\%, so that
national income was about 5-10\% larger than domestic output.

At the world level, wealth-income and capital-output ratios always coincide
(by definition). The long-run ratio is governed by the steady-state
condition $\beta =s/g$. In the very long run, if the growth rate slows down
at the global level (in particular due to the possible stabilization of
world population), then the global $\beta $ might rise. We report on figure
5.1 one possible evolution of the world wealth-income ratio in the 21st
century, assuming that the world income growth rate stabilizes at about
1.5\% and world saving rate at about 12\%. Under these (arguably specify and
uncertain) assumptions, the world $\beta $ would rise to bout 700-800\% by
the end of the 21st century.

\subsection{The steady-state capital share: $\protect \alpha =r\cdot \protect%
\beta =a\cdot \protect \beta ^{\frac{\protect \sigma -1}{\protect \sigma }}$}

How does the evolution of the capital-income ratio $\beta $ relate to the
evolution of the capital share $\alpha _{t}=r_{t}\cdot \beta _{t}$ (where $%
r_{t}$ is the average rate of return)? All depends on whether the
capital-labor elasticity of substitution $\sigma $ is larger or smaller than
one.

Take a CES production function $Y=F(K,L)=(a\cdot K^{\frac{\sigma -1}{\sigma }%
}+(1-a)\cdot L^{\frac{\sigma -1}{\sigma }})^{\frac{\sigma }{\sigma -1}}$.
The rate of return is given by $r=F_{K}=a\cdot \beta ^{-1/\sigma }$ (with $%
\beta =K/Y$), and the capital share is given by $\alpha =r\cdot \beta
=a\cdot \beta ^{\frac{\sigma -1}{\sigma }}$. If $\sigma >1$, then as $\beta
_{t}$ rises, the fall of the marginal product of capital $r_{t}$ is smaller
than the rise of $\beta _{t}$, so that the capital share $\alpha
_{t}=r_{t}\cdot \beta _{t}$ is an increasing function of $\beta _{t}$.
Conversely, if $\sigma <1$, the fall of $r_{t}$ is bigger than the rise of $%
\beta _{t}$, so that the capital share is a decreasing function of $\beta
_{t}$.\footnote{%
Because we include all forms of capital assets into our aggregate capital
concept $K$, the aggregate elasticity of substitution $\sigma $ should be
interpreted as resulting from both supply forces (producers shift between
technologies with different capital intensities) and demand forces
(consumers shift between goods and services with different capital
intensities, including housing services vs. other goods and services).}

As $\sigma \rightarrow \infty $, the production function becomes linear,
i.e. the return to capital is independent of the quantity of capital: this
is like a robot economy where capital can produce output on its own.
Conversely, as $\sigma \rightarrow 0$, the production function becomes
putty-clay, i.e. the return to capital falls to zero if the quantity of
capital is slightly above the fixed proportion technology.

A special case if when the capital-labour elasticity of substitution $\sigma 
$ is exactly equal to one: changes in $r$ and in $\beta$ exactly compensate
each other so that the capital share is constant. This is the Cobb-Douglas
case $F(K,L)=K^{\alpha }L^{1-\alpha }$. The capital share is entirely set by
technology: $\alpha _{t}=r_{t}\cdot \beta _{t}=\alpha $. A higher
capital-output ratio $\beta _{t}$ is exactly compensated by a lower capital
return $r_{t}=\alpha /\beta _{t}$, so that the product of the two is
constant.

There is a large literature trying to estimate the elasticity of
substitution between labor and capital, reviewed in Antras (2004) and
Chirinko (2008); see also Karabarbounis and Neiman (2014). The range of estimates is wide.
Historical evidence suggests that the elasticity of substitution $\sigma $
may have risen over the development process. In the 18$^{th}$-19$^{th}$
centuries, it is likely that $\sigma $ was less than one, particularly in
the agricultural sector. An elasticity less than one would explain why
countries with large quantities of land (e.g., the US) had lower aggregate
land values than countries with little land (the Old World). Indeed, when $%
\sigma <1$, price effects dominate volume effects: when land is very
abundant, the price of land is extremely low, and the product of the two is
small. An elasticity less than 1 is exactly what one would except in an
economy in which capital takes essentially one form only (land), as in the
18th and early 19th century. When there is too much of the single capital
good, it becomes almost useless.

Conversely, in the 20th century, capital shares $\alpha $ have tended to
move in the same direction as capital-income ratios $\beta $. This fact
suggests that the elasticity of substitution $\sigma $ has been larger than
one. Since the mid-1970s, in particular, we do observe a significant rise of
capital shares $\alpha _{t}$ in rich countries (figure 5.2). Admittedly, the
rise in capital shares $\alpha _{t}$ was less marked than the rise of
capital-income ratios $\beta _{t}$ -- in other words, the average return to
wealth $r_{t}=\alpha _{t}/\beta _{t}$ has declined (figure 5.3). But this
decline is exactly what one should expect in any economic model: when there
is more capital, the rate of return to capital must go down. The interesting
question is whether the average return $r_{t}$ declines less or more than $%
\beta _{t}$ increases. The data gathered by Piketty and Zucman (2014)
suggest that $r_{t}$ has declined less, i.e., that the capital share has
increased, consistent with an elasticity $\sigma >1$. This result is
intuitive: an elasticity larger than one is what one would expect in a
sophisticated economy with different uses for capital (not only land, but
also robots, housing, intangible capital, etc.). The elasticity might even
increase with globalization, as it becomes easier to move different forms of
capital across borders.

Importantly, the elasticity does not need to be hugely superior to one in
order to account for the observed trends. With an elasticity $\sigma $
around 1.2-1.6, a doubling of capital-output ratio $\beta $ can lead to a
large rise in the capital share $\alpha $. With large changes in $\beta $,
one can obtain substantial movements in the capital share with a production
function that is only moderately more flexible than the standard
Cobb-Douglas function. For instance, with $\sigma =1.5$, the capital share
rises from $\alpha =28\%$ to $\alpha =36\%$ if the wealth-income ratio jumps
from $\beta =2.5$ to $\beta =5$, which is roughly what has happened in rich
countries since the 1970s. The capital share would reach $\alpha =42\%$ in
case further capital accumulation takes place and the wealth-income ratio
attains $\beta =8$. In case the production function becomes even more
flexible over time (say, $\sigma =1.8$), the capital share would then be as
large as $\alpha =53\%$.\footnote{%
With $a=0.21$ and $\sigma =1.5$, $\alpha $ $=a\cdot \beta ^{\frac{\sigma -1}{%
\sigma }}$goes from $28\%$ to $36\%$ and $42\%$ as $\beta $ rises from $2.5$
to $5$ and $8$. With $\sigma =1.8$, $\alpha $ rises to $53\%$ if $\beta =8$.}
The bottom line is that we certainly do not need to go all the way towards a
robot economy ($\sigma =\infty $) in order to generate very large movements
in the capital share.

\subsection{The steady-state level of wealth concentration: $Ineq=Ineq(%
\overline{r}-g)$}

The possibility that the capital-income ratio $\beta $ -- and maybe the
capital share $\alpha $ -- might rise to high levels entails very different
welfare consequences depending on who owns capital. As we have seen in
section 3, wealth is always significantly more concentrated than income, but
wealth has also become less concentrated since the 19th-early 20th century,
at least in Europe. The top 10\% wealth holders used to own about 90\% of
aggregate wealth in Europe prior to World War 1, while they currently own
about 60-70\% of aggregate wealth.

What model do we have in order to analyze the steady-state level of wealth
concentration? There is a large literature devoted to this question. Early
references include Champernowne (1953), Vaughan (1979), and Laitner (1979).
Stiglitz (1969) is the first attempt to analyze the steady-state
distribution of wealth in the neoclassical growth model. In his and similar
models of wealth accumulation, there is at the same time both convergence of
the macro-variables to their steady state values and of the distribution of
wealth to its steady state form. Dynamic wealth
accumulation models with random idiosyncratic shocks have the additional property that a higher $%
\overline{r}-g$ differential (where $\overline{r}$ is the net-of-tax rate of return to
wealth and $g$ is the economy's growth rate) tends to magnify steady-state
wealth inequalities. This is particularly easy to see in dynamic model with
random multiplicative shocks, where the steady-state distribution of wealth
has a Pareto shape, with a Pareto exponent that is directly determined by $%
\overline{r}-g$ (for a given structure of shocks).

