IMES DISCUSSION PAPER SERIES

Digitalization, Entrepreneurship, and Wealth Inequality

Ichiro Muto, Fumitaka Nakamura, and Makoto Nirei
Discussion Paper No. 2024-E-1

INSTITUTE FOR MONETARY AND ECONOMIC STUDIESBANK OF JAPAN
2-1-1 NIHONBASHI-HONGOKUCHOCHUO-KU, TOKYO 103-8660
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IMES Discussion Paper Series 2024-E-1
March 2024

Digitalization, Entrepreneurship, and Wealth Inequality
Ichiro Muto*, Fumitaka Nakamura**, and Makoto Nirei***
Abstract
What are the main drivers of the recent increase in wealth concentration in the U.S.?
This paper quantifies the role played by digitalization using a tractable model withheterogeneous agents with risk aversion. The model combines (1) digital capital that
substitutes for labor in the production process and (2) households’ investments inrisky digital assets to replicate the asset growth of the wealthy since the 1990s. In
the equilibrium, a small number of prosperous households with low risk aversion,i.e., digital entrepreneurs, hold most of the risky digital capital, whereas a large
number of risk-averse households rely mainly on labor income. Hence, when
digitalization advances, these risk-tolerant households enjoy higher returns fromdigital capital, further accumulating digital capital disproportionately. Based on themodel calibrated to the U.S. economy, we show that digitalization (an increase indigital productivity by 21-43 percent) has contributed to more than about 50 percentof the increase in the share of wealth of the top 1 percent of households and morethan about 80 percent of that of the top 0.1 percent of households observed over thelast 30 years. Moreover, it explains about 20-40 percent increase in the annual
savings of the top 1 percent of households. Finally, the comparative statics on themacroeconomic variables show that while advances in digitalization decrease thelabor share by 3-5 percentage points, which is in line with the empirical literature, italso increases wages, meaning that risk-averse households, who rely mainly on labor
earnings, also gain some benefits from digitalization.
Keywords: Digitalization; Entrepreneurship; Wealth inequality; Savings inequality
JEL classification: E21, E22, E24, E25* General Manager, Aomori Branch, Bank of Japan (E-mail: ichirou.mutou@boj.or.jp)** Director, Institute for Monetary and Economic Studies, Bank of Japan (currently,
International Monetary Fund, E-mail: fumitaka.nakamura@boj.or.jp)*** Professor, Graduate School of Economics, University of Tokyo (E-mail: nirei@e.utokyo.ac.jp)
The authors would like to thank David Hofman, Jesper Lindé, Roland Meeks, Jerome
Vandenbussche, and participants at the CEBRA annual meeting, seminar at the IMF MCM, andBOK/ERI - BOJ/IMES workshop for their helpful comments. The views expressed in this paper
are those of the authors and do not necessarily reflect the official views of the Bank of Japan, theIMF, its Executive Board, or IMF management.

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

1 I NTRODUCTION
Wealth inequality is a salient feature of the modern U.S. economy, and elucidating its mechanisms has become one of the most critical research agendas in macroeconomics. Saez andZucman (2016, 2020) show that wealth concentration in the U.S. economy has increased substantially in recent decades, with most of this increase being driven by the wealthiest classes ofthe population. In other words, wealth inequality has arisen as a thicker tail of the asset distribution. Gomez (2023) decomposes the growth of the top percentile of the wealth distribution
and finds that more than half of the rise in top wealth inequality is explained by the rise in thefortunes of individuals who have newly appeared in the top percentile.
Evidence suggests that many of the recent wealthiest are entrepreneurs who have actively
invested in the development of digital technologies. According to the Federal Reserve Board’s
Distributional Financial Accounts presented in Table 1, as of 2022, private business accounts
for 31.8% of the assets of the top 0.1% of households in the wealth distribution, second only
to corporate equities and mutual fund shares at 41.6%. This means that the wealthiest part ofthe population includes not only investors who earn large returns on their financial assets but
also entrepreneurs who run their own businesses. Consistent with this finding, using U.S. administrative tax data, Smith et al. (2019) report that a large portion of the income of the top 1%
in the U.S. as of 2014 is attributable to the human capital of entrepreneurs. The Forbes list of
the richest people in the U.S. in Table 2 shows more directly that most of the wealthiest are entrepreneurs who have actively developed and used digital technologies. Digital entrepreneurs
now make up the majority of the top tier of U.S. billionaires, and the expansion of wealth inequality in the U.S. has been driven by the rise of these digital entrepreneurs.
Table 1: Asset composition by wealth percentile group in 2022:Q4
Income Source

Bottom 50 %10-50 %

1-10 %

0.1-1.0%

Top 0.1%

Real estate

56.9

38.1

24.1

16.6

8.2

Consumer durable goods

18.9

6.7

2.9

1.9

2.4

Corporate equities and mutual fund shares

2.1

7.5

21.6

40.3

41.6

Defined benefit pension entitlements

3.2

18.0

15.9

1.7

0.3

Defined contribution pension entitlements

5.5

9.6

9.8

3.1

0.8

Private businesses

1.7

4.2

8.8

16.8

31.8

Other assets

11.7

15.9

16.8

19.7

14.8

Source: Federal Reserve Board Distributional Financial Accounts (DFA).

1M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 2: Forbes, The Ten Richest Americans in 2023
Rank

Name

Net Worth (dollar, billion)

Source of Wealth1

Elon Musk

251

Tesla, SpaceX2

Jeff Bezos

161

Amazon3

Larry Ellison

158

Oracle4

Warren Buffett

121

Berkshire Hathaway

5

Larry Page

114

Google6Bill Gates

111

Microsoft7

Sergey Brin

110

Google8Mark Zuckerberg

106

Facebook

9

Steve Ballmer

101

Microsoft

10

Michael Bloomberg

96.3

Bloomberg LP

Source: Forbes.

Against this backdrop, this paper investigates how the development of digital technology
has had a significant impact on wealth inequality. To this end, we construct a tractable dynamic stochastic general equilibrium model with heterogeneous agents. In an otherwise standard neoclassical growth model, our model includes three key ingredients. First, we assume
that firms use three types of inputs for production: labor, traditional capital, and digital capital.
Following recent studies on labor replacement with new technology, such as Alonso et al. (2022)
and Eden and Gaggl (2018), we assume that digital capital is a substitute for labor, while labor is
a complement of traditional capital. In addition, we assume that digital capital is complementary to traditional capital, as in Berg et al. (2018).1 Second, we assume that investment in digital
capital is risky: Autor et al. (2020) argue that high-tech industries have a "winner take most"nature with network effects and scale-biased innovations that provide a very high return on investment for winners but little to nothing for losers. Brynjolfsson et al. (2020) also point out thatthe value of digital capital is more volatile than that of traditional capital because it is intangible,
less traded on secondary markets, and its value may be more closely tied to a particular firm.
This high-risk/high-return characteristic of digital technology firms can be empirically confirmed in U.S. data (Rajgopal et al., 2022). Third, to introduce digital entrepreneurs, we assume
that households are heterogeneous in preference parameters (Gârleanu and Panageas, 2015),
and decide how much to invest in assets with different risks depending on their risk aversion
1

Berg et al. (2018) refer to what we call digital capital as "robot capital," which is "the combination of computers, artificial intelligence, big data and the digitalization of information, networks, sensors and servos that areemphasized in the literature on the new machine age (Brynjolfsson and McAfee (2014))."2

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

and time preference. A small fraction of households who are sufficiently risk-tolerant to invest
in risky digital assets are interpreted as entrepreneurs in our model. Their investment in risky
digital assets builds up digital capital, an input to the production side of the economy.
The contribution of this study is three-fold. First, we develop a parsimonious heterogeneous agent model replicating the U.S. wealth distribution well. In our heterogeneous settings,
the households with low risk aversion actively invest in risky digital assets because their returnis higher, whereas households with high risk aversion refrain from investing in such assets dueto their uncertain returns. Thus, in the equilibrium, successful entrepreneurs willing to takerisks accumulate high-return digital wealth, while risk averse households mainly rely on labor
earnings and accumulate a moderate amount of wealth, creating wealth inequality. Comparing
the wealth distribution between the data and the model, we show that the stationary equilibrium of the benchmark economy calibrated to the U.S. economy replicates the wealth distribution reported by Saez and Zucman (2020) and the Survey of Consumer Finances (SCF). This
demonstrates that our parsimonious setting of the model, which incorporates heterogeneity inhousehold preferences on risk aversion and risky digital assets in the production function, arecapable of capturing the wealth inequality in the U.S.
Second, we evaluate the importance of advances in digital technology in explaining the rise
in wealth inequality in the past 30 years. To capture the effects of digitalization, we conductcomparative statics on the impact of increased productivity of digital capital. We obtain theanalytical result that, under certain conditions, an increase in the productivity of digital capital widens wealth inequality and lowers the labor share. Our model also replicates the wealthconcentration in the U.S. economy from 1989 to 2019 reported by Saez and Zucman (2020) andthe SCF well. In particular, quantitative analysis based on our model confirm that digitalization
has significantly increased the share of households with low risk aversion in the top 0.1% and
1% of the wealth distribution. This result indicates that the concentration of risky digital capitalin risk-tolerant households due to advances in digital technology can explain the magnitude of
the rise in U.S. wealth inequality in the last three decades.
Third, we validate our heterogeneous agent model by cross-checking the testable implications of the model along several dimensions of macroeconomic data. Savings and investing
decisions of households play a key role in our analysis. Mian et al. (2021) elucidate the savings behavior of households over the past 40 years based on an empirical study, pointing out
a significant rise in savings by the top 1% of the wealth distribution. Fagereng et al. (2020) use
Norwegian administrative panel data and conclude that capital gains are the crucial factor in
explaining the differences in savings rates between the rich and the poor. Compared to thesestudies, we use a model with risky digital capital to assess the importance of changes in household behavior and changes in the wealth distribution in explaining the increase in the savings

