Making Sense of Piketty’s ‘Fundamental Laws’ in a Post-Keynesian Framework: The Transitional Dynamics of Wealth Inequality
Stefan
Ederer
Miriam
Rehm
AC15183353
181
MATERIALIEN ZU WIRTSCHAFT UND GESELLSCHAFT978-3-7063-0747-5
WORKING PAPER-REIHE
DER AK WIEN
MAKING SENSE OF PIKETTY’S ‘FUNDAMENTAL LAWS’
IN A POST-KEYNESIAN FRAMEWORK:
THE TRANSITIONAL DYNAMICS OF WEALTH INEQUALITY
Stefan Ederer
Miriam Rehm
WPR_181_MakingSenseofPickettys.indd 1 28.09.18 12:03
Materialien zu Wirtschaft
und Gesellschaft Nr. 181
Working Paper-Reihe der AK Wien
Herausgegeben von der Abteilung Wirtschaftswissenschaft und Statistik
der Kammer für Arbeiter und Angestellte
für Wien
Making Sense of Piketty’s ‘Fundamental Laws’ in a Post-
Keynesian Framework: The Transitional Dynamics of
Wealth Inequality
Stefan Ederer, Miriam Rehm
Oktober 2018
Die in den Materialien zu Wirtschaft und Gesellschaft
veröffentlichten Artikel geben nicht unbedingt die
Meinung der AK wieder.
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Abstract
If Piketty’s main theoretical prediction (? > ? leads to rising wealth inequality) is taken to its
radical conclusion, then a small elite will own all wealth if capitalism is left to its own devices.
We formulate and calibrate a Post-Keynesian model with an endogenous distribution of
wealth between workers and capitalists which permits such a corner solution of all wealth
held by capitalists. However, it also shows interior solutions with a stable, non-zero wealth
share of workers, a stable wealth-to-income ratio, and a stable and positive gap between the
profit and the growth rate determined by the Cambridge equation. More importantly,
simulations show that the model conforms to Piketty’s empirical findings during a transitional
phase of increasing wealth inequality, which characterizes the current state of high-income
countries: The wealth share of capitalists rises to over 60%, the wealth-to-income ratio
increases, and income inequality rises. Finally, we show that the introduction of a wealth tax
as suggested by Piketty could neutralize this rise in wealth concentration predicted by our
model.
Zusammenfassung
Wenn Pikettys theoretische Hauptaussage (? > ? führt zu steigender Vermögens-
ungleichheit) radikal zu Ende gedacht wird, dann wird eine kleine Elite das gesamte Vermögen
besitzen wird, wenn diesen inhärenten Tendenzen des Kapitalismus nicht entgegengewirkt
wird. Wir formulieren und kalibrieren ein Post-Keynesianisches Modell mit einer endogenen
Vermögensverteilung zwischen ArbeitnehmerInnen und KapitalistInnen, das Pikettys
Ecklösung ermöglicht, in der KapitalistInnen das gesamte Vermögen besitzen. Es beinhaltet
aber auch andere Optionen, in denen ArbeitnehmerInnen einen stabilen, positiven
Vermögensanteil besitzen, das Vermögens-Einkommens-Verhältnis stabil ist, und eine stabile
und positive Lücke zwischen der Wachstums- und der Profitrate besteht, die durch die
Cambridge Gleichung bestimmt wird. Vor allem zeigen Simulationen, dass das Modell in einer
Übergangsphase Pikettys empirische Resultate repliziert: Der Vermögensanteil der
KapitalistInnen steigt auf über 60%, das Vermögens-Einkommens-Verhältnis steigt an, und die
Einkommensungleichheit ebenso. Schließlich zeigen wir, dass die Einführung einer
Vermögenssteuer, wie von Piketty vorgeschlagen, diesen Anstieg der
Vermögenskonzentration neutralisieren könnte.
Keywords: Post-Keynesian, model, wealth, saving, inequality, Piketty, simulation
JEL: C63, D31, E12, E21
Inhalt:
1. Introduction 1
2. Piketty and the Post-Keynesians 2
3. A Model of the Wealth Distribution 5
4. Transitional dynamics 11
5. Model extensions 12
6. Simulating the dynamics 14
7. The effects of a wealth tax 17
8. Conclusion 19
9. Bibliography 21
10. Appendix 24
1
1. Introduction
Thomas Piketty’s best-selling book ‘Capital in the 21st Century’ (Piketty 2014) triggered a
renewed interest in empirical research regarding the accumulation and distribution of wealth,
and a lively debate about their causes and consequences. Wealth determines income, power
and opportunities, and lies at the very heart of economic inequalities. Understanding the
dynamics of wealth accumulation and distribution is thus crucial to tackle these inequalities.
In a nutshell, Piketty’s (2014) theoretical argument is that, since the profit rate is usually higher
than the growth rate in an economy (an empirical regularity which he finds for most countries
and time periods for which his detailed archival work provides data), wealth increases over
time faster than income. This entails a more unequal distribution of income, because the share
of profits increases and wealth ownership and capital income are more concentrated than
labour income. A rising income inequality finally feeds back into a more unequal distribution
of wealth, so that wealth will be ever-increasingly concentrated in the hands of a small elite.
Piketty (2014) nevertheless carefully balances such a radical interpretation of his analysis with
theoretical counter tendencies historical analysis, and empirical work.
Empirically, Piketty (2014) provides extensive data on the historical evolution of wealth-to-
income ratios, wealth, and the personal income distribution. He shows that the wealth-to-
income ratio has risen, and that wealth and income have become more unequally distributed
in high-income countries since about the 1980s. Regarding the profit rate and the growth rate,
he argues that they have been largely stable over the long run, but that the former is
empirically higher than the latter.
The reception of the book in Post-Keynesian economics has been mixed. On the one hand,
Post-Keynesian economists recognize the empirical contributions of the book: the collection
of historical data and the carving out of observable patterns therein (Rowthorn 2014, Rehm
and Schnetzer 2015, King 2017). On the other hand, Piketty’s neoclassical theoretical
framework by which he explains the dynamics of wealth and income inequality has attracted
the criticism of Post-Keynesian economists, in whose theoretical frameworks distribution has
long played a major role (e.g. Galbraith 2014, Palley 2014, López-Bernardo et al. 2016a). Based
on the Cambridge equation (Pasinetti 1962), they point out that the wealth distribution in the
long run can be stable, a statement that is clearly in contradiction to a reading of Piketty that
takes his neoclassical theory to its radical logical conclusions.
Both Piketty and Post-Keynesians, however, consider the transition phase – the world in which
this and the next generation lives – as the relevant reference point for economic analysis. The
development of income and wealth inequality over the next decades matters. The goal of this
paper is to show that Post-Keynesian theory is capable of explaining this short-run dynamic of
wealth accumulation and distribution, for which Piketty (2014) presents abundant empirical
evidence. In the ‘transitional phase’, i.e. when the wealth share of capitalists is below its long-
run equilibrium value, a rising wealth-to-income ratio and increasingly unequal distributions
of wealth and income can be described well by Post-Keynesian theory.