\subsubsection{An illustrative example with closed-form formulas}

In order to illustrate this point, consider the following model with
discrete time $t=0,1,2,...$. The model can be interpreted as an annual model
(with each period lasting $H=1$ year), or a generational model (with each
period lasting $H=30$ years), in which case saving tastes can be interpreted
as bequest tastes. Suppose a stationary population $N_{t}=[0,1]$ made of a
continuum of agents of size one, so that aggregate and average variables are
the same for wealth and national income: $W_{t}=w_{t}$ and $Y_{t}=y_{t}$.
Effective labor input $L_{t}=N_{t}\cdot h_{t}=h_{0}\cdot (1+g)^{t}$ grows at
some exogenous, annual productivity rate $g$. Domestic output is given by
some production function $Y_{dt}=F(K_{t},L_{t})$.

We suppose that each individual $i\in \lbrack 0,1]$ receives the same labor
income $y_{Lti}=y_{Lt}$ and has the same annual rate of return $r_{ti}=r_{t}$%
. Each agent chooses $c_{ti}$ and $w_{t+1i}$ so as to maximize a utility
function of the form $V(c_{ti},w_{ti})=c_{ti}^{1-s_{ti}}w_{ti}^{s_{ti}}$,
with wealth (or bequest) taste parameter $s_{ti}$ and budget constraint $%
c_{ti}+w_{t+1i}\leq y_{Lt}+(1+r_{t})\cdot w_{ti}$. Random shocks only come
from idiosyncratic variations in the saving taste parameters $s_{ti}$, which
are supposed to be drawn according to some i.i.d. random process with mean $%
s=E(s_{ti})<1$.\footnote{%
For a class of dynamic stochastic models with more general structures of
preferences and shocks, see Piketty and Saez (2013).}

With the simple Cobb-Douglas specification for the utility function, utility
maximization implies that consumption $c_{ti}$ is a fraction $1-s_{ti}$ of $%
y_{Lt}+(1+r_{t})\cdot w_{ti}$, the total resources (income plus wealth)
available at time $t$. Plugging this formula into the budget constraint, we
have the following individual-level transition equation for wealth:

\begin{equation}  \label{microtransition}
w_{t+1i}=s_{ti}\cdot \lbrack {y}_{Lt}+(1+r_{t})\cdot w_{ti}]
\end{equation}

At the aggregate level, since by definition national income is equal to $%
y_{t}=y_{Lt}+r_{t}\cdot w_{t}$, we have

\begin{equation}  \label{macrotransition}
w_{t+1}=s\cdot \lbrack {y}_{Lt}+(1+r_{t})\cdot w_{t}]=s\cdot \lbrack {y}%
_{t}+w_{t}]
\end{equation}

dividing by $y_{t+1}\approx (1+g)\cdot y_{t}$ and denoting $%
\alpha_{t}=r_{t}\cdot \beta_{t}$ the capital share and $(1-\alpha_{t}) =
y_{Lt}/y_{t}$ the labor share, we have the following transition equation for
the wealth-income ratio $\beta _{t}=w_{t}/y_{t}$:

\begin{equation}  \label{betatransition}
\beta _{t+1}=s\cdot \frac{1-\alpha_{t}}{1+g}+s\cdot \frac{1+r_{t}}{1+g}\cdot
\beta_{t}= \dfrac{s}{1+g}\cdot (1+\beta _{t})
\end{equation}

In the open-economy case, the world rate of return $r_{t}=r$ is given. From
the above equation one can easily see that $\beta _{t}$ converges towards a
finite limit $\beta $ if and only if 
\begin{equation*}
\omega =s\cdot \dfrac{1+r}{1+g}<1
\end{equation*}

In case $\omega >1$, then $\beta _{t}\rightarrow \infty $. In the long run,
the economy is no longer a small open economy, and the world rate of return
will have to fall so that $\omega <1$.

In the closed-economy case, $\beta _{t}$ always converges towards a finite
limit, and the long-run rate of return $r$ is equal to the marginal product
of capital and depends negatively upon $\beta $. With a CES production
function, for example, we have: $r=F_{K}=a\cdot \beta ^{-1/\sigma }$ (see
section 5.3 above).

Setting $\beta_{t+1}=\beta_{t}$ in equation \ref{betatransition}, we obtain
the steady-state wealth-income ratio:

\begin{equation*}
\beta _{t}\rightarrow \beta =s/(g+1-s)=\widetilde{s}/g
\end{equation*}

where $\widetilde{s}=s(1+\beta )-\beta $ is the steady-state saving rate
expressed as a fraction of national income.

Noting $z_{ti}=w_{ti}/w_{t}$ the normalized individual wealth, and dividing
both sides of equation \ref{microtransition} by $w_{t+1}\approx (1+g)\cdot
w_{t}$, the individual-level transition equation for wealth can be rewritten
as follows:\footnote{%
Note that $y_{Lt}=(1-\alpha )\cdot y_{t}$, where $\alpha =r\cdot \beta
=r\cdot s/(1+g-s)$ is the long-run capital share. Note also that the
individual-level transition equation given below holds only in the long run
(i.e. when the aggregate wealth-income ratio has already converged).}

\begin{equation}  \label{normalized}
z_{t+1i}=\frac{s_{ti}}{s}\cdot \lbrack (1-\omega )+\omega \cdot z_{ti}]
\end{equation}

Standard convergence results (e.g., Hopenhayn and Prescott, 1992, Theorem 2,
p.1397) then imply that the distribution $\psi _{t}(z)$ of relative wealth
will converge towards a unique steady-state distribution $\psi (z)$ with a
Pareto shape and a Pareto exponent that depends on the variance of taste
shocks $s_{ti}$ and on the $\omega $ coefficient.

For instance, assume simple binomial taste shocks: $s_{ti}=s_{0}=0$ with
probability $1-p$, and $s_{ti}=s_{1}>0$ with probability $p$ (with $s=p\cdot
s_{1}$ and $\omega<1<\omega /p$). The long run distribution function $1-\Psi _{t}(z)=proba(z_{ti}\geq z)$
will converge for high $z$ towards

\begin{equation*}
1-\Phi(z)\approx \left( \dfrac{\lambda }{z}\right) ^{a}
\end{equation*}

with a constant term $\lambda$:

\begin{equation*}
\lambda =\dfrac{1-\omega }{\omega -p}
\end{equation*}

a Pareto coefficient $a$:

\begin{equation}  \label{pareto}
a=\dfrac{\log (1/p)}{\log (\omega /p)}>1
\end{equation}

and an inverted Pareto coefficient $b$:

\begin{equation*}
b=\dfrac{a}{a-1}=\dfrac{\log (1/p)}{\log (1/\omega )}>1
\end{equation*}

To see this, note that the long-run distribution with $\omega <1<\omega /p$
looks as follows: $z=0$ with probability $1-p$, $z=\dfrac{1-\omega }{p}$
with probability $(1-p)\cdot p$, ..., and $\ z=z_{k}=\dfrac{1-\omega }{%
\omega -p}\cdot \lbrack (\dfrac{\omega }{p})^{k}-1]$ with probability $%
(1-p)\cdot p^{k}$. As $k\rightarrow +\infty $, $z_{k}\approx \dfrac{1-\omega 
}{\omega -p}\cdot $ $(\dfrac{\omega }{p})^{k}$. The cumulated distribution
is given by: $1-\Phi (z_{k})=proba(z\geq z_{k})=\sum \limits_{k^{\prime
}\geq k}(1-p)\cdot p^{k^{\prime }}=p^{k}$. It follows that as $z\rightarrow
+\infty $, $\log [1-\Phi (z)]\approx a\cdot \lbrack \log (\lambda )-\log (z)]
$, i.e. $1-\Phi (z)\approx (\lambda/z) ^{a}$. In case $\omega /p<1$, then $%
z_{k}=\dfrac{1-\omega }{p-\omega }\cdot \lbrack 1-(\dfrac{\omega }{p})^{k}]$
has a finite upper bound $z_{1}=\dfrac{1-\omega }{p-\omega }$.\footnote{%
See Piketty and Saez (2013, working paper version, p.51-52).}