3

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITYof the rich and investment in digital assets. In particular, we decompose the change in savings
into two components —changes in the savings policy function and changes in the distributionof wealth —to quantify the driving forces behind the rise in savings inequality. Our decomposition suggests that the change in the distribution of wealth is the predominant factor in thechange in savings, meaning that the wealthy group accelerated savings mainly because theirwealth position grew. We also explore other distributional implications of digitalization. Advancement in digitalization increases output through improvements in productivity, but the increase in output is disproportionately distributed to risk-tolerant households. In particular, wefind a rise in the return on digital capital and a decline in the labor share. Thus, entrepreneurs
who own most of the digital assets benefit from the advances in digital technology, whereas therisk-averse households who rely mainly on labor earnings are mostly excluded from the benefits, leading to a rise in wealth inequality. Nevertheless, the wage slightly increases, implyingthat risk-averse households also benefit moderately from digitalization.
R ELATED L ITERATURE
This article is related to a recent body of work studying the impacts of automation. Acemoglu
and Restrepo (2018, 2020) show that automation would reduce employment and wages, incorporating a task-based model in the spirit of Zeira (1998) into a representative agent framework.
Extending the task-based model to a heterogeneous agent framework, Moll et al. (2022) andHémous and Olsen (2022) show that advances in automation could be a key driver of the observed trends in income and wealth inequality. Specifically, Moll et al. (2022) analyze the impactof automation on increasing capital income inequality, while Hémous and Olsen (2022) investigate the impact of automation on increasing labor income inequality through an expanding
skill premium, using an endogenous growth model.
Our model differs from theirs because we aim to study the impact of digitalization, building on Berg et al. (2018) rather than a task-based model by incorporating digital capital as acomplement to traditional capital while being a substitute for labor in the production function.
Therefore, our model potentially captures the positive impact of digitalization on traditional
capital accumulation. In addition, our model introduces the high-risk, high-return characteristic of digital capital that distinguishes it to from traditional capital. In this respect, our model
departs from Berg et al. (2018) by embedding their approach into a standard heterogeneousagent framework, as in Aoki and Nirei (2017). Based on previous studies (Cagetti (2003), Gomes
and Michaelides (2008), Calvet et al. (2021)), we introduce heterogeneities in time preference
and risk aversion, enabling households to decide how much to invest in assets with different
risks depending on their risk aversion and time preference. This model framework illustrates ina parsimonious way the channel through which an advancement in digitalization leads to the4

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

concentration of wealth at the top of the distribution.
This paper is also related to the extensive literature studying the forces behind rising wealthinequality. Similar to our study, two relatively early studies on wealth inequality in the UnitedStates, Quadrini (1999) and Cagetti and De Nardi (2006) focus on the presence of entrepreneursin the top tier of the wealth distribution. However, they consider the possibility that the presence of borrowing constraints on entrepreneurs increases their savings rate and affects wealthinequality, which differs in perspective from our analysis, which focuses on digital capital andits risk characteristics. De Nardi and Fella (2017) list several possible factors behind the risein U.S. wealth inequality and examine the relevance of each, concluding that the presence ofentrepreneurs is one important determinant. İmrohoroğlu and Zhao (2022) demonstrate that,in a period of lower interest rates, entrepreneurs benefit from lower borrowing costs and workers suffer lower returns, thus increasing wealth inequality. However, these studies do not pay
special attention to digital capital or digital entrepreneurs. Gomes et al. (2024) investigate how
automation affects the accumulation of household wealth. They show that households reduce
their exposure to the stock market in response to the rise in human capital risks due to increased
automation. These studies differ from ours in that they focus on the portfolio choices of ordinary households rather than entrepreneurs. Lee (2021, 2023) and Straub (2019) examine theimplications of increased wealth or income inequality on macroeconomic variables such as aggregate demand and real interest rates. However, these studies do not analyze the relationshipbetween wealth inequality and digitalization.
The rest of the paper is organized as follows. In section 2 we develop the model to study
the quantitative predictions of digitalization, and present a basic analytical result on inequality.
Section 3 describes the calibration strategy by explaining how we match the model to data in the
U.S. Section 4 discusses the quantitative results on how digitalization affects inequality, savings
behavior, and macroeconomic variables. Section 5 concludes.

2 M ODEL
In this section, we develop a parsimonious and tractable model to quantify the impact of digitalization on the concentration of wealth and savings in the top portion of the wealth distribution.
We use a continuous time setting so that we explicitly obtain a double-power law stationary
distribution of wealth using the result of Toda (2014).
There are two key features of the model. First, households are heterogeneous in relative risk
aversion γi and time preference ρ i . With this setup, we allow households to selectively choose
entrepreneurial investments depending on their preferences, thereby generalizing the modelof household wealth distribution by Aoki and Nirei (2017). In this way, digitalization affects not
5

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

only production technology but also the differential savings choices of households.
Second, we introduce risky labor-substituting capital in addition to traditional capital, following the formulation of Berg et al. (2018). We interpret the labor-substituting capital as digital
capital. Digital capital is risky, meaning that the volatility of the return is higher than that of safe
assets. Risk-averse households have the incentive to invest in risky digital capital when its mean
return is higher than that of the safe asset.
The underlying uncertainty that drives the ex-post dispersion of household wealth stems
from distributional shocks to the return to digital capital. We posit that an individual investorof digital assets is exposed to an uninsurable idiosyncratic risk to the rate of return, whereas
the mean return to digital capital is not affected by the idiosyncratic risk. For example, ownersof digital capital are subject to stochastic depreciation of the capital, which makes their asset
holdings volatile. These distributional shocks aid our model analysis by simplifying the transmission mechanism.
2.1 H OUSEHOLDS
There is a continuum of households of measure 1, indexed by i ∈ [0, 1]. They differ in risk aversion γi as well as time preference ρ i . We use a perpetual youth model where each household
dies at Poisson hazard rate ζ and is replaced by a newborn household with the same preferences. Households participate in a public pension program that seizes all their assets upon
death while living households receive ζ times their financial wealth. Thus, if a household dies,
all of its assets except for human capital are distributed to other living households.
Households have the following Epstein-Zin recursive utility, where we can separate relativerisk aversion from the elasticity of intertemporal substitution.
J ti = max Et

·Z∞
f
t

i

(C s , J si )d s¸.

f i is a normalized aggregator of current consumption C and continuation utility J i as follows,·f (C , J ) = (ρ i + ζ)(1 − γi )J log(C ) −
i

i

i

³
´¸
1
i
log (1 − γi )J,
1 − γiwhere the stochastic death rate ζ is added to the time preference discount rate ρ. In our setting,the elasticity of intertemporal substitution is assumed to be one.
Households inelastically supply a unit of labor and receive labor income ω. The portfolioof households consists of risk-free assets and risky digital assets. Risk-free assets consist of
traditional capital K and human wealth H . Household i chooses its portfolio depending on its
level of risk aversion and time preference. Let θi be the fraction of total wealth Wi invested in6

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
digital capital and 1 − θi be the fraction invested in risk-free capital. The net rate of return totraditional capital is equal to the risk-free rate r , and the depreciation rate of traditional capitalis denoted by δ. Human wealth H t is defined by
Z∞
Ht =

t

Rs

ωs e − t (ζ+r u )d u d s.

The steady-state value of human wealth is H = ω/ (ζ + r ).
In addition to the risk-free asset, households can invest in a risky asset, that is, digital assets
A t . A unit of output can be converted to a unit of digital assets, which is used by firms as digital
capital to produce goods. A unit of digital assets generates a mean rate of return r A . Moreover,
we assume that digital assets are subject to a household-specific rate-of-return risk σd Z t , whereZ t follows the Wiener process. That is, the aggregate return of digital capital is deterministic
and given by r A , whereas digital assets held by each household generate stochastic returns withdiffusion parameter σ, making this investment risky since the realized return to individuals canbe volatile. Therefore, the instantaneous rate of return to digital assets is written asr A d t + σd Z t .
Our motivation for introducing the term σd Z t is to capture the highly volatile nature of investment in digital assets. As described in previous studies (Autor et al. (2020), Brynjolfsson
et al. (2020), and Rajgopal et al. (2022)), the high-tech industry tends to show high-risk, highreturn properties: successful investment yields significant profits, whereas households lose adisproportionate amount of wealth if a negative shock is realized. To tractably incorporate therate of return risk into our model, we consider the following setup. The digital asset A t in aggregate generates deterministic return r A,t , which is equal to the deterministic marginal product of
A t minus its depreciation. However, an individual household is exposed to household-specificdistributional shocks at each instant, meaning that the distribution of its yield is different acrossinvestors.2 In this setup, each household faces an idiosyncratic rate of return risk when it holds
digital assets. In our economy with heterogeneous risk aversion, risk-tolerant households invest disproportional amounts of their wealth in digital capital. Thus, in the stationary equilibrium, successful risk-tolerant households dominate the wealthy class, as they accumulate theirwealth through prosperous investment projects. Since the successful risk-tolerant households
generate their income mostly from their digital assets rather than risk-free assets, including labor earnings, they represent entrepreneurs in our model.
Households maximize the Epstein-Zin recursive utility by optimally choosing sequences of
2

Thus, the aggregated excess return on digital assets, θi W σd Z t , is always equal to zero.