2
To do so, we build a Post-Keynesian model in the tradition of Bhaduri and Marglin (1990)
which incorporates an endogenous wealth distribution. We extend the model by blended
incomes of workers and capitalists, differential rates of return, and capital gains. We show
that a stable wealth share is a likely outcome in the long run, and both the euthanasia and the
triumph of the rentier are special cases, and thus reiterate the critique of Piketty’s hypothesis
of an ever-increasing wealth concentration. Furthermore, we use the model to explain a
‘transitional dynamic’ that resembles the empirical evidence presented by Piketty and his
projections to the future. A rising wealth-to-income ratio, rising wealth and income inequality
and a profit rate that is higher than the growth rate of the capital stock (and thus income) are
all consistent with our extended Post-Keynesian model.
Piketty, like many Post-Keynesians, is concerned with the factors that might disturb this long-
run capitalist path towards an equilibrium. He worries that the world of increasing inequality,
which his data describes, might be prone to upheavals, social unrest, and war. This is why
Piketty focuses on finding democratic, political solutions – such as a wealth tax – to the ever-
rising importance of inherited wealth over wealth acquired through work. We follow his
suggestion and implement a wealth tax in the model and show that this stabilizes both wealth
and income inequality.
The structure of the paper is as follows. The literature review in section 2 discusses both
Piketty and his Post-Keynesian critics, as well as the Post-Keynesian models of distribution.
Section 3 describes the model and its extensions in detail. Section 4 presents a numerical
simulation of both short-run dynamics and the long-run equilibrium. Section 5 discusses the
effects of a wealth tax. Section 6 concludes.
2. Piketty and the Post-Keynesians
Piketty (2014)’s theoretical framework, formally elaborated in Piketty and Zucman (2014),
consists of two fundamental laws and a fundamental contradiction of capitalism. Piketty’s first
law states that the share of income from capital in total national income is equal to the rate
of profit times the wealth-income ratio, or ? = ? × ?, an accounting identity (Piketty 2014:
52). The second law is the Harrod-Domar equation (Piketty and Zucman 2014: 1274) that the
ratio of wealth to income equals the ratio of the saving rate to the growth rate. In Piketty’s
denotation, ? = ? ?? (Piketty 2014: 166). Piketty’s fundamental contradiction argues that
wealth (which he treats synonymously to capital) increases faster than income if the profit
rate is higher than the growth rate of GDP, i.e. if the oft-cited formula ? > ? holds (Piketty
2014: 571). Piketty’s analysis is based on a neoclassical production function, it assumes that
the elasticity of substitution between labour and capital is higher than one (Piketty and
Zucman 2014: 1271), and the rate of profit is technologically determined (Piketty and Zucman
2014: 1270).
Empirically, Piketty’s work is ground-breaking in the breadth and depth of his coverage of both
wealth and income time series. In particular, he documents an increasingly unequal
3
distribution of wealth (Piketty 2014: 332f), rising wealth-to-income ratios (ibid.: 164f), and an
increasing concentration of the personal income distribution (ibid.: 290f) in Europe and the
U.S., roughly since the 1980s. He also documents a slowing growth rate (ibid.: 93f) and a
declining rate of return (ibid: 199f) over that time horizon, even though he sees both as
roughly stable over the long run. Crucially for his argument, he finds his ‘fundamental
contradiction’ of capitalism to be empirically valid for most economies and historical periods,
i.e. the profit rate to be higher than the growth rate (ibid: 350f). Piketty sees a number of
positive feedback mechanisms, which lead to an ever-increasing inequality in the distribution
of income and wealth. Barring political intervention and notwithstanding his detailed
discussion of institutional and systemic factors in the wealth distribution, following this train
of thought to its logical conclusion means that Piketty’s main prediction is that all wealth will
be concentrated in the hand of a very small elite.
The Post-Keynesian and structuralist reception of Piketty’s work has been largely sceptical.
King’s (2016) comprehensive review summarizes the arguments (of neoclassical, Austrian and
Post-Keynesian economists) in nine over-arching points of critique. These range from
differences in opinion regarding the relevance of wealth inequality and the suggested policy
conclusions, but also empirical disagreements and theoretical critiques, the latter including
the importance of institutions and of low-income countries. Relevant for this paper are the
two critiques that 1) “Piketty uses the wrong (neoclassical) theory” (King 2016: 7) and 2) that
the “predicted increase in wealth is implausible” (King 2016: 3).
Regarding the first point, many Post-Keynesian and structuralist economists criticized Piketty
for using a neoclassical framework. Piketty (2014: 230f) misrepresents the Cambridge capital
controversy of the 1950s and 1960s (Galbraith, 2014; Palley, 2014; López-Bernardo et al.
2016a), which showed that neoclassical economics is logically inconsistent in its valuation of
capital since the value of capital is simultaneously determined by and determines the rate of
profit (Harcourt, 1972; Felipe and McCombie, 2013). Connected to this is the critique of
Piketty’s conflation of wealth and capital. In particular, Post-Keynesians and Marxists objected
to treating housing wealth as productive capital, measuring wealth at highly volatile market
prices untethered from ‘fundamental’ values (Duménil and Levy 2014, Rowthorn, 2014).1
The bulk of Post-Keynesian criticism was directed at the second point (López-Bernardo et al.,
2016; Michl, 2014; Rowthorn, 2014; Taylor, 2014; van Treek, 2015). As discussed, from
Piketty’s fundamental laws and in particular the inequality ? > ? follows a long-run dynamics
of wealth accumulation that leads to an ever-increasing share of wealth concentrated in the
hands of capitalists. Treating the variables as independent, van Treek (2015) numerically
illustrates that the profit rate exceeding the growth rate is not a sufficient condition for the
wealth-to-income ratio and inequality continue to rise indefinitely. Differential saving rates of
households with different positions in the income distribution are a necessary precondition
1 We briefly address this ‘valuation problem’ in the model section, and take the larger point regarding productive
(rather than total) capital on board in the formulation of our model.
4
for Piketty’s conclusion to hold, and the wealth distribution is highly sensitive to changes in
these differential saving rates.
Based on Pasinetti’s Cambridge equation ??? = ? (i.e., the equilibrium growth rate equals the
saving rate of capitalists times the profit rate), Taylor (2014) argues that the two corner
solutions euthanasia and the triumph of the rentier, as well as an interior solution, are all
possible results. He focuses on the interaction between the wealth share of capitalists,
capacity utilization and the profit share, and points out that the feedback mechanisms
between these variables can stabilize or destabilize the system, depending on the structure of
the economy. He shows that a rising wealth share entails the possibility of a rising profit share
and chronic underutilisation of production capacities and stagnation in the long run, if the
profit share responds strongly positively to an increase in the wealth share.
