ISSN (Print) 0473-453X

Discussion Paper No. 1223

ISSN (Online) 2435-0982

THE PREFERENCE
FOR WEALTH AND INEQUALITY:TOWARDS A PIKETTY THEORYOF WEALTH INEQUALITY

Jean-Baptiste Michau
Yoshiyasu Ono
Matthias Schlegl

November 2023

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Preference for Wealth and Inequality:
Towards a Piketty Theory of Wealth Inequality *Jean-Baptiste Michau† Yoshiyasu Ono‡ Matthias Schlegl§
November 2023

Abstract
What are the consequences of the preference for wealth for the accumulation ofcapital and for the dynamics of wealth inequality? Assuming that wealth per se isa luxury good, inequality tends to rise whenever the interest rate is larger than theeconomic growth rate. This induces the economy to converge towards an equilibrium with extreme wealth inequality, where the capital stock is equal to the golden
rule level. Far from immiseration, this equilibrium results in high wages and in the
golden rule level consumption for ordinary households. We then introduce shocks
to the preference for wealth and show that progressive wealth taxation prevents
wealth from being held by people with high saving rates. This permanently reduces the capital stock, which is detrimental to the welfare of future generation ofworkers. This also raises the interest rate, to the benefit of the property-owning
upper-middle class. By contrast, a progressive consumption tax successfully andpersistently redistributes welfare from the very rich to the poor.
Keywords: Capital accumulation, Progressive wealth tax, Wealth inequality,
Wealth preference
JEL Classification: D31, E21, E22, H20* Financial supports from the JSPS KAKENHI Grants (JP20H05631 and 21K13261) and from the

Joint Usage/Research Center for ”Behavioral Economics” at ISER of Osaka University are gratefully
acknowledged.
†
Ecole Polytechnique, France; jean-baptiste.michau@polytechnique.edu.
‡
Institute of Social and Economic Research, Osaka University, Japan; ono@iser.osaka-u.ac.jp.
§
Sophia University, Japan; m-schlegl-4t5@sophia.ac.jp.

1

The immense accumulations of
fixed capital which, to the great
benefit of mankind, were built up
during the half-century before thewar, could never have come about
in a society where wealth was
divided equitably.
John Maynard Keynes
The Economic Consequences of thePeace (1919)

1

Introduction

Under capitalism, the distribution of wealth is highly unequal, much more so thanthe distribution of labor income. The neoclassical growth model with heterogeneous
labor earnings cannot account for the extent of this wealth inequality. Instead, thedesire to accumulate wealth, or to leave bequests, appears to be an essential driverof the saving behavior of wealthy households, who consume much less than theirpermanent income, thereby accumulating ever more wealth. The preference for wealthhas therefore become a common ingredient of models of wealth inequality such as toaccount for the thickness of the upper tail of the wealth distribution.
The goal of this paper is to investigate the implications of the preference for wealthfor the accumulation of capital, for the dynamics of wealth inequality, and for the interplay between the two. Our analysis relies on a continuous-time neoclassical growthmodel with a rate n of demographic growth and rate g of productivity growth. We add
a preference for wealth, assuming that wealth is a luxury good. We also impose a borrowing constraint (i.e. a minimum wealth level). We initially consider that all households have identical preferences. Throughout our analysis, we interpret infinitelylived households as dynasties.
We begin by characterizing the behavior of households in partial equilibrium, witha fixed capital stock. We show that, whenever the interest rate r is greater than theeconomic growth rate n + g, wealth trajectories diverge. With r > n + g, there is a
threshold level of personal wealth above which wealth diverges to infinity and below
which it converges to a low level. Conversely, when r < n + g, the wealth of allhouseholds converges to the same level.
In his magnum opus, Piketty (2014) has argued that the main driver of wealth2

inequality over the long-run is the gap between r and n + g.1 He explained that, whenr > n + g, households only need to consume a bit less than their permanent income
to see their wealth inexorably grow over time (relative to the size of the economy).
Moreover, wealthy households are the most likely to take advantage of r > n + g
since their income is so large that they can afford to consume only a small fraction ofit. But, why would wealthy households consume less than their permanent income?
Because the accumulation of wealth per se is a luxury good! We therefore provide themicrofoundation that induces the saving behavior of households to be in line with
Piketty (2014)’s theory of wealth inequality. Not only does it result in rising inequality
when r > n + g, but also in falling inequality when r < n + g.2
We then investigate the general equilibrium consequences of these results. With
endogenous capital accumulation, the economy can either converge to an egalitarian
steady state or to a (degenerate) inegalitarian steady state. The former is characterized
by an equal distribution of wealth (for all households with identical labor earnings).
The latter by a degenerate distribution of wealth, where a vanishing fraction of households hold arbitrarily large wealth while all the others converge to a low wealth level.
The interest rate r tends to n + g and the capital stock to the golden rule level. Far
from immiseration, this equilibrium is characterized by high wages and the golden
rule level of consumption (which maximizes steady state aggregate consumption) for
wealth-poor households. The total consumption of the rich asymptotically has zero
mass, which implies that extreme wealth inequality is not detrimental to the long-term
prosperity of the population.
The egalitarian steady state always exists when the preference for wealth is sostrong that, even with an equal distribution of wealth, the capital stock is above thegolden rule level. Conversely, the inegalitarian steady state only exists when the preference for wealth is weaker, in which case wealth inequality is necessary to bring thecapital stock to the golden rule level. This shows that the preference for wealth isan essential driver of capital accumulation, which prevents the capital stock from remaining much below the golden rule level. This complements the finding from ourprevious research that, whenever the egalitarian steady state has a capital stock above
the golden rule level, the preference for wealth can generate a rational bubble that prevents the over-accumulation of capital, exactly as in an OLG economy (Michau, Ono,and Schlegl, 2023). Thus, the preference for wealth creates a fundamental tendency for
1

Piketty (2014) denotes the growth rate of the economy by g, which corresponds to n + g within our
analysis. We carefully distinguish demographic growth n from productivity growth g, since they are notinterchangeable. We find that the long-run dynamics of wealth inequality is nonetheless determined by
n + g, consistently with Piketty (2014)’s argument.
2
Appendix A provides quotations from Piketty (2014) summarizing his r vs. n + g theory. His
narrative exactly corresponds to the forces at play within our analysis.

3the capital stock to approach the golden rule level. There is also a range of parameters
for which both steady states exist. However, a thick upper tail to the initial wealthdistribution induces convergence to the inegalitarian steady state, which makes thisoutcome empirically more likely whenever both are possible.
In practice, dynasties are unlikely to sustain a preference for wealth forever. We
therefore subsequently allow for shocks to the preference for wealth. We assume that
only a fifth of households have a preference for wealth and that it is only sustained on
average for three generations. We rely on a mean field game formulation à la Achdou,Han, Lasry, Lions, and Moll (2022) to run numerical simulations. Our calibration satisfies r > n + g and matches the thickness of the upper tail of the wealth distribution.
We obtain an equilibrium where dynasties with a preference for wealth get richer overtime, until they lose this preference, which induces them to gradually spend their accumulated savings. The average saving rate is strongly increasing in wealth.
We then rely on simulations to investigate a number of tax reforms. We first consider a 20% levy on wealth above $10 million financing a lump-sum transfer to poor
households. Recall that, within the neoclassical growth model, a one-off lump-sum
redistribution of wealth has no impact on the aggregate variables of the economy. By
contrast, with a preference for wealth, the poor quickly spend the wealth that would
have been saved by the rich. This leads to a consumption boom and to an investmentbust. The resulting decline in the capital stock raises the interest rate and reduces
wages. This induces middle-class households, who are not directly affected by the policy, to reduce their consumption. These general equilibrium effects generate a welfare
gain for middle-class households with a preference for wealth, whose saving behavior
is rewarded by a higher interest rate, and a welfare loss for poorer households who do
not qualify for the lump-sum transfer and who suffer from lower wages.
We then simulate a progressive wealth tax at rate 2% per year above $10 million
to finance redistribution to poor households. The consequences are similar as for thewealth levy, except that rich households respond by sharply increasing their consumption. Their saving plummets, which allows the policy to persistently compress wealthinequality, to such an extent that the tax revenue continuously decreases over time.
By preventing the concentration of wealth within the hands of few households withhigh saving rates, this policy permanently reduces the capital stock. This entails sizeable welfare losses for future generations of workers. This policy also substantially
increases the wealth of the upper-middle class, who benefit from a higher interest
rate without being hurt by the wealth tax. Hence, over the long-run, the progressive
wealth tax redistributes wealth and welfare from the very rich to the property-owning
upper-middle class.
Finally, we consider a progressive consumption tax with a rate of 50% for consump4

tion above $200 000 per year. It turns out to be a more efficient redistribution tool since
it targets wealthy dynasties who have lost their preference for wealth and who therefore have a low saving rate. The policy slightly raises the capital stock. While wealthinequality remains almost unchanged, consumption inequality falls. The progressive
consumption tax successfully redistributes consumption and welfare from rich to poor
households, even over the long-run.
Related literature. The relationship between the distribution of wealth and the accumulation of capital has long been a controversial topic. As the opening quotation ofthis paper indicates, Keynes (1919) viewed the high saving rate of the capitalist classas an essential driver of the accumulation of capital. Moreover, he believed that thethrifty behavior of wealthy households was due to a preference for wealth: ”Saving
was for old age or for your children; but this was only in theory,–the virtue of the cake was that
it was never to be consumed, neither by you nor by your children after you”.
Kaldor (1955, 1957) emphasized the different saving rates of capitalists and workers
as critical to the accumulation of capital. Pasinetti (1962) showed that, in the long-run,the accumulation of capital is uniquely determined by the saving behavior of capitalists. Samuelson and Modigliani (1966) established that this result can be reversed if
the saving rate of workers is sufficiently high, in which case they end up owning theentire capital stock of the economy. This early literature shows the importance of heterogeneity in saving rates across people for the determination of both the distributionof wealth and the accumulation of capital.
Instead of assuming class-dependent saving behavior, Bourguignon (1981) assumed
scale-dependent saving, whereby the saving rate is an increasing function of income.
He showed that, when the interest rate is larger than the growth rate of the economy,both an egalitarian steady steady state and an inegalitarian steady state can exist. Our
work generalizes this insight by providing a microfoundation, by allowing for morethan two types, and by taking transitional dynamics into account.
While these early contributions assume exogenous saving functions, the literature
now provides microfoundations that endogenize the heterogeneity in saving rates.
Yaari (1964) was the first to assume a bequest motive for saving, modeled as utility
from bequeathed wealth. Atkinson (1971) then showed that, if the utility from bequeathed wealth is less convex than the utility from consumption, then bequests areluxury goods and the saving rate is increasing in wealth. This is directly related to
our work since the preference for wealth of an infinitely-lived household can be interpreted as a warm-glow bequest motive within an altruistic dynasty. And we doassume throughout our analysis that the utility from wealth is less convex than theutility from consumption, resulting in wealth being a luxury good.
The related empirical literature provides strong support for the forces at play within
5

our analysis. At the microeconomic level, the saving rate is increasing in wealth, whichis inconsistent with the permanent-income hypothesis. The preference for wealth or
the bequest motive appears to be a key driver of this heterogeneity (Carroll, 2000;
Dynan, Skinner, and Zeldes, 2004; Lockwood, 2018; Straub, 2019; Fagereng, Holm,and Natvik, 2021b; De Nardi, French, Jones, and McGee, 2021).3 At the macroeconomic level, the wealth distribution has a thick upper tail. Again, heterogeneity insaving rates appears to be a major driver of this wealth inequality (Saez and Zucman, 2016; Garbinti, Goupille-Lebret, and Piketty, 2021; Ozkan, Hubmer, Salgado, andHalvorsen, 2023). Turning to dynamics, it was estimated that the cumulated stockof inherited wealth accounts for more than 50% of the total wealth of the economy
and that this number is likely to rise to over 80% by the end of the century (Kotlikoff
and Summers, 1981; Piketty, 2011, 2014; Piketty and Zucman, 2015). This shows that
wealth accumulation is an intergenerational process, which is consistent with our dynastic perspective (whereby our infinitely-lived households should be interpreted as
dynasties).
Our paper contributes to a sizeable literature in quantitative macroeconomics onthe determinants of wealth inequality.4 An important challenge is to be able to account for the thickness of the upper tail of the wealth distribution. In a seminal contribution, that is closely related to our work, De Nardi (2004) has shown that the bequest
motive, and the resulting high saving rate of wealthy households, can account for
top-end wealth inequality. Our contribution is to investigate the range of equilibrium
possibilities induced by this bequest motive or preference for wealth, emphasizing thepossibility of rising or falling inequality depending on the relative magnitude of theinterest rate and the economic growth rate. We also carefully analyze the impact of the
distribution of wealth on the accumulation of capital over the short and long term.
An alternative way to generate heterogeneity in saving rates is to assume heterogeneity in discount rates. Ramsey (1928) had hypothesized that, under differencesin discount rates, ”equilibrium would be attained by a division of society into two classes,
the thrifty enjoying bliss and the improvident at the subsistence level”. This conjecture wasformally verified by Becker (1980). Assuming that the discount rate is stochastic cangenerate a thick upper tail to the wealth distribution (Krusell and Smith, 1998; Carroll,Slacalek, Tokuoka, and White, 2017; Toda, 2019). A preference for wealth and a low
discount rate have similar effects: in the absence of shocks, they lead to extreme inequality; with stochastic shocks, the persistence over time of a high saving rate within
3

Fagereng, Holm, Moll, and Natvik (2021a) find that, in Norway, the saving rate (as a fraction ofincome) is increasing in wealth if income includes capital gains, and is flat otherwise. Our work does
not distinguish capital gains from dividends.
4
See De Nardi and Fella (2017), De Nardi, Fella, and Yang (2017), and Benhabib and Bisin (2018) for
excellent surveys.