7

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
consumption, C t , and an asset portfolio, θi , under the following budget constraint.
dWi = [(θi r A + (1 − θi )r + ζ)Wi −C ] d t + θi Wi σd Z .

(1)Here, Wi denotes the “total wealth" of households, which is the sum of traditional capital,
digital assets, and human wealth.3 Assuming that return on assets r , r A and labor earnings ω
are constant over time, the Hamilton-Jacobi-Bellman equation is given by½
0 = max f
θi ,C

i

1 i
2
2 2
i
(C , J ) + J W((θi r A + (1 − θi )r + ζ)Wi −C ) + J W
W θi W i σ
i

2

¾
.

(2)

2.2 F IRMS
The production side of the model follows Berg et al. (2018). Firms use three types of inputs for
production: labor L t , traditional capital K t , and digital capital A t . Following recent studies onlabor replacement with new technology, such as Alonso et al. (2022) and Eden and Gaggl (2018),
we assume that digital capital is a substitute for labor, while labor is a complement of traditional
capital. In addition, we assume that digital capital is complementary to traditional capital, as
in Berg et al. (2018) and Alonso et al. (2022).
More specifically, the firm produces a final good Y t according to a CES production function£¤1
Y t = λk K tα + λV Vtα α .

(3)Vt denotes the composite of labor L t and digital capital A t ,
h
i1
φ φ
,
Vt = λ A (b A t )φ + λL L t(4)where b represents digital capital-augmenting productivity. λk , λV , λ A , and λL denote CES
weight parameters for traditional capital K t , composite Vt , digital capital A t , and labor L t , respectively. The elasticity of substitution between K t and Vt and between A t and L t are denotedby α and φ, respectively. We assume α < 0 to capture the idea that K and V are complements,
while 0 < φ < 1 so that digital capital A is a substitute of labor L.
3

Given that human wealth H is the same across all households, we exclude this term when showing the resultsin later sections. In other words, “wealth" in the results section represents the sum of traditional capital and digitalassets.

8

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

From (3) and (4), the competitive factor prices of K t , A t , and L t are derived as,r = λK K α−1 Y 1−α − δ,(5)r A = Y 1−α λV V α−φ λ A b φ A φ−1 − δ A ,(6)

ω = Y 1−α λV V α−φ λL L φ−1 ,(7)where δ and δ A denote the rate of depreciation of traditional capital and digital capital, respectively. We will discuss the relationship between δ and δ A in the calibration section in moredetail.
2.3 S TATIONARY E QUILIBRIUM
Here, we define the equilibrium of the model. To simplify the analysis, we exclusively focus on
the steady-state solutions and omit the time subscript t henceforth. A competitive equilibrium
can be defined as a set of household variables, {C i , θi }, price variables {r, r A , ω}, and aggregate
variables {K , A, L, Y } such that:• Households maximize their utility subject to budget constraints and taking prices as given.
• Firms maximize their profits. That is, the competitive factor prices satisfy equations (5),(6), and (7).
• Goods, traditional capital, digital capital, and labor markets clear.
First, we provide the solution for the households’ problem. Household i maximizes its utility
by choosing its consumption C i and asset portfolio θi subject to the budget constraint given thewage and interest rates. Household behavior is characterized by the following equations:Ci= ρ i + ζ,WirA −rθi = 2 ,σ γidWi = ((θi r A + (1 − θi )r + ζ)Wi −C i ) d t + θi σWi d B i t ,
ω
H=
.
r +ζ
The first two equations are the linear consumption and portfolio choices familiar from the Merton problem. The first equation shows that the consumption of each household depends only
on its time preference and the size of its wealth. The second equation shows that the asset portfolio (share of risky digital assets) is proportional to the risk premium (return on digital asset r A
9M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

minus risk-free rate r ), whereas it is inversely proportional to the household’s risk aversion andvolatility of the rate of return in digital assets σ2 . The third equation shows the stochastic law of
motion for wealth. The fourth equation is the definition of human wealth.
Market clearing conditions for digital capital, traditional capital, labor, and goods markets
yield the following set of equations:
Z1A=

θi W i d i ,

0

Z1
K=

0

(1 − θi )Wi d i − H ,L = 1,F (b, K , A, L) = C + δK + δ A A,
where in the last equation, C denotes the aggregated consumption demand,
Z1
C=

Ci d i .

0Note that in the steady state, the investment demand for traditional capital is equal to the depreciated amount δK , and the same applies to A as well because we assume that a good isconverted to one unit of digital capital. Also note that, in the labor market, we assume that one
unit of labor is supplied inelastically by a unit continuum of households.
2.4 S TATIONARY DISTRIBUTION : ANALYTICAL SOLUTIONSIn this model, the stationary distribution of wealth can be derived analytically for each type ofhousehold. In the following, we show that the wealth distribution follows a double Pareto distribution, as discussed in Toda (2014). First, from the budget constraint, household i ’s wealth,conditional on its survival, evolves according toÃd logWi t = µwi −

σ2wi
2

!
d t + σwi d B i t

whereµwi = θi r A + (1 − θi )r − ρ i ,(8)

σwi = θi σ.

(9)

10

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Household i dies at Poisson rate ζ, in which event Wi t is reset to the initial wealth H . We can obtain the wealth distribution of households by integrating households’ wealth over time.4 Specifically, the stationary distribution of the logarithm of Wi t is described byµ¶κi ζ Wi −ψ1,if (logWi ) =, if Wi ≥ H
χi Hµ¶κi ζ Wi ψ2,i
, if Wi < H
f (logWi ) =χi H
where H is the initial wealth of households, κi is the normalization parameter defined shortly,and χi , η i , ψ1,i , and ψ2,i are parameters determined as follows:
ψ1,i =
ψ2,i =

χi − η i
σ2wi
χi + η i

,

(10)

,
σ2wi
q
χi = 2ζσ2wi + η2i ,η i = µwi −

σ2wi
2

.

(11)
(12)
(13)

Thus, by a change of variables, we can obtain the stationary distribution of wealth asµ¶κi ζ Wi −ψ1,if (Wi ) =, if Wi ≥ H
χi W i Hµ¶κi ζ Wi ψ2,if (Wi ) =, if Wi < H
χi W i H

(14)
(15)

The normalization parameter κi is determined from the following relationship:
Z1 Z∞
0

0

½
¾
Z1
κi ζ
1
1
+
di =
κi d i = 1
f (Wi )dWi d i =ψ1,i ψ2,i
0
0 χi
Z1

Therefore, for each κi , we can compute the fraction of each type of household. Thus, the derived
wealth distribution follows a double-Pareto distribution whose Pareto exponent at the upper
tail is ψ1i . We will describe the comparative statics of this Pareto exponent as a function ofdigital productivity b in the next section. Specifically, we analytically show that an increase in b
will lead to a rise in inequality under certain conditions:
4

See Aoki and Nirei (2017) for a more detailed explanation of this derivation.

11

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

The average level of wealth is derived as follows:
aver ag e
Wi=Z∞
Wi f (Wi )dWi .

0

Using the derived stationary distribution with respect to wealth above, the average level of
wealth can be obtained when ψ1,i > 1, from the following explicit-form expression:
κi ζHaver ag e
Wi
=
χi

½

¾
1
1
+
.
ψ2,i + 1 ψ1,i − 1

We can also compute the savings rate for each wealth bin, defined as the ratio of savings toincome in the bin. To this end, we first define τυ as the top υ percentile of the wealth distribution
and determine τυ implicitly by the following equation:5
X Z∞
τυ

i

f (Wi )dWi =X κi ζH ψ1,iχi ψ1,ii(τυ )−ψ1,i = υ.

(16)

Using Ito’s lemma, the savings of the top υ%, ∆SR υ , can be calculated from the following equations:∆SR υ =

X Z∞
i

τυ

∆Wi f (Wi )dWi =X
i(e µW − 1)

κi ζH ψ1,i(τυ )−ψ1,i +1 ,χi (ψ1,i − 1)(17)

where ∆Wi is the savings (change in wealth), which can be calculated from the budget constraint and the formula for geometric Brownian motion. Also, µw is defined in the following
equation:µw = θi r A + (1 − θi )r − ρ i .

(18)

Similarly, income ∆I υ can be characterized from the following equation:∆I υ =

Xi(e µI − 1)

κi ζH ψ1,i(τυ )−ψ1,i +1 ,χi (ψ1,i − 1)

(19)

where
µI = θi r A + (1 − θi )r + ζ.

(20)

υUsing these relationships, the savings rate of the top υ% group is computed as ∆SR∆I υ .

5This formula holds when υ is small enough so that the wealth level is greater than the human capital (Wi ≥ H ).
A slightly modified version of the equation applies to the case when households’ wealth is less than human capital.