As $\omega $ rises, $a$ declines and $b$ rises, which means that the
steady-state distribution of wealth is more and more concentrated.\footnote{%
A higher inverted Pareto coefficient $b$ (or, equivalently, a lower Pareto
coefficient $a$) implies a fatter upper tail of the distribution and higher
inequality. On the historical evolution of Pareto coefficients, see
Atkinson, Piketty and Saez (2011, p.13-14 and 50-58).} Intuitively, an
increase in $\omega =$ $s\cdot \dfrac{1+r}{1+g}$ means that the
multiplicative wealth inequality effect becomes larger as compared to the
equalizing labor income effect, so that steady-state wealth inequalities get
amplified.

In the extreme case where $\omega \rightarrow 1^{-}$ (for given $p<\omega $%
), $a\rightarrow 1^{+}$ and $b\rightarrow +\infty $ (infinite inequality).
That is, the multiplicative wealth inequality effect becomes infinite as
compared to the equalizing labor income effect. The same occurs as $%
p\rightarrow 0^{+}$ (for given $\omega >p$): an infinitely small group gets
infinitely large random shocks.\footnote{%
In the binomial model, one can directly compute the \textquotedblleft
empirical\textquotedblright \ inverted Pareto coefficient $b^{\prime }=\dfrac{%
E(z\mid z\geq z_{k})}{z_{k}}\rightarrow \dfrac{1-p}{1-\omega }$ as $%
k\rightarrow +\infty $. Note that $b^{\prime }\simeq $ $b$ if $p,{\omega }%
\simeq 1$ but that the two coefficients generally differ because the true
distribution is discrete, while the Pareto law approximation is continuous.}
Explosive wealth inequality paths can also occur in case the taste parameter 
$s_{ti}$ is higher on average for individuals with high initial wealth.%
\footnote{%
Kuznets (1953) and Meade (1964) were particularly concerned about this
potentially powerful unequalizing force.}

\subsubsection{Pareto formulas in multiplicative random shocks models}

More generally, one can show that ll models with multiplicative random
shocks in the wealth accumulation process give rise to distributions with
Pareto upper tails, whether the shocks are binomial or multinomial, and
whether they come from tastes or other factors. For instance, the shock can
come from the rank of birth, such as in the primogeniture model of Stiglitz
(1969),\footnote{%
With primogeniture (binomial shock), the formula is exactly the same as
before. See, e.g., Atkinson-Harrison (1978, p. 213), who generalize the
Stiglitz (1969) formula and get: $a=log(1+n)/log(1+sr)$, with $s$ the saving
rate out of capital income. This is the same formula as $a=log(1/p)/log(%
\omega /p)$: with population growth rate per generation $=1+n$, the
probability that a good shock occurs -- namely, being the eldest son -- is
given by $p=1/(1+n)$. Menchik (1980), however, provides evidence on estate
division in the U.S. showing equal sharing is the rule.} or from the number
of children (Cowell, 1998),\footnote{%
The Cowell result is more complicated because families with many children do
not return to zero (unless infinite number of children), so there is no
closed form formula for the Pareto coefficient $a$, which must solve the
following equation: $\sum \frac{p_{k}\cdot k}{2}(\frac{2\cdot \omega }{k}%
)^{a}=1$, where $p_{k}=$ \  \ fraction of parents who have k children, with $%
k=1,2,3$,etc., and $\omega =$ average generational rate of wealth
reproduction.} or from rates of return (Benhabib, Bisin and Zhu, 2011, 2013;
Nirei, 2009). Whenever the transition equation for wealth can be rewritten
so as take a multiplicative form

\begin{equation*}
z_{t+1i}=\omega _{ti}\cdot z_{ti}+\varepsilon _{ti}
\end{equation*}

where $\omega _{ti}$ is an i.i.d. multiplicative shock with mean $\omega
=E(\omega _{ti})<1$, and $\varepsilon _{ti}$ an additive shock (possibly
random), then the steady-state distribution has a Pareto upper tail with
coefficient $a$, which must solve the following equation:

\begin{equation*}
E(\omega _{ti}^{a})=1
\end{equation*}

A special case is when $p\cdot (\omega /p)^{a}=1$ , that is $%
a=log(1/p)/log(\omega /p)$, the formula given in equation \ref{pareto}
above. More generally, as long as $\omega _{ti}>1$ with some positive
probability, there exists a unique $a>1$ such that $E(\omega _{ti}^{a})=1$.
One can easily see that for a given average $\omega =E(\omega _{ti})<1$, $%
a\rightarrow 1$ (and thus wealth inequality tends to infinity) if the
variance of shocks goes to infinity, and $a\rightarrow \infty $ if the
variance goes to zero.

Which kind of shocks have mattered most in the historical dynamics of the
distribution of wealth? Many different kinds of individual-level random
shocks play an important role in practice, and it is difficult to estimate
the relative importance of each of them. One robust conclusion, however, is
that for a given variance of shocks, steady-state wealth concentration is
always a rising function of $r-g$. That is, due to cumulative dynamic
effects, relatively small changes in $r-g$ (say, from $r-g=2\%$ per year to $%
r-g=3\%$ per year) can make a huge difference in terms of long-run wealth
inequality.

For instance, if we interpret each period of the discrete-time model
described above as lasting $H$ years (with $H=30$ years = generation
length), and if $r$ and $g$ denote instantaneous rates, then the
multiplicative factor $\omega $ can be rewritten as:

\begin{equation*}
\omega =s\cdot \dfrac{1+R}{1+G}=s\cdot e^{(r-g)H}
\end{equation*}

with $1+R=e^{rH}$ the generational rate of return and $1+G=e^{gH}$ the
generational growth rate. If $r-g$ rises from $r-g=2\%$ to $r-g=3\%$, then
with $s=20\%$ and $H=30$ years, $\omega =$ $s\cdot e^{(r-g)H}$ rises from $%
\omega =0.36$ to $\omega =0.49$. For a given binomial shock structure $p=10\%
$, this implies that the resulting inverted Pareto coefficient $b=(\log
(1/p))/(\log (1/\omega))$ shifts from $b=2.28$ to $b=3.25$. This corresponds
to a shift from an economy with moderate wealth inequality (say, with a top
1\% wealth share around 20-30\%) to an economy with very high wealth
inequality (say, with a top 1\% wealth share around 50-60\%).

Last, if we introduce taxation into the dynamic wealth accumulation model,
then one naturally needs to replace $r$ by the after-tax rate of return $%
\overline{r}=(1-\tau )\cdot r$, where $\tau $ is the equivalent
comprehensive tax rate on capital income, including all taxes on both flows
and stocks. That is, what matters for long-run wealth concentration is the
differential $\overline{r}-g$ between the net-of-tax rate of return and the
growth rate. This implies that differences in capital tax rates and tax
progressivity over time and across countries can explain large differences
in wealth concentration.\footnote{%
For instance, simulation results suggest that differences in top inheritance
tax rates can potentially explain a large fraction of the gap in wealth
concentration between countries such as Germany and France (see Dell (2005)).%
}

\subsubsection{On the long-run evolution of $\overline{r}-g$}

The fact that steady-state wealth inequality is a steeply increasing
function of $\overline{r}-g$ can help explaining some of the historical
patterns analyzed in section 3.