6some dynasties results in a thick upper tail. The main difference is that the preference for wealth endogenously generates scale-dependence in the saving behavior ofhouseholds, while the heterogeneity in discount rates only induces type-dependence.
A related source of wealth concentration is stochastic aging, whereby older households have had more time to accumulate wealth (Wold and Whittle, 1957; Castaneda,
Diaz-Gimenez, and Rios-Rull, 2003; Benhabib, Bisin, and Zhu, 2016). Again, this is related to the persistence over time of the preference for wealth within some dynasties.5
A completely different explanation for the thick upper tail of the wealth distribution is stochastic returns to capital (Champernowne, 1953; Benhabib, Bisin, and Zhu,
2011, 2015; Toda, 2014; Nirei and Aoki, 2016; Cao and Luo, 2017).6 This is particularly
relevant for entrepreneurs whose portfolios cannot be properly diversified (Quadrini,
2000; Cagetti and De Nardi, 2006). Our work abstracts from this source of wealth dispersion, which could amplify the diverging wealth dynamics that we find, especially
if wealthy households obtain higher returns on their saving (Bach, Calvet, and Sodini,
2020; Fagereng, Guiso, Malacrino, and Pistaferri, 2020; Best and Dogra, 2023).
While much of the literature (including our own contribution) focuses on one or
two mechanisms in isolation, a few papers have combined these different sources of
wealth dispersion to assess quantitatively their ability to account for wealth inequality in the United States (De Nardi and Yang, 2016; Benhabib, Bisin, and Luo, 2019;
Kaymak, Leung, and Poschke, 2022; Gaillard, Hellwig, Wangner, and Werquin, 2023).
They all find that a non-homothetic bequest motive à la De Nardi (2004), whereby
bequests are luxury goods, is an essential ingredient to account for top-end wealthinequality.7
Piketty (2014) forcefully argues that wealth inequality tends to rise whenever r >n + g, which he denotes r > g. To provide a theoretical justification for this conjecture, a number of microfoundations were found leading to a Pareto upper tail to thewealth distribution, with a thickness that is increasing in r − g. Piketty and Zucman(2015) proposed a generational model (where one period corresponds to one generation) with a stochastic bequest motive, but no altruism. An alternative is to either rely
on a stochastic aging model (Jones, 2014; Hiraguchi, 2019) or on stochastic returns to
wealth (Acemoglu and Robinson, 2015; Cao and Luo, 2017). This yields a Pareto tail5

Castaneda, Diaz-Gimenez, and Rios-Rull (2003) also assumes earnings inequality, which feeds into
wealth inequality. However, empirically, the upper tail of the wealth distribution is much thicker thanthat of the earnings distribution. Hence, earnings inequality alone cannot account for top-end wealthinequality, which was formally shown by Stachurski and Toda (2019). For clarity and simplicity, weshut down this channel by assuming that labor earnings are identical across all individuals.
6
Interestingly, Ma and Toda (2021) have shown that stochastic returns to capital can be consistent
with a saving rate that is increasing in wealth.
7
An exception is Hubmer, Krusell, and Smith (2021) who rely on stochastic discount rates. But, aspreviously mentioned, the effects of the bequest motive and of stochastic discount rates are comparable.

7

with thickness proportional to r−g−M P C, where M P C denotes the marginal propensity to consume out of income. To the best of our knowledge, we are the first to offera microfoundation where inequality is rising whenever r > g and falling wheneverr < g, which is more in line with the narrative of Piketty (2014).
Building on Chamley (1986) and Judd (1985), there is a large literature on the taxation of the capital owned by infinitely-lived households. Rather than derive optimal
taxation formulas, we emphasize the non-trivial general equilibrium effects of wealthtaxation, emphasizing the sizeable impact on people who are not directly affected by
the policy. Some of our main insights could not be obtained within the neoclassicalgrowth model, on which much of the literature on capital taxation relies.8
Finally, our work is related to the growing literature on Heterogeneous Agent New
Keynesian (HANK) models, which investigates the macroeconomic consequences ofheterogeneity in marginal propensities to consume (Kaplan, Moll, and Violante, 2018;
Auclert and Rognlie, 2020; Auclert, Rognlie, and Straub, 2023). In particular, under apreference for wealth, wealth inequality can be rising over time resulting in a fall inaggregate demand (Ono, 1994; Kumhof, Rancière, and Winant, 2015; Illing, Ono, andSchlegl, 2018; Mian, Straub, and Sufi, 2021). While similar dynamics are at work within
our framework, the absence of nominal rigidities implies that aggregate demand isfully determined by the supply side of the economy, i.e. by the capital stock throughthe production function.
The rest of the paper proceeds as follows. Section 2 introduces our model. Section
3 characterizes the behavior of households in partial equilibrium, with a fixed capitalstock. General equilibrium results are presented in Section 4. Shocks to the preference
for wealth are introduced in Section 5, which relies on a numerical simulation of ourmodel. Section 6 investigates a wealth levy, a progressive wealth tax, and a progressive
consumption tax. The paper ends with a conclusion.

2

Model

Time is continuous. There is a unit mass of infinitely-lived households. Each household should be interpreted as a dynasty where the current generation is altruistic towards future generations. The size of any household at time t is equal to Nt = ent ,which implies that population grows at rate n within each household. Technologicalprogress is characterized by a labor-augmenting level of productivity Gt = egt , which
8

Saez and Stantcheva (2018) have derived insightful formulas for the optimal taxation of capital
under a preference for wealth. However, their analysis either assumes that wealth or bequests are notluxury goods (since utility from consumption is linear) or they take convergence to steady state for
granted, which our analysis shows is only justified when r < n + g.

8grows at an exogenous rate g. Thus, at time t, each household is endowed with Nt Gt
efficiency units of labor, which grows at rate n + g.

2.1

Firms

A representative firm uses capital Kt and labor Nt to produce output Yt using a constant returns to scale neoclassical production functionYt = Gt Nt f

Kt
Gt Nt


,

(1)

where f (·) is increasing and concave and satisfies the Inada conditions. At each point
in time, the firm must choose how much capital Kt to rent from households at price Rt
and how much labor Nt to employ at wage w̃t such as to maximize its profits
Gt Nt f

Kt
Gt Nt


− Rt Kt − w̃t Nt .

(2)

The real interest rate rt is equal to the rental cost of capital Rt net of depreciation δ,i.e. rt = Rt − δ. In equilibrium, each factor of production must be paid its marginal
product, which gives
rt = f ′ (kt ) − δ,(3)wt = f (kt ) − kt f ′ (kt ),(4)

where kt = Kt /(Gt Nt ) denotes capital per efficiency unit of labor and wt = w̃t /Gtdenotes the wage rate relative to labor productivity.

2.2

Households

Each household is indexed by i ∈ [0, 1]. Let ait and cit denote the wealth and consumption of household i per efficiency unit of labor, respectively. The initial distribution of
wealth ai0 across households is exogenously given by an initial density function ϕ(·, 0).
The household’s flow of funds is given byȧit = (rt − n − g) ait + wt − cit .

(5)

We impose the borrowing constraint
ait ≥ a,9(6)

where a ≤ 0.9
Individuals within household i derive utility from their consumption level per
capita cit Gt . They also derive utility from their holding of wealth per capita ait Gt relative to their labor productivity Gt .10 We assume balanced growth preferences à la King,
Plosser, and Rebelo (1988). This entails( exp (1−σ) ln ci G +v ai −1([ ( t t ) ( t )])u(cit Gt , ait ) =1−σln (cit Gt ) + v (ait )if σ ̸= 1,
if σ = 1(7)where v (·) is increasing and concave and σ ∈ (0, ∞). Following Kumhof, Rancière,and Winant (2015) and Michau (2023), we consider that the subutility of wealth v (ait )is characterized by constant relative risk aversion, relative to a reference point ζ ≤ 0,1−µ(ai −ζ ) −1
which gives v (ait ) = γ t 1−µ. Households discount the future at rate ρ, with ρ >n + (1 − σ)g. Their intertemporal utility is therefore given by
exp (1 − σ)

Z ∞
e

−ρt



1−µ

(ai −ζ ) −1
gt + ln (cit ) + γ t 1−µ

Nt

1−σ

0


−1
dt.

(8)

For these preferences to be homothetic (as in Benhabib, Bisin, and Zhu (2011) and Benhabib, Bisin, and Zhu (2016)), we would need to have both ζ = 0 and µ = 1. We instead
impose µ < 1, which implies that wealth is a luxury good (as in Atkinson (1971) andDe Nardi (2004)). When preferences are additively separable between consumption
and wealth, i.e. when σ = 1, our preference for wealth can be interpreted as a warmglow bequest motive within a stochastic aging model (as in Mian, Straub, and Sufi
(2021)).
The household maximizes its intertemporal utility (8) subject to its wealth accumulation equation (5), with ai0 given, and to its borrowing constraint (6).11 By the
maximum principle, the solution to household i’s problem is characterized by the consumption Euler equation

"
#
−µ

ċit
1
γ (ait − ζ)−µ i
+ (1 − σ)γ ait − ζ≥
rt − ρ − σg +ȧt and ait ≥ a,
−1
i
i
ct
σ
(ct )
9

(9)

Note that this borrowing
constraint (6) cannot be binding if a is below the natural borrowing limit
R∞ Rs
given at time t by − t e− t (ru −n−g)du ws ds.
10
An alternative is to assume that individuals derive utility from their holding of wealth per capita,i
at Gt , relative to output per capita, Yt /Nt = f (kt )Gt , or relative to the average wealth per capita,
Kt /Nt = kt Gt . These alternative formulations would introduce an externality from wealth accumulation, but would not fundamentally modify our results on the dynamics of wealth inequality.
11
The borrowing constraint (6) is the only limit that we are imposing on indebtedness. In particular,
our formulation does not rule out Ponzi schemes.

10with complementary slackness, together with the transversality condition
lim e−(ρ−n−(1−σ)g)t+(1−σ)γ(ait −ζ)

1−µ

−1

cit

1−µ

t→∞

−σ  iat − a = 0.

(10)

The preference for wealth adds a new term to the consumption Euler equation: theratio of the marginal utility of wealth to the marginal utility of consumption. This new
term introduces heterogeneity in saving behaviors across wealth levels. As we shallsee, this dramatically affects the dynamics of wealth inequality.

2.3

Market clearing

The goods market clearing condition imposes that total output must be equal to thesum of investment and consumption
Gt Nt f

Kt
Gt Nt



h

Z 1

i

= δKt + K̇t +cit Gt Nt di.

(11)

0

As k̇t = K̇t /(Gt Nt ) − (n + g)kt , we must have
Z 1
k̇t = f (kt ) − (δ + n + g) kt −cit di.

(12)0Households’ supply of savings must be equal to firms’ demand for capital, which
gives the asset market clearing conditionZ 1

ait di = kt .

(13)

0

2.4

Equilibrium

Let us now define the equilibrium of the economy.
1

Definition 1 The equilibrium of the economy ((cit , ait )i=0 , kt , rt , wt )∞t=0 is characterized by:• The profit maximizing behavior of firms, which determines the real interest rate (3) andthe wage rate (4);• The solution to the household’s problem, which is jointly given by the flow of funds
constraint (5), the Euler equation with the borrowing constraint (9), the transversality
condition (10), and the distribution ϕ(·, 0) of initial wealth ai0 across households;• The asset market clearing condition (13).
By Walras’ law, the goods market clearing condition (12) can be deduced from theother equilibrium conditions of the economy. Appendix B defines this equilibrium as11

a “mean field game”. This alternative, but equivalent, exposition is particularly usefulto characterize numerically the evolution over time of the wealth distribution, denotedby ϕ(a, t).

3

Fixed capital stock

Before turning to the general equilibrium of the economy, we now characterize thedynamics of wealth accumulation at the household level for a given capital stock k,real interest rate r = f ′ (k) − δ, and wage rate w = f (k) − kf ′ (k). We consider that thereal interest rate r = f ′ (k) − δ is strictly smaller than ρ + σg as, otherwise, by the Euler
equation (9), we cannot have a steady state with constant consumption and wealth per
efficiency unit of labor.
If we ignore the borrowing constraint (6), the dynamics of consumption and wealthaccumulation are jointly determined by the household’s flow of funds constraint (5)and by the Euler equation (9) with equality. Any steady state {c, a} is characterized byc = (r − n − g)a + w,
ρ + σg − rc=(a − ζ)µ ,
γ

(14)
(15)

where the first equation ensures constant wealth and the second constant consumption. Recall that we assume µ < 1 throughout our analysis. Equation (14) implies that
there are two cases to consider: r > n + g and r ≤ n + g.
Let us start by focusing on the former case, where r > n + g. In Figure 1, equation(14) is represented by the ȧt = 0 curve and equation (15) by the ċt = 0 curve. A
first possibility is that the wage rate w is so high that there is no steady state, withevery household’s consumption and wealth (per efficiency unit of labor) diverging to
infinity. Let us now focus on the more interesting situation where the wage rate issufficiently low to ensure the existence of a steady state. When ζ ≥ −w/(r − n − g),there exists two solutions {c, a} to the system given by equation (14) and (15). We
denote by aU the upper solution for wealth and by aS the lower one, as displayed in
Figure 1. When ζ < −w/(r − n − g), only one solution exists, which we denote by aU .
The following lemma, proved in Appendix C.1, gives the corresponding equilibrium
possibilities taking into account the borrowing constraint (6) and the non-negativityof consumption.
Lemma 1 When r > n + g, if the wage rate w is not too high, the steady state with wealth aU
always exists and is unstable. Hence,• if a0 ∈ (aU , ∞), then wealth diverges to infinity;
12


o
n

−w
−w
U
• if a0 ∈ max a, r−n−g , a , then wealth converges to max aS , a when ζ ≥ r−n−g
o
n
−w
−w
when ζ < r−n−g.
and to max a, r−n−ghThis establishes that a critical determinant of wealth dynamics is whether initial wealthis above or below aU . If it is above, then wealth diverges to infinity. If it is below, thenwealth either converges to aS , or to the natural debt limit −w/(r − n − g), or to theborrowing constraint a, whichever exists and is higher. Wealth inequality is therefore
rising over time.

Figure 1: Phase Diagram for r > n + g

Figure 1 displays the corresponding phase diagram for the case where ζ > −w/(r−n − g) and max{aS , a} = aS . For any given level of initial wealth, a household must
be on the trajectory represented by the solid line. Alternative trajectories, represented
by dashed lines can be ruled out. They either converge to zero consumption, which
violates the transversality condition (10), or have exploding consumption and falling13

wealth, which either eventually violates the borrowing constraint (6) if a > ζ or reach
infinite consumption in finite time and result in a < ζ if a ≤ ζ. The lower panel of
Figure 1 shows the increase in wealth ȧt as a function of wealth at .
These diverging wealth dynamics are resulting from the fact that, when µ < 1, beyond a certain wealth level, the accumulation of savings becomes an increasing function of wealth (as can be seen from the lower panel of Figure 1). Intuitively, whenµ < 1, the marginal utility of consumption declines faster than the marginal utilityof wealth, resulting in rising wealth accumulation as households get richer. In otherwords, when µ < 1, wealth is a luxury good. This saving behavior of very wealthy
households is consistent with the empirical evidence provided by Carroll (2000), Dynan, Skinner, and Zeldes (2004), Lockwood (2018), Straub (2019), Fagereng, Holm, andNatvik (2021b), and De Nardi, French, Jones, and McGee (2021).
Let us now turn to the case where r ≤ n + g. There must now be a unique solution{c, a} to the system given by equations (14) and (15). We denote the corresponding
wealth level by aS . The following lemma is proved in Appendix C.2.
Lemma 2 When r ≤ n + g, the steady state with wealth aS is saddle-path stable. Hence,
wealth converges to max{aS , a}.
This establishes that, when r ≤ n + g, inequality is falling over time with all households (with identical labor income) converging to the same wealth level max{aS , a},irrespective of their initial wealth. In fact, once r ≤ n + g, the upper steady state ofLemma 1 ceases to exist, leaving the economy with the lower steady state, which is
now guaranteed to exist provided that aS ≥ a. Figure 2 displays the corresponding
phase diagram.
Lemma 1 and 2 formalize Piketty (2014)’s insight that, when r > n + g, wealthy
households get wealthier over time, resulting in rising inequality; while the opposite
occurs when r ≤ n + g. These dynamics of wealth inequality are induced by thepreference for wealth.
Let us now endogenize the capital stock to investigate how the real interest rate
and the wage rate respond to these dynamics of wealth accumulation. Throughout thefollowing section, we shall consider that the steady state levels of individual wealthaU and aS are functions of k.