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

2.5 C OMPARATIVE STATICS ON INEQUALITY
The stationary distribution (14, 15) shows that the wealth distribution follows a double Pareto
distribution with upper-tail index ψ1,i . Moreover, from (8), (9), and (10), the upper-tail indexof ψ1,i depends on the productivity of digital capital, b. That is, changes in the productivityof digital capital have a distributional impact. Our parsimonious model allows for analyticalresults on comparative statics of wealth inequality, the labor share, and capital returns when
digital productivity b changes. In this section, we derive the comparative statics in the case ofhomogeneous agents κ = 1, where all households have common preferences (ρ, γ).
As we show in Appendix A, the stationary equilibrium conditions boil down to the following
two equations that simultaneously determine (r, θ).
!µ
¶ −1
¶
r − ρ + (σθ)2 γ
r + δ α−1−δ− θ − δA θ
ρ +ζ =
λK
ζ

 α
(
Ã
µ
¶ α
µ
¶ 1 !)1−φ α−φ
2
α−1
α−1
r +δ
r + δ A + σ γθθr +δ
1 = λK+ λV 
1−δ
φλK
ρ + ζ + δA θλK
λ A λV b
Ãµ

(21)

(22)

Given the optimal investment strategy θ = (r A −r )/(σ2 γ), the two equations can be equivalently
seen as determining the pair (r, r A ). The first equation (21) shows a relationship between r andθ which must hold to clear the market for traditional capital K , whereas the second equation(22) indicates the market-clearing condition for digital capital A.
As depicted in Figure 1, the stationary equilibrium (r, θ) is determined by the intersection of
curves corresponding to (21, 22). The market-clearing schedule for digital capital (22) is downward sloping. When r increases, the firm’s demand for K decreases, as in (5), which negatively
affects the demand for digital capital per output, A/Y . Meanwhile, the stationary output-towealth ratio Y /W is negatively related to r as we show in Appendix A. Hence, demand for A/Wmust decrease even further. To meet the decreased demand for A/W , r A − r decreases so that
supply θ decreases.
The curve for A/W (22) shifts to the right when digital capital productivity b increases, because the productivity gain raises firms’ demand for digital capital for any fixed (r, r A ). Sinceb does not affect the market clearing condition for traditional capital (21), we conclude thatan increase in b unambiguously increases the stationary digital capital share θ of wealth andrisk premium r A − r as long as the slope of (21) exceeds that of (22). A sufficient condition forthis is obtained when (21) is upward sloping, which holds if the digital asset share is sufficiently
small, in particular if θ < ζ/(2σ2 γ). This condition is satisfied at the equilibrium value in ourbenchmark calibration.
We state this result on θ as well as other comparative statics in the following proposition.
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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Figure 1: Market clearing conditions for traditional capital K /W and digital capital A/W .
r
λ1/α
K −δ
Market clearing for K /W (21)Market clearing for A/W at higher bMarket clearing for A/W (22)
0

ζ
2σ2 γ

1

θ

Note: When the productivity of digital capital b increases, the A/W schedule shifts to the right. The market
clearing condition for K /W is sloping upward in region 0 < θ < ζ/(2σ2 γ). Since θ is proportional to the risk
premium, the horizontal axis can be also considered a linear function of r A − r .

Proposition 1 (Comparative Statics of b). Suppose that a stationary equilibrium r > 0, r A > r ,ω, Y , ψ1 > 1, and θ ∈ (0, 1) exists for a given b. Then, a stationary equilibrium exists for digital
capital productivity in the neighborhood of b and has the following properties.
(a) If θ < ζ/(2σ2 γ) holds, then d θ/d b > 0, d (r A − r )/d b > 0, and d r /d b > 0.
(b) If θ < ζ/(2σ2 γ) and ψ1 < 2 hold, then d ψ1 /d b < 0.
(c) If |d r /d b| is sufficiently small, then d (ωL/Y )/d b < 0.
Proof: See Appendix A.
Item (a) states that both θ and r are increasing in b if the digital capital to total wealth ratioθ is sufficiently small in equilibrium. The condition θ < ζ/(2σ2 γ) is equivalent to saying thathousehold supply of K /W is increasing in r A for any fixed r . That is, an increase in the return
to digital capital induces a greater increase in the supply of digital and traditional capital combined than the increase in digital capital alone. This condition is satisfied under our benchmark
calibration where the equilibrium digital wealth share is sufficiently small.
The slope of (21) tends to be close to zero under our calibrated parameters. This is interpreted as follows. An increase in the risk premium r A − r raises the stationary investment ofdigital capital and the output-wealth ratio, leading to a greater demand for K per unit of wealthby firms (see (36) and (39) in Appendix A). At the same time, households respond to a higherr A − r by raising θ as well as traditional capital under the condition θ < ζ/(2σ2 γ), while shrinking the share of human wealth. This leads to an increased supply of traditional capital. Hence,
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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

an increase in r A − r raises both demand and supply of K /W , leaving r little affected in order tokeep the capital market balanced.
The proposition presumes that a stationary equilibrium exists uniquely. Establishing theexistence of a stationary equilibrium is generally difficult. However, we note that at θ = 0, equa¯¯ ¯tion (22) is solved by r¯ := λ1/αK −δ. At θ = 0 and r = r , the right-hand side of (21) equals r (r −ρ)/ζ,which exceeds the left-hand side value ρ + ζ for a wide range of parameter values. In that casethe market-clearing r for K /W at θ = 0 is below r¯, which is necessary for a solution θ > 0 to exist. Furthermore, if a stationary equilibrium exists satisfying 0 < θ < ζ/(2σ2 γ), it exists uniquely
because (21) is upward sloping.
Item (b) states that the stationary Pareto exponent ψ1 decreases when b increases if theconditions of item (a) and ψ1 < 2 are satisfied. That is, digitalization increases inequality inthe upper tail of the household wealth distribution. Empirically estimated Pareto exponents
are consistently below 2 for U.S. households after the mid-1980s (see Aoki and Nirei (2017)),
satisfying the condition ψ1 < 2. The driving force of a small ψ1 is the heavy investment by risktolerant households in risky digital capital when the digital capital productivity increases.
Finally, item (c) states that an increase in b lowers the labor share ωL/Y when the impactof b on the stationary risk-free rate is negligible. In our calibration scenarios, the change in therisk-free rate when b changes is smaller than 10 basis points, which justifies ignoring the effectthrough r . In that case, the labor share is mainly driven by a change in θ. Since digital capitalis a substitute for labor (i.e., φ > 0), an increase in b shifts factor demand away from labor todigital capital, leading to a depressed wage income share in the labor market equilibrium.
Thus, our model captures two forces of digital capital advancement that drive inequality:
first, digitalization enriches households who are willing and able to invest in digital capital, andsecond, digital capital substitutes firms’ factor demand away from labor and lowers the labor
share of income.

3 CALIBRATIONThis section describes how we map our framework to the data. We calibrate our model by
matching the household wealth distribution, their savings behavior, and various properties ofdigital capital in the U.S.
3.1 C ALIBRATION IN STEADY STATE
Table 3 shows our parameter values. Most of the parameters are based on statistical data or theliterature, while other parameters are jointly calibrated by solving the model to match wealthshares in the U.S.

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 3: Parameter Values
Parameter

Value

Target/Source

Data Source

ζ

1/40

σ

0.5

Business income share in 1989

Kuhn and Rios-Rull (2020)

δ

0.06

Literature

δA

0.18

Depreciation rate of digital capital

BEA

α

-0.72

Capital stock to output ratio in 1989

Feenstra et al. (2015)

φ

0.5

Literature

λK

0.5

Aggregate saving in 1989

Saez and Zucman (2016)

λL

0.83

Labor share in 1989

Grossman and Oberfield (2022)

ρH

0.032

Internally calibrated

L

0.024

Internally calibrated

γH

6.5

Internally calibrated

L

0.16

Internally calibratedργFirst, ζ is set to be 1/40, implying that households accumulate their wealth for 40 years on
average after they start working. We use the depreciation rate of traditional capital δ = 0.06,
which is the standard value. For the depreciation rate of digital capital, we use δ A = 0.18, which
is higher than the traditional capital depreciation rate δ.6 To calibrate δ A , we first specify digital
capital in our model as the “information processing equipment” and “nonresidential intellectual property products” in private fixed assets from the statistics of the Bureau of Economic
Analysis (BEA), following Nordhaus (2021). Then we set δ A to be the weighted average of depreciation rates based on the stock of capital in 1989.7
Our choice for the value of φ, the elasticity of substitution between labor and digital capital,
is set based on the following literature. Alonso et al. (2022) indicate that φ may lie somewhere
between 0.33 and 0.67. Eden and Gaggl (2018) conclude that the elasticity of substitution between labor and ICT-related capital has risen sharply since the late 90s, and thus, φ rises from
0.6 to 0.7. Berg et al. (2018), who calibrate the elasticity of substitution of a production function
similar to ours, find that φ is 0.52 based on the elasticity of substitution between ICT capital andunskilled labor observed in the U.S. data. We set φ to 0.5 which is in line with these previous
estimates.
6

Li and Hall (2020) also discusses the estimation of capital depreciation, stating that the depreciation rate for
digital capital is generally higher than traditional capital.
7
Eden and Gaggl (2018) also estimate the depreciation rates of digital capital. Although their definition of digital
capital is not exactly the same as ours, the depreciation rates lie between 0.15 and 0.20, which is in line with thevalue used in this paper.