López-Bernardo et al. (2016a) also focus on the long run and the Cambridge equation. They
argue that the profit rate must necessarily always be greater than the growth rate, provided
that capitalists’ savings rate is less than unity (that is, they consume a certain share of their
income). Piketty’s empirical finding that r is greater than g is thus fully consistent with a stable
income distribution and, by extension, a stable wealth distribution. Only if the profit rate
exceeds the ratio of the growth rate and the saving rate of capitalists will there be
redistribution from wages to profits, and thus a rising profit share and ultimately, a rising
concentration of wealth. Furthermore, although individual capitalists can increase their rate
of accumulation by saving more, capitalists as a whole cannot, because an increase in the
saving rate would inevitably entail a fall in the profit rate. López-Bernardo et al. (2016a) argue
that Piketty overlooked this ‘fallacy of composition’, even though he explicitly discusses (but
dismisses by referring to technology) its relevance (Piketty 2014: 215f). However, they point
out that Piketty’s cumulative feedback mechanism between accumulation and wealth
concentration is indeed valid when it comes to personal income distribution, in particular
among capitalists, because they have no access to wage income in a simple formulation.2
Post-Keynesian models are well suited for the analysis of both short- and long-run dynamics
of wealth accumulation and distribution. They have long been used to investigate the
distribution between labour and capital in the wage-/profit-led debate (Bhaduri and Marglin
1990, Stockhammer and Ederer 2008, Stockhammer et al. 2009, Lavoie and Stockhammer
2013, Barbosa and Taylor 2006, Kiefer and Rada 2015) and the distribution of personal income
(Lavoie 2009, Carvalho and Rezai 2016, Palley 2017a). It is therefore indeed vexing that
Piketty’s writing shows no reception of the extensive Post-Keynesian body of literature on the
theory of distributions.
Two seminal Post-Keynesian models of the distribution of wealth are Palley (2012, 2017b) and
Taylor et al. (2015). In Taylor et al.’s (2015) model, workers receive blended wage and profit
income. The model incorporates capital gains due to its empirical relevance in the US. Taylor
2 Meade (1964) points out this disadvantage. Pasinetti (1974) argues that the thriftiest group among capitalists
will eventually dominate the others, as noted by Taylor (2014). We return to this point in the following section.
5
et al. (2015) simulate a wealth concentration ratio of about 60% - the top 1% owns roughly
60% of total wealth. Wealth is split between “the rich” and the “middle class-workers”, as the
“bottom 60%” empirically do not own any wealth. Palley’s (2017b) model also includes an
endogenous wealth distribution. The paper permits doubly blended income sources – both
workers and capitalists receive both work and profit income in a Post-Keynesian framework.
The parameters influencing the personal income distribution (and growth) are the functional
income distribution, the distribution of the wage bill between workers and capitalists, and the
distribution of wealth between workers and capitalists. The latter, in turn, depends on the
differential propensities to save.
Post-Keynesian models have paid less attention to the transitory dynamics for the phase
during which the wealth share (and, indeed, the functional and personal income distribution)
differ from their (long-run) equilibria. The differential equations often used by the profit-led
strand of the literature (and their concomitant VAR analyses) lend themselves to investigating
these transition phases. Indeed, Taylor (2014) derives a differential equation for the wealth
share. However, the paper then focuses on stable and instable equilibria, while pointing to
possible feedback mechanisms between the distribution of wealth with the profit share and
capacity utilization.
This paper extends the model of Ederer and Rehm (2017), which follows the Post-Keynesian
tradition along the lines of Palley (2017b) and Taylor et al. (2015) with an endogenous wealth
distribution in a two-class economy. The question we are asking is, whether Piketty’s empirical
evidence regarding a rising wealth share for decades can be integrated into a Post-Keynesian
model, which permits for interior solutions for the wealth distribution. The main aim is to
investigate the transition phase between short- and long-run dynamics of wealth
accumulation and distribution, and thereby contribute to closing this gap in the Post-
Keynesian literature of growth models.
3. A Model of the Wealth Distribution
The model is a standard two-class, Post-Keynesian formulation in the tradition of Bhaduri and
Marglin (1990), drawing on Palley (2017b) and Taylor et al. (2015) and closely following Ederer
and Rehm (2017).3 We introduce four innovations: (1) Wealth is accumulated through saving;
(2) blended wage and capital income goes to both workers and capitalists; (3) beyond
differential savings rates, workers and capitalists also have differential returns on their assets;
(4) firms save and accumulate wealth, which is passed on to owners of shares via capital gains.
Section 3a introduces a basic version of the model, and section 3b includes extensions (2) to
(4).
3 The model is briefly restated here for the convenience of the reader, since we discuss the solution and the
relevance of our analytical assumptions relating to Piketty’s theoretical and empirical definitions in detail.
6
a. Basic model
In the basic formulation of the model, income Y is divided between total profits R and the
wage bill W according to the (exogenous) functional income distribution ? (the profit share).
? = ?? (1)
? = (1 ? ?)? (2)
All wages accrue to workers in the simple version of the model. Income of capitalists (subscript
r) thus amounts to profits R on their share of productive wealth z. Workers (denoted by
subscript w) also receive a part of profits R proportional to their share of wealth ownership (1
– z), which together with wages make up total income of workers ??.
?? = ?? (3)
?? = ? + (1 ? ?)? (4)
We follow Post-Keynesian convention by assuming a positive differential between savings
rates of capitalists ?? and workers ??. Per definition, consumption propensities of workers and
capitalists multiply with their respective incomes for total consumption C.
? = (1 ? ??)?? + (1 ? ??)?? (5)
The investment equation is formulated according to the standard Post-Keynesian functional
form in the Bhaduri-Marglin tradition, i.e. growth of the capital stock K depends on capacity
utilization u and the profit share ?. This formulation allows for both wage-led and profit-led
demand growth regimes, depending on the values of the parameters ?1 and ?2 and the saving
rates.
? = (?0 + ?1? + ?2?)? (6)
The aggregate goods market is in equilibrium, output equals demand. Since we abstract from
all sectors other than households and firms, total demand consists of consumption and
investment.4
? = ? + ? (7)
In order to de-trend income, profits, and investment, we follow convention by normalizing
them to the capital stock. This yields stable solutions for capacity utilization u, the profit rate
r and the growth rate of the capital stock g.
? =
?
?
(8)
? =
?
?
= ?? (9)
? =
?
?
(10)
4 The abstraction from government and an external sector is in line with Piketty (2014), who in much of his work
omits them on empirical grounds following careful examination of their importance and valuation problems.
7
The only asset in the model is productive wealth V owned by households, which is equal to
the capital stock K:
? = ? (11)
The ownership of (productive) capital entitles to the receipt of the corresponding share in
profits. Both workers and capitalists accumulate wealth through savings. The level of savings
is the difference between income and consumption.
In focusing on productive capital, we take an important strand of criticism of Piketty’s
neoclassical basis, discussed in section 2, on board, even though we appreciate Piketty’s
argument that wealth categories are to a certain degree fungible. Departing from total wealth
as measured in microeconomic surveys introduces a potential wedge between Piketty’s and
our empirical results, since the critiques were not solely based on theoretical arguments
regarding unproductive capital, but also revolved around a rising empirical importance of
housing wealth as a share of total private wealth (Duménil and Levy, 2014). Our model is
conceptually based on capital (rather than wealth), since it is adapted from a framework which
focuses on productive investment and growth. Because productive wealth is distributed more
unequally (i.e., housing is distributed more equally), our simulations are on the conservative
side as we are more likely to find Piketty’s extreme inequality.
In order to trace the behaviour of the model compared to Piketty’s empirical evidence, we
include the above-mentioned wealth share of capitalists ?, the wealth-to-income ratio ?
(which is the inverse of capacity utilization u), and the ratio of capitalists’ to workers’ income,
as a measure for the personal class-based income distribution ? in the model.