First, it is worth emphasizing that during most of history, the gap $%
\overline{r}-g$ was large, typically of the order of 4-5\% per year. The
reason is that growth rates were close to zero until the industrial
revolution (typically less than 0.1-0.2\% per year), while the rate of
return to wealth was generally of the order of 4-5\% per year, in particular
for agricultural land, by far the most important asset.\footnote{%
In traditional agrarian societies, e.g. in 18$^{th}$ century Britain or
France, the market value of agricultural land was typically around 20-25
years of annual land rentn which corresponds to a rate of return of about
4-5\%. Returns on more risky assets such as financial loans were sometime
much higher. See Piketty (2014).} We have plotted on figure 5.4 the world
GDP growth rates since Antiquity (computed from Maddison, 2010) and
estimates of the average return to wealth (from Piketty, 2014). Tax rates were negligible prior
to the 20$^{th}$ century, so that after-tax rates of return were virtually
identical to pre-tax rates of return, and the $\overline{r}-g$ gap was as
large as the $r-g$ gap.

The very large $\overline{r}-g$ gap until the late 19$^{th}$-early 20$^{th}$
century is in our view the primary candidate explanation as to why the
concentration of wealth has been so large during most of human history.
Although the rise of growth rates from less than 0.5\% per year before the 18%
$^{th}$ century to about 1-1.5\% per year during the 18$^{th}$-19$^{th}$
centuries was sufficient to make a huge difference in terms of population
and living standards, itt had a relatively limited impact on the $\overline{r%
}-g$ gap : $\overline{r}$ remained much bigger than $g$.\footnote{%
It is also possible that the rise of the return to capital during the 18$%
^{th}$-19$^{th}$ centuries was somewhat larger than the lower-bound
estimates that we report on figure 5.4, so that the $r-g$ gap maybe did not
decline at all. See Piketty (2014) for a more elaborate discussion.}

The spectacular fall of the $\overline{r}-g$ gap in the course of the 20$%
^{th}$ century can also help understand the structural decline of wealth
concentration, and in particular why wealth concentration did not return to
the extreme levels observed before the world wars. The fall of the $%
\overline{r}-g$ gap during the 20$^{th}$ century has two components: a 
large rise in $g$, and a large decline in $\overline{r}$. Both, however,
might well turn out to be temporary.

Start with the rise in $g$. The world GDP growth rate was almost 4\% during
the second half of the 20$^{th}$ century. This is due partly to a general
catch up process in per capita GDP levels (first in Europe and Japan between
1950 and 1980, and then in China and other emerging countries starting
around 1980-1990), and partly to unprecedented population growth rates
(which account for about half of world GDP growth rates over the past
century). According to UN demographic projections, world population growth
rates should sharply decline and converge to 0\% during the second half of
the 21$^{st}$ century. Long run per capita growth rates are notoriously
difficult to predict: they might be around 1.5\% per year (as posited on
figure 5.4 for the second half of the 21$^{st}$ century), but some authors
-- such as Gordon (2012) -- believe that they could be less than 1\%. In any
case, it seems plausible that the exceptional growth rates of the 20$^{th}$
century will not happen again -- at least regarding the demographic
component -- and that $g$ will indeed gradually decline during the 21st
century.

Looking now at $\overline{r}$, we also see a spectacular decline during the
20th century. If we take into account both the capital losses (fall in
relative asset prices and physical destructions) and the rise in taxation,
the net-of-tax, net-of-capital-losses rate of return $\overline{r}$ fell
below the growth rate during the entire 20$^{th}$ century after world war I.

Other forms of capital shocks could occur in the 21$^{st}$ century. But
assuming no new shock occurs, and assuming that rising international tax
competition to attract capital leads all forms of capital taxes to disappear
in the course of the 21$^{st}$ century (arguably a plausible scenario,
although obviously not the only possible one), the net-of-tax rate of return 
$\overline{r}$ will converge towards the pre-tax rate of return $r$, so that
the $\overline{r}-g$ gap will again be very large in the future. Other
things equal, this force could lead to rising wealth concentration during
the 21$^{st}$ century.

The $\overline{r}-g$ gap was significantly larger in Europe than in the U.S.
during the 19th century (due in particular to higher population growth in
the New World). This fact can contribute to explain why wealth concentration
was also higher in Europe. The $\overline{r}-g$ gap dramatically declined in
Europe during the 20th century -- substantially more than the US --, which
can in turn explain why wealth has become structurally less concentrated
than in the US. The higher level of labor income inequality in the US in
recent decades, as well as the sharp drop in tax progressivity, also
contribute to higher wealth concentration in the US (see Saez and Zucman,
2014). Note, however, that the US is still characterized by higher
population growth (as compared to Europe and Japan), and that this tends to
push in the opposite direction (i.e., less wealth concentration). So whether
the wealth inequality gap with Europe will keep widening in coming decades
is very much an open issue at this stage.

More generally, we should stress that although the general historical patten
of $\overline{r}-g$ (both over time and across countries) seems consistent
with the evolution of wealth concentration, other factors do also certainly
play an important role in wealth inequality.

One such factor is the magnitude of idiosyncratic shocks to rates of return $%
r_{ti}$, and the possibility that average rates of return $%
r(w)=E(r_{ti}|w_{ti}=w)$ vary with the initial wealth levels. Existing
evidence on returns to university endowments suggests that larger endowments
indeed tend to get substantially larger rates of returns, possibly due to
scale economies in portfolio management (Piketty 2014, chapter 12). The same
pattern is found for the universe of U.S. foundations (Saez and Zucman,
2014). Evidence from Forbes global wealth rankings also suggests that higher
wealth holders tend to get higher returns. Over the 1987-2013 period, the
top fractiles (defined in proportion to world adult population) of Forbes
global billionaire list have been growing on average at about 6-7\% per year
in real terms, when average adult wealth at the global level was rising at
slightly more than 2\% per year (see table 5.1).

Whatever the exact mechanism might be, this seems to indicate that the world
distribution of wealth is becoming increasingly concentrated, at least at
the top of the distribution. It should be stressed again, however, that
available data is of relatively low quality. Little is known about how the
global wealth rankings published by magazines are constructed, and it is
likely that they suffer from various biases. They also focus on such a
narrow fraction of the population that they are of limited utility for a
comprehensive study of the global distribution of wealth. For instance, what
happens above one billion dollars does not necessarily tell us much about
what happens between \$10 and 100 million. This is a research area where a
lot of progress needs to be made.

\subsection{The steady-state level of the inheritance share: $\protect%
\varphi =\protect \varphi (g)$}

\subsubsection{The impact of saving motives, growth and life expectancy}

The return of high wealth-income ratios $\beta $ does not necessarily imply
the return of inheritance. From a purely logical standpoint, it is perfectly
possible that the steady-state $\beta =s/g$ rises (say, because $g$ goes
down and $s$ remains relatively high, as we have observed in Europe and
Japan over the recent decades), but that all saving flows come from
lifecycle wealth accumulation and pension funds, so that the inheritance
share $\varphi $ is equal to zero. Empirically, however, this does not seem
to be the case. From the (imperfect) data that we have, it seems that the
rise in the aggregate wealth-income ratio $\beta $ has been accompanied by a
rise in the inheritance share $\varphi $, at least in Europe.

This suggests that the taste for leaving bequests (and/or the other reasons
for dying with positive wealth, such as precautionary motives and imperfect
annuity markets) did not decline over time. Empirical evidence shows that
the distribution of saving motives varies a lot across individuals. It could
also be that the distribution of saving motives is partly determined by the
inequality of wealth. Bequests might partly be a luxury good, in the sense
that individuals with higher relative wealth also have higher bequest taste
on average. Conversely, the magnitude of bequest motives has an impact on
the steady-state level of wealth inequality. Take for instance the dynamic
wealth accumulation model described above. In that model we implicitly
assume that individuals leave wealth to the next generation. If they did
not, the dynamic cumulative process would start at zero all over again at
each generation, so that steady-state wealth inequality would tend to be
smaller.