4

General equilibrium

We now solve for the general equilibrium of the economy, as implied by Definition 1.
We begin by characterizing the egalitarian steady state whereby all households have

14

Figure 2: Phase Diagram for r < n + g
the same wealth level, equal to k E . This steady state cE , k E is determined bycE = f (k E ) − (δ + n + g)k E ,
′

cE =

(16)

E

µ
ρ + σg + δ − f (k ) E
k −ζ ,
γ(17)

which is obtained by substituting the expression for the interest rate (3), the wage rate(4), and a = k E into the partial equilibrium steady state conditions (14) and (15) of theprevious section.
Let us now introduce an assumption that we will maintain throughout our analysis.
Assumption 1 The equation γ (k + α − ζ)−µ [f (k) − (δ + n + g)k] = [ρ + σg + δ − f ′ (k)]defines α as a decreasing function of k for all admissible values of k.
15

This assumption implies that, in the egalitarian steady state, an exogenous increase
α in wealth reduces households’ propensity to save and therefore reduces the corresponding steady state capital stock k. Not only is it the natural economic scenario
to expect, but it is also technically a very mild condition. Totally differentiating thisequation with respect to k reveals that Assumption 1 must be satisfied for any given
calibration provided that ρ is sufficiently close to n + (1 − σ)g.
The following lemma is proved in Appendix C.3.

Lemma 3 An egalitarian steady state equilibrium cE , k E always exists and, under Assumption 1, it must be unique. This steady state is locally stable if and only if
rE < n + g +µcE,
kE − ζ

(18)

where rE = f ′ (k E ) − δ.
We now consider throughout our analysis that Assumption 1 is satisfied (but we explicitly refer to it in the proofs whenever it is being used).
From the perspective of any household taking the capital stock k E as fixed, theegalitarian steady state corresponds to the stable steady state of the previous section,
which implies aS (k E ) = k E . When rE > n + g, we know by Lemma 1 that, under afixed capital stock equal to k E , there also exists an unstable upper steady state, withwealth level aU (k E ) ∈ (k E , ∞). Hence, if initially the aggregate capital stock is in theneighborhood of k E and the wealth distribution has an upper bound that is much below aU (k E ), then the economy converges to the egalitarian steady state. Conversely, if
the initial wealth distribution has an unbounded support, with a strictly positive massof households with wealth well above aU (k E ), then wealth inequality is growing overtime, which pulls the economy away from the egalitarian steady state. This implies
that a one-off lump-sum redistribution of wealth can have a permanent impact on the
dynamics of wealth inequality.
Let k ∗ denote golden rule level of the capital stock, which maximizes aggregate
consumption in steady state. From the capital accumulation equation (12), it is determined by f ′ (k ∗ ) − δ = n + g. We now characterize the alternative steady state towards
which the economy can converge, which we call the inegalitarian steady state. The
following lemma is proved in Appendix C.4.
Lemma 4 If and only if rE > n + g, the economy can converge to an inegalitarian steady statewith the aggregate capital stock equal to k ∗ , the real interest rate equal to n + g, a mass one ofhouseholds holding wealth max aS (k ∗ ), a , and a zero mass of arbitrarily wealthy households.

16

Clearly, while the aggregate variables converge to a steady state, the wealth distribution is degenerate with a vanishing fraction of households accumulating arbitrarily
large fortunes. This corresponds to a situation of extreme wealth inequality.
As wealthy households accumulate ever more wealth, the capital stock converges
to the golden rule level and the threshold aU (k) diverges (formally limk→k∗− aU (k) =∞). This progressively reduces the mass of wealthy households, which converges tozero. In the limit, the real interest rate is equal to n + g.
As the economy approaches the golden rule, the natural debt limit becomes infinitely low. Hence, consistently with Lemma 1, the wealth of poor households must
converge to max aS (k ∗ ), a . Note that, for the average wealth of households to beequal to k ∗ , we must have aS (k ∗ ) < k ∗ . The proof of Lemma 4 establishes that aS (k ∗ ) <k ∗ if and only if rE > n+g, which turns out to be the necessary and sufficient condition
for the existence of the inegalitarian steady state.
What does extreme wealth inequality imply for consumption? By the household’s
flow of funds constraint (5) with r = n + g, the consumption of poor households with
wealth max aS (k ∗ ), a must be equal to their labor income w(k ∗ ) = f (k ∗ ) − k ∗ f ′ (k ∗ ) =f (k ∗ )−(δ +n+g)k ∗ . Despite extreme wealth inequality, and a possibly very low wealthlevel max aS (k ∗ ), a , this consumption level is higher than in the egalitarian steadystate, where by equation (16) consumption is equal to f (k E ) − (δ + n + g)k E .12 This
paradoxical result is due to the behavior of the richest households who accumulate
ever larger fortunes to satisfy their preference for wealth. This induces the capitalstock to approach the golden rule level, which raises the steady state consumption ofordinary households. In addition, the consumption share of the wealthiest households
tends to zero. On the one hand the richest are getting ever wealthier, but on the other
hand their mass converges to zero and their propensity to consume also converges tozero.
We have therefore established that, over the long-run, the economy can either converge to an egalitarian steady state or to a degenerate inegalitarian steady state. Moreprecisely, we have the following possibilities:• When rE ∈ (−δ, n + g), the economy converges to the egalitarian steady state;E• When rE ∈ n + g, n + g + kµc
E −ζ , depending on the initial wealth distribution,
the economy either converges to the egalitarian or to the inegalitarian steadystate;
E• When rE ∈ n + g + kµc,∞, the economy converges to the inegalitarian steadyE −ζ
state.
12

Recall that k ∗ maximizes f (k) − (δ + n + g)k.

17

In the middle case, if the initial wealth distribution has a thick upper tail, the economy
will converge to the inegalitarian steady state. A one-off redistribution of wealth thatcuts this upper tail can set the economy on an egalitarian path. This shows that, evenover the very long-run, the distribution of wealth can be history dependent.
What are the welfare consequences of these alternative equilibrium possibilities? Ifhouseholds have a preference for holding physical capital and if the borrowing constraint is not binding, then the first fundamental theorem applies and, for any given
initial wealth distribution, the general equilibrium must be Pareto efficient. If households have a preference for holding financial wealth, rather than physical capital, thenasset prices exert a pecuniary externality that could make the equilibrium inefficient.
In fact, whenever rE < n + g, there could exists a rational bubble on an infinitelylived asset (Michau, Ono, and Schlegl, 2023). Such a bubble would generate a Pareto
improvement, by fulfilling households’ preference for wealth without accumulating
capital beyond the golden rule level k ∗ .13
Hence, with a preference for wealth, either of two forces that can push the realinterest rate towards n + g: a thick upper tail to the initial wealth distribution whenrE > n + g and a rational bubble when rE < n + g. This shows that, in a market
economy, the preference for wealth can always induce convergence to the golden rulelevel of the capital stock k ∗ , which maximizes steady state consumption. However,Ewhenever rE < n + g + kµc
E −ζ , it is also possible that the economy converges to the egalitarian steady state without any bubble, resulting in a capital stock above the golden
rule when rE < n + g and below when rE > n + g.
In Appendix D, we show that the possibility of a rational bubble can easily be incorporated within our framework by introducing an infinitely-lived asset that is intrinsically worthless. We illustrate how, when rE < n + g, the bubble can simultaneously
raise total wealth and reduce the capital stock such as to bring it down to the golden
rule level.
Our inegalitarian steady state is degenerate, with a vanishing fraction of the population acquiring a growing share of the capital stock. However, in practice, the preferences of wealthy dynasties might change as their wealth diverges to infinity. To
capture this, we now introduce shocks to the preference for wealth.

5

Preference for wealth shocks

We now assume a two-state Poisson process for the preference for wealth, where j ∈{W, N } denotes a state. In state W the household has a preference for wealth, while in13

This is similar to a dynamically inefficient OLG economy, where a rational bubble can fulfill young
households’ desire to save for retirement without over-accumulating capital.

18

state N it does not. Let λW denote the rate at which households lose their preference for
wealth, i.e. the transition rate from state W to N , and λN the rate at which households
acquire a preference for wealth, i.e. the transition rate from state N to W .
An infinitely lived household should be interpreted as a dynasty. Whenever a parent has a preference for wealth, sometimes known in the literature as a ”capitalist
spirit”, she transmits this trait to her child with probability 1 − λW . Conversely, thedaughter of a parent who does not have a preference for wealth nonetheless has aprobability λN of acquiring it from some role model in society.14
Let ϕj (a, t) denote the density of the wealth distribution in state j ∈ {W, N } atR
Rtime t, where ϕW (a, t) da + ϕN (a, t) da = 1. Appendix E relies on a ”mean fieldgame” formulation of this economy to provide a formal definition of equilibrium inthe presence of the preference for wealth shocks.
We now rely on numerical simulations to investigate the properties of this economy. In this section, we first calibrate the model, before analyzing the properties ofour simulated economy. In the following section, we will consider a number of tax
experiments. Our numerical algorithm, which builds on Achdou, Han, Lasry, Lions,and Moll (2022), is described in Appendix F.

5.1

Calibration

Our model is calibrated to the U.S. economy. We assume a Cobb-Douglas production
function f (k) = Ak α . The capital share α is set equal to 0.3. To match the U.S. GDP
per household in 2012, which was equal to $132 000, the scale parameter A is set equal
to 2885.15 The depreciation rate δ is calibrated such as to match an average wealth per
household in the U.S. in 2012 of $343 000, as reported in Table 1 from Saez and Zucman(2016). This gives δ = 7.55%.
Population grows at n = 1% per year and productivity at g = 2% per year, whichcorresponds to the U.S. averages over the past century. For simplicity, we assume thathouseholds cannot borrow, so a = 0. This is a natural assumption within a dynastic
context, where parents cannot pass on debt to their children. We set the discount rate ρequal to 4% per year and assume additively separable preferences between consumption and wealth, which gives σ = 1.
1−µ −1
Recall that the subutility of wealth is given by v(a) = γ (a−ζ). The reference
1−µwealth level ζ is set equal to 6 months worth of GDP per household.16 The intensity γ
14This narrative is reminiscent of cultural transmission à la Bisin and Verdier (2001). However, endogenizing the process of cultural transmission is beyond the scope of our analysis.
15
The World Bank reports U.S. GDP per capita for 2012 as $51 780, while the U.S. Census Bureau
estimates an average household size of 2.55 for the same year. This implies that GDP per household
was around $132 000.
16
Neither the other parameters of the calibration nor the non-targeted moments from our simulation

19

and the curvature µ of the preference for wealth are calibrated such as to match moments of the U.S. wealth distribution reported in Table 1 of Saez and Zucman (2016).
At the lower end of the wealth distribution, we target the average wealth of the bottom
90% of the distribution to be equal to $84 000; while, at the upper end, we target the
average wealth of the top 1% to be equal to $13.84 million. This yields γ = 7.65 · 10−6
and µ = 0.23.
Regarding the distribution of preferences, we assume that the fraction of households with a preference for wealth is always in steady state, and is therefore equal to
λN /(λN + λW ). We consider that the preference for wealth persists on average for three
generations, 75 years, which gives λW = 1/75. And we assume that only 20% of households have a preference for wealth, which implies that the absence of a preference for
wealth persists on average for 12 generations, 300 years, yielding λN = 1/300. While
we could easily assume a higher share of households with a preference for wealth,this calibration illustrates that only a fairly small share is sufficient to have dramaticconsequences on the dynamics of wealth inequality.
Finally, the initial wealth distribution is partly endogenously determined. Moreprecisely, we start our simulations from a double Pareto-lognormal distribution that
is identical in state W and N and we run our model for 100 years before matching
the moments described in the previous paragraphs. Following the survey by Benhabib and Bisin (2018), the parameter determining the thickness of the upper tail of
the Pareto distribution is set equal to 1.5. Our numerical simulations rely on a wealthgrid going up to $5 billion. We impose an upper bound of $3.5 billion to our initial
Pareto-lognormal distribution, which is rescaled such as to account for the missing
mass. Full details are provided in Appendix F.
As we shall see below, this economy does not necessarily reach a stationary state.
However, the aggregate variables of the economy, such as output or the capital stock,converge after about 50 years and remain almost constant thereafter. Over the following 100 years (i.e. from year 100 to 200 after our initial double Pareto-lognormal distribution), output increases by 0.2%, the capital stock by 0.8%, and the average wealthof the top 1% decreases by 3.9% (less than 0.04% per year). While the wealth distribution keeps evolving, it does so sufficiently slowly to consider that the economy isin a quasi-stationary state 100 years after the beginning of the simulations. Moreover,wealth dynamics over thousands of years are of little empirical relevance to analyse
wealth inequality within our modern world. The calibration is summarized in Table1.
are very sensitive to the choice of ζ. Choosing instead one or two years worth of GDP would hardly
modify our findings.