? =
??
?
(12)
? =
?
?
=
1
?
(13)
? =
??
??
(14)
8
Table 1: Stocks and flows in the Post-Keynesian model with endogenous wealth distribution
Households Firms
Workers Capitalists Current Capital Total
Consumption ??? ??? +? 0
Investment +? ?? 0
Wages +? ?? 0
Profits +?? +?? ?? 0
Wealth ???? ???? +?? 0
Total 0 0 0 0 0
Source: own elaboration.
Table 1 shows the stock-flow matrix of the model. A plus sign denotes a source of funds, a
minus sign is a use of funds, and rows and columns sum to zero. Firms pay wage income to
workers (row 3), and distribute profits (row 4), which both workers and capitalists receive.
Workers and capitalists consume their income (row 1), and firms invest (row 2). Both workers
and capitalists save and thus accumulate wealth in the form of productive capital (row 5).
Households’ savings finance firms’ investment (column 4).
b. Short-run equilibrium
The short-run solution of the model assumes wealth shares to be constant, since wealth
accumulates over long time periods. Capacity utilization, the profit rate and the growth rate
adjust simultaneously, so that the equilibrium in the aggregate goods market (i.e. the IS-
condition) is fulfilled for any wealth share of capitalists z. The short-run solutions for these
three variables thus depend on the distribution of wealth:
?? =
?0 + ?2?
?? + (?? ? ??)?? ? ?1
(15)
?? =
?0? + ?2?
2
?? + (?? ? ??)?? ? ?1
(16)
?? =
(?0 + ?2?)[?? + (?? ? ??)?? ]
?? + (?? ? ??)?? ? ?1
(17)
In particular, a higher wealth share of capitalists lowers capacity utilization, the profit rate and
the growth rate of the capital stock. The reason is that a higher wealth share of capitalists
transfers profit income to capitalists, which depresses total consumption due to capitalists’
higher saving rate. The effect of the wealth distribution on these variables hinges on the
differential saving rates.
Contrary to a rise in the profit share, which can either raise or reduce the growth rate
(depending on whether growth is wage-led or profit-led), a rise in the wealth share
unambiguously depresses growth, since consumption decreases and there is no counteracting
9
effect on investment. The latter is determined by the profitability of firms, not by the
distribution of profits among workers and capitalists. Note that the profit rate is more sensible
to the wealth share than the growth rate, because its effect on the latter works indirectly via
capacity utilization. This is the reason why the profit rate decreases faster than the growth
rate when the wealth share rises, and the two variables will eventually fulfil the Cambridge
equation (see below).
Since the wealth-to-income ratio x is the inverse of capacity utilisation, it increases the more
wealth is concentrated in the hands of capitalists. Likewise, the personal income distribution
as measured by the ratio of income of capitalists to the income of workers, ?, is more unequal
when the wealth share of capitalists is high, because a higher share of profits accrues to
capitalists in that case.
?? =
?? + (?? ? ??)?? ? ?1
?0 + ?2?
(18)
?? =
??
1 ? ??
(19)
c. Long-term equilibrium
Over time, both capitalists and workers accumulate wealth until the wealth share adjusts to
its long-term equilibrium. Pasinetti’s (1962) Cambridge equation ??? = ? implies that the
wealth share is stable when capitalists save exactly the amount that corresponds to their share
in the increase in total capital (Palley, 2012, 2017b; Taylor, 2014). Consequently, Palley
(2017b) points out that the Cambridge equation should be interpreted as an ‘ownership
equilibrium condition’, since capitalists must save just enough to maintain their ownership
share.
Since both the profit rate and the growth rate depend on the wealth share of capitalists z, we
get a long-run equilibrium value for the wealth distribution:
??? =
??? ? ??
(?? ? ??)?
(20)
As long as difference between the saving rates of workers and capitalists is sufficiently high,
the equilibrium value for z is positive. The (long-run) distribution of wealth only depends on
the differential saving rates and the profit share. Capitalists’ long-run equilibrium wealth share
z** is higher: (1) the higher the profit share, (2) the higher the saving rate of capitalists, and
(3) the lower the saving rate of workers. It does however not depend on the growth rate (i.e.
on the parameters of the investment equation), which contradicts Piketty’s argument. In the
long-run equilibrium, in our model the wealth share is constant, so each class has to save the
exact amount that keeps their wealth share constant, independent of how slowly or how fast
the economy grows.
10
There are two corner solutions for the wealth share: (1) If workers do not save at all, all wealth
will (naturally) be concentrated in the hand of capitalists (? = 1). In this case, the share of
total income going to profits is the only determinant of the distribution of income. (2) If
capitalists do not save, it is self-evident that eventually all wealth will belong to workers (? =
0). Furthermore, if the two saving rates are equal (?? = ??), the model has no meaningful
solution for z. All wealth would be concentrated in the hand of workers, if capitalists have no
access to wage income.5
The long-run solution for capacity utilisation, the profit rate and the growth rate are:
??? =
?0 + ?2?
??? ? ?1
(21)
??? =
(?0 + ?2?)?
??? ? ?1
(22)
??? =
??(?0 + ?2?)?
??? ? ?1
(23)
Note that they are now independent from workers’ saving rate. The Kaleckian principle, that
‘capitalists earn what they spend’ (Kalecki, 1971) applies here in a broader sense, i.e. that a
higher saving rate of capitalists would diminish capacity utilisation, the profit rate and the
growth rate in equilibrium.
The long-run solutions for the wealth-to-income ratio and personal income distribution are
??? =
??? ? ?1
?0 + ?2?
(24)
??? =
??? ? ??
??(1 ? ?)
(25)
The personal income distribution also does not depend on the growth rate, but only on the
profit share and the two saving rates. The wealth-to-income ratio on the other hand is higher,
the lower the growth parameter ?0. This is in line with Piketty’s argument that slow economic
growth would raise x. Nevertheless, it has no impact on the distribution of wealth.
For the wealth-to-income ratio, Piketty’s second fundamental ‘law’ (which they refer to as an
accounting identity, Piketty and Zucman (2014: 1274) and call the “Harrod-Domar-Solow
formula” (Piketty and Zucman 2014: 1257)) is fulfilled:
??? =
?
?
(26)
where ? = ??[(1 ? ?) + ?(1 ? ?)] + ???? is the aggregate saving rate of the economy.
6 In
contrast to Piketty, however, as noted above, in our model the aggregate saving rate and the
growth rate adjust endogenously through the change in the wealth share until they reach the
long-run equilibrium.
5 Meade (1964) points this out, see also Taylor (2014).
6 The mathematical proof is available upon request.
11
In the Post-Keynesian model, the wealth-to-income ratio and the personal income distribution
are thus determined simultaneously with all the other (short-term) variables for any wealth
share z. This is in contrast to Piketty’s model where the wealth-to-income ratio is the result of
the developments of the (constant) profit rate and the (constant) growth rate over time.
It should be mentioned that one of Piketty’s main theoretical repercussion of a rising wealth-
to-income ratio is that the profit share falls. In the Post-Keynesian model, in contrast, we
follow convention and treat the profit share as exogenous. In contrast to Piketty’ model, it is
the profit rate that varies according to changes in capacity utilisation. For Post-Keynesians, the
profit share is determined by the power of workers and capitalists, which has significantly
shifted in the direction of the latter since the 1980s. The deregulation of trade and capital
flows as well as of financial markets and institutional changes has brought labour in the
defensive; the result was a secular fall of the profit share.