Now, assume that we take as given the distribution of bequest motives and
saving parameters. Are there reasons to believe that changes in the long-run
growth rate $g$ or in the demographic parameters (such as life expectancy)
can have an impact on the inheritance share $\varphi $ in total wealth
accumulation?

This question has been addressed by a number of authors, such as Laitner
(2001) and DeLong (2003).\footnote{%
See also Davies et al. (2012, p.123-124).} According to DeLong, the share of
inheritance in total wealth accumulation should be higher in low-growth
societies, because the annual volume of new savings is relatively small in
such economics (so that in effect most wealth originates from inheritance).
Using our notations, the inheritance share $\varphi =\varphi (g)$ is a
decreasing function of the growth rate $g$.

This intuition is interesting (and partly correct) but incomplete. In low
growth societies, such as preindustrial societies, the annual volume of new
savings -- for a given aggregate $\beta $ -- is indeed low in steady-state: $%
s=g\cdot \beta $. In contrast, the flow of inheritances is given by: $%
b_{y}=\mu \cdot m\cdot \beta $ (see section 4 above). Therefore for given $%
\mu $ and $m$, inheritance flows tend to dominate saving flows in low-growth
economies, and conversely in high-growth economies.

For instance, if $\mu =1$, $m=2\%$ and $\beta =600\%$, the inheritance flow
is equal to $b_{y}=12\%$. The inheritance flow $b_{y}$ is 4 times bigger
than the saving flow $s=3\%$ if $g=0.5\%$, it is equal to the saving flow $%
s=12\%$ if $g=2\%$, and it is 2.5 times smaller than the saving flow $s=30\%$
if $g=5\%$. Therefore -- the argument goes -- inherited wealth represents
the bulk of aggregate wealth in low-growth, preindustrial societies; makes
about half of aggregate wealth in medium-growth, mature economies; and a
small fraction of aggregate wealth in high-growth, booming economies.

This intuition, however, is incomplete, for two reasons. First, as we
already pointed out in section 4, saving flows partly come from the return
to inherited wealth, and this needs to be taken into account. Next, the $\mu 
$ parameter, i.e. the relative wealth of decedents, is endogenous and might
well depend on the growth rate $g$, as well as on demographic parameters
such as life expectancy and the mortality rate $m$. In the pure lifecycle
model where agents die with zero wealth, $\mu $ is always equal to zero, and
so is the inheritance share $\varphi $, independently of the growth rate $g$%
, no matter how small $g$ is. For given (positive) bequest tastes and saving
parameters, however, one can show that in steady-state $\mu =\mu (g,m)$
tends to be higher when growth rates $g$ and mortality rates $m $ are lower.

\subsubsection{A simple benchmark model of aging wealth and endogenous $%
\protect \mu $}

In order to see this point more clearly, it is necessary to put more
demographic structure into the analysis. Here we follow we follow a
simplified version of the framework introduced by Piketty (2011).

Consider a continuous-time, overlapping-generations model with a stationary
population $N_{t}=[0,1]$ (zero population growth). Each individual $i$
becomes adult at age $a=A$, has exactly one child at age $a=H$, and dies at
age $a=D$. We assume away inter vivos gifts, so that each individual
inherits wealth solely when his or her parent dies, i.e., at age $a=I=D-H$.

For example, if $A=20$, $H=30$ and $D=60$, everybody inherits at age $%
I=D-H=30$ year-old. But if $D=80$, then everybody inherits at age $I=D-H=50$
year-old.

Given that population $N_{t}$ is assumed to be stationary, the (adult)
mortality rate $m_{t}$ is also stationary, and is simply equal to the
inverse of (adult) life expectancy: $m_{t}=m=\dfrac{1}{D-A}$.\footnote{%
It is more natural to focus upon adults because minors usually have very
little income or wealth (assuming that $I>A$, i.e. $D-A>H$, which is the
case in modern societies).}

For example, if $A=20$ and $D=60$, the mortality rate is $m=1/40=2.5\%$. If $%
D=80$, the mortality rate is $m=1/60=1.7\%$. That is, in a society where
life expectancy rises from 60 to 80 year-old, the steady-state mortality
rate among adults is reduced by a third. In the extreme case where life
expectancy rises indefinitely, the steady-state mortality rate becomes
increasingly small: one almost never dies.

Does this imply that the inheritance flow $b_{y}=\mu \cdot m\cdot \beta $
will become increasingly small in aging societies? Not necessarily: even in
aging societies, one ultimately dies. Most importantly, one tends to die
with higher and higher relative wealth. That is, wealth also tends to get
older in aging societies, so that the decline in the mortality rate $m$ can
be compensated by a rise in relative decedent wealth $\mu $ (which, as we
have seen, has been the case in France).

Assume for simplicity that all agents have on average the same uniform
saving rate $s$ on all their incomes throughout their life (reflecting their
taste for bequests and other saving motives such a precautionary wealth
accumulation) and a flat age-income profile (including pay-as-you-go
pensions). Then one can show that the steady-state $\mu =\mu (g)$ ratio is
given by the following formula:

\begin{equation*}
\mu (g)=\dfrac{1-e^{-(g-s\cdot r)(D-A)}}{1-e^{-(g-s\cdot r)H}}=\dfrac{%
1-e^{-(1-\alpha )g(D-A)}}{1-e^{-(1-\alpha )gH}}
\end{equation*}

With: $\alpha =r\cdot \beta =r\cdot s/g$ = capital share in national income

In other words, the relative wealth of decedents $\mu (g)$ is a decreasing
function of the growth rate $g$ (and an increasing function of the rate of
return $r$ or of the capital share $\alpha $).\footnote{%
This steady-state formula applies both to the closed-economy and
open-economy cases. The only difference is that the rate of return $r$ is
endogenously determined by the marginal product of domestic capital
accumulation in the closed economy case (e.g. $r=F_{K}=a\cdot \beta
^{-1/\sigma }$ with a CES production function), while it is a free parameter
in the open economy setup (in which case the formula can be viewed as $\mu
=\mu (g,r)$).} If one introduces taxes into the model, one can easily show
that $\mu $ is a decreasing function of the growth rate $g$ and an
increasing function of net-of-tax rate of return $\overline{r}$ (or the
net-of-tax capital share $\overline{\alpha }$).\footnote{%
With taxes, $\overline{r}$ also becomes a free parameter in the
closed-economy model, so the formula should always be viewed as $\mu =\mu (g,%
\overline{r})$.}

The intuition for this formula, which can be extended to more general saving
models, is the following. With high growth rates, today's incomes are large
as compared to past incomes, so the young generations are able to accumulate
almost as much wealth as the older cohorts, in spite of the fact that the
latter have already started to accumulate in the past, and in some cases
have already received their bequests. Generally speaking, high growth rates $%
g$ are favorable to the young generations (who are just starting to
accumulate wealth, and who therefore rely entirely on the new saving flows
out of current incomes), and tend to push for lower relative decedent wealth 
$\mu $. High rates of return $\overline{r}$, by contrast, are more favorable
to older cohorts, because this makes the wealth holdings that they have
accumulated or inherited in the past grow faster, and tend to pusher for
higher $\mu $.

In the extreme case where $g\rightarrow \infty $, then $\mu \rightarrow 1$
(this directly follows from flat saving rates and age-labor income profiles).

Conversely, in the other extreme case where $g\rightarrow 0$, then $\mu
\rightarrow \overline{\mu }=\dfrac{D-A}{H}>1$.

