A Schumpeterian Exploration of Gini and Top/Bottom
Income Shares
Tetsugen HARUYAMA
September 2021
Discussion Paper No. 2125

GRADUATE SCHOOL OF ECONOMICS
KOBE UNIVERSITYROKKO, KOBE, JAPANA Schumpeterian Exploration of Gini and Top/Bottom
Income Shares
Tetsugen HARUYAMA∗
September 2021

Abstract
Data show that an increase in the Gini coefficient is associated with a falling bottom pB %income share and an increasing top pT % income share where, e.g. pB = 40 and pT = 1. This
relationship, which we call the X inequality relationship, is pervasive in the sense that it isobserved in many countries, including the U.S., the U.K., France and others. The purpose ofthis paper is (i) to construct a Schumpeterian growth model to explain the relationship, and(ii) to identify/quantify factors behind it via calibration of the U.S economy. Our model gives
rise to a double-Pareto distribution of income as a result of entrant and incumbent innovations.
Its advantage is that it allows us to develop iso-Gini loci and iso-income share schedules ina tractable way. Using a double-Pareto distribution as an approximation of an underlying
income distribution, calibration analysis revels that a declining business dynamism and fiscal
policy changes in the past decades played a significant role in generating the X inequality
relationship in the U.S.

∗

Graduate School of Economics, Kobe University. Helpful comments were received at Japanese Economic
Association Spring Meeting 2021 and Society for the Advancement of Economic Theory 2021 Meeting, for which I
am grateful. This research is funded by Grant-in-Aid for Scientific Research C (18K01509).

1

1 Introduction

3

2 A Schumpeterian Profit Distribution with Pareto Tails
2.1 The Basic Model Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Step I: Pareto Distribution of Profit Income . . . . . . . . . . . . . . . . . . . . . .
2.3 Step II: The Entire Distribution of Profit Income . . . . . . . . . . . . . . . . . . .
2.4 Double-Pareto Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5 The Number of Monopoly Industries . . . . . . . . . . . . . . . . . . . . . . . . . .

8
8
11
12
15
15

3 Inequality Measures
3.1 Iso-Gini Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Top/Bottom Income Shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3 Data and a Double-Pareto Approximation . . . . . . . . . . . . . . . . . . . . . . .

16
16
17
19

4 Endogenous Growth
4.1 Consumers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Demand for Intermediate Products . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Profits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 R&D Technology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5 R&D Decisions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Labour Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

20
20
20
22
22
23
24

5 Steady State Equilibrium
5.1 Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.3 Comparative Statics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

24
24
25
26

6 Calibration
6.1 Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Calibrated Values and the Model Fit . . . . . . . . . . . . . . . . . . . . . . . . . .
6.3 Quantifying Factors for X Inequality Relationship . . . . . . . . . . . . . . . . . .
6.4 Declining Business Dynamism and Policy Changes . . . . . . . . . . . . . . . . . .