20

Parameter
Capital intensity
Scale parameter
Depreciation rate
Population growth rate
Productivity growth rate
Borrowing limit
Discount rate
Complementarity parameter
Reference wealth level
Concavity parameter
Intensity parameterTransition rate from W to N
Transition rate from N to W

Calibration
α = 0.3
A = 2885
δ = 7.55%
n = 1%
g = 2%a=0
ρ = 4%σ=1
ζ = −66000µ = 0.23
γ = 7.65 · 10−6
λW = 1/75
λN = 1/300

MomentCapital share
U.S. GDP per household in 2012
Average household wealth in 2012
Long-run U.S. averageLong-run U.S. average
No borrowing·
Additively separable preferences
6 months worth of GDP per household
Average wealth of bottom 90%
Average wealth of top 1%
Average persistence of state W of 75 years
20% of households in state W

Table 1: Calibration

5.2

Simulation

Let us now investigate the properties of our simulated economy. Throughout this section we focus on the equilibrium at the point in time used for the calibration (which is
100 years after the double Pareto-lognormal distribution used to initiate our simulation).
Table 2 contrasts the moments from our simulation with their empirical counterparts in the U.S., where the matched moments are highlighted in bold. Our model successfully matches the thickness of the upper tail of the U.S. wealth distribution. Our
calibration strategy forces our model to match the average wealth of the bottom 90%of the wealth distribution. However, our model induces 72% of households to be at
the borrowing limit, while in the data only 20% of U.S. households have non-positive
wealth (Wolff, 2021). This gross overestimation of the share of households with zero
wealth is clearly due to the absence of precautionary saving. However, such precautionary saving of poor households would not fundamentally affect the dynamics ofwealth inequality that we emphasize in this paper.
Recall that in a neoclassical growth model, without a preference for wealth, the interest rate converges to ρ+σg = 6% (which follows from the Euler equation (9) without
a preference for wealth, i.e. with γ = 0). In the previous section, we have shown thatwith a representative household who has a preference for wealth, in the inegalitarian
steady state, the interest rate converges to n + g = 3%.17 In our calibration, the interest
17

Our calibration implies that, in the absence of wealth preference shocks, the economy would converge to the inegalitarian steady state since the initial wealth distribution has a thick upper tail and wehave rE > n + g, where rE = f ′ (k E ) − δ with k E and cE jointly defined by (16) and (17).

21Moment
U.S. GDP per household in 2012
Average wealth per household in 2012
Average wealth: Bottom 90%
Average wealth: Top 10%
Average wealth: Top 1%
Average wealth: Top 0.1%
Average wealth: Top 0.01%
Top 10% wealth percentile
Top 1% wealth percentile
Top 0.1% wealth percentile
Top 0.01% wealth percentile
Wealth share: Bottom 90%
Wealth share: Top 10%
Wealth share: Top 1%
Wealth share: Top 0.1%
Wealth share: Top 0.01%
Share of households with a ≤ 0

Empirical
13200034300084000
2.56 mil.
13.84 mil.
72.8 mil.
371 mil.
6600003.96 mil.
20.6 mil.
111 mil.
0.228
0.772
0.418
0.220
0.112
0.198

Model
13200034200084000
2.67 mil.
13.82 mil.
84.06 mil.
348.67 mil.
1.04 mil.
2.69 mil.
22.7 mil.
185 mil.
0.221
0.779
0.404
0.246
0.102
0.722

Table 2: Comparison of moments
Sources: GDP per household in 2012 is obtained by combining data for GDP per capita from the WorldBank with the average household size in 2012 from the U.S. Census Bureau. Data for average wealth,wealth percentiles and wealth shares in 2012 are from Table 1 of Saez and Zucman (2016). Data on theshare of households with non-positive wealth is taken from Table 1 of Wolff (2021), where we report the
average value over 2001-2019.

rate is equal to 4.02%. Hence, even though only 20% of households have a preference
for wealth, their saving behavior substantially raises the capital stock relative to theneoclassical growth model. Note that, at the golden rule, the capital stock would be
13.8% higher and output 4.0% higher; while, in the absence of preference for wealth,
the steady state capital stock would be 20.3% lower and output 6.6% lower.
Figure 3 displays the saving and consumption behavior of households. The upper
two panels depict the increase in wealth ȧj0 at time 0 of households of type j ∈ {W, N }as a function of their wealth aj0 . The upper left panel focuses on the saving behaviorof households with wealth up to $20 million, while the upper right panel raises thatthreshold to $500 million.18 Wealth accumulation in both states exhibits a U-shaped
pattern, similar to the one depicted in the lower panel of Figure 1 for the partial equilibrium case. Under our calibration, households with a preference for wealth always
accumulate wealth (as shown by the solid line). By contrast, households without apreference for wealth dissipate wealth, unless their own wealth level is above $250
18

Our wealth grid goes up to $5 billion but, under our simulation, less than 0.0016% of households
owning 2.99% of aggregate wealth are above $500 million.

22

million, beyond which they also accumulate wealth (as shown by the dashed line).
This saving behavior of wealthy households of type N is due to their anticipation thatthey might eventually regain the preference for wealth.

Figure 3: Saving and consumption behavior

The fact that wealth diverges above a certain threshold regardless of the statej ∈ {W, N } suggests that, even with wealth preference shocks, the economy neverconverges to a stationary state. However, this divergence of very wealthy households
appears to be of limited empirical relevance over time horizons of interest. At time 0(which is 100 years after the double Pareto-lognormal distribution), less than 0.0051%
of households of type N own more than $250 million and their cumulative wealthamounts to 4.8% of the total wealth of the economy. Our simulation over several centuries shows that the dynamics of wealth inequality is primarily driven by the rising
wealth of households of type W and the falling wealth of households of type N , notby the rising wealth of a very small number of households of type N . Thus, comparedto the previous section, the wealth preference shocks considerably slow down the diverging wealth accumulation dynamics, to such an extent that the aggregate variablesof the economy are almost constant over time.19
19

Alternatively, to have a stationary wealth distribution, we could assume that households who do
not have a preference for wealth either fail to foresee that some of their descendants might have a

23

The lower left panel of Figure 3 plots the consumption functions, where cj0 denotes
the consumption at time 0 of households of type j ∈ {W, N } as a function of theirwealth aj0 . Households with a preference for wealth choose to consume much less andsave much more for any given wealth level than households who do not directly value
wealth.
The last panel of Figure 3 depicts the saving rates sj0 at time 0 of households oftype j ∈ {W, N } as a function of their wealth aj0 . Let Ajt denotes real wealth, with
ajt = Ajt /(Gt Nt ) being wealth per efficiency unit of labor. The household’s flow offunds given by (5) can equivalently be written as Ȧjt = rt Ajt + wt Gt Nt − cjt Gt Nt . Hence,the saving rate should be defined as sjt = Ȧjt /(rt Ajt + wt Gt Nt ), which can equivalently
be written asȧj + (g + n)ajt
sjt = t.
(19)
rt ajt + wt
As the economy is growing over time, some households of type N can simultaneously
have a positive saving rate, i.e. sNt > 0, and a decreasing level of wealth per efficiency
Nunit of labor, i.e. ȧt < 0. This shows that, in a growing economy, the dynamics ofwealth inequality is primarily driven by changes in wealth per efficiency unit of laborȧjt rather than by the saving rate sjt , despite the later being commonly emphasized inpolicy discussions.
As households of type W always accumulate wealth, none of them is borrowing
constrained. By contrast, 90.2% of households of type N are borrowing constrained.
The average wealth of households is equal to $1.23 million for type W and to $120 000
for type N (and to $1.22 million for the 9.8% of households of type N who are notborrowing constrained). Thus 72.12% of the wealth of the economy is owned by the
20% of households who are of type W . However, among the wealthiest households, a
sizeable fraction is of type N . Households of type N account for 10.9% of households
within the top 10% of the wealth distribution, 32.0% of households within the top 1%,
and 50.4% of households within the top 0.1%. Many of these wealthy individuals oftype N belong to dynasties that were formally of type W . Despite this composition
effect, the average saving rate is increasing in wealth: the saving rate reaches 41.5%
for the top 1%, 30.9% for the following 9%, and only 2.1% for the bottom 90%.
preference for wealth or never value the preference for wealth of their descendants (which corresponds
to ”imperfect empathy” à la Bisin and Verdier (2001)). Both interpretations imply that the HamiltonJacobi-Bellman equation of households of type N should be written with λN = 0.

24

6Wealth taxation

Let us now rely on numerical simulations to investigate the effects of the taxation of
wealth. We first consider a wealth levy, before turning to a progressive tax on wealth,and finally to a progressive consumption tax.

6.1

Wealth levy

A wealth levy is a one-off lump-sum redistribution of wealth. In 1945, after World War
II, France implemented a wealth levy of up to 20% (and up to 100% on war profits). In
1947, about 40% of Japan’s cultivated land was redistributed, from landlords to theirtenants. While such wealth levies are exceptional, they offer a good way to analyze thegeneral equilibrium effects of wealth redistribution within our model. We therefore
consider a 20% levy on wealth above a $10 million threshold, redistributed lumpsum to borrowing constrained households with zero wealth. This levy raises $16 026,
amounting to 4.7% of the total wealth of the economy. It finances a lump-sum transfer
of $22 199 for each borrowing-constrained household, amounting to 24.0% of theirannual labor income.
Figure 4 shows the impact of the wealth levy on the aggregate variables of theeconomy. For each variable, the dashed line displays the trajectory that the economy
would have followed in the absence of the levy, while the solid line shows the trajectory following the implementation of the policy. Borrowing constrained households,
who are of type N , spend their lump-sum transfer within less than 6 years. This results
in a consumption boom. However, on impact, output remains equal to f (k0 ). Hence,the 5.3% surge in consumption leads to a 14.0% drop in investment. This induces acontraction of the capital stock of up to 3.1%. Hence, output and wages fall, while theinterest rate rises. Once the recipients have spent the transfer, aggregate consumption
falls. This allows investment, and subsequently the capital stock, to recover.
Changes in future interest rates and wages induce strong general equilibrium effects. Hence, households who are not directly affected by the wealth levy modify their
behavior. In particular, households with positive but limited wealth respond to higher
interest rates and lower wages by reducing their consumption and raising their saving,
thereby limiting the contraction in investment.
Figure 5 displays the consumption-equivalent welfare gain of households at time
0 as a function of their type j ∈ {W, N } and of their wealth aj0 (before the implementation of the levy). As expected, the mechanical impact of the policy induces a welfare
gain for households with zero wealth, who benefit from the transfer, and a welfare
loss for very wealthy households, who are subject to the levy. More interestingly, thegeneral equilibrium effects generate a sizeable welfare gain for moderately wealthy25

Figure 4: Effects of a 20% wealth levy on macroeconomic aggregateshouseholds of type W with wealth between $152 000 and $11.7 million and of type Nwith wealth between $825 000 and $11.7. Their saving is rewarded by a higher interest
rate, which is particularly beneficial to households of type W who enjoy accumulating
wealth. Conversely, the capital levy is detrimental to low wealth households of type
W who own less than $152 000 and of type N who own less than $825 000. Having
non-zero wealth, they do not receive the transfer and they lose from the reduction in
the wage rate induced by the fall in the capital stock.
These general equilibrium effects are the consequence of the heterogeneity in saving rates. In the neoclassical growth model, i.e. without any preference for wealth, theinterest rate converges to ρ + σg and all households consume their permanent income(r − n − g)a + w. Hence, the marginal propensity to consume out of wealth is identical,
and equal to r − n − g, for all households. This implies that a wealth levy does not have
any impact on capital accumulation, or on any other macroeconomic aggregate of theeconomy. Hence, the middle class, who neither pay the levy nor receive the transfer,
are not affected by this policy.
By contrast, under a preference for wealth, the tax levy redistributes resources fromrich households with a high saving rate to poor households with a low saving rate,

26

Figure 5: Consumption-equivalent welfare gain from the wealth levy

resulting in a fall in aggregate saving and, hence, in investment.20 To be more precise,an individual with wealth a has income ra+w. So, when one dollar is taken away froma wealthy individual, her income falls by r dollars and her consumption by (1 − s)r,which is typically very small since her saving rate s is high, especially if she is of typeW (as can be seen from the lower left panel of Figure 3). For instance, with r = 4%
and s = 75%, a dollar taken from a wealthy individual reduces her consumption by
only 1 cent. By contrast, poor households at the borrowing constraint have a very highmarginal propensity to consume out of wealth and therefore choose to spend much ofthat dollar of transfer over a short period of time. So, the increase in the consumptionof the poor is an order of magnitude larger than the decline in the consumption of therich. Aggregate consumption is therefore bound to surge, and investment bound tocontract.
In a nutshell, the levy transforms the stock wealth into a flow of consumption.
While this redistribution policy trivially satisfies the government budget constraint,
it cannot bypass the aggregate resource constraint, which entails a meaningful tradeoff: the consumption boom of the poor occurs at the expense of investment and of
capital accumulation. This is detrimental to poor households who do not receive theR∞
N
Formally, a [sWt (a)[rt a + wt ]ϕW (a, t) + st (a)[rt a + wt ]ϕN (a, t)]da = it − δkt , where it denotes theflow of investment per efficiency unit of labor. This expression follows from the capital accumulation
equation (12), which gives it = k̇t + (δ + n + g)kt , together with the asset market clearing equation (13),and the definition of the saving rate (19).
20

27

transfer. Moreover, the tax levy results in permanently less wealth in the hands of richhouseholds with very high saving rates, leading to a permanently lower capital stock.
Under a dynastic interpretation of our infinitely-lived households, this is detrimental
to future generations of poor households who will suffer from lower wages.21
While a levy on initial wealth used to finance redistribution to poor households isa free lunch within the neoclassical growth model, such is not the case under a preference for wealth. This is fundamentally due to the heterogeneity in saving behavior
across households with different wealth levels. Hence, these insights are valid under alternative microfoundations leading to heterogeneity in saving behavior, such as
differences in discount rates.

6.2

Progressive wealth tax

Wealth inequality in the U.S. is high. This has induced some politicians to advocate
for a progressive wealth tax. In 2019, Elizabeth Warren proposed to tax at 2% per year
wealth above $50 billion, and at 3% per year wealth above $1 billion. Bernie Sanders
proposed an even more progressive tax, starting at a rate of 1% above $32 million andrising to 8% above $10 billion. Saez and Zucman (2019a,b) even considered a ”radical
wealth tax”, with a constant marginal rate of 10% above $1 billion. The aim of thispolicy would not be to collect revenue, but to ”deconcentrate wealth”.
Until 2017, France has had a progressive wealth tax with a marginal tax rate of
0.7% above $1.4 million, rising up to 1.5% above $10 million. However, ”professional
wealth” (defined as wealth related to the primary source of labor income) was exempt,
allowing many wealthy entrepreneurs to avoid this tax. In 2018, this tax became a
progressive real estate tax, with similar rates and rules.
While progressive wealth taxes are rather uncommon, bequests taxes are widespread.
They are typically progressive, with a maximum rate that was equal in 2013 to 45% in
France, 40% in the U.K., 35% in the U.S., and 30% in Germany (Piketty, 2014). From the
1950s to the 1970s, this maximum rate in the U.K. and the U.S. rose above 70%. Recall
that the preference for wealth of an infinitely-lived household can be interpreted as abequest motive within an altruistic dynasty. This makes our framework particularly
suitable to investigate the taxation of dynastic wealth. Note that when the interest rateis equal to 4%, a 40% bequest tax levied once a generation, i.e. once every 25 years, isroughly equivalent to a 2.1% annual wealth tax, since 1.0425 (1 − 0.4) = (1.04 − 0.021)25 .
In light of these policies, we simulate a progressive wealth tax with a 2% marginal
rate per year above a threshold of $10 million. The proceeds are redistributed to house21

This is also detrimental to the future selves of the receivers, but this is not detrimental from awelfare perspective since they could have chosen to save the money and chose not to. By contrast,
future generations of not-yet-born individuals could not have made that choice.