It is worth noting that this maximal value $\overline{\mu }$ rises in
proportion to life expectancy $D-A$ (for given generation length $H$).
Intuitively, with $g\approx 0$ and uniform saving, most of wealth originates
from inheritance, so that young agents are relatively poor until inheritance
age $I=D-H$, and most of the wealth concentrates between age $D-H$ and $D$,
so that relative decedent wealth $\mu \approx \overline{\mu }=\dfrac{D-A}{H}$%
.\footnote{%
In the extreme case where young agents have zero wealth and agents above age 
$I=D-H$ have average wealth $\overline{w}$ , then average wealth among the
living is equal to $w=\dfrac{(D-I)\cdot \overline{w}}{D-A}$ and , so that $%
\overline{\mu }=\dfrac{\overline{w}}{w}=\dfrac{D-A}{H}$. See Piketty (2011),
Propositions 1-3.}

That is, as life expectancy $D-A$ rises, wealth gets more and more
concentrated at high ages. This is true for any growth rate, and all the
more for low growth rates. In aging societies, one inherits later in life,%
\footnote{%
Although in practice, this is partly undone by the rise of inter vivos
gifts, as we have seen above.} but one inherits bigger amounts. With $%
g\approx 0$ , one can see that both effects exactly compensate each other,
in the sense that the steady-state inheritance flow $b_{y}$ is entirely
independant of life expectancy. That is, with $m=\dfrac{1}{D-A}$ and $%
\overline{\mu }=\dfrac{D-A}{H}$, we have $b_{y}=\overline{\mu }\cdot m\cdot
\beta =\dfrac{\beta }{H}$ , independently from $D-A$. For a given
wealth-income ratio $\beta =600\%$ and generation length $H=30 $ years, the
steady-state annual inheritance flow is equal to $b_{y}=$ $20\%$ of national
income, whether life expectancy is equal to $D=60$ years or $D=80 $ years.

Strictly speaking, this is true only for infinitely small growth $g\approx 0$%
. However by using the above formula one can see that for low growth rates
(say, $g\approx 1-1.5\%$) then the steady-state inheritance flow is
relatively close to $b_{y}=\dfrac{\beta }{H}$ and is almost independent of
life expectancy. It is only for high growth rates -- above 2-3\% per year --
that the steady-state inheritance flow is reduced substantially.

\subsubsection{Simulating the benchmark model}

Available historical evidence shows that the slowdown of growth is the
central economic mechanism explaining why the inheritance flow seems to be
returning in the early 21$^{st}$ century to approximately the same level $%
b_{y}\approx 20\%$ as that observed during the 19$^{th}$ and early 20$^{th}$
centuries.

By simulating a simple uniform-saving model for the French economy over the
1820-2010 period (starting from the observed age-wealth pattern in 1820, and
using observed aggregate saving rates, growth rates, mortality rates,
capital shocks and age-labor income profiles over the entire period), one
can reasonably well reproduce the dynamics of the age-wealth profile and
hence of the $\mu $ ratio and the inheritance flow $b_{y}$ over almost two
centuries (see figure 5.6).

We can then use this same model to simulate the future evolution of the
inheritance flow in coming decades. As one can see on figure 5.6, a lot
depends on future values of the growth rate $g$ and the net-of-tax rate of
return $\overline{r}$ over the 2010-2100 period. Assuming $g=1.7\%$ (which
corresponds to the average growth rate observed in France between 1980 and
2010) and $\overline{r}=3.0\%$ (which approximatively corresponds to
net-of-tax average real rate of return observed in 2010), then $b_{y}$
should stabilize around 16-17\% in coming decades. If growth slows $g=1.0\%$ and the net-of-tax rate of return rises to $%
\overline{r}=5.0\%$ (e.g., because of a rise of the global capital
share and rate of return, or because of a gradual repeal of capital taxes), $b_{y}$
would keep increasing towards 22-23\% over the course of the 21$^{st}$
century. The flow of inheritance would approximately return to its 19$^{th}$
and early 20$^{th}$ centuries level.

In figure 5.7, we use these projections to compute the corresponding share $%
\varphi $ of cumulated inheritance in the aggregate wealth stock (using the
PPVR definition and the same extrapolations as those described above). In
the first scenario, $\varphi $ stabilizes around 80\%; in the second
scenario, it stabilizes around 90\% of aggregate wealth.

These simulations, however, are not fully satisfactory, first because a lot
more data should be collected on inheritance flows in other countries, and
next because one should ideally try to analyze and simulate both the flow of
inheritance and the inequality of wealth. The computations presented here
assume uniform saving and solely attempt to reproduce the age-average wealth
profile, without taking into account within-cohort wealth inequality. This
is a major limitation.

\section{Concluding comments and research prospects}

In this article, we have surveyed the empirical and theoretical literature
on the long run evolution of wealth and inheritance in relation to output
and income. The magnitude and concentration of wealth and inheritance
(relative to national income) were very high in the 18$^{th}$-19$^{th}$
centuries up until World War 1, then dropped precipitously during the 20$%
^{th}$ century following World War shocks, and have been rising again in the
late 20$^{th}$ and early 21$^{st}$ centuries. We have showed that over a
wide range of models, the long run magnitude and concentration of wealth and
inheritance are an increasing function of $\overline{r}-g$, where $\overline{%
r}$ is the net-of-tax rate of return to wealth and $g$ is the economy's
growth rate, and we have argued that these predictions are broadly
consistent with historical patterns. These findings suggest that current
trends toward rising wealth-income ratios and wealth inequality might
continue during the 21$^{st}$ century, both because of the slowdown of
population and productivity growth, and because of increasing international
competition to attract capital.

We should stress, however, that this is an area where a lot of progress
still needs to be made. Future research should particularly focus on the
following issues. First, it becomes more and more important to study the
dynamics of the wealth distribution from a global perspective.\footnote{%
See the important, pioneering work of Davies et al (2010, 2012).} In order
to do so, it is critical to take into account existing macro data on
aggregate wealth and foreign wealth holdings. Given the large movements in
aggregate wealth-income ratios across countries, such macro-level variations
are likely to have a strong impact on the global dynamics of the
individual-level distribution of wealth. It is also critical to use existing
estimates of offshore wealth and to analyze how much tax havens are likely to
affect global distributional trends (see Zucman, 2014).
Next, a lot more historical and international data needs to be collected on
inheritance flows. Last, there is a strong need of a
better articulation between empirical and theoretical research. A lot more
work has yet to be done before we are able to develop rigorous and credible
calibrations of dynamic theoretical models of wealth accumulation and
distribution.

\renewcommand{\baselinestretch}{1.16}\normalfont

\clearpage

\noindent {\Large \textbf{References}}\vspace{0.1cm}

\noindent \textbf{Antras, Paul} \textquotedblleft Is the U.S. Aggregate
Production Function Cobb-Douglas? New Estimates of the Elasticity of
Substitution,\textquotedblright \  \textit{The B.E. Journal of Macroeconomics}%
, 4(1), 2004.

\noindent \textbf{Atkinson, A.}, \textit{The Economics of Inequality},
Clarendon Press, 1983, 330p.

\noindent \textbf{Atkinson, Anthony B.,} \textquotedblleft Wealth and
Inheritance in Britain from 1896 to the Present,\textquotedblright \ Working
paper, 2013.

\noindent \textbf{A}t\textbf{kinson, A. \& A.J. Harrison}, \textit{%
Distribution of Personal Wealth in Britain, 1923-1972}, Cambridge University
Press, 1978, 330p.

\noindent \textbf{Atkinson, Anthony B., Thomas Piketty and Emmanuel Saez,}
\textquotedblleft Top Incomes in the Long Run of History,\textquotedblright
\  \textit{Journal of Economic Literature}, 2011, vol.49(1), pp.3-71.

\noindent \textbf{Benhabib, J., A. Bisin and S. Zhu,} \textquotedblleft The
Distribution of Wealth and Fiscal Policy in Economies with Finitely Lived
Agents",\  \textit{Econometrica}, 2011, vol.79(1), p.123-157

\noindent \textbf{Benhabib, J., A. Bisin and S. Zhu,} \textquotedblleft The
Wealth Distribution in Bewley Models with Investment Risk", NYU, 2013

\noindent \textbf{B}l\textbf{inder, A.}, ``Comments on Modigliani and
Kotlikoff-Summers'' , in Modelling the Accumulation and Distribution of
Wealth, D. Kessler \& A. Masson eds., pp.68-76, Oxford University Press, 1988

\noindent \textbf{Boisguillebert, Pierre, }\textit{Le d\'{e}tail de la
France, }Paris, 1695, 215p. Reprinted in E. Daire, \textit{Economistes
financiers du 18}$^{\mathit{e}}$\textit{\ si\`{e}cle}, Paris: Guillaumin,
1843.