Nevertheless, the profit share can also be endogenously determined within the model. Taylor
(2014) briefly discusses the stabilising and destabilising mechanisms in such a model. The
outcome however is ambiguous. While the profit share in such a model increases along with
a rising wealth share, its long-run stability depends on the parameters of the model. For a
wide range of values, a stable long-run equilibrium wealth share is very likely, so that the
results of the abovementioned analysis remain valid. However, there is also the possibility of
an ‘explosive trajectory’ where at its end all wealth is concentrated in the hand of capitalists.
This would obviously validate Piketty (2014)’s predictions. To examine the dynamics of such a
model is however beyond the scope of this paper.
4. Transitional dynamics
Another way to derive the Cambridge equation is to take the derivatives of the wealth share
z with respect to time and rewrite the differential equation (recall that ? = ?? ?? ):
??? =
????? ? ?????
?2
= (
????
??
?
???
?
)
??
?
= (??? ? ?)? (27)
It immediately follows that the wealth share is only stable if the Cambridge equation is
fulfilled.7 Furthermore, it is obvious that the wealth share z rises if ??? > ?. If this inequality
holds, the (percentage) increase in capitalists’ wealth is higher than the (percentage) increase
in total wealth. Piketty’s famous ‘fundamental contradiction of capitalism’ ? > ? would thus
be a special case of this inequality when ?? = 1, i.e. when capitalists accumulate all their
income.8 The only (albeit very important) difference is, that in the Post-Keynesian model the
profit rate and the growth rate are not exogenous and constant, as in Piketty’s model, but are
7 Another (trivial) solution of the differential equation is ? = 0.
8 Piketty (2014, p. 26, emphasis added) writes that „…when the rate of return significantly exceeds the growth
rate of the economy…“, which can be interpreted along the lines of the abovementioned inequality (López-
Bernardo et al., 2016).
12
both endogenously determined and adjust simultaneously until the long-run equilibrium is
reached.
From the short-run solutions for r* and g* we see that both the profit rate and the growth
rate decrease unambiguously when the wealth share rises. A higher wealth share reduces
aggregate demand and thus also the profit rate and the growth rate. As long as ? < ???, both
rates are higher than their long-run equilibria, i.e. ?? > ??? and ?? > ???. For low wealth
shares, the increase in capitalists’ wealth is higher than the one in total wealth, and the
inequality ??? > ? holds. Consequently, capitalists’ wealth share rises.
Piketty’s ‘fundamental contradiction’ of capitalism, that the wealth concentration increases if
and because ? > ?, is therefore valid in the Post-Keynesian model when z is below its long-
term equilibrium. If r is “significantly greater” than g, the wealth share of capitalists will rise.
Nevertheless, in contrast to Piketty’s view, this can only be a temporary situation. When the
wealth share reaches its equilibrium value, the Cambridge equation is fulfilled and the wealth
share remains constant.
The profit rate and the growth rate are not the only variables that depend on the wealth share,
as discussed in section 3.b. As long as z goes up, capacity utilisation decreases. Furthermore,
the wealth-to-income ratio, which is the inverse of the former, rises. The personal income
distribution becomes more unequal, since a higher wealth share shifts profits from workers
to capitalists.
In the ‘transitional phase’, i.e. as long as the wealth share is below its long-run equilibrium,
the Post-Keynesian model thus predicts a development of the variables in line with Piketty’s
(2014) empirical data. Both wealth and income distributions become more unequal, the
wealth-to-income ratio increases, and economic growth weakens. However, the mechanisms
behind these developments differ from Piketty’s, and not only will the wealth share eventually
reach its stable long-run equilibrium, but all variables are determined within the model and
adjust simultaneously.
Finally, let us briefly look at the effects of a slowdown in growth. In Piketty’s view (2014: 233),
a lower growth rate raises the wealth-to-income ratio, which entails a more unequal income
distribution. In the Post-Keynesian model, a fall in the growth rate (which is represented by a
lower value for ?0), reduces both the long-run profit rate and the long-run growth rate, so
that the Cambridge equation is unaffected. It has thus no effect on the distribution of wealth
and income. However, it reduces capacity utilisation and raises the wealth-to-income ratio. In
short, while a higher wealth share entails a reduction in the growth rate (due to changes in
the saving rates or the profit share), the reverse is not true.
5. Model extensions
The simple version of the model above is analytically tractable, but it arguably does not yet
capture the spirit of Piketty’s many-faceted analysis. This section therefore discusses three
13
possible extensions to the simple model, which nonetheless retains the Post-Keynesian
properties that were derived in the previous section. We introduce (1) blended income for
capitalists, (2) differential rates of returns across workers and capitalists, and (3) capital gains.
First, in the basic version of the model, all wages accrued to workers. However, Piketty
emphasizes that wages play an important (if diminishing) role even at the very top of the
income distribution. Empirically, the share of wages received by capitalists lies between 5 and
10 per cent for the US and most EU-countries (Taylor et al. 2015, Ederer and Rehm 2017). In
the vein of Palley (2017b), the model can thus be extended by blending not just workers’, but
also capitalists’ income sources, i.e. distributing wages between workers and capitalists. As
Ederer and Rehm (2017) show, the personal income distribution is then more skewed towards
capitalists than in the simple case since workers now receive less (wage) income.
Second, Piketty points to differential rates of return – the higher the wealth owned, the higher
the returns on this wealth (Piketty 2014: 447f). The reasons for differential returns across the
wealth distribution might lie, among others, in more professional wealth management at
higher wealth levels, the ability to take higher risk, or a higher likelihood of insider knowledge.
Empirical analysis (Ederer and Rehm 2017) finds that the composition of wealth varies
between workers and capitalists in particular, with the former holding a larger share of their
wealth in low-yield asset classes (in particular bank deposits). The implication for the
distribution of profits is that capitalists receive higher capital income and thus benefit more
from the compound interest effect. We thus distinguish between two asset types within
productive wealth: deposits, which we assume for simplicity to be non-interest bearing, and
profit-generating assets, which yield profit income.
Third, capital gains are an important avenue through which retained earnings of firms are
distributed to the owners. Even though Piketty does not emphasize them in his theoretical
considerations, his empirical results show that they are highly relevant. These capitals gains
solely depend on the saving rate of firms and can be integrated into a Post-Keynesian model
(López-Bernardo et al., 2016b; Taylor et al., 2017). Since capitalists typically hold a larger share
of their wealth in profit-generating assets, a higher saving rate of firms skews the distribution
of wealth and income towards them. Furthermore, a higher saving rate of firms can be
expected to reduce demand, capacity utilisation and growth, since firms by definition have a
saving rate equal to one and therefore a higher saving rate than capitalists and workers.