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
As for σ, α, λK , and λL , we match the following statistics consistent with the U.S. data in
1989, which we refer to as the benchmark economy. First, we calibrate σ so that the stationary
business income share in our model is 0.11. This value is taken from Kuhn and Rios-Rull (2020),
which uses the SCF for the estimation. α is set to −0.72 so that the capital stock to output ratiois 3.8. We set λK to 0.5 so that the aggregate savings rate becomes 0.11, consistent with Saez andZucman (2016). They calculated the savings rate based on the tax returns micro-data, which
are from NBER micro-files and from internal IRS files.8 In addition, we calibrate λL so that thelabor share is 0.62, which is in line with Grossman and Oberfield (2022), as we discuss later.
For computational simplicity, we assume that there are two types of households in termsof risk aversion: households with high risk aversion (γH ) and those with low risk aversion (γL ).
In addition, households of each level of risk aversion are divided equally into two types of time
preference: impatient households (ρ H ) and patient households (ρ L ). In total, we have fourtypes of households: (γL , ρ L ), (γL , ρ H ), (γH , ρ L ), and (γH , ρ H ). Following Quadrini (1999), we
assume that 10% of the population has low risk aversion. That is, the percentage of each household (ρ L , γL ), (ρ H , γL ), (ρ L , γH ), (ρ H , γH ) is (0.05, 0.05, 0.45, 0.45) respectively. To calibrateρ H , ρ L , γH , and γL , we use the simulated method of moments to choose the remaining fourparameters, assuming that the U.S. economy achieved a stationary equilibrium in 1989. Specifically, our targets are the wealth shares of the distribution at the top 0.1, top 1 percent, top 5
percent, and top 10 percent.9 The resulting values of the preference parameters are ρ H = 0.032,
ρ L = 0.027, γH = 6.5, and γL = 0.16.
Table 4: Calibration targets: Model vs Data

Calibration Targets

Data in 1989

Model

Labor share

0.62

0.62

Capital stock to output ratio

3.8

3.3

Business income share

0.11

0.13

Aggregate saving rate

0.11

0.10

Note: The data for the labor share are from Grossman and Oberfield (2022) and the capital stock to output ratio is
calculated based on the data in Feenstra et al. (2015). For the business income share, we refer to Kuhn and(r A +δ A )∗A
Rios-Rull (2020). In the model, we define A+K
as the capital stock to output ratio and business
Y andY

income share, respectively. As for the aggregate savings, the data come from Saez and Zucman (2016).

8

They shed light on wealth inequality by focusing on capitalized income tax data. For example, if the stock ofclaims of households is equal to 50 times the flow of interest income in tax data, they estimate the size of the claims
by multiplying 50 by the corresponding income amount.
9
The data of the wealth distribution is provided by Moritz Kuhn. These data underlie the analysis by Kuhn and
Rios-Rull (2020).

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 5: Wealth Distribution: Data vs Model
SCF 1989 (%)

SZ 1989 (%)

Benchmark (%)

Bottom 90 %

32.9

35.3

28.9

Top 10 %

67.1

64.7

71.1

Top 5 %

54.2

50.6

54.0

Top 1 %

29.9

28.6

27.2

Top 0.1 %

10.5

12.1

10.8

Note: The SCF data come from Kuhn and Rios-Rull (2020). SZ shows the data from Saez and Zucman (2020).

Table 4 presents a comparison of key statistics of the stationary equilibrium in the benchmark economy and their empirical counterparts. This table indicates that the model’s outputs
match their empirical counterparts very well. The labor share in the model is 0.62, which is inline with the data. The capital stock to output ratio is 3.3 in the model, which is close to the data
(3.8). The business income share and aggregate savings rate are 0.13 and 0.11 in the model,
respectively, which are also consistent with the data.
The distribution of wealth is shown in Table 5. The second column shows the distribution
observed in the 1989 Survey of Consumer Finances (SCF). The third column shows the datafrom Saez and Zucman (2016). The fourth column reports the corresponding wealth share estimated in the model for the benchmark calibration. The table reveals that the model generates
a wealth distribution that is similar to that obtained from the data, since the degree of wealthconcentration in the benchmark economy is remarkably consistent with the wealth shares inthe data in 1989. This demonstrates that our parsimonious model with four types of household
preferences (ρ, γ) reasonably captures the observed patterns of U.S. wealth concentration in1989.
3.2 P ROPERTIES OF THE B ENCHMARK E CONOMY
Table 6 summarizes the business income share for households of various wealth groups. As we
explained earlier, business income, defined as (r A +δY A )∗A originates from the risky investment
and is closely related to the return on digital assets. The first column refers to percentiles of thewealth distribution. In the second column, data from Kuhn and Rios-Rull (2020) are presented,
and the corresponding statistics calculated from the benchmark calibration are reported in thethird column. This table makes it clear that households with greater wealth tend to have higher
shares of business income, and overall the model matches the data well, especially for the top
income households. This result indicates that our simple model of risky capital investment
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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

can quantitatively replicate the salient patterns of the business income distribution that drives
wealth inequality, as we argue in the following sections.
Table 6: Business income share by percentile of the wealth distribution: Model vs Data

Data (%)

Benchmark (%)

Top 10 %

17.7

20.8

Top 5 %

19.1

24.3

Top 1 %

34.4

33.6

Note: The data for business income share are based on Kuhn and Rios-Rull (2020), using the 1989 data.

3.3 DIGITALIZATION SCENARIOS
We capture recent advances in digital technology by an exogenous increase in b, the productivity of digital capital. In this paper, we prepare two scenarios given that it is not straightforward
to assess the advance in digitalization. In scenario A, following Nordhaus (2021), we use stockbased statistics: we match the changes in the ratio KA consistent with the BEA data. In thisapproach, digital capital is defined as the “information processing equipment” and “nonresidential intellectual property products” in private fixed assets from the statistics of the BEA. Thisdefinition is consistent with our calibration of δ A . According to this dataset, the ratio of digitalassets to traditional capital KA increases by 19 percent, which corresponds to an increase of bfrom 1 to 1.21 in our model specification.
In scenario B, we use flow-based statistics. Specifically, we match the digital capital productivity so that the digital investment to GDP ratio is consistent with that in Corrado and Hulten(2010).10 Based on the database, the digital investment to GDP ratio increased by four percentage points from 1989 to 2017. Here, digital capital is defined as software, research anddevelopment, organization capital, and computer and network design consulting.11 This gives
us b = 1.43. Thus, scenario A suggests moderate advances in digitalization, while scenario B
implies greater advances. These two scenarios provide us with a relevant range of magnitudes
on the productivity impacts of digitalization, which allows for various scopes and methods for
10

Given the digital capital depreciation rate δ A and stock of the digital capital at the steady state, we can calculate
the annual digital investment. We increase b to match the four percentage points rise in the digital investment to
GDP ratio.
11
Since digital investment can also be related to the broader category of investment, we also check the robustnessof our results by including other types of investment. For example, even if we include brand and design (industrial),
this also gives a four percentage point increase. Moreover, when we include all types of intangible investment, theincrease is less than 5 percentage points. Thus, the increase in digital investment does not change even if we use
alternative definitions.

19M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

measurement.

4 QUANTITATIVE R ESULTS
In this section, we numerically solve for a stationary equilibrium and investigate the quantitative impacts of digitalization. We conduct comparative statics, showing that the change indigital capital technology explains the concentration of wealth at the top of the distribution byraising returns on digital capital because rich entrepreneurs disproportionately benefit from its
higher return. In addition, we quantify the macroeconomic consequences of digitalization on
output, the labor share, and the interest rate.
4.1 W EALTH DISTRIBUTION
In this section, we evaluate how digitalization alters the savings distribution and wealth inequality in the model economy by numerically conducting the comparative statics of the stationary equilibrium when digital productivity b increases.
Table 7 and Table 8 compare the wealth distributions implied by the model and those fromthe data. The first column in Table 7 presents percentiles of the wealth distribution. The second
and the third columns show the data from the 1989 and 2019 Survey of Consumer Finances(SCF).12 The fifth and sixth columns show the data from Saez and Zucman (2020). The “dif"column shows the difference between the data in 2019 and 1989.
Table 8 shows the wealth distribution of the stationary equilibrium of the model. Baseline,
Case A, and Case B show the computed results in the benchmark economy and in the digitalization scenarios A and B described in Section 3.3, respectively. Column “dif" shows the difference
between the results in the digitalization scenario and those in the benchmark economy.
12

Bricker et al. (2021) augment the SCF data with the wealth of the Forbes 400 families. Based on their calculations, the wealth share of the top 1% of households was 25.1% in 1989 and 31.7% in 2019, representing an increaseof 6.6 percentage points during this period.

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 7: Wealth Distribution (Data)
Data SCF

Data SZ

1989 (%)

2019

dif

1989

2019

dif

Bottom 90 %

32.9

23.6

-9.3

35.3

28.6

-6.7

Top 10 %

67.1

76.4

+9.3

64.7

71.4

+6.7

Top 5 %

54.2

64.9

+10.7

50.6

57.9

+7.3

Top 1 %

29.9

37.2

+7.3

28.6

34.9

+6.3

Top 0.1 %

10.5

14.1

+3.6

12.1

17.6

+5.5

Note: The second and the third columns show the data from the Survey of Consumer Finances based on Kuhn
and Rios-Rull (2020). SZ shows the data from Saez and Zucman (2020). The “dif" column shows the difference
between the data in 2019 and 1989.