\noindent \textbf{Bowley, Arthur L.}, \textit{The Change in the Distribution
of National Income, 1880--1913}, Oxford: Clarendon Press, 1920.

\noindent \textbf{Carroll Christopher D.,} \textquotedblleft Why Do the Rich
Save so much?\textquotedblright , in \textit{Does Atlas Shrug? The Economic
Consequences of Taxing the Rich}, J. Slemrod ed., Harvard University Press,
2000.

\noindent \textbf{Carroll Christopher D., Jody Overland and David N. Weil,}
``Saving and Growth with Habit Formation,'' \textit{American Economic Review}%
, 2000, vol.90(3), pp.341-355.

\noindent \textbf{Champernowne, D.G.} ``A Model of Income Distribution,'' 
\textit{Economic Journal} 63(250), 1953, pp.318-351.

\noindent \textbf{Chirinko, R.} \textquotedblleft $\sigma$: The Long and
Short Of It,\textquotedblright \  \textit{Journal of Macroeconomics}, 30,
2008, pp.671-686.

\noindent \textbf{Colquhoun, Patrick}, \textit{A Treatise on the Wealth,
Power and Resources of the British Empire, }London, 1815, 561p.

\noindent \textbf{Colson, Cl\'{e}ment}, \textit{Cours d'\'{e}conomie
politique,} Paris: Gauthier-Villars, 1903, 1918, 1927 (several editions).

\noindent \textbf{Cowell, Frank}, "Inheritance and the Distribution of
Wealth", LSE, 1998.

\noindent \textbf{Davies, James B., and Anthony F. Shorrocks}
\textquotedblleft The Distribution of Wealth," \textit{Handbook of Income
Distribution,} vol. 1, A.B. Atkinson and F. Bourguignon (eds), 1999, chap.
11, 605-675.

\noindent \textbf{Davies, James B., Susanna Sandstr\"{o}m, Anthony Shorrocks
and Edward N. Wolff} \textquotedblleft The Level and Distribution of Global
Household Wealth," \textit{Economic Journal,} 2010, vol.121, pp.223-254.

\noindent \textbf{Davies, James B., Rodrigo Lluberas and Anthony Shorrocks,}
\textquotedblleft Global Wealth Databook", Credit Suisse, 2012

\noindent \textbf{Dell, Fabien,} \textquotedblleft Top Inccomes in Germany
and Switzerland over the Twentieth Century", J\textit{ournal of the European
Economic Association}, 2005, vol.3,, p.412-421

\noindent \textbf{DeLong, J. B.}, \textquotedblleft Bequests: An Historical
Perspective,\textquotedblright \ in \textit{The Role and Impact of Gifts and
Estates}, ed. A. Munnell (Washington, DC: Brookings Institution, 2003).

\noindent \textbf{Feinstein, Charles,} \textquotedblleft Capital Formation
in Great Britain, 1760-1860," in \textit{The Cambridge Economic History of
Europe}, vol.7, Part 1, Chapter 2, Cambridge: Cambridge University Press,
1978, pp.28-96.

\noindent \textbf{Fisher, I., }"Economists in Public Service", \textit{%
American Economic Review}, 1920

\noindent \textbf{Foville, Alfred de}, ``The Wealth of France and of Other
Countries," \textit{Journal of the Royal Statistical Society}, vol.56(4),
1893, pp.597-626.

\noindent \textbf{Giffen, Robert,} \textit{The Growth of Capital}, London,
1889, 169p.

\noindent \textbf{Goldsmith, Raymond W.,} \textit{Comparative National
Balance Sheets: A Study of Twenty Countries, 1688-1978, }Chicago: The
University of Chicago Press, 1985, 353p.

\noindent \textbf{Goldsmith, Raymond W.,} \textit{Pre-Modern Financial
Systems, }Cambridge: Cambridge University Press, 1991, 348p.

\noindent \textbf{Gordon, Robert J.}, ``Is U.S. Economic Growth Over?
Faltering Innovation Confronts the Six Headwinds", NBER working paper 18315,
2012.

\noindent \textbf{Greenwood, Daphne}, ``An Estimation of U.S. Family Wealth
and its Distribution from Microdata, 1973", \textit{Review of Income and
Wealth}, 29, 1983, pp.23-43.

\noindent \textbf{Hopehnayn, Hugo and Edward Prescott}, \textquotedblleft
Stochastic Monotonicity and Stationary Distributions for Dynamic
Economies,\textquotedblright \  \textit{Econometrica}, 60(6), 1992,
pp.1387-1406.

\noindent \textbf{Jones, Charles I. and Paul M. Romer}, \textquotedblleft
The New Kaldor Facts: Ideas, Institutions, Population, and Human
Capital,\textquotedblright \  \textit{American Economic Journal:
Macroeconomics}, 2010 vol.2(1), pp.224-245.

\noindent \textbf{Kaldor, Nicholas,} \textquotedblleft Capital Accumulation
and Economic Growth,\textquotedblright \ in \textit{The Theory of Capital,}
F.A. Lutz and D.C. Hague (eds), New York: Saint Martins Press, 1961.

\noindent \textbf{Karabarbounis, Loukas and Brent Neiman,} ``The Global
Decline of the Labor Share," \textit{Quarterly Journal of Economics},
129(1), 2014, pp.61-103.

\noindent \textbf{King, Gregory,} \textit{Natural and Political Observations
and Conclusions Upon the State and Condition of England}, 1696, 45p.,
Published in Chalmers 1804, pp.29-73.

\noindent \textbf{King, Willford Isbell, }\textit{The Wealth and Income of
the People of the United States}, New York: McMillan Press, 1915, 278p.

\noindent \textbf{King, Willford Isbell, }``Wealth Distribution in the
Continental United States at the Close of 1921", \textit{Journal of the
American Statistical Association}, 22, June 1927, pp.135-153.

\noindent \textbf{Kaplan, Steven N., and Joshua Rauh,} \textquotedblleft
Family, Education, and Sources of Wealth among the Richest Americans,
1982--2012.\textquotedblright , \textit{American Economic Review}, 2013,
103(3): 158 -- 166

\noindent \textbf{Karagiannaki, Eleni,} ``Recent trends in the size and the
distribution of inherited wealth in the UK", CASE paper 146, London School
of Economics, 2011

\noindent \textbf{Kopczuk, Wojciech and Lena Edlund, }"Women, wealth and
mobility", \textit{American Economic Review}, 2009, 99(1), 146-78.

\noindent \textbf{Kopczuk, Wojciech and Joseph Lupton,} \textquotedblleft To
Leave or Not To Leave: the Distribution of Bequest
Motives,\textquotedblright \  \textit{Review of Economic Studies}, 2007,
vol.74(1), pp.207-235.

\noindent \textbf{Kopczuk, Wojciech and Emmanuel Saez,} "Top Wealth Shares
in the United States, 1916-2000: Evidence from Estate Tax Returns", \textit{%
National Tax Journal}, 57(2), 2004, 445-487

\noindent \textbf{Kotlikoff, L.}, \textquotedblleft Intergenerational
Transfers and Savings\textquotedblright , \textit{Journal of Economic
Perspectives}, 1988, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
2(2), pp.41-58

\noindent \textbf{Kotlikoff, L. \& L. Summers}, \textquotedblleft The Role
of Intergenerational Transfers in Aggregate Capital
Accumulation\textquotedblright , \textit{Journal of Political Economy},
1981, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
89, pp.706-732

\noindent \textbf{Kuznets, S.}, \textit{Shares of Upper Income Groups in
Income and Savings, 1913-1948}, National Bureau of Economic Research, 1953,
707p.