28

holds at the borrowing constraint (or just above such as to avoid taxing the wealth ofvery poor households at more than 100%). Initially, 0.23% of households have wealthabove $10 million. They own 30.0% of the capital stock of the economy. The tax finances a transfer of $2 216 per household with zero wealth (and a transfer of size
$2 216 − a for households with initial wealth a below $2 216). At the borrowing constraint, this amounts to 2.4% of annual labor income.
Figure 6 displays the impact of this policy on the main macroeconomic aggregatesof the economy. Unlike the wealth levy, this wealth tax yields a continuous flow ofincome to its recipients. Hence, the transfers are all spent instantly by borrowing constrained households, who therefore remain borrowing constrained. The continuous
flow of transfers therefore induces a much more persistent boom in consumption than
the wealth levy. The flip side is a more persistent decline in investment, resulting ina contraction of the capital stock of up to 3.1% (relative to the trajectory without thetax). Output falls by up to 0.9% and wages by up to 0.9%.

Figure 6: Effects of progressive wealth tax on macroeconomic aggregates

To fully account for these macroeconomic effects, we need to understand how
different households respond to this progressive wealth tax. Figure 7 displays theincrease in saving ∆ȧj0 and in consumption ∆cj0 at time 0 for households of typej ∈ {W, N } as a function of their wealth aj0 . By each household’s flow of funds con29

straint (5), the sum of these two responses must be equal to the transfer received netof the tax paid.22 Borrowing constrained households raise their consumption by theamount of the transfer and their wealth remains equal to zero. Type N householdswith wealth below $578 000 raise their consumption to deplete their wealth such as
to qualify for the transfers. Very wealthy households who pay the tax respond by reducing their saving, but also by raising their consumption. Higher consumption isthe consequence of the tax-induced distortion to their intertemporal allocation of consumption. Type W households who are just below the $10 million threshold are nolonger expecting to cross it. They therefore anticipate their future saving to be lower
and their future consumption to be higher. This induces them to raise their current
consumption even though, in equilibrium, they will never pay the tax. Householdswith smaller wealth respond to the higher interest rates and the lower wages by reducing their consumption and raising their saving, as was the case under the tax levy.

Figure 7: Effects of progressive wealth tax on impact

The sharp decline in saving above the $10 million threshold implies that, over time,
the tax successfully compresses wealth inequality. This is shown by the top three panels of Figure 8. The average wealth of the top 1% shrinks by 60.8% over a century.
As the capital sock falls by a much smaller amount, the average wealth of the top
10% to 1% increases considerably, by 46.7% over a century. This sharp reduction in
wealth inequality has an ambiguous impact on consumption inequality, as shown by
Formally, for any wealth level a, we have ∆ȧj0 (a) + ∆cj0 (a) = max{T0 − a, 0} − τ max{a − aT , 0},
where T0 denotes the initial transfer, τ that tax rate, and aT the exemption threshold.
22

30

the bottom three panels of Figure 8. Initially both the top 1% and bottom 90% raise
their consumption, while the remaining 9% are induced to consume less by generalequilibrium effects, which allows them accumulate more wealth. Over the long-run,the smaller wealth of the top 1% induces them to consume less, while the much higher
wealth of the following 9% allows them to consume more.

Figure 8: Redistributive effects of progressive wealth tax

The fall in the average wealth of the top 1% implies that tax revenue contracts overtime. Hence, the magnitude of the transfer to the poor converges to zero, as shown by
the left panel of Figure 9. Hence, the consumption of borrowing constrained households eventually converges back to their labor income, which is reduced by the progressive wealth tax due to a smaller capital stock. The right panel of Figure 9 displays
the sum of the wage rate and of the transfer, which corresponds to the consumption ofborrowing-constrained households. It shows that, 38.8 years after its implementation,
the progressive wealth tax results in lower consumption for poor households. Overthe long-run, the progressive wealth tax benefits the upper-middle class, rather thanthe poor.
In the neoclassical growth model, i.e. without the preference for wealth, such a
progressive wealth tax would initially entail comparable general equilibrium effects,
31

Figure 9: Effects of progressive wealth tax on the transfer to the poor

since the tax would also induce wealthy households to reduce their saving. However,as the wealth of these households falls below the $10 million threshold, the interest
rate would converge back to ρ + σg and the aggregate variables of the economy would
return to the same steady state as before. Hence, the progressive wealth tax would permanently reduce wealth inequality, without reducing the long-term capital stock. By
contrast, with a preference for wealth, the progressive wealth tax prevents the concentration of wealth in the hands of few dynasties that are so wealthy that their propensity
to save is very high. This permanently reduces the capital stock.
Figure 10 displays the consumption-equivalent welfare gain of households at time
0 as a function of their type j ∈ {W, N } and of their wealth aj0 . The result are qualitatively similar as for a wealth levy: very wealthy households loose from the tax,
borrowing-constrained households gain from the transfer, while moderately wealthy
households benefit from a higher interest rate that rewards their high propensity tosave, especially if they are of type W and therefore enjoy accumulating wealth.
Social welfare. An infinitely-lived household should be interpreted as a dynasty. By
focusing on welfare at time 0, we only attach some weight to the welfare of future generations through the altruism of their parents. Phelan (2006) and Farhi and Werning(2007) have forcefully, and correctly, argued that the social planner can also directly
care about future generations of not-yet-born individuals, independently of the altruism of their parents. Let θ ∈ [0, ∞) denote the social discount factor. Over a time horizonof length T , social welfare can therefore be defined as
Z T
W (θ) =0θe−θtv̄t dt,1 − e−θT(20)

where v̄t denote social welfare for the generation alive at time t.23 When θ = ∞,23

Such a formulation is more natural in a discrete time setting where one time period corresponds

32

Figure 10: Consumption-equivalent welfare gain at time 0

the planner does not directly care about future generations and we have W = v̄0 .
And when θ = 0, the planner cares equally about all future generations and we haveRT
W = (1/T ) 0 v̄t dt.24 Social welfare within generation t can be utilitarian, which givesR∞v̄t = a [vW (a, t)ϕW (a, t) + vN (a, t)ϕN (a, t)]da, where vj (a, t) denote the welfare at time
t of a household of type j with wealth a. Alternatively, social welfare at t can be
Rawlsian with v̄t = vN (a, t).
Figure 11 displays the consumption-equivalent social welfare gain from the implementation of the progressive wealth tax as a function of the social discount rate θ.25
The solid line corresponds to the utilitarian case, and the dashed line to the Rawlsiancase. As the social discount rate θ goes down, the planner cares more about future generations and, in the Rawlsian case, social welfare is lower. In fact, whenever θ is below
0.045, the progressive wealth tax generates a social welfare loss. This results from theadverse effect of the progressive wealth tax on the capital stock, on wages and, hence,
to one generation. Our formulation can be seen as the limit as the life span of each generation tends
to zero. As we focus on the long-term evolution of inequality, this simplification has no bearing on thequalitative insights from our analysis.
24
While in theory the length T of the horizon can be arbitrarily large, our simulations take T = 300
years.
25
Given that we have logarithmic utilityR of consumption, the consumption-equivalent social welfare∞
gain x is simply determined by ∆W (θ) = 0 e−(ρ−n)s ln(1 + x)ds, where ∆W (θ) is the increase in social
welfare due to the tax policy. This gives x = e(ρ−n)∆W (θ) − 1. It corresponds to the proportional increasein consumption within the pre-reform allocation that yields the same social welfare as obtained after
the implementation of the tax policy.

33

on the welfare of future generations of poor households. In particular, the Rawlsianplanner with θ = 0, which Piketty and Saez (2013) emphasize as the ”meritocartic
Rawlsian steady state” benchmark, rejects our progressive wealth tax.

Figure 11: Social welfare gain

Figure 12 shows the consumption-equivalent social welfare gain for the generation
alive at time t.26 The Rawlsian case shows that the progressive tax policy fails to raise
the welfare of poor households born more than 16.5 years after the implementation of
the policy. This explains the social welfare loss displayed in Figure 11 for θ sufficiently
small.
Figure 11 shows that, in the utilitarian case, the progressive wealth tax always enhances social welfare. Moreover, the social welfare gain is larger if the planner cares
more about future generations. Figure 12 shows that this is driven by the benefit thathouseholds of type W derive from the policy. Indeed, the progressive wealth tax induces a more egalitarian distribution of wealth, which benefits upper-middle classhouseholds of type W . Future generations of households who value owning some assets (such as their house) are the main beneficiary of the progressive wealth tax thatprevents the concentration of wealth within the top 1% of the population.
26

Formally, it displays e(ρ−n)∆v̄t −1 as a function of t, where ∆v̄t denotes the increase in social welfare
due to the tax policy for the generation alive at time t.

34

Figure 12: Consumption-equivalent social welfare gain for each generation

6.3

Progressive consumption tax

The problem with the progressive wealth tax is that it penalizes wealthy householdsof type W who did not intend to spend their wealth. A progressive consumption
tax seems preferable since, conditional on wealth, it targets the big spenders of type
N rather than the savers of type W . Let us therefore investigate such a progressive
consumption tax. Note that, in practice, a non-linear consumption tax can be implemented by taxing total income ra + w net of saving ȧ + (n + g)a. As the saving entails
a tax deduction, households have no incentive to under-report them.
We consider a 50% tax on consumption in excess of a threshold of $200 000 per year.
As before, the proceeds are redistributed to households at the borrowing constraint (orjust above, as for the progressive wealth tax). At time 0, this tax is paid by 0.25% ofhouseholds, which is comparable to the 0.23% subject to the progressive wealth tax inthe previous section. Initially, the transfer amounts to $426. While this is only a fifthof the initial transfer under the progressive wealth tax, as we shall see, the amount of
the transfer is substantially more persistent over time.27
Figure 13 displays the impact of the progressive consumption tax on the main
macroeconomic aggregates of the economy. Initially, households with very low wealthdeplete their wealth such as to qualify for the transfer, which results in a small consumption boom. Wealthier households, who would have consumed more than $200 000
in the absence of the tax, significantly reduce their consumption. This leads to a protracted fall in consumption, which raises investment, the capital stock, and output.
While the magnitude of the effect is moderate, this results in higher wages, which is
the opposite of the effect of the progressive wealth tax.
Figure 14 shows that the progressive consumption tax hardly compresses wealth
27
The rate of our progressive consumption tax is below the peak of the Laffer curve. Reducing themarginal tax rate above $200 000 to 49% reduces tax revenue at time 0.

35

Figure 13: Effects of consumption tax on macroeconomic aggregatesinequality. However, it sharply and persistently reduces consumption inequality. Figure 15 shows that the policy results in a permanent transfer of wealth from rich to poor
households, unlike the progressive wealth tax of the previous section.
Figure 16 shows that, in both the utilitarian and the Rawlsian case, the progressive
consumption tax generates a positive social welfare gain for any value of the social discount rate θ. The social welfare gain is even larger in the Rawlsian case since the policy
is particularly beneficial to the poor. Over the long-run, the progressive wealth tax redistributes welfare from the very rich to the upper-middle class, while the progressive
consumption tax redistributes from the rich to the poor. This is due to the eliminationof adverse general equilibrium effects that hinder the accumulation of capital.
While we have taken the preference for wealth as exogenously given, a consumption tax could alter that preference. In particular, it could reduce the extent to whichhouseholds of type W enjoy accumulating wealth, even if they had no intention of
spending it. This could greatly diminish the merits of a progressive consumption tax.
While these considerations are important, the endogeneity of the preference for wealthto the tax system is beyond the scope of our analysis and therefore left for future research.
Finally, relaxing the assumption of perfectly competitive markets, it is interesting36

Figure 14: Effects of progressive consumption tax on macroeconomic aggregates

Figure 15: Effects of progressive consumption tax on the transfer to the poor

to think about the special case of the luxury industry and of its interaction with wealthinequality. Rich dynasties that have lost their preference for wealth are likely to spend
lavishly on luxury goods. Indeed, over the previous decades, the rise in wealth inequality throughout the world has fueled an unprecedented expansion of the luxury
industry. Given the power of branding, this industry is characterized by very highmark-ups. Hence, this conspicuous consumption shifts wealth to the owners of the37

Figure 16: Social welfare gain

luxury brands, such as Bernard Arnault (the owner and CEO of LVMH) who has become in 2023 the wealthiest individual on earth and who appears to have a strong
preference for wealth. This suggests that the luxury industry may, to some extent, act
as a progressive consumption tax that redistributes its proceeds to billionaires with apreference for wealth, which contributes to the accumulation of capital. Our analysis
suggests that, however frivolous, this conspicuous consumption is not as wasteful asfirst meets the eye.28

7Conclusion

The preference for wealth captures the essence of the spirit of capitalism, which entails frugality and thriftiness. According to Weber (1930): “When the limitation of consumption is combined with this release of acquisitive activity, the inevitable practical result
is obvious: accumulation of capital through ascetic compulsion to save”. Our analysis has
shown that this spirit of capitalism can indeed be an essential driver of the enduringprosperity of a capitalist economy.
If the overwhelming majority of the population has a high discount rate and a weak28

In The Theory of Moral Sentiments (1759), Adam Smith argues that the ”invisible hand” operates
through the consumption of luxuries, which redistributes the excesses of wealth inequality. By contrast,the ”invisible hand” of The Wealth of Nations (1776) emphasizes the social usefulness of investment andcapital accumulation. For an interesting discussion of these two ”invisible hands”, see Brewer (2009).