Table 8: Wealth Distribution (Model)
Baseline

Case Adif (Case A)

Case Bdif (Case B)

Bottom 90 %

28.9

27.2

-1.7

25.5

-3.4

Top 10 %

71.1

72.8

+1.7

74.5

+3.4

Top 5 %

54.0

56.5

+2.5

59.0

+5.0

Top 1 %

27.2

30.6

+3.4

34.1

+6.9

Top 0.1 %

10.8

13.9

+3.1

17.5

+6.7

Note: Baseline, Case A, and Case B show the model results in the benchmark economy and in the digitalization
scenarios A and B described in Section 3.3, respectively. The “dif" column shows the difference between the
results of the “advances in digitalization" scenario and those in the benchmark economy.

Comparing Table 7 and Table 8 highlights that digitalization could be one of the main driversof wealth concentration from 1989 to 2019. Our model is capable of reproducing the wealthconcentration in the U.S. economy from 1989 to 2019. Saez and Zucman (2020) report thatthe increase in the share of wealth of the top 0.1% and 1% of households over this period isabout 5.5 and 6.3 percentage points, respectively. The SCF data indicates that the increase in
the share of the wealth of the top 0.1% and 1% of households is 3.6 and 7.3 percentage points.
Our model predicts that the increase in the share of the wealth of the top 0.1% and 1% is 3.1 and
3.4 percentage points in Case A, and 6.7 and 6.9 percentage points in Case B, respectively. These
results indicate that over the past 30 years, digitalization has contributed to more than about 50

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

percent of the increase in the share of wealth of the top 1 percent of households and more thanabout 80 percent of that of the top 0.1 percent of households.
The intuition behind this concentration in top-wealth households is as follows. In equilibrium, risk-tolerant households behave like entrepreneurs and hold more risky digital capital. They enjoy higher returns from risky investment in digital capital as digitalization becomes
more advanced and the realized return on digital capital higher. Thanks to this additional return, they can accumulate more wealth, implying wealthy households disproportionately end
up holding a larger share of wealth. On the other hand, risk-averse households become workers. Since they hold little in risky assets, they do not benefit from digital capital investments.
Moreover, the risk-averse households rely on labor income relatively more. Labor is substituted
by digital capital as an input of production as digitalization advances. Even though the wage
rate increases slightly due to the accumulation of traditional and digital capital, as we describein our discussion of the macroeconomic consequences of digitalization in section 4.3, the effectof an increasing wage is dominated by the substitution effect, leading to a decline in the labor
share. Thus, the bottom 90% of households do not benefit from the increased capital income
from digitalization and suffer from the declining labor share caused by the labor-substitution
effects of digitalization. Hence, digitalization widens the wealth disparity in our model by treating successful risk-taking households favorably.
4.2 S AVINGS BEHAVIOR
Table 9 shows the increase in annual savings of the top 1% of the wealth distribution over the
past 30 years, indicating that the wealthiest 1% contributed significantly to the increase in theaggregate saving-output ratio. The second and the third columns show the stationary savingoutput ratio in the benchmark scenario and in the digitalization scenario. The fourth column
shows the difference between the two: savings of the top 1% of households increase by 0.3% of
output through digitalization in Case A and 0.6% in Case B. The third row shows the estimate
reported by Mian et al. (2021) for the increase in the savings-to-GDP ratio that accrued to the
top 1% of households from 1983–1997 to 2008–2016.13 As can be seen from Table 9, the increasein savings of the top 1% of households in the model accounts for 21%-43% of the estimatedincrease in savings in the U.S. This indicates that our model captures an important portion of
the savings increase of the rich, but leaves room for other factors. For example, the effect of
capital gains caused by the productivity change, which may have contributed significantly tothe savings of the rich as suggested by Fagereng et al. (2021), is not incorporated in our model.
13

We cite an estimate using the income-less-consumption approach of the CBO, but other estimates reported byMian et al. (2021) give similar results.

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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 9: Savings in the top 1% increase compared to output: Model vs Data

Benchmark (%)

Advance in digitalization (%)

Saving increase (% point)

Case A

0.9

1.2

0.3

Case B

0.9

1.5

0.6

Data

1.4

Note: This table shows the annual savings of the top 1% of households scaled by output. The second and thirdcolumns show the savings to output ratio of the top 1% of households in the benchmark model and thedigitalization scenarios, respectively. The fourth column shows the difference between the two rows, describing
the increase in savings due to digitalization. The third row cites the data from Mian et al. (2021).

The aggregate savings rate is an integral of household savings rates weighted by the wealthdistribution. Thus, the increase in the savings-to-output ratio can be decomposed into the contribution of the shift in savings rates and the contribution of the increase in wealth amounts.
Specifically, for each top v percent group, the increase in savings of the wealth group is decomposed as:∆SR v ∆SR v Wv
=
,
Y
Wv Y

(23)

where Y denotes aggregate output, ∆SR v the increase in the savings rate of group v definedin equation (17), and Wv the wealth holdings of households in group v. The first component
captures the shift in the savings policy function, which represents the propensity to save out ofwealth. The second component captures the effect of wealth concentration that changes thecomposition of the set of policy functions across households.
Table 10 shows the savings decomposition for each wealth percentile group. In the table,
the overall change shows the change in savings in each percentile of wealth divided by output,
corresponding to the left hand side of the equation. The contribution of the savings policy function and the contribution of wealth holdings correspond to the change in the first term and thesecond term, respectively. The correlated term is the residual coming from the second-ordercontribution. The numbers in parentheses show the share of contribution to overall change.
There are two findings from this exercise. First, richer households tend to increase overall savings as digitalization advances. For example, in Case A, the overall change in savings is 9 percent
for the top 10 percent of households while it is more than 45 percent for the top 0.1 percent ofhouseholds. The difference becomes striking in Case B: the change in savings is more than 100
percent for the top 0.1 percent of households, which is almost 6 times the savings of the top 10
percent of households. This shows that richer households tend to save large amounts of wealth
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M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

because of their higher propensity to save and the sheer amount of wealth they possess, even
without capital gains taken into consideration.
Second, the driver of savings is different among the wealth percentiles. For instance, in Case
B, the contribution of the savings function is around 30 percent while the contribution from thewealth distribution is 63 percent for the top 10 percent of households. On the other hand, thesavings policy function explains less than 20 percent and the wealth distribution accounts foraround 70 percent of the savings for the top 0.1 percent of households. Thus, the contributionof the change in the savings behavior of the rich is modest. Rather, the increase in the wealthlevel of the high-wealth group significantly contributes to the overall increase in savings.
In sum, the productivity gain in digital capital in our model can account for about 20%40% of the increase in aggregate savings. Aggregate savings are affected by both the change insavings behavior and wealth concentration, and we find wealth concentration to be the major
factor, accounting for about 70% of the aggregate savings change.

24

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 10: Savings Decomposition (percent)Case ATop 10 %Top 5 %
Top 1 %
Top 0.1 %Overall change

Savings policy

Wealth dist.

Correlated term

8.6

2.6

5.8

0.2

(30.6)(67.6)

(1.8)

4.0

8.1

0.3

(32.0)(65.4)

(2.6)

7.1

16.0

1.2

(29.3)(66.0)

(4.7)

9.7

32.8

3.2

(21.2)(71.9)

(6.9)

12.4
24.3
45.6

Case BTop 10 %Top 5 %
Top 1 %
Top 0.1 %Overall change

Savings policy

Wealth dist.

Correlated term

18.8

6.3

11.8

0.7

(33.2)(62.9)

(3.9)

9.2

16.7

1.5

(33.6)(60.8)

(5.6)

15.9

33.8

5.4

(28.8)(61.4)

(9.8)

20.5

72.4

14.8

(19.1)(67.2)(13.8)

27.4
55.1
107.7

Note: Overall change shows the change in savings in each percentile of wealth divided by output. “Savings policy"
and “Wealth dist." show contributions of the savings policy function (savings divided by wealth), andcontribution of wealth (wealth divided by output), respectively. Numbers in parentheses show the share ofcontribution to overall change. The calculation for the decomposition is described in the main text.

Table 11 shows the share of each type of household in various wealth percentiles. In ourmodel, we have four types of preferences depending on parameters of risk aversion and thetime-discount rate, namely (γL , ρ L ), (γL , ρ H ), (γH , ρ L ), and (γH , ρ H ). The table in the baseline
section shows the share in percent, while those in Case A and Case B describe the increase/decrease
compared to the baseline results. There are three findings from this table.
First, the table highlights that in the baseline economy, risk-averse and patient households(γH , ρ L ) constitute 82.5 percent of mildly rich households (top 10 percent of the wealth distribution). Households of type (γH , ρ H ) are concentrated around the median level of wealth. In ourmodel, 90 percent of households are risk-averse, and they have little chance of achieving the10th percentile of the wealth distribution if they are also impatient. However, risk-averse but
patient households accumulate some wealth, a portion of which is invested into risky assets,
25

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

and successful ones amount to more than 80 percent of the top 10 percent group.
Second, households with low risk aversion, and especially those with a low time-discountrate, (γL , ρ L ), account for the majority of households in the top 0.1 percent of the wealth distribution. In our model, the savings policy function depends on their preferences. Householdswith lower time-discount rates save more, and those with lower risk aversion tend to invest in
risky digital capital. Thus, low risk aversion and low time-discount rate households in particular have a large incentive to invest in risky digital assets. Some of those households draw highreturns and grow their wealth remarkably, which we interpret as successful entrepreneurs in
our model.
Third, as digitalization advances, the share of high risk aversion households decreases, whilethat of low risk aversion households increases. Especially, among the top 0.1 percent of households in Case B, households with γL increase by almost 10 percent compared to the baseline
scenario, while households with γH decrease by the same amount. This is also intuitive because these risk-tolerant households benefit from advances in digitalization, which leads to ahigher return of risky digital capital, enabling them to accumulate more wealth. In particular,
the share of (γL , ρ H ) increases remarkably compared to (γL , ρ L ) in the top wealth percentile (top0.1 percent). Households with (γL , ρ H ) are characterized by low risk aversion and a high timediscount rate, implying that their propensity to save is low compared to ρ L households, ceterisparibus. Thus, in the baseline economy, households with (γL , ρ L ) comprise the majority of therich since they have a higher propensity to save, and more incentives to invest in risky digital
capital. On the other hand, in Case B, the benefits of holding risky digital capital increase alsofor ρ H households as the risk premium increases. Moreover, as we can see in Table 10, the contribution from wealth holdings plays an important role, indicating that they can accumulate
their wealth even if their marginal propensity to save is low. Thus, the increase in the share of(γL , ρ L ) is relatively subdued, leading to a change in the composition of wealthy households inthe digitalization scenario.