28
28
28
31
36

7 Conclusion

37

A Derivation of (22)40

B Derivation of (28) and Iso-Gini Contours

41

C Derivation of (29) and (30)42

D Relative Slopes of Iso-Gini, Iso-SB and Iso-ST Curves

43

E Growth Rate

43

F Endogenizing gI

44

2

1

Introduction

Bottom 40% Share
Bottom 40% Share

Bottom 40% ShareTop 1% ShareTop 1% ShareTop 1% ShareIn the literature on inequality, the Gini coefficient and the top/bottom income shares areoften used to show how inequality evolves over Figure 1: The X inequality relationship in the U.S.,
the U.K. and France. Data Source: World Income
time and to make comparison among counInequality Database.
tries. Although they show different aspects
of inequality, they seem to move in a certain
Top 1%
systematic way. A clue is provided by Leigh
Bottom 40% (right axis)(2007) who demonstrates that the Gini and the
top income shares in particular have a strong
positive relationship in 13 countries.1 Using
Panel (a): Income Shares in US
data, Atkinson, Piketty and Saez (2011) also
0.12
0.18
show that the top income shares can have siz1960
0.11
able impacts on the Gini coefficient for the
2019
0.16
0.10
whole economy, despite that the number of in0.14
come earners in the top 1% is very small rel0.09
ative to the total population. Such a close
0.12
0.08
link between the Gini and the income shares
0.10
is intuitive. However, it is not clear what
0.46 0.48 0.50 0.52 0.54 0.56 0.58
Gini Coefficienteconomic forces drive them to move in a way
data show. Viewed this way, several interesting questions arise. Are such co-movements
Panel (b): Income Shares in UK 0.140
inevitable? What economic mechanisms areworking to make their relationship so strong?
0.14
2019
0.135
0.13
What about the bottom income shares? Do
0.130
0.12
they move along with the Gini coefficient as
1980
0.125
0.11
well? If so (it is indeed as shown below),
0.10
0.120
how one can explain the triangle relationship
0.09
0.115
among the Gini coefficient and the top/bottom
0.08
0.110
income shares? In addition, top incomes are
0.07
0.42
0.44
0.46
0.48
0.50
known to follow a Pareto distribution. What
Gini Coefficientrole does it play in forming such relationships? The present paper represents an effort
Panel (c): Income Shares in France
to approach those questions from the Schum0.145
peterian perspective, pioneered by Aghion and
0.11
0.140
Howitt (1992) and others, with a focus on the
0.135
2
2019
role of innovation.
0.10
0.130
1960
Panel (a) of Figure 1 shows how the Gini
0.125
0.09
coefficient is related to the top 1% and bottom
0.120
30% income shares in the U.S. The former is
0.115
0.08
0.110
about twice as large as the latter. The starting
0.42
0.44
0.46
0.48
0.50
and end years, 1962 and 2019, are located near
Gini Coefficient
the left and right axes, respectively, meaningthat the Gini coefficient increases in that pe1
The author even argues that the income shares are “a good substitute” of the Gini coefficient if the latter is
not available.
2
See Aghion, Akcigit and Howitt (2014) for a survey on the literature.

3

4

Table 1: Correlation coefficients with the Gini coefficients. Negative values are shown in red. Grayed cells indicate that the null hypothesisof zero correlation cannot be rejected at a 5% significance level. The bordered cells correspond to Figure 1. Data Source: World Inequality
Database.

riod with a dip in early years. The top 1% income share moves along with the Gini, so that the
scatter plots over the period shows a positive trend. In sharp contrast, the bottom 30% income
share is negatively and more tightly related to the Gini coefficient. Putting them together, what
we may call the X inequality relationship clearly emerges. The U.K. data are shown in Panel (b)of Figure 1 over the 1980-2019 period. The X inequality relationship exits, though not as strongly
as in the U.S. This result may not be surprising, given that those English-speaking countries are
known to share an increasing trend in the Gini and the top income shares in particular. Whatis somewhat surprising is the case of France in Panel (c), which is often referred to as a contrasting case. Inequality measures of some continental European countries, including France, are
known to move differently from those of the Anglo-Saxon countries. Despite this, the X inequality
relationship is clearly visible.
To explore the relationship further, Table 1 shows correlation coefficients between the Gini
coefficient and the top/bottom income shares for 24 countries. Data periods are given in column(1), and column (2) indicates whether the linear trend of the Gini coefficient over the period is
positive or negative. Columns (3)-(7) show correlation coefficients between the Gini coefficientand the top income shares. Correlation between the Gini coefficient and the bottom income sharesare given in columns (8)-(12). Negative values are shown in red. Grayed cells indicate that thenull hypothesis of zero correlation cannot be rejected at a 5% significance level. Inspecting thetable, four observations can be made. First, it is immediately clear that correlation is dominantly
positive for the top income shares and negative for the bottom income shares, implying the X
inequality relationship. It is so even if grayed cells are ignored. Second, the bordered cells for
the U.S., the U.K. and France correspond to Figures 1. There are many countries with the X
inequality relationship as strong as or even stronger than that of the three countries. This suggeststhat the X relationship is a widely observed phenomenon. Third, there is no grayed cells and“wrongly” signed cells in the bottom income shares. It implies that a negative correlation in thebottom income shares is more likely to occur than a positive correlation in the top income shares.3
Fourth, the absolute values of correlation coefficients get higher, as the income share increases.
While it is because the Gini coefficient is susceptible to changes in the middle income range,
correlation coefficients in the bottom 10% are more than 0.8 in absolute value except for Greece.
At the backdrop of these observations, the present paper makes several contributions. First,
we develop a Schumpeterian growth model which can account for the X inequality relationship.
In the model, entrant and