38

propensity to save, how will the economy accumulate the capital it needs for long-term
prosperity? By concentrating wealth in the hands of few households who, thanks totheir spirit of capitalism and to their immense wealth, have high saving rates! Our
analysis therefore extends the scope of the invisible hand. Indeed, without a preference for wealth, the market mechanism fails to ensure the well-being of the not-yetborn descendants of non-altruistic parents. Thanks to the preference for wealth, thegreed of wealthy households serves the interests of these future generations of workers, even though this was not part of their intention. The spirit of capitalism ensuresthat markets are not only efficient, but also caring about future generations. Moreover, wealth inequality only expands when it is needed to foster capital accumulation.
Indeed, if thriftiness is sufficiently widespread, we end up with r < n + g, which
spontaneously induces convergence towards an egalitarian distribution of wealth.29
Empirically, the saving rate is strongly increasing in wealth. This implies that redistribution from rich to poor is bound to reduce the supply of savings, which is detrimental to future generations. An alternative would be for the government to implement a progressive wealth tax and to save, rather redistribute, the proceeds. However,in addition to hindering the incentives to work hard to accumulate wealth, such a policy would reduce the welfare that thrifty households derive from holding wealth andit would probably deteriorate the quality of investment as rich households are likely to
be more careful when investing their own money than a government agency is wheninvesting public money.
A progressive wealth tax induces general equilibrium effects that benefit the propertyowning upper-middle class. A progressive consumption tax targets wealthy households with a low propensity to save, which makes it preferable for redistribution to
the poor. In many countries, wealthy households already face a progressive consumption tax: the personal income tax is progressive and wealthy households can avoid
it by reinvesting their income from capital within their firm. Our work provides ajustification for this tax practice, which is widespread albeit controversial.
While we have assumed a neoclassical economy, our results can be overturned in a
Keynesian setup, at least in the short-run. When aggregate demand is deficient, wealthredistribution from rich to poor households raises consumption, output, and, hence,
investment and capital accumulation. This a consequence of the paradox of thrift: adecline in the aggregate propensity to save raises the volume of aggregate savings.30
In fact, in practice, the existence of nominal rigidities implies that progressive wealth29

Our earlier research shows that, in this case, the preference for wealth can also be a source of efficiency by generating a rational bubble that prevents an over-accumulation of capital (Michau, Ono, andSchlegl, 2023).
30
Such Keynesian insights can be obtained under a preference for wealth by simply introducing
money and assuming a downward nominal wage rigidity (Ono, 1994, 2001; Michau, 2018).

39

taxation is likely to create a spending boom and, hence, an investment boom in the
short-run. This suggests that the adverse consequences of progressive wealth taxation that we have emphasized throughout this paper are likely to be hard to identify
empirically following tax reforms, even if they are predominant over the long-run.
Clearly, our analysis has abstracted from major political economy problems causedby an extreme concentration of wealth. Billionaires can buy influence and distort thepolitical process to their advantage, as the robber barons showed in the U.S. in the late
19th century. More generally, excessive wealth inequality can trigger revolutionary
forces that should be avoided through progressive wealth taxation (Farhi, Sleet, Werning, and Yeltekin, 2012; Farhi and Werning, 2014). Restraining wealth inequality may
therefore be necessary to preserve a healthy balance of power in society, even if thisreduces the accumulation of capital.

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46

A

Piketty (2014)’s theory of wealth inequality

In his introduction (page 34), Piketty (2014) argues that:
In slowly growing economies, past wealth naturally takes on disproportionate importance, because it takes only a small flow of new savings toincrease the stock of wealth steadily and substantially.
If, moreover, the rate of return on capital remains significantly above thegrowth rate for an extended period of time, then the risk of divergence inthe distribution of wealth is very high.
When the rate of return on capital significantly exceeds the growth rate of
the economy (as it did through much of history until the nineteenth century
and as is likely to be the case again in the twenty-first century), then it logically follows that inherited wealth grows faster than output and income.
People with inherited wealth need save only a portion of their income from
capital to see that capital grow more quickly than the economy as a whole.
Under such conditions, it is almost inevitable that inherited wealth will
dominate wealth amassed from a lifetime’s labor by a wide margin, andthe concentration of capital will attain extremely high levels.
In chapter 10 (page 442), he further explains:
The primary reason for the hyperconcentration of wealth in traditional
agrarian societies and to a large extent in all societies prior to World War I isthat these were low-growth societies in which the rate of return on capital
was markedly and durably higher than the rate of growth.
For example, if g = 1% and r = 5%, saving one-fifth of the income from capital (while consuming the other four-fifths) is enough to ensure that capitalinherited from the previous generation grows at the same rate as the economy. I one saves more, because one’s fortune is large enough to live wellwhile consuming somewhat less of one’s annual rent, then one’s fortune
will increase more rapidly than the economy, and inequality of wealth will
increase even if one contributes no income from labor. For strictly mathematical reasons, then, the conditions are ideals for an ”inheritance society”
to prosper – where by ”inheritance society” I mean a society characterized
by both a very high concentration of wealth and a significant persistence oflarge fortunes from generation to generation.
Piketty (2014) hardly delves into the general equilibrium consequences of the r > g
dynamics of wealth inequality. In chapter 10 (page 455), he nonetheless briefly men47

tions ”the forces that can avoid an indefinite inegalitarian spiral and stabilize inequality of wealth”, emphasizing that:
if the fortunes of wealthy individuals grow more rapidly than average income, the capital / income ratio will rise indefinitely, which in the long run
should lead to a decrease in the rate of return of capital. Nevertheless, thismechanism can take decades to operate.
By contrast, we show that, unless a high propensity to save is so widespread that theeconomy converges to an egalitarian steady state, the economy reaches an equilibrium
where r asymptotically converges to g, the capital stock converges to the golden rulelevel, and wealthy inequality grows without bounds. Hence, this process does neither”avoid an indefinite inegalitarian spiral” nor ”stabilize inequality of wealth”.

B

Mean field game

This appendix offers a ”mean field game” representation of the equilibrium given by
Definition 1.
The household’s problem consists in maximizing intertemporal utility (8) subject
to the flow of funds constraint (5) and the borrowing constraint (6), for a given level of
initial wealth. The following lemma defines the detrended value function v(a, t) thatwill be constant in steady state.
Lemma A1 The Hamilton-Jacobi-Bellman equation for the detrended value function v(a, t) is
given by(ρ − n − (1 − σ)g) v(a, t) = max
c

i

h
1−µ
−1
−1
exp (1 − σ) ln (c) + γ (a−ζ)
1−µ1−σ+va (a, t) [(rt − n − g) a + wt − c] + vt (a, t), (A1)

where va (a, t) = ∂v (a, t) /∂a and vt (a, t) = ∂v (a, t) /∂t.
Proof. The Hamilton-Jacobi-Bellman equation corresponding to the household’s problem is given by(ρ − n) ṽ (a, t) = max
c


h
i
1−µ
−1
exp (1 − σ) gt + ln (c) + γ (a−ζ)
−1
1−µ
1−σ
+ṽa (a, t) [(rt − n − g) a + wt − c] + ṽt (a, t) ,48

where ṽ (a, t) denotes the value function for the original (non-detrended) household’s
problem. Let c (a, t) denote the corresponding policy function. Using the intertemporal utility function (8), we know that the value function is defined as
Z ∞ṽ (a, t) =e−(ρ−n)(s−t)


h
i
1−µ
−1
exp (1 − σ) gs + ln (c(a, s)) + γ (a−ζ)
−1
1−µ

t

Z ∞

ds,
1−σ

h
i
(a−ζ)1−µ −1
exp (1 − σ) ln (c(a, t)) + γ 1−µ−1e−(ρ−n−(1−σ)g)(s−t)
ds
1−σ
t
Z
Z ∞
e(1−σ)gt ∞ −(ρ−n−(1−σ)g)(s−t)
1
+
e
ds −
e−(ρ−n)(s−t) ds,
1−σ t
1−σ t
e(1−σ)gt111−,= e(1−σ)gt v(a, t) +
ρ − n − (1 − σ)g 1 − σ
ρ−n1−σ(1−σ)gt= e

where v(a, t) denotes the detrended value function. We therefore have ṽa (a, t) =ge(1−σ)gte(1−σ)gt va (a, t) and ṽt (a, t) = (1 − σ)ge(1−σ)gt v(a, t) + e(1−σ)gt vt (a, t) + ρ−n−(1−σ)g. Substituting these expressions into the above Hamilton-Jacobi-Bellman equation for ṽ (a, t)
and multiplying both sides by e−(1−σ)gt yields
1 e−(1−σ)gt11−(ρ − n) v(a, t) +
ρ − n − (1 − σ)g 1 − σ ρ − n 1 − σ
i

h
1−µ
−1
− e−(1−σ)gt
exp (1 − σ) ln (c) + γ (a−ζ)
1−µ
= max
c
1−σ
+va (a, t) [(rt − n − g) a + wt − c] + (1 − σ)gv(a, t) + vt (a, t) +g.
ρ − n − (1 − σ)g

After simplification, we obtain the Hamilton-Jacobi-Bellman equation of the lemma
for the detrended value function v (a, t).
The first-order condition yields(c (a, t))−σ e(1−σ)γ(a−ζ)1−µ −1
1−µ= va (a, t) ,where c (a, t) is the policy function.
The borrowing constraint at ≥ a implies that, when at = a, we must haveȧt ≥ 0.

49This implies(rt − n − g) a + wt − c (a, t) ≥ 0,
(rt − n − g) a + wt ≥ c (a, t) ,((rt − n − g) a + wt )−σ ≤ (c (a, t))−σ ,((rt − n − g) a + wt )−σ e(1−σ)γ(a−ζ)1−µ −1
1−µ≤ va (a, t) ,(A2)which is the state constraint boundary condition.
Let ϕ (a, t) denote the density of the wealth distribution at time t, where the initial
wealth distribution ϕ (a, 0) is exogenously given. The Kolmogorov forward equationis∂ [(rt − n − g) a + wt − c (a, t)] ϕ (a, t) ,∂a
= − [rt − n − g − ca (a, t)] ϕ (a, t)

ϕt (a, t) = −− [(rt − n − g) a + wt − c (a, t)] ϕa (a, t) ,(A3)

where ϕa (a, t) = ∂ϕ (a, t) /∂a, ϕt (a, t) = ∂ϕ (a, t) /∂t, and ca (a, t) = ∂c (a, t) /∂a.
We now have all the elements to define the equilibrium of the economy as a ”meanfield game”.
Definition A1 The general equilibrium of the economy is characterized by the HamiltonJacobi-Bellman equation (A1), the state constraint boundary condition (A2), the Kolmogorov
forward equation (A3), the given initial wealth distribution ϕ (a, 0), the asset market clearing
equationZ∞aϕ (a, t) da,

kt =

(A4)

a

and the factor market pricing equations rt = f ′ (kt ) − δ and wt = f (kt ) − kt f ′ (kt ).
This definition of equilibrium is equivalent to the one given by Definition 1.31
31

Note that, in an economy without capital, we can consider that the wage rate w is given by a fixedparameter and the interest rate rt at time t is determined by the asset market clearing equation, whichsimplifies toZ∞0=aϕ (a, t) da.
a

50

C

ProofsC.1

Proof of Lemma 1

Any state equilibrium must satisfy c ≥ 0. Hence, by equation (14), we must havea ≥ −w/(r − n − g) and, by equation (15), we must have a ≥ ζ. To prove the lemma,
we first characterize the solution to the system given by equations (14) and (15) fora ∈ [ζ, ∞) before establishing the consequences of the borrowing limit a ≥ a and thenatural debt limit a ≥ −w/(r − n − g).
Let us define the function h(a) as the difference between equation (14) and equation(15):h(a) = (r − n − g)a + w −ρ + σg − rγ(a − ζ)µ .

(A5)

In the absence of borrowing constraints, a steady state equilibrium is characterized by
any wealth level a ∈ [ζ, ∞) that solves h(a) = 0. The function h(·) is continuous over[ζ, ∞). We have′h (a) = r − n − g − µρ + σg − rγ(a − ζ)−(1−µ) ,with h′′ (a) > 0, lima→ζ h′ (a) = −∞, and lima→∞ h′ (a) = r − n − g > 0. Let â ∈ [ζ, ∞)denote the unique solution to h′ (â) = 0. We clearly have h′ (a) < 0 for all a < â andh′ (a) > 0 for all a > â.
The function h(·) therefore has the following four properties:• h′′ (a) > 0;
• h′ (a) < 0 for all a < â and h′ (a) > 0 for all a > â;• lima→∞ h(a) = ∞ > 0, since µ < 1;• h(ζ) = (r − n − g)ζ + w > 0 if and only if ζ > −w/(r − n − g).
It immediately follows that we have the following possibilities:• if ζ < −w/(r − n − g), then h(ζ) < 0, implying that there must exist a uniquesteady state with a ∈ (â, ∞) and h′ (a) > 0;
• if ζ ≥ −w/(r − n − g) and h(â) < 0, then h(ζ) ≥ 0, implying that there must exist
two steady states, the lower one with a ∈ [ζ, â) and h′ (a) < 0 and the upper one
with a ∈ (â, ∞) and h′ (a) > 0;
• if ζ > −w/(r −n−g) and h(â) > 0, then h(ζ) > 0, implying that there is no steadystate.
51

Note that â, as defined by h′ (â) = 0, is independent of w. Hence, if w is sufficiently
low (or ”not too high”), we must have h(â) < 0, which rules out this last possibility.
A steady state with h′ (a) > 0 must be unstable, while a steady state with h′ (a) < 0
must be saddle-path stable. This can be readily seen from Figure 1. This can also
be formally established by linearizing the differential equations (5) and (9) around thesteady state. When they exist, the unstable steady state is denoted by aU and the stable
one by aS .
Finally, let us impose the borrowing limit a ≥ a and the natural debt limit a ≥−w/(r − n − g). Assuming that a < aU , these borrowing limits are only relevant if
a0 < aU . Let us therefore consider that a0 ∈ [max{a, −w/(r − n − g)}, aU ). Two cases
should be considered:• when ζ ≥ −w/(r − n − g) and h(â) < 0, wealth converges to max{aS , a};• when ζ < −w/(r − n − g), wealth converges to max{a, −w/(r − n − g)}.

C.2

Proof of Lemma 2

Let h(a) be defined as in the proof of Lemma 1. We are now assuming that r ≤ n + g.
We therefore have h(ζ) = (r − n − g)ζ + w > 0, since ζ ≤ 0. We also have lima→∞ h(a) =−∞ < 0. Finally, h(·) is continuous over [ζ, ∞) with h′ (·) < 0. Hence, there must exists
a unique steady state, characterized by h(a) = 0. As this steady state satisfies h′ (a) < 0,
it must be saddle-path stable. It is denoted by aS .
For any initial wealth, the economy must therefore either converge to aS or to theborrowing limit a, whichever is higher.