26

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 11: Share of each preference type (percent)Baseline(γL , ρ L )(γL , ρ H )(γH , ρ L )(γH , ρ H )Top 10 %

11.0

6.6

82.5

0.0

Top 5 %

14.9

8.0

77.1

0.0

Top 1 %

28.8

11.7

59.4

0.0

Top 0.1 %

58.7

15.2

26.1

0.0

Case A (Change from the baseline)(γL , ρ L )(γL , ρ H )(γH , ρ L )(γH , ρ H )Top 10 %

-0.2

+0.4

-0.3

0.0

Top 5 %

0.0

+0.8

-0.8

0.0

Top 1 %

+1.0

+2.4

-3.2

0.0

Top 0.1 %

+1.7

+4.2

-5.9

0.0

Case B (Change from the baseline)(γL , ρ L )(γL , ρ H )(γH , ρ L )(γH , ρ H )Top 10 %

-0.4

+0.6

-0.3

0.0

Top 5 %

-0.1

+1.3

-1.3

0.0

Top 1 %

+1.5

+4.1

-5.4

0.0

Top 0.1 %

+2.2

+7.4

-9.6

0.0

Note: These tables show the share of each type of household in each wealth percentile. While the table of thebaseline shows the level of the share, those of Case A and Case B describe the change compared to the baseline
results.

4.3 M ACROECONOMIC C ONSEQUENCES OF DIGITALIZATION
Table 12 summarizes the steady-state impacts of increases in b on key macroeconomic variables. Four findings emerge. First, digitalization increases output, as shown in the second row
in Table 12. As the productivity of digital capital increases, output increases from 1.78 to 1.93 inCase A and 2.12 in Case B, which implies that the digitalization considered here contributes toaggregate economic growth by 8 to 19 percentage points.
Second, the return on digital capital increases as digitalization advances, which is also discussed in the previous section. The fourth row in Table 12 presents the return on digital capital,r A . In the benchmark economy, the return on digital capital is 0.044. As the productivity ofdigital capital increases, the return on digital capital increases to 0.046 in Case A and to 0.048
in Case B. This result suggests that digitalization permanently increases returns to wealth for27

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

households who hold digital capital. On the other hand, the return on traditional capital is almost constant, as seen in the table. Thus, the return on digital capital increases, while that on
traditional capital remains the same, indicating that the risk premium rises due to digitalization.
This result confirms that the gains from the high productivity of digital capital are distributed
to risk-tolerant households, whereas risk-averse households gain little in their capital income.
Also, since the returns to digital capital are volatile, the stationary distribution of wealth amongrisk-tolerant households becomes more dispersed as their portfolio shifts to digital assets. The
wealth disparity within risk-tolerant households is manifested by the low Pareto exponent ψ1of this type of household.14 Also, the lack of response in r , observed in this table, validates thepremise of Proposition 1(c): if |d r /d b| is sufficiently small, then d (ωL/Y )/d b < 0. We find thatthe response of the risk-free rate is sufficiently small when the productivity of digital capital increases, which confirms the assumption of this proposition in our quantitative exercise. This
indicates that the labor share, which is the main income source for ordinary households, declines as b increases. We describe the implications of the decline in the labor share in moredetail next.
Table 12: Macroeconomic Consequences

BenchmarkCase A

Case B

Output

1.78

1.93

2.12

Wage

1.11

1.16

1.21

Return on digital (risky) capital

0.044

0.046

0.048

Return on risk-free capital

0.032

0.032

0.033

Labor share

0.62

0.60

0.57

Digital capital/Output

0.58

0.69

0.80

Third, as the sixth row of the table shows, the labor share ωL/Y decreases due to digitalization. The labor share is 62 percent in the benchmark economy, and it falls to 60 percent in Case
A and to 57 percent in Case B. We note that the magnitude of decrease in Case B is comparable to the decline in the labor share of the U.S. nonfarm business sector shown by Grossman
and Oberfield (2022) (from 0.62 in 1989 to 0.57 in 2019). Although digitalization increases theaggregate pie of the economy (output), the labor share decreases as the return to digital capital increases. This benefits households with low risk aversion disproportionately because these
14

Indeed, ψ1 of (γL , ρ L ) households decreases from 1.48 to 1.36 in Case A and to 1.28 in Case B. That of (γL , ρ H )decreases from 1.90 to 1.69 in Case A and to 1.55 in Case B. Note that the Pareto tail index of overall wealth distribution coincides with the lowest ψ1 among household types.

28

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

households hold considerable amounts of digital capital, whereas most of the risk averse households rely mainly on labor earnings, which is consistent with the literature.15
Finally, wages increase in the long-run in response to digitalization. In our quantitative exercise, the wage increases by five percent in Case A and nine percent in Case B. As we mentioned
in the model section, the composite of labor and digital capital V and traditional capital K arecomplements, and thus K (scaled by output) increases by 9 percent in Case A and 19 percent inCase B. Even though labor and digital capital are substitutes, this increase in traditional (riskfree) capital eventually raises labor demand and the real wage, which is consistent with Berget al. (2018). Reflecting the increase in the real wage, Table 13 indicates that digitalization increases consumption for every type of household in both Case A and Case B. Specifically, while
low risk-averse households increase their spending significantly (by 20–30 percent) in case B
compared to the baseline, risk averse households also increase their consumption by around
10 percent. Although this is still less than the increase in aggregate output, reflecting the risein inequality, these results demonstrate that working households with little digital capital also
benefit from wage increases due to digitalization.
Table 13: Consumption increase for each preference type (percent)(γL , ρ L )(γL , ρ H )(γH , ρ L )(γH , ρ H )Aggregate output

Case A

15.1

10.6

5.1

4.7

8.8

Case B

34.8

23.3

10.8

10.0

19.2

Note: This table shows the increase in consumption compared to the baseline for each type of household. As areference, it also shows the increase in aggregate output, Y.

5 C ONCLUSION
This study investigates the role played by digitalization in the increased wealth concentration in
the U.S. over the last three decades using a tractable model with agents that are heterogeneousin risk aversion. The key feature is introducing risky digital capital that can substitute for labor.
A small number of prosperous households with low risk aversion, i.e., digital entrepreneurs, actively invest in risky digital assets because their return is higher, whereas households with highrisk aversion refrain from investing in such assets due to their uncertain return. In the equilibrium, successful entrepreneurs willing to take risks accumulate high-return digital wealth,15

Berg et al. (2018), Acemoglu and Restrepo (2018) and Moll et al. (2022) analyze the impact of digitalization on
inequality. Although the model settings are different across the literature (Berg et al. (2018) uses a similar production function to ours, while Acemoglu and Restrepo (2018) and Moll et al. (2022) employ a task-based approach),they all point out the decrease in the labor share that accompanies digitalization.

29

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

while risk averse households mainly rely on labor earnings with a moderate amount of wealth,creating wealth inequality. When digitalization advances, these risk-tolerant households enjoy
higher returns from digital capital, further accumulating digital capital disproportionately.
Using comparative statics, we analytically show that the increase in the productivity of digital capital widens wealth inequality and lowers the labor share. Based on a numerical exercise
using a model calibrated to match the U.S. economy, we demonstrate that our model replicates
the wealth concentration in the U.S. economy from 1989 to 2019 reported by Saez and Zucman(2016, 2020) and the SCF well. In particular, quantitative analysis based on our model confirm
that digitalization has significantly increased the share of households with low risk aversion in
the top 0.1% and 1% of the wealth distribution. This result indicates that the concentration ofrisky digital capital among risk-tolerant households due to advances in digital technology can
explain the magnitude of the rise in U.S. wealth inequality in last three decades. We also findthat the large increase in the savings of the rich in the U.S. reported by Mian et al. (2021) can
also be considered a result of digitalization through the concentration of wealth, while otherfactors such as the effect of capital gains also contribute. Finally, comparative statics describethat while digitalization increases output and decreases the labor share, which is in line with theempirical literature, it also increases wages, implying that risk-averse households who mainly
rely on labor earnings also gain some benefits from digitalization.