With these extensions, the analytical solution of the model becomes more complicated than
in the basic version (see Appendix B). Capitalists’ wealth share now depends not only on the
saving rates of workers and capitalists and on the profit share, but furthermore on the
distribution of wages between workers and capitalists, on their respective shares of wealth
held as profit-generating assets, and on the saving rate of firms. The higher capitalists’ share
of the wage bill, the higher is their wealth share in the long run. The higher the share of
capitalists’ wealth held in the form of profit-generating assets, the higher is their wealth share
(the same holds for workers). The higher the saving rate of firms, the higher is the wealth
14
share of capitalists. This extended model nevertheless exhibits the same short- and long-run
dynamic as its basic version.
6. Simulating the dynamics
As section 3.d showed, even though the Post-Keynesian wealth model does not corroborate
Piketty’s theory of a corner solution for the wealth concentration, his empirical analysis is
consistent with a transitional phase during which the wealth share of capitalists is below its
long-term equilibrium. We therefore focus on the transitional dynamics next in order to
illustrate the behaviour of the extended model in this phase.
The model is calibrated using parameters from the empirical literature (Ederer and Rehm
2017). As discussed in section 3e, the relevant parameters are the saving rates of workers and
capitalists, the share of wealth held as profit-generating assets by workers and by capitalists,
the profit share, the distribution parameter for wages (between workers and capitalists) and
the saving rate of firms. For details on the parameter values and sources, see Table A1 in the
Appendix.
Figure 1 shows a benchmark simulation using the average parameter values of all ten
European countries9 for which data is available. It depicts the dynamics of the model for
capacity utilization ? in the top left panel and for the wealth-income ratio ? (which, as is clearly
visible in the graphic, is the inverse of capacity utilization) at the top right. On the bottom, it
shows the profit rate ? and the growth rate ? on the left, and wealth concentration ? and
income concentration ? (as above, measured as the ratio of capitalists wealth and income to
worker’s wealth and income, respectively).
9 These countries comprise Austria, Belgium, Cyprus, Spain, Finland, France, Greece, Malta, Portugal, and
Slovakia.
15
Figure 1: Short- and long-run dynamics of the Post-Keynesian wealth model for Europe
Source: own elaboration.
Note: This graph shows the dynamic in capacity utilisation ?, the wealth-to-income ratio ?,
the profit rate ?, the growth rate ?, capitalists’ wealth share ?, and the ratio of capitalists’
income to workers’ income (personal income distribution) ?. For parameter starting values,
see Appendix A.
There are three main findings: First, the profit rate is always higher than the growth rate.
However, the differential decreases over time – if slightly – as the wealth share rises until it
reaches an equilibrium. Second, the model clearly approaches an equilibrium in the long run;
the wealth-to-income ratio, wealth and income inequality do not rise indefinitely. Third,
however, during a transitional phase over the next 50 to 100 years, the model points to a
steep rise in these variables, and a concomitant fall in capacity utilization.
The first result confirms Piketty’s ‘first law’ of capitalism, ? > ?. This, however, holds by
definition in a Post-Keynesian model due to Pasinetti’s ‘Cambridge equation’, ??? = ? (except
for the unrealistic case in which capitalists do not consume, so that their saving rate is equal
to one). The direction of change, however, contrasts with Piketty, who suggests that the
differential between the two rates, if anything, will widen.
The second finding is in clear contradiction to Piketty’s view (2014: 361), namely that “[t]he
fact that the return on capital is distinctly and persistently greater than the growth rate is a
powerful force for a more unequal distribution.” Leaving aside historical contingencies and
caveats, which Piketty (2014: 361f) is careful to discuss and include in his theoretical
.19
.20
.20
.21
.21
.22
.22
25 50 75 100 125 150 175 200
u
4.4
4.5
4.6
4.7
4.8
4.9
5.0
5.1
25 50 75 100 125 150 175 200
x
.00
.02
.04
.06
.08
.10
25 50 75 100 125 150 175 200
r g
0.00
0.20
0.40
0.60
0.80
1.00
25 50 75 100 125 150 175 200
z theta
16
predicionts, Piketty’s bare-bones model predicts an ever-increasing wealth share, and thus
eventually a corner solution for the concentration of wealth, in which a tiny elite owns all
wealth. In contrast, our calibrated simulation confirms the theoretical finding from our
analytical model, namely that the wealth share held by capitalists usually stabilizes at an
interior solution. In the Post-Keynesian world, both workers and capitalists own a stable share
of wealth.
Our third finding bridges the gap between a Post-Keynesian analysis and Piketty’s empirical
findings, or, put differently, it provides an analytical underpinning to Piketty’s work. During
the transition phase from the current state towards the long-run equilibrium, all variables
exhibit the development described by Piketty: The share of wealth owned by capitalists, the
Post-Keynesian equivalent to Piketty’s ‘elite’ defined by percentiles of the wealth distribution,
rises unequivocally and by significant levels from under 50% to more than 60%. Similarly, the
wealth-to-income ratio increases to around 5. In addition, personal income inequality
increases, i.e. the ratio of capitalists’ income to workers’ income rises from just below 40% to
roughly 50%. As with wealth, Piketty measures these as income share of percentiles of the
income distribution. To conclude, these are the main messages of Piketty’s (2014) book: the
wealth-to-income ratio, the wealth concentration, and personal income inequality will all
increase, if capitalism is left to its own devices.
Figure 2 shows the effect of a rise in ‘animal spirits’, i.e. the exogenous component of the
investment function ?0. Piketty does not focus on the growth regime in too much detail, but
here, too, our findings from the calibrated model are in line with his broad predictions – and,
as expected, with Post-Keynesian models: If autonomous investment increases, the profit rate
and the growth rate rise, as does capacity utilization, so the wealth-to-income ratio falls.
However, the simulations also make clear that a change in autonomous investment has only
transient effects on the distribution of wealth. In the long-run, capitalists’ wealth share
approaches the same value as in the baseline simulation. The equilibrium value for the wealth
share does not depend on the growth rate since the profit rate also increases endogenously,
so that the Cambridge equation is fulfilled at the same value for z as before.
17
Figure 2: Effects of an increase in the growth rate on short- and long-run dynamics of the Post-
Keynesian wealth model in Europe
Source: own elaboration.
Note: This graph shows the dynamic in the profit rate ?, the growth rate ?, the wealth-to-
income ratio ?, and capitalists’ wealth share ? following an increase in autonomous
investment, compared to a baseline scenario. For parameter starting values, see Appendix A.
7. The effects of a wealth tax
Piketty’s (2014: 532) solution for his predicted increasing wealth concentration and rising
wealth-to-income ratio is a global wealth tax. Although Post-Keynesian criticize the proposal
for being utopian (Palley 2014), it is nevertheless interesting to investigate the effects of a
wealth tax in the Post-Keynesian model.
Piketty (2014: 571) (tentatively) suggests a tax of 0.1% on wealth up to 200.000 Euro, 0.5% up
to 1 million Euro, 1% up to 5 million Euro, 2% up to 1 billion Euro, and 5% (to 10%) on wealth
greater than a billion Euro. For simplicity, we use the average for the lower two brackets (0.3%)
as workers’ average tax rate and the average of the higher three tax brackets (4.3%) as
capitalists’ tax rate. Furthermore, we formulate the tax a flat rate ? on the total wealth of both
workers and capitalists and we assume that tax earnings are spent as government
.00
.02
.04
.06
.08
.10
.12
25 50 75 100 125 150 175 200
r (baseline) r (growth rate +1%)
.00
.02
.04
.06
.08
.10
.12
25 50 75 100 125 150 175 200
g (baseline) g (growth rate +1%)
3.2
3.6
4.0
4.4
4.8
25 50 75 100 125 150 175 200
x (baseline) x (growth rate +1%)
0.00
0.20
0.40
0.60
0.80
1.00
25 50 75 100 125 150 175 200
z (baseline) z (growth rate +1%)
18
consumption to maintain the stock-flow consistency of the model.10 Income of capitalists and
workers becomes
?? = ?? ? ????