C.3

Proof of Lemma 3

Let us define the following two functions
c1 (k) = f (k) − (δ + n + g)k,c2 (k) =ρ + σg + δ − f ′ (k)(k − ζ)µ .
γ

Clearly, by equations (16) and (17), a steady state equilibrium k E is the solution toc1 (k E ) = c2 (k E ).
Let us define k̃ and k ∗ by f ′ (k̃) = ρ + δ + σg and f ′ (k ∗ ) = δ + n + g, respectively. We
have f ′ (k̃) − δ = ρ + σg > n + g = f ′ (k ∗ ) − δ, which implies k̃ < k ∗ . Let us also define k̂
by f (k̂) = (δ+n+g)k̂ with f ′ (k̂) < δ+n+g. We therefore have f ′ (k ∗ ) = δ+n+g > f ′ (k̂),
and hence k ∗ < k̂. This establishes that k̃ < k ∗ < k̂.
52

We clearly have c1 (k) > 0 for all k ∈ [k̃, k̂) with c1 (k̂) = 0. We also have c′2 (k) > 0
for all k ∈ [k̃, k̂] with c2 (k̃) = 0. Hence, as c1 (·) and c2 (·) are both continuous within[k̃, k̂], there must exist at least one solution to c1 (k) = c2 (k) within (k̃, k̂).
Let us now show that, under Assumption 1, this solution must be unique. We havec′1 (k) − c′2 (k)

′′

(k)
µ
(k − ζ)µ −c2 (k),γk−ζ
"#µk − ζ c′1 (k) f ′′ (k) (k − ζ)µ+1=
c2 (k)+−1 .
k−ζ
µ c2 (k)µγc2 (k)=fc′1 (k) +From Assumption 1, we can compute dα/dk at the equilibrium where α = 0. It exactly
corresponds to the term in the square bracket, and we know from Assumption 1 thatit must be strictly negative. This establishes that any solution to c1 (k) = c2 (k) mustsatisfy c′1 (k) < c′2 (k). By continuity of c1 (·) and c2 (·), it follows that under Assumption
1 the egalitarian steady state must be unique.
For the egalitarian steady state to be locally stable when capital stock is equal to k E
and the real interest rate to rE , no household should want to individually deviate fromholding that wealth level. Hence, by the proof of Lemma 1, we must have h′ (k E ) < 0(with the h(·) function defined in the proof of Lemma 1). This givesEr <n+g+µρ + σg − rEγkE − ζ

−(1−µ).

Using the steady state condition (17), this inequality can be written as
rE < n + g +µcE,
kE − ζ

(A6)

which is the condition from Lemma 3.
Note that a stronger requirement for stability is that, starting from the steady state,households do not want to collectively deviate from it. To check the stability of thesteady state to a joint deviation, we impose ait = kt and cit = ct for all i into the flowof funds constraint (5) and the consumption Euler equation (9). We then linearize theresulting system around the steady state. As ct is a jump variable, while kt is not, atleast one of the two eigenvalues should be negative. A sufficient condition for this isgiven byrE < n + g +EµµcE′′ E (k − ζ)
−
f
(k
)
.
kE − ζ
γ(A7)

As f ′′ (k E ) < 0, this condition is milder than (A6). Hence, if (A6) is satisfied, then sois (A7). Decreasing returns to scale reduces the attractiveness of a collective deviation,
relative to an individual deviation.
53

C.4

Proof of Lemma 4

Before proving the main result about the inegalitarian steady state, we establish thefollowing lemmas. Recall that k ∗ denotes the capital stock at the golden rule, such thatf ′ (k ∗ ) − δ = n + g.
Lemma A2 aS (k ∗ ) exists. aU (k) exists for k in the neighborhood of k ∗ with k < k ∗ and it
satisfies limk→k∗− aU (k) = ∞.
Proof. Let us define the function h(a, k) by substituting r = f ′ (k) − δ and w = f (k) −kf ′ (k) into equation (A5) from the proof of Lemma 1. This yields

′

′



h(a, k) = (f (k) − δ − n − g)a + f (k) − kf (k) −ρ + σg + δ − f ′ (k)γ(a − ζ)µ . (A8)

Let us denote h1 (a, k) = ∂h(a, k)/∂a and h2 (a, k) = ∂h(a, k)/∂k. The function aU (k) is
defined by h(aU (k), k) = 0 and h1 (aU (k), k) > 0; while aS (k) is defined by h(aS (k), k) =0 and h1 (aS (k), k) < 0.
We have
∗

∗

∗ ′

∗



h(a, k ) = f (k ) − k f (k ) −ρ − n − (1 − σ)gγ(a − ζ)µ .

Recall that we have assumed ρ > n + (1 − σ)g. It immediately follows that aS (k ∗ ) exists
such that h(aS (k ∗ ), k ∗ ) = 0 with h1 (aS (k ∗ ), k ∗ ) < 0.
By continuity, for k in the neighborhood of k ∗ , there must also exist some aS (k)such that h(aS (k), k) = 0 with h1 (aS (k), k) < 0. Also, when k satisfies ρ + σg + δ >f ′ (k) > n + g + δ (which implies k < k ∗ ), by equation (A8) with µ < 1, we also havelima→∞ h(a, k) = ∞. Having h(aS (k), k) = 0, h1 (aS (k), k) < 0, and lima→∞ h(a, k) = ∞for k < k ∗ implies that aU (k) ∈ (aS (k), ∞) must exist such that h(aU (k), k) = 0 withh1 (aU (k), k) > 0. By the proof of Lemma 1, aU (k) must be unique.
Let â(k) be defined by h1 (â(k), k) = 0. From equation (A8), we have
ρ + σg + δ − f ′ (k)(â(k) − ζ)µ−1 ,f (k) − δ − n − g = µ
γ
 
 1
µ ρ + σg + δ − f ′ (k) 1−µâ(k) = ζ +.
γ f ′ (k) − δ − n − g′Clearly, we have limk→k∗− â(k) = ∞. But, by the proof of Lemma 1, we know thataU (k) > â(k). It follows that limk→k∗− aU (k) = ∞.
Lemma A3 We have aS (k ∗ ) < k ∗ if and only if rE > n + g.
54

Proof. From equation (A8) with k = k ∗ , where f ′ (k ∗ ) − δ = n + g, aS (k ∗ ) is determinedbyµρ + σg + δ − f ′ (k ∗ ) S ∗a (k ) − ζ .
f (k ∗ ) − k ∗ f ′ (k ∗ ) =γThis impliesS∗a (k ) = ζ += ζ+

γ[f (k ∗ ) − k ∗ f ′ (k ∗ )]
ρ + σg + δ − f ′ (k ∗ ) µ1,

γ[f (k ∗ ) − (δ + n + g)k ∗ ]ρ + σg + δ − f ′ (k ∗ ) µ1.

Note that, by equations (16) and (17), k E is the solution tokE = ζ +

γ[f (k E ) − (δ + n + g)k E ]ρ + σg + δ − f ′ (k E ) µ1.

Let us define the following function
b(k) = ζ +γ[f (k) − (δ + n + g)k]ρ + σg + δ − f ′ (k) µ1.

By the above expressions, we have b(k ∗ ) = aS (k ∗ ) and b(k E ) = k E . By definition of
α(k) in Assumption 1, we have b(k) = k + α(k). Assumption 1 implies α′ (k) < 0 and,hence, b′ (k) = 1 + α′ (k) < 1. Thus, whenever k E ̸= k ∗k E − aS (k ∗ )b(k E ) − b(k ∗ )=< 1.
kE − k∗kE − k∗It follows that aS (k ∗ ) < k ∗ if and only if k E < k ∗ or, equivalently, if and only if rE >n + g.
By Lemma 1, an inegalitarian steady state can exist under a fixed capital stockprovided that r = f ′ (k) − δ > n + g or, equivalently, that k < k ∗ . How is this possibility
affected by endogenizing the capital stock?
Clearly, the capital stock cannot converge to k > k ∗ as, by Lemma 2, the steadystate distribution of wealth is egalitarian whenever k ≥ k ∗ . We therefore focus on thepossibility that k < k ∗ . To establish that there exists a steady state with the capitalstock converging to k ∗ , we henceforth consider that k is in the (left-)neighborhood ofk ∗ . By Lemma A2, we know that aU (k) exists.
R∞
Assuming that the wealth distribution has an unbounded support, i.e. â ϕ(a, t)da >0 for any â, if the capital stock converges to some fixed k with k < k ∗ , we would eventually have k̇ > 0, since the wealth of households with wealth above aU (k) would55

diverge to infinity. Thus, the only possibility is for the capital stock k to converge tothe golden rule level k ∗ .
We have limk→k∗− −w/(r − n − g) = −∞. Hence, the wealth of poor households
cannot converge to the natural debt limit and, by Lemma 1, it must instead converge tomax{aS (k ∗ ), a}. If aS (k ∗ ) > a, we consider that in the neighborhood of the inegalitarian steady state there is no household with wealth below the neighborhood of aS (k ∗ )(i.e. the wealth of poor households has effectively converged to the neighborhood ofaS (k ∗ )). Hence, as k converges to k ∗ , the capital stock can only increase because ofhouseholds with wealth above aU (k). But, as limk→k∗− aU (k) = ∞, k cannot exceed k ∗as the economy converges to the inegalitarian steady state, since aU (k) would reach infinity before k becomes higher than k ∗ (thereby preventing k̇ from being positive oncek = k ∗ ).32 As k converges to k ∗ , aU (k) diverges and the mass of wealthy households
converges to zero.
Finally, this steady state possibility exists if and only if the wealth of poor households max{aS (k ∗ ), a} is smaller than k ∗ as, otherwise, the average wealth in the economy would be strictly above k ∗ . By Lemma A3, we have aS (k ∗ ) < k ∗ if and only if
rE > n + g. Hence, rE > n + g is a necessary and sufficient condition for the existenceof this inegalitarian steady state.

D

Rational bubble

In our previous work (Michau, Ono, and Schlegl, 2023), we have shown that a rational
bubble can exist under a preference for wealth whenever rE < n + g. Building on
this, we can easily incorporate the possibility of rational bubbles within the present
framework.
Let us introduce an infinitely-lived asset with price (per efficiency unit of labor)
equal to bt . For simplicity, we assume that this asset does not yield any dividends andis therefore intrinsically worthless.33 In the absence of bubble, its price would triviallybe equal to zero. Under a rational bubble, the return on this asset must be the same as
If, however, aS (k ∗ ) > a and many poor households have wealth much below aS (k ∗ ), then we cansimultaneously have k = k ∗ and k̇ > 0, since the increase in the capital stock is then driven by poor
households. However, this is not possible in the neighborhood of the inegalitarian steady state, once
the wealth of poor households has converged to aS (k ∗ ).
33
In Michau, Ono, and Schlegl (2023), we allow for the more general case where the bubble can beon an asset that does yield dividends. This does not fundamentally modify the nature of the rational
bubble.
32

56

the market rate of return, which impliesḃt = (rt − n − g)bt ,= (f ′ (kt ) − δ − n − g)bt .

(A9)In the presence of the infinitely-lived asset, which introduces the possibility of arational bubble, the equilibrium remains characterized by Definition 1 with the market
clearing condition (13) replaced byZ 1

ait di = kt + bt ,0

and with the dynamics of bt given by equation (A9). Importantly, the initial price ofthe bubble b0 is an equilibrium object that is not specified from the outset. We only
require that b0 ≥ 0 since the asset can be freely disposed of.
The infinitely-lived asset can also be introduced within our mean field game definition of equilibrium from Appendix B. The equilibrium is still characterized by Definition A1 with the goods market clearing condition (A4) replaced byZ ∞aϕ (a, t) da,

kt + bt =

(A10)

a

and with the dynamics of bt given by equation (A9).
Figures A1 illustrates the equilibrium consequences of a rational bubble.34 Startingfrom a given initial wealth distribution ϕ(·, 0), the equilibrium path without a bubble
is represented by the black lines, while the path converging to a bubbly steady stateis represented by the blue lines.35 As rE < n + g, in the absence of bubble, the capitalstock converges to a level that is above the golden rule, while the real interest rate
converges to rE which is below n + g. The bubble induces the capital stock to converge
to the golden rule and the interest rate to converge to n + g.
The bubble raises total wealth, as shown by the dashed-blue line of the left panel(while total wealth without the bubble is just equal to the capital stock shown by theblack line). By raising wealth, the bubble reduces the marginal utility of wealth, which
reduces households’ propensity to save, thereby crowding out capital. Note that, on
the left panel, the magnitude of the bubble corresponds to the difference between thedashed-blue line and the solid-blue line.
34

This illustrative simulation was performed under the following calibration ρ = 0.1, n = 0.02, g =0.01, δ = 0.05, α = 0.3, A = 1, σ = 1, µ = 0.5, γ = 0.236, ζ = −2.47, a = 0.
35
Our bubbly and bubble-less economy start with the same initial distribution of wealth but different
capital stocks. Alternatively, we could have started with the same capital stock in both cases, but higher
wealth in the bubbly equilibrium. The main insights from this exercise would have been unchanged.

57

Figure A1: Transition dynamics with a rational bubble

E

Preference for wealth shocks

We now assume that, at any point in time, a household can be in either of two states. In
state W the household has a preference for wealth, while in state N it does not. Let λW
denote the rate at which households lose their preference for wealth, i.e. the transition
rate from state W to N . Similarly, let λN denote the rate at which households get apreference for wealth, i.e. the transition rate from state N to W .
Let j ∈ {W, N } denote a state, and −j the other state. Thus, if j = W , then −j = N ,and vice versa. Let Ij be the indicator function such that Ij = 1 if j = W and Ij = 0
if j = N . The Hamilton-Jacobi-Bellman equation for the detrended value function is
given byih(a−ζ)1−µ −1−1
exp (1 − σ) ln (c) + Ij γ 1−µ(ρ − n − (1 − σ)g) vj (a, t) = maxc
1−σ+vj,a (a, t) [(rt − n − g) a + wt − c] + λj [v−j (a, t) − vj (a, t)] + vj,t (a, t) .(A11)

The first-order conditions is(cj (a, t))−σ e(1−σ)Ij γ(a−ζ)1−µ −1
1−µ= vj,a (a, t) .

(A12)

The state constraint boundary condition is((rt − n − g) a + wt )−σ e(1−σ)Ij γ(a−ζ)1−µ −1
1−µ≤ vj,a (a, t) .

(A13)

The distribution of households across wealth and states is jointly given by ϕW (a, t) andRR
ϕN (a, t), where ϕW (a, t) da + ϕN (a, t) da = 1. The Kolmogorov forward equationis∂∂ϕj (a, t) = − [((rt − n) a + wt − cj (a, t)) ϕj (a, t)]−λj ϕj (a, t)+λ−j ϕ−j (a, t) . (A14)
∂t
∂a
58

Relying on this ”mean field game” formulation, we can now provide a formal definition equilibrium.
Definition A2 The general equilibrium of the economy with preference for wealth shocks ischaracterized by the Hamilton-Jacobi-Bellman equation (A11), the state constraint boundary
condition (A13), the Kolmogorov forward equation (A14), the initial wealth distribution given
by ϕW (a, 0) and ϕN (a, 0), the asset market clearing equation
Z ∞a[ϕW (a, t) + ϕN (a, t)]da,

kt =

(A15)

a

and the factor market pricing equations
rt = f ′ (kt ) − δ(A16)wt = f (kt ) − kt f ′ (kt ).