\noindent \textbf{Lampman, R.J.}, \textit{The share of top wealth-holders in
national wealth 1922-1956}, Princeton University Press, 1962

\noindent \textbf{Mallet, B.}, ``A Method of Estimating Capital Wealth from
the Estate Duty Statistics'' , \textit{Journal of the Royal Statistical
Society}, 1908, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
71(1), pp.65-101

\noindent \textbf{Mallet, B.}, \textbf{H.C. Strutt}, \textquotedblleft The
Multiplier and Capital Wealth\textquotedblright ,\textit{\ Journal of the
Royal Statistical Society}, 1915, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
78(4), pp.555-599

\noindent \textbf{Laitner, John}, ``Bequest Behaviour and the National
Distribution of Wealth", \textit{Review of Economic Studies}, 46(3), 1979,
pp.467-483.

\noindent \textbf{Laitner, John}, ``Secular Changes in Wealth Inequality and
Inheritance", \textit{Economic Journal}, 111(474), 2001, pp.691-721.

\noindent \textbf{Lindert, Peter}, "Unequal English Wealth Since 1688", 
\textit{Journal of Political Economy}, 1986

\noindent \textbf{Maddison, Angus,} ``Historical Statistics of the World
Economy: 1-2008 AD," updated tables, 2010 (www.ggcc.net/maddison).

\noindent \textbf{Menchik, Paul L.} , "Primogeniture, Equal Sharing, and the
U.S. Distribution of Wealth", \textit{Quarterly Journal of Economics} 94(2),
1980, 299-316.

\noindent \textbf{Modigliani, F.}, \textquotedblleft Life Cycle, Individual
Thrift and the Wealth of Nations\textquotedblright , \textit{American
Economic Review}, 1986, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
76(3), pp.297-313

\noindent \textbf{Modigliani, F.}, ``The Role of Intergenerational Transfers
and Lifecyle Savings in the Accumulation of Wealth'' , \textit{Journal of
Economic Perspectives}, 1988, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
2(2), pp.15-40

\noindent \textbf{Morigushi C. and E. Saez} , "The Evolution of Income
Concentration in Japan, 1886-2005: Evidence from Income Tax Statistics", 
\textit{Review of Economics and Statistics} 90(4), 2008, 713-734.

\noindent \textbf{Nirei, Makoto ,} \textquotedblleft Pareto Distributions in
Economic Growth Models", working paper, Hitotsubashi University, 2009

\noindent \textbf{Ohlsson, H., J. Roine, D. Waldenstrom}, "Inherited Wealth
Over the Path of Development: Sweden 1810-2010", mimeo, 2013

\noindent \textbf{Petty, William, }\textit{Verbum Sapienti}, 1664, 26p.
Published as an addendum to W. Petty, \textit{The Political Anatomy of
Ireland}, London: 1691, 266p. Reprinted in \textit{The Economic Writings of
Sir William Petty,} C.H. Hull ed., 2 vol., Cambridge University Press, 1989.

\noindent \textbf{Piketty, Thomas,} \textquotedblleft On the Long-Run
Evolution of Inheritance: France 1820-2050,\textquotedblright \  \textit{%
Quarterly Journal of Economics}, 2011, vol.126(3), pp.1071-1131.

\noindent \textbf{Piketty, Thomas,} \textit{Capital in the 21st Century},
Harvard University Press, 2014

\noindent \textbf{Piketty, Thomas, Gilles Postel-Vinay and Jean-Laurent
Rosenthal,} \textquotedblleft Wealth Concentration in a Developing Economy:
Paris and France, 1807-1994\textquotedblright , \textit{American Economic
Review}, 2006, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
96(1), pp.236-256

\noindent \textbf{Piketty, Thomas, Gilles Postel-Vinay and Jean-Laurent
Rosenthal,} \textquotedblleft Inherited vs. Self-Made Wealth: Theory and
Evidence from a Rentier Society (1872-1927)", \textit{Explorations in
Economic History}, 2013.

\noindent \textbf{Piketty, Thomas and Emmanuel Saez,} \textquotedblleft A
Theory of Optimal Inheritance Taxation,\textquotedblright \  \textit{%
Econometrica,} 2013, vol.81(5), pp.1851-1886

\noindent \textbf{Piketty, Thomas and Gabriel Zucman,} \textquotedblleft
Capital is Back: Wealth-Income Ratios in Rich Countries
1700-2010,\textquotedblright \ CEPR Working Paper, 2013.

\noindent \textbf{Piketty, Thomas and Gabriel Zucman,} \textquotedblleft
Capital is Back: Wealth-Income Ratios in Rich Countries
1700-2010,\textquotedblright \ 2014, \textit{Quarterly Journal of Economics}%
, forthcoming.

\noindent \textbf{Roine, J. and D. Waldenstrom}, \textquotedblleft Wealth
Concentration over the Path of Development: Sweden,
1873-2006\textquotedblright , \textit{Scandinavian Journal of Economics},
2009, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
111, pp.151-187

\noindent \textbf{Saez, Emmanuel and Gabriel Zucman} ``The Distribution of
Wealth, Capital Income, and Rates of Returns in the U.S. Since 1913'', 2014,
working paper.

\noindent \textbf{Schinke, Christoph,} \textquotedblleft Inheritance in
Germany 1911 to 2009,\textquotedblright \ 2013, PSE Working Paper.

\noindent \textbf{S\'{e}aill\`{e}s, J.}, La r\'{e}partition des fortunes en
France, Editions Felix Alcan, 1910, 143p.

\noindent \textbf{Stiglitz, J.E.}, ``Distribution of Income and Wealth Among
Individuals", \textit{Econometrica}, 1969, vol.37(3), pp. 382-397

\noindent \textbf{Stamp, J.C.}, \textquotedblleft The Wealth and Income of
the Chief Powers\textquotedblright , \textit{Journal of the Royal
Statistical Society}, 1919, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
82(4), pp.441-507

\noindent \textbf{Stewart, Charles}, ``Income Capitalization as a Method of
Estimating the Distribution of Wealth by Size Group", in \textit{Studies in
Income and Wealth}, New York: National Bureau of Economic Research, 1939,
vol. 3.

\noindent \textbf{Strutt, H.C.}, \textquotedblleft Notes on the Distribution
of Estates in France and the United Kingdom\textquotedblright , \textit{%
Journal of the Royal Statistical Society}, 1910, n%
%TCIMACRO{\U{b0}}%
%BeginExpansion
${{}^\circ}$%
%EndExpansion
73(6), pp.634-644

\noindent \textbf{Vauban, S\'{e}bastien, }\textit{Projet d'une dixme royale,}%
\textbf{\ }Paris, 1707, 267p. Reprinted in E. Daire, \textit{Economistes
financiers du 18}$^{\mathit{e}}$\textit{\ si\`{e}cle}, Paris: Guillaumin,
1843.

\noindent \textbf{Vaughan, R.N.,} \textquotedblleft Class Behaviour and the
Distribution of Wealth \textquotedblright , \textit{Review of Economic
Studies}, 46(3), 1979, pp.447-465.

\noindent \textbf{Waldenstrom, Daniel}, ``Lifting All Boats? The Evolution
of Income and Wealth Inequality over the Path of Development", PhD
Dissertation, Lunds universitet, 2009.

\noindent \textbf{Wolff, Edward and Maury Gittleman}. \textquotedblleft
Inheritances and the Distribution of Wealth Or Whatever Happened to the
Great Inheritance Boom?\textquotedblright , \textit{Journal of Economic
Inequality}, 2013.

\noindent \textbf{Zucman, Gabriel,} \textquotedblleft The Missing Wealth of
Nations: Are Europe and the U.S. Net Debtors or Net Creditors?", \textit{%
Quarterly Journal of Economics}, 128(3), 2013, pp.1321-1364.

\noindent \textbf{Zucman, Gabriel,} \textquotedblleft Tax Evasion on Offshore Profits and Wealth", \textit{Journal of Economic Perspectives}, 2014, forthcoming.

\includepdf[pages={-}, landscape=true]{PikettyZucman2014HIDFigures.pdf} 
\includepdf[pages={-}]{PikettyZucman2014HIDTable.pdf}

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