?? = ? + (1 ? ?)? ? ????
For the dynamic equations, see Appendix B.
Figure 3 shows that the wealth tax redistributes income from capitalists (with a higher saving
rate) to workers. It therefore raises capacity utilisation and simultaneously reduces the
wealth-income ratio, while the growth rate and the profit rate also rise. Most importantly
however, from Piketty’s point of view, the wealth share of capitalists would fall noticeably,
from roughly 0.62% in the baseline scenario to about 50%. That is, a wealth tax as suggested
by Piketty would neutralize the rise in wealth inequality predicted by the model. This
reduction in the wealth share is permanent, as long as the tax is imposed periodically.
Conversely, one-off levies only have a transitory effect as the wealth concentration returns to
its long-run equilibrium.
Figure 3: Effects of a wealth tax on short- and long-run dynamics of the Post-Keynesian wealth
model in Europe
Source: own elaboration.
Note: This graph shows the dynamic in the wealth-to-income ratio ?, and capitalists’ wealth
share ? for a wealth tax of 0.3% on workers’ and 4.3% of capitalists’ wealth, compared to a
baseline scenario. For parameter starting values, see Appendix A.
Other taxes that reduce the income of capitalists and thus dampen their ability to accumulate
wealth, such as an inheritance tax or income taxes for capitalists, are fungible to a wealth tax;
they will lead to the same effects as the ones described in Figure 4. In fact, in the Post-
10 Another way to balance the budget would be to spend all tax revenue on monetary transfers. Transfers directly
increase the income of households and have a distributive effect themselves given that workers and capitalists
benefit differently from them. Public consumption on the other hand has no direct distributive effect in the
model, which allows us to isolate the sole effect of Piketty’s tax rates.
0.00
0.20
0.40
0.60
0.80
1.00
25 50 75 100 125 150 175 200
z (baseline) z (wealth tax)
3.6
3.8
4.0
4.2
4.4
4.6
4.8
5.0
25 50 75 100 125 150 175 200
x (baseline) x (wealth tax)
19
Keynesian model, these different taxes can easily be calibrated to yield identical results. If
wealth is passed on to the next generation every 25 years, an inheritance tax of 60% has on
average the same incidence as a general (yearly) wealth tax of 2.4% (which is our average tax
rate). Similarly, since capital income is a share of wealth, taxing capital income is equivalent
to taxing wealth directly. Introducing a tax on capital income of approximately 30% is
consequently roughly equivalent to a yearly wealth tax of 2.4%, given that the profit rate in
the simulation is about 8%.11
8. Conclusion
This paper bridges the gap between a Post-Keynesian analysis and Piketty’s (2014) empirical
insights. We develop a Post-Keynesian model with an endogenous distribution of wealth
between workers and capitalists, and extend it by including blended wage and capital income
of both workers and capitalists, differential returns on assets between workers and capitalists,
and capital gains. Since the transition phase to the long-run equilibrium matters in the real
world, we focus on the short- and long-run dynamics of the model, and evaluate Piketty’s
proposal of a wealth tax.
If Piketty’s main theoretical prediction (? > ? leads to rising wealth inequality) is taken to its
radical conclusion, then a small elite will own all wealth if capitalism is left to its own devices.
Our model permits such a corner solution of all wealth held by capitalists, but usually
economies will show a stable long-run wealth distribution in which workers have a positive
wealth share. In such an interior equilibrium, the wealth-to-income ratio is stable, and there
is a (stable and positive) gap between the profit rate and the growth rate, which is given by
the Cambridge equation. The specific level of the equilibrium wealth distribution between
workers and capitalists depends on their saving rates, the profit share, the share of wage
income that accrues to capitalists, the differential returns on wealth for the two household
groups, and the saving rate of firms.
We therefore provide an alternative to the simple version of Piketty’s theoretical long-run
equilibrium. However, like Piketty (2014), Post-Keynesians are interested in the real world of
living generations; we therefore show that the model has a transitional phase, i.e. when the
wealth share of capitalists is below its long-term equilibrium. During this transition phase, the
model behaves according to Piketty’s (2014) empirical findings for high-income countries since
the 1980s. In this situation, the wealth share of capitalists increases endogenously, the wealth-
to-income ratio rises, the differential between the profit rate and the growth rate gradually
decreases (but is always higher than the long-term gap), and income inequality rises.
Consistent with Keynesian logic, a rising wealth share reduces aggregate demand and thus
capacity utilization and growth.
11 However, it should be noted that this back-of-the-envelope calculation does not take into account that an
income tax does not capture capital gains.
20
For this transition phase, the paper thus provides theoretical foundations to Piketty’s
abundant and convincing empirical findings, and it permits us to model the long-run evolution
of inequality. Concretely, our simulations show that wealth inequality in Europe – as measured
by capitalists’ wealth share – would rise from well under 50% to more than 60%. This level
differs substantially across individual countries, as Ederer and Rehm (2017) show.
Piketty (2014) emphasizes both the determinants and the effects of these pernicious empirical
developments in detail, and carefully weighs counter effects to his theoretical finding of
capitalism’s inherent tendency to an increasing wealth concentration. Piketty, like many Post-
Keynesians, worries that the world of increasing inequality, which his data describes, might be
prone to upheavals, social unrest, and war. This is why Piketty (2014) suggests a wealth tax to
address the increasing concentration of wealth. We find that the introduction of a permanent
wealth tax at levels suggested by Piketty (or, equivalently, a suitable inheritance tax or capital
income tax) can indeed neutralize the rise in wealth inequality predicted by the model. It
reduces the equilibrium value for the wealth share owned by capitalists in Europe – and thus
the wealth concentration – to 50%. It also reduces the wealth-to-income ratio and dampens
income inequality.
There are a number of interesting avenues for future work. First, expanding the analysis to
other countries, such as the U.S., is an obvious next step. Second, endogenizing the profit
share and working through the stability aspects of such a model might provide valuable insight
into potential ‘Piketty dynamics’ in a Keynesian framework. Third, delving into the policy
research might yield more detailed information on the relative merits of a wealth tax versus
an inheritance tax or capital income taxes. Finally, investigating the dynamics of the
distribution of wealth globally – as well as the redistributive effects of a global wealth tax –
would be very interesting, but is unfortunately severely constrained by data availability.