(A17)

and

F

Numerical resolution of the model

In this appendix, we outline the algorithm that we use to numerically solve the modelwith preference for wealth shocks of Section 5 and 6. From Appendix E, the mean fieldgame definition of equilibrium is given by equations (A11), (A13), (A14), (A15), (A16),(A17), together with the initial wealth distribution ϕW (a, 0) and ϕN (a, 0). To solve thisequilibrium numerically, we rely on the finite difference method proposed by Achdou,Han, Lasry, Lions, and Moll (2022).
We approximate the detrended value functions vj (a, t), where j ∈ {W, N } denotes
the type of the household, at i = 1, ..., I discrete points in the state space dimension, i.e.
{ai }Ii=1 , using a non-uniform grid where ∆ai ≡ ai+1 − ai denotes the distance between
grid points i + 1 and i. Note that a1 constitutes the borrowing limit, i.e. a1 = a. Startingfrom an initial distribution ϕ(a, 0), we simulate the evolution of the wealth distribution
over T years, subdivided into m = 1, ..., M equally spaced discrete points in the time
space dimension, such that each time step represents δt = T /M years.
Step 1: Initial guess for the long-run capital stock kM
Let kM denote the long-run value of the capital stock at the end of the simulation
period M (i.e. after T years). The numerical algorithm starts with an initial guess forkM , which is subsequently updated. The subscript s = 0, 1, ... indicates the numbersof steps in this iteration process so that kM
denotes the guess for the long-run capital59

sstock in iteration loop s. The associated long-run values for the real interest rate rMsand the real wage wM
follow from the factor pricing equations (A16) and (A17).

Step 2: Numerical resolution of the associated long-run value functionsss
Given kM
, rM
and wM, we solve the discretized version of the Hamilton-JacobiBellman (HJB) equation (A11) for the associated long-run value functions vj (a, M )sunder the assumption vj,m (a, m) = 0 for m = M .
This requires another iteration process. Let q = 0, 1, ... denote the associated steps
and vj (a, M )s,0 for j ∈ {W, N } the initial guess of the long-run value functions overthe state space. The updating process requires an approximation of the derivative
vj,a (a, t) in (A11). We use an upwind scheme to approximate this derivative with eitherv (a,M )s,q −v (a ,M )s,qs,q
, or backwards difference approximation,a forward, vj,a,F(a, M ) ≈ j i+1 ∆ai j is,q −v (a
s,q
v
(a
,M
)
,M
)
s,q
j i−1
, depending on the sign of the associated optimal
vj,a,B(a, M ) ≈ j i∆ai−1
saving function.
Specifically, the optimal consumption functions cs,qj,p (a, M ) satisfies the first-ordercondition (A12), where the subscript p ∈ {F, B} indicates the forward (F) and backwards (B) difference approximations respectively. The associated saving functionss,qss
are ss,qj,p (a, M ) = (rM − n − g) ai + wM − cj,p (a, M ). If the value function is concave
s,q
s,q
s,q
in a, it holds that vj,a,F(a, M ) < vj,a,B(a, M ) and hence ss,qj,F (a, M ) < sj,B (a, M ). We
therefore use a forward difference approximation when ss,qj,F (a, M ) > 0, while a backs,qwards difference approximation is used when sj,B (a, M ) < 0. For grid points withs,qss,qj,F (a, M ) ≤ 0 ≤ sj,F (a, M ), we set savings equal to zero and the derivative of thevalue function equal to the corresponding consumption level. At the first grid point
a1 we implement the state constraint boundary condition (A13), which is enforced
whenever the forward difference would result in negative savings and hence prevents
households from violating their borrowing limit. A similar state constraint boundary
condition needs to be implemented at the last grid point aI on the forward difference.
(aI −ζ )1−µ −1s,qss −σ (1−σ)Ij γ
1−µ.
Specifically we set vj,a,F(aI , M ) = ((rM− n − g) aI + wM
) e
Using this upwind scheme for the choice of the difference approximation, we iterate on the long-run value function until convergence according to

v

s,q+1


=


−1 

1
1 s,q
s,q
s,q
+ ρ − n − (1 − σ)g I2·I − A
u + v
,
∆
∆

(A18)

where ∆ denotes the step size in the iteration process, I2·I is the identity matrix ofdimension 2 · I, us,q and v s,q denote the stacked (by wealth level i and household typej) 2 · I vectors of utility and value functions and As,q denotes the 2 · I x 2 · I transition60

matrix consisting of saving functions and transition probabilities λj .36 The converged
value function vj (a, M )s will be imposed as terminal condition on the time-dependent
Hamilton-Jacobi-Bellman equation in step 4.
Step 3: Initial guess for the time path of the capital stocks
Given the long-run capital stock kMand the value functions vj (a, M )s , we next
solve for the dynamics over the time space. For this purpose, we make an initial guess
for the dynamics of the capital stock, which is subsequently updated. Let l = 0, 1, ...
s,ldenote the steps in this iteration process and km
the time path of the capital stock in
s,0
s,0 M
iteration l. For each iteration s, the initial guess is denoted by km≡ {km}m=1 , where
the capital stock k1s,0 is determined by the initial wealth distribution ϕ(a, 0), and hences,0independent of s, and the capital stock kM
is given by our guess for the long-run values,0s
in step 1, i.e. kM≡ kM. The associated dynamics of the real interest rate and the reals,ls,lwage are denoted by rm and wm
respectively.
Step 4: Solving the Hamilton-Jacobi-Bellman equations,l, we solve the discrete approximation of the HJB equation (A11) backGiven km
ward in time for the terminal condition vj (a, M )s derived in step 2. The choice of thederivative vj,a (a, t) in (A11) follows an equivalent upwind scheme using the optimalsaving functions calculated based on the forward and backwards difference approximations. In addition, the two state boundary conditions are implemented at the eachtime step m.
Starting from the terminal condition vj (a, M )s , we solve for the value function
backward in time as
v

s,l,m



−1 

1 s,l,m+1
1s,l,m+1
s,l,m+1
I2·I − Au+ v,=
ρ − n − (1 − σ)g +
δt
δt

(A19)

where v s,l,m and us,l,m are 2 · I vectors of stacked value and utility functions respectively, i.e. v s,l,m ≡ [vW (a1 , m)s,l ...vW (aI , m)s,l vN (a1 , m)s,l ...vN (aI , m)s,l ]′ , I2·I denotes theidentity matrix with dimension 2 · I, δt the step size of the time grid, and As,l,m the 2 · I
x 2 · I time-dependent transition matrix consisting of saving functions and transition
probabilities λj .
Step 5: Solving the Kolmogorov Forward equation
Given the optimal consumption and saving functions of step 4, we use the discrete
approximation of the Kolmogorov Forward equation (A14) to solve for the dynamics
of the wealth distribution, starting from the initial distribution ϕ(a, 0). We approximate
36

For details on the construction and composition of A, we refer to Achdou et al. (2022)

61the derivative [sj (ai , m)ϕj (ai , m)]′ by forward and backward differences following thesame scheme as in step 4. Starting from the initial wealth distribution, equation (A14)
is solved forward in time as−1 m
ϕm+1 = I2·I − (As,l,m )T δt
ϕ .

(A20)

where ϕm is the 2 · I vector of stacked densities over wealth space and types j, i.e.
ϕm ≡ [ϕW (a1 , m)...ϕW (aI , m)ϕN (a1 , m)...ϕN (aI , m)]′ , and (As,l,m )T is the transpose of
the transition matrix As,l,m from step 4.
Note that before solving (A20), the density ϕm is transformed in order to preservei−1
and subsequently
mass under a non-uniform grid as ϕ̃j (ai , m) = ϕj (ai , m) ∆ai +∆a2
recovered.37s,lStep 6: Updating the guess for km
The dynamics of the wealth distribution from the previous step imply a corres,l,new
given by asset market clearing condition
sponding time path of the capital stock km(A15). We then use a relaxation mechanism to update the guess of the time path of thes,ls,l,new
capital stock kmfrom step 3 with the new value km
according to
s,l+1
s,l
s,l,newkm= ψkm+ (1 − ψ)km
,

(A21)

s,l+1
where ψ ∈ (0, 1) determines the weight on the initial guess. So km
is the updated
guess for the time path of the capital stock with corresponding paths for the real inters,l+1
s,l+1.
and the real wage wm
est rate rms,lStep 7: Convergence of km
Steps 4 to 6 are repeated for the updated guess of the capital stock dynamics untils,ls,Lconvergence of km, with l = 0, 1, .... We denote the converged time path by km.

Step 8: Update of the long-run capital stocks,Ls,Limplied by km
The long-run value of the capital stock kM
is endogenously determined by the evolution of the wealth distribution and in most cases not consistentswith the initial guess for the long-run capital stock kM
in step 1. Hence, we update theinitial guess using a similar relaxation mechanism as in step 6, i.e.
s,L
s+1
s
kM
= ψ̃kM+ (1 − ψ̃)kM
.

(A22)

where ψ̃ ∈ (0, 1) determines the weight on the initial guess.
37

This procedure is described in detail in section 7 of the numerical appendix of Achdou et al. (2022).

62

sStep 9: Convergence of kM
Steps 2 to 8 are repeated for the updated guess of the long-run capital stock untilsconvergence of kM, with s = 0, 1, ....

This completes the description of the numerical algorithm. The following remarks
discuss some additional considerations.
Specification of the wealth and time grid: We set the number of grid points to I = 10 000
and choose an upper limit of aI = $5 billion for the wealth grid.38 We simulate themodel for T = 400 years, which are equally spread across M = 1600 grid points on
the time dimension. Hence, each iteration step represents δt = T /M = 0.25 years. At
the beginning and end of the wealth grid, we implement the state constraint boundary
conditions. The lower state space constraint is enforced whenever the forward difference approximation would result in negative savings and hence prevents households
from violating their borrowing limit. In contrast, the upper state space is enforced
whenever the backward difference approximation would result in positive savingsand hence prevents households from accumulating wealth above aI . While the lower
constraint has an economic rationale, the upper constraint is a technical necessity that
distorts the behavior of households towards the end of the wealth grid, preventing
them from further wealth accumulation.39 When specifying the initial distribution,
we do not allocate any mass above $3.5 billion. Hence, the mass of households above$3.5 billion is virtually zero (up to 10 digits) after 100 years of simulations (i.e. the time
we conduct the tax experiments). This implies that these distortions at the upper endof the wealth grid seem to be of little relevance for our simulations, which are very
reliable up to at least $3.5 billion.
Initial distribution: We start our simulations from a double Pareto-lognormal distribution that is identical in state W and N and we run our model for 100 years before matching the moments described in Section 5.1. The probability density functionf P L (·) of the double Pareto-lognormal distribution is given by
αP L β P L
ϕ̃f P L (a) = P L(α + β P L )awith x1 = αP L σ P L − ln(a)−m
σP L

PL



ln(a) − mP L
σP L



1 − Φ̃(x1 ) 1 − Φ̃(x2 )+ϕ̃(x1 )
ϕ̃(x2 )and x2 = β P L σ P L − ln(a)−m
σP L

38

PL

!
,

where ϕ̃(·) and Φ̃(·) denote

We choose a non-uniform grid to put more grid points at the lower end such that 20% of the grid
points are below $1 million.
39
The effects of the upper state boundary condition are equivalent those of a 100% wealth tax above
the upper threshold.

63

the pdf and the cdf of the standard normal distribution. We set αP L = 1.5, β P L = 10,mP L = 15 and σ P L = 0.6. We do not allocate any mass at grid points above $3.5
billion (see previous remarks on the specification of the wealth grid) and rescale the
distribution for the remaining grid points to mass 1.
Wealth levy: The 20% wealth levy of Section 6.1 is implemented 100 years after the
initial double Pareto-lognormal distribution (which we denote as year 0 in the text, as
this is the year when the moments are matched to calibrate the model, as described in
Section 5.1). The wealth distribution is modified consistently with the tax levy. The
density of households above the $10 million threshold is reallocated using a linear
interpolation. The mass of households at the borrowing limit is reallocated to the two
grid points around their post-transfer wealth level, where the allocation is determinedsuch as to maintain aggregate wealth unchanged. We then simulate the model for 300
years, without any modification to the solution algorithm.
Wealth tax: The 2% wealth tax of Section 6.2 is implemented via the household’s
flow of funds constraint asȧt = (rt − n − g)at + wt − ct + max{Tt − at , 0} − τ max{at − aT , 0},
where τ = 0.02 is the tax rate, aT = $10 million is the deduction, and Tt is the transfer
payment. This affects the value function via the savings function within the HamiltonJacobi-Bellman equation (A11), which shows up in the transition matrix A of the numerical algorithm. However, this does not affect the first-order condition (A12). Otherwise, the structure is unaffected. The budget-neutral path of the transfer Tt is endogenously determined by the evolution of the wealth distribution such as to balance
the government’s budgetZ ∞
max{Tt − a, 0} − τ max{a − aT , 0} [ϕW (a, t) + ϕN (a, t)]da = 0.

aThis is implemented at the end of step 3 in the algorithm: For l = 0, we make a guesson the time path of the transfer Tms,l . For the following iterations l > 0, we use thesimulated path of the wealth distribution from the previous iteration l − 1 to calculates,ls,Lthe associated budget-neutral transfer Tms,l . As the capital stock km
converges to km,s,L
the budget-neutral transfer converges to Tm as well.
Consumption tax: The 50% consumption tax of Section 6.3 affects the household’s

64flow of funds constraint asȧt = (rt − n − g)at + wt − ct + max{Ttc − at , 0} − τc max{ct − cT , 0},
where τc = 0.5 is the tax rate, cT = $200 000 is the deduction, and Ttc is the transfer
payment. This is implemented in the same way as for the progressive wealth tax. An
important difference, however, is that the progressive consumption tax does affect thefirst-order condition (A12) from the Hamilton-Jacobi-Bellman equation (A11), which
is now given by(cj (a, t))−σ e(a−ζ)1−µ −1(1−σ)Ij γ
1−µ(=vj,a (a, t)(1 + τc )vj,a (a, t)

if cj (a, t) < cT,
if cj (a, t) ≥ cT

(A23)

where cj (a, t) is the policy function. We implement this in the numerical algorithm insteps 2 and 4 when deriving the optimal consumption function based on the forward
and backwards approximations of the value functions. Specifically, we calculate eachof the two policy functions cj (a, m)0 and cj (a, m)τ for the marginal tax rate equal to
0 and τc , respectively, for both the forward and backward difference approximations
based on the two cases in (A23). For each point on the wealth grid, the optimal consumption function is then derived as
0 cj (ai , m)cj (ai , m) =cj (ai , m)τ
 Tc

if cj (ai , m)0 < cT
if cj (ai , m)τ ≥ cT ,
else

(A24)

which implies the possibility of bunching at cT . The choice of the approximation isthen determined by the same upwind scheme based on the signs of the saving functions associated with (A24). The same selection criterion is then used to recover thepolicy functions cj (a, m) from the approximation of vj,a (a, m), which appear in boththe transition matrix A (via the saving function) and the utility u in steps 2 and 4.
Otherwise, the structure of the algorithm and the iteration process is unaltered.

65