30

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34

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITYA D ERIVATION AND PROOF FOR S ECTION 2.5
Stationary equilibrium conditions

Here we assume homogeneous households: κ = 1 andcommon ρ and γ. We define the risk premium as r p := r A −r . Labor supply is fixed at L = 1. The
steady state (r, r p , ω, Y , K , A,W,C , H , θ) satisfies the following conditions:r + δ = λK

µ

K
Y

¶α−1
(24)

¶
µ ¶
V α−φ A φ−1
r + r p + δ A = λ A λV bYYµ ¶α−φ µ ¶φ−1VL
ω = λL λVYYφµY = ωL + (r + δ)K + (r + r p + δ A )A
Y = C + δK + δ A A
rp
θ= 2
σ γ
C
= ρ +ζ
W
A
=θ
W
K +H
= 1−θ
W
ω
H=
L
r +ζ
Equilibrium human wealth

(25)
(26)
(27)
(28)
(29)
(30)
(31)
(32)
(33)

We obtain the following equilibrium relationship:ζH= ζ − µw .
W

(34)This is derived using (27, 28, 30, 33) as follows:µ¶C(r + ζ)H + r K + r A A (r + ζ)HH
ζHρ +ζ =+ rp θ ===+r 1−+ r + r p θ.
W
W
W
W
W
This leads to (34), using µw = θr A + (1 − θ)r − ρ = r + θr p − ρ.
Households’ supply of digital capital θof K /W :Equations (32, 34) imply the following supply functionr − ρ + (σθ)2 γ
K
=
− θ.
W
ζ

35

(35)

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
The supply of A/W is simply θ = r p /(σ2 γ). Thus, the pair (r, r p ) determines the households’
supply of K /W and A/W .
From (24), firms’ demand for capital per total wealth, given

Firms’ demand for digital capital
Y /W , isµ¶ 1
r + δ α−1 Y
K
=
W
λK
W

(36)

which is decreasing in r . Also, our production function requires V /Y to satisfy 1 = λK (K /Y )α +λV (V /Y )α . These relationships imply that V /Y is increasing in r . Substituting out V /Y fromthe demand function for A/Y in (25) and using (36), we obtain a demand function for A/W
given Y /W in an implicit form as
1 = λK

µ

µ
¶ α
µ
¶
¶ α
r + δ α−1r + δ A + θσ2 γ A W 1−φ α−φ
+ λV
.
λK
W Y
λ A λV b φ(37)

Derivation of (21) and (22) that determine stationary equilibrium prices (r, r p )We have Y =C + δK + δ A A from (28). Dividing both sides by W we obtainYK= ρ + ζ + δ + δ A θ.
W
W

(38)

¶ 1 !
Y
r + δ α−1δ= ρ + ζ + δ A θ.
1−
λK
W

(39)

Combining with (36), we haveÃµHence, we have a set of equations (31, 35, 36, 37, 39) that determines r and θ (or r p ) along withK /W , A/W , and Y /W .
Equating the supply and demand for K /W (35, 36) and substituting out Y /W using (39), we
obtain an equilibrium relationship between r and θ in the market for traditional capital K :ρ +ζ =

Ãµ

!µ
¶ −1
¶
r + δ α−1r − ρ + (σθ)2 γ−δ− θ − δ A θ.
λK
ζ

(40)

Similarly, equating the supply and demand for A/W (31, 37) using (39), we obtain a relationshipbetween r and θ in the market for digital capital A:

1 = λKµ¶ α

r + δ α−1r + δ A + σ γθ+ λV λK
λ A λV b φ
2

(

Ã

¶ 1

θ
r + δ α−11−δρ + ζ + δA θ
λK

36

µ

α
!)1−φ  α−φ



.

(41)

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY

Thus we have derived (21) and (22).
Proof of Proposition 1(a)Now we inspect the properties of the stationary equilibrium. Here

we assume that the stationary equilibrium exists uniquely. In particular, all equilibrium conditions are satisfied with non-negativity conditions for A, K ,W, Y , H .
In (40), the expression in the last pair of brackets on the right-hand side is equal to K /W (35)and thus positive. Hence the expression in the first pair of brackets must also be positive. Then,
we observe that the right-hand side is increasing in r .
In (41), the right-hand side is strictly increasing in r since 0 < φ < 1 and since we have established that ((r +δ)/λK )−1/(α−1) −δ ≥ 0. Also, the right-hand side is strictly increasing in θ. Hence,d r /d θ < 0 holds for (41).
Moreover, an increase in b lowers the right-hand side of (41) since φ > 0. At any fixed r , anincrease in b must raise θ in (41) to clear the market for A/W . Hence, an increase in b shifts thecurve (41) to the right in Figure 1.
In (35), we have d (K /W )/d θ ≤ 0 if θ ≤ ζ/(2σ2 γ). The supply function for K /W is decreasing
in r p given r for this region of θ. When this holds, the right-hand side of (40) is decreasing in θ.
Thus, d r /d θ ≥ 0 must hold for (40).
In summary, the stationary equilibrium pair (r, θ) is determined as shown in Figure 1. If thestationary equilibrium exists in the region θ < ζ/(2σ2 γ), equation (40) is upward sloping at θ.
Moreover, an increase in b does not affect (40) while it shifts (41) to the right. Therefore, an
infinitesimal increase in b increases both r p and r . This completes the proof for Item (a).
Item (b): Impact of b on Pareto exponent ψ1

We know from the stationary distribution of

wealth that

µ
¶
1
1
ζH
+
(42)
W=
χ ψ2 + 1 ψ1 − 1qwhere ψ1 = (χ − η)/σ2w , ψ2 = (χ + η)/σ2w , χ = 2ζσ2w + η2 , η = µw − σ2w /2, µw = θr A + (1 − θ)r −ρ = r + θr p − ρ, and σw = θσ. Note that (ψ1 , −ψ2 ) are the two roots of the quadratic equation
0 = σ2w ψ2 + 2ηψ − 2ζ. By Vieta’s formulas, ψ1 − ψ2 = −2η/σ2w and −ψ1 ψ2 = −2ζ/σ2w . Thus, ψ1satisfies
0 = f (ψ1 , θ, r ) := ψ1 −

2ζ
r −ρ
+
− 1 + 2γ.
2ψ1 (σθ)(σθ)2Note that ∂ f /∂ψ1 > 0 and ∂ f /∂r > 0. Also, we assume an environment where the equilibrium
satisfies θ < ζ/(2σ2 γ) and therefore we have d r /d b ≥ 0 and d θ/d b ≥ 0 from Item (a). Hence, we
obtain d ψ1 /d b < 0 if ∂ f /∂θ > 0. Condition ∂ f /∂θ > 0 is equivalent tor − ρ − 2ζ/ψ1 < 0.
37

(43)

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Note that ψ1 > 1 must hold in equilibrium
because¶the household wealth distribution has aµq
finite mean. Since ψ1 = (χ − η)/σ2w =2ζσ2w + η2 − η /σ2w , the condition ψ1 > 1 is equivalentto 2(ζ − η) > σ2w . Using η = r − ρ + σ2w (γ − 1/2), the condition is equivalently transformed to
ζ−γσ2w > r −ρ. Hence, (43) holds if 2ζ/ψ1 > ζ−γσ2w . This implies that a sufficient condition for(43) is ψ1 < 2.
In summary, if the stationary equilibrium satisfies ψ1 < 2, we obtain d ψ1 /d b < 0.
Item (c): Impact of b on labor share ωL/Y

We use(µ¶ 1 )r + δ α−1ω (r + ζ)H W ζ + r ζ + ρ − r − (σθ)2 γ
=
=
1−
δ .
Y
W
Y
ζ
ζ + ρ + δA θ
λK
The derivative of the right-hand side with respect to r is at most finite. Thus we investigate theimpact of θ on ω/Y , given r is unchanged.
2r p
2σ2 θγδAδA∂ log(ω/Y )
=−
−
=
−
−
2
∂θ
ζ + ρ − r − (σθ) γ ζ + ρ + δ A θ
ζH /W (C + δ A A)/W
<0
Hence, we obtain the desired result.

B A LTERNATIVE SCENARIO
In this appendix, we study an alternative scenario of digitalization. In the main text, we only
increase b, the productivity of digital capital, to capture the advances in digitalization. In thisalternative scenario, we capture recent advances in digital technology by an exogenous increasein φ, the elasticity of substitution between digital capital and labor, in addition to b. Specifically,
we change φ from 0.5 to 0.6, which is in line with the analysis shown in Eden and Gaggl (2018).
Given this increase, we match the digital capital productivity so that the digital investment to
GDP ratio rises by 4 percentage points, corresponding to Case B. Table 14 present the wealthdistribution for alternative scenarios. The table highlights that even if we change both b and φ,
the change in wealth distribution is very similar to the results in Case B. This result indicates thateven if we also change the substitutability between digital capital and labor in addition to the
productivity of digital capital, the results are very similar as long as we use the same calibration
target.

38

M UTO, N AKAMURA & N IREI : D IGITALIZATION AND I NEQUALITY
Table 14: Wealth Distribution in alternative scenario
Data SCF

Data SZ

Model

1989 (%)

2019

1989

2019

Bench

AD (II)

Bottom 90 %

32.9

23.6

35.3

28.6

28.9

25.5

Top 10 %

67.1

76.4

64.7

71.4

71.1

74.5

Top 5 %

54.2

64.9

50.6

57.9

54.0

59.0

Top 1 %

29.9

37.2

28.6

34.9

27.2

34.1

Top 0.1 %

10.5

14.1

12.1

17.6

10.8

17.4

Second and third columns show the data from the Survey of Consumer Finances based on Kuhn and Rios-Rull(2020). SZ shows the data coming from Saez and Zucman (2020). Bench, AD(II) show the results based on thequantitative analysis in benchmark economy, scenario where both b and φ increase, respectively.

39