21
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24
Appendix A: Parameter values
Table A.1: Parameter values for the model calibration, average over countries
Parameter Value Source
?? 0.07 Ederer/Rehm 2018, Data: EHBS
?? 0.24 Ederer/Rehm 2018, Data: EHBS
? 0.39 Ederer/Rehm 2018, Data: HFCS
? 0.06 Ederer/Rehm 2018, Data: HFCS
?? 0.49 Ederer/Rehm 2018, Data: HFCS
?_? 0.91 Ederer/Rehm 2018, Data: HFCS
Note: Rows refer to: (1) ?? saving rate of workers, (2) ?? saving rate of capitalists, (3) ? profit
share, (4) ? share of capitalists in the wage bill, (5) ?? share of workers’ wealth held in profit-
generating assets, (6) ?? share of capitalists’ wealth held in profit-generating assets. In order
to reproduce the empirical results of Ederer and Rehm (2017), we set the parameter for the
saving rate of firms equal to zero. Since the empirical value of this parameter is usually higher,
the long-run values of the simulations are on the conservative side.
Appendix B: Extended model
Table B.1: Transaction flow matrix in the extended model
Households Firms Banks
Workers Capitalists Current Capital Capital Total
Consumption ??? ??? +? 0
Investment +? ?? 0
Wages +?? +?? ?? 0
Profits +?? +?? ?? +?? 0
Equity ???? ???? +?? 0
Deposits ???? ???? +?? 0
Loans +?? ??? 0
Total 0 0 0 0 0 0
Source: own elaboration.
25
Table B.2: Balance sheet matrix in the extended model
Households
Workers Capitalists Firms Banks Total
Capital +? +?
Equity +?? +?? ?? 0
Deposits +?? +?? ?? 0
Loans ?? +? 0
Wealth ??? ??? ??? ??
Total 0 0 0 0 0
Source: own elaboration.
Retained profits:
?? = ??
Disposable income:
?? = (1 ? ?)? +
??(1 ? ?)
??(1 ? ?) + ???
(1 ? ?)?
?? = ?? +
???
??(1 ? ?) + ???
(1 ? ?)?
Wealth dynamic:
??? =
????
?
? ?? = {?? [(1 ? ?)? +
???
??(1 ? ?) + ???
(1 ? ?)?] +
???
??(1 ? ?) + ???
??} ? ? ??
Appendix C: Basic model with a (progressive) wealth tax
Table C.1: Stocks and flows in the model with a wealth tax
Households Firms State
Workers Capitalists Current Capital Current Total
Consumption ??? ??? +? +?? 0
Investment +? ?? 0
Wages +? ?? 0
Profits +?? +?? ?? 0
Taxes ??? ??? +? 0
Wealth ???? ???? +?? 0 0
Total 0 0 0 0 0 0
Source: own elaboration.
26
Wealth dynamic:
??? = [??(? ? ??) ? ?]?
?? =
(?0 + ?2?)(??? ? ??) + ?1(???? ? ????) ? ????(?? ? ???)
(?0 + ?2?)(?? ? ??)? + ?1(???? ? ????) ? ?????(?? ? ??)
Unsere AutorInnen:
Miriam Rehm, ist Mitarbeiter der Abteilung Wirtschaftswissenschaft in der Kammer für Arbeiter und
Angestellte für Wien
Stefan Ederer ist Ökonom am Österreichischen Institut für Wirtschaftsforschung (WIFO) und arbeitet
zu den Themen Makroökonomie, Verteilung, Wirtschaftspolitik und Finanzmärkten.
„Materialien zu Wirtschaft und Gesellschaft"
Die Working Paper-Reihe der AK Wien
sind unregelmäßig erscheinende Hefte, in denen aktuelle Fragen der Wirtschaftspolitik behandelt
werden. Sie sollen in erster Linie Informationsmaterial und Diskussionsgrundlage für an diesen
Fragen Interessierte darstellen.
Ab Heft 80 sind die Beiträge auch als pdf-Datei zum Herunterladen im Internet
http://wien.arbeiterkammer.at/service/studien/MaterialienzuWirtschaftundGesellschaft/index.html
Heft 159 Markus Knell Überlegungen zur fairen und nachhaltigen Ausgestaltung
eines Pensionskontensystems, Oktober 2016
Heft 160 Miriam Rehm Different but equal? Classes, wealth, and perceptions in
Europe, Oktober 2016
Heft 161 Tobias Schweitzer Budgetanalyse, November 2016
Heft 162 Reinhold Russinger Volkswirtschaftliche Gesamtrechungen 1996-2015,
Oktober 2016
Heft 163 Michael Mies Extension of the Empirical Stock-Flow Consistent Model for
Austria, Nov. 2016
Heft 164 Xavier Timebau The Elusive Recovery. Independent Annual Growth Survey
2017, Nov. 2016
Heft 165 Markus Griesser Verankerung wohlstandsorientierter Politik, Dezember 2016
Heft 166 Michael Mesch Erwerbs- und Einkommenschancen in Österreich im Kontext
der intergenerationellen Einkommenspersistenz,
August 2017
Heft 167 Benjamin Ferschli Vermögenskonzentration in Österreich und Europa,
September 2017
Heft 168 Julia Groiß Vermögensunterschiede nach Geschlecht in Österreich und
Deutschland: Eine Analyse auf der Personenebene,
September 2017
Heft 169 Benjamin Herr Riding in the gig-economy: An in-depth study of a branch in
the app-based on-demand food delivery industry,
November 2017
Heft 170 Reinhold Russinger Volkswirtschaftliche Gesamtrechnungen 1995-2016,
Dezember 2017
Heft 171 Xavier Timbeau Independent Annual Growth Survey 2018, Dezember 2017
Heft 172 Stefan Ederer Will wealth become more concentrated in Europe?,
Dezember 2017
Heft 173 Stefan Vogtenhuber Arbeitskräfteangebot und -nachfrage: Verdrängung durch
Bildungsexpansion?, Jänner 2018
Heft 174 Romana Brait Budgetanalyse 2018-2022, April 2018
Heft 175 Georg Feigl Wohlstandsbericht 2018, Mai 2018
Heft 176 Stefan Jestl Inheritance Tax Regimes, Juli 2018
Heft 177 Sebastian Leitner Factors driving wealth inequaltiy in European countries,
Juli 2018
Heft 178 Vasily Astrov Die Lohnentwicklung in den mittel- und osteuropäischen
Mitgliedsländern der EU
Heft 179 Joachim Becker Neo-Nationalismus in der EU: sozio-ökonomische
Programmatik und Praxis
Heft 180 Oliver Picek Kann nationale Konjunkturpolitik noch Beschäftigung
schaffen?
Heft 181 Stefan Ederer Making Sense of Piketty’s ‘Fundamental Laws’ in a Post-
Keynesian Framework: The Transitional Dynamics of
Wealth Inequality
Eigentümer, Verleger, Herausgeber und Vervielfältiger: Kammer für Arbeiter und Angestellte für
Wien; alle: 1041 Wien, Prinz Eugen-Straße 20-22, Postfach 534
181
MATERIALIEN ZU WIRTSCHAFT UND GESELLSCHAFT978-3-7063-0747-5
WORKING PAPER-REIHE
DER AK WIEN
MAKING SENSE OF PIKETTY’S ‘FUNDAMENTAL LAWS’
IN A POST-KEYNESIAN FRAMEWORK:
THE TRANSITIONAL DYNAMICS OF WEALTH INEQUALITY
Stefan Ederer
Miriam Rehm
WPR_181_MakingSenseofPickettys.indd 1 28.09.18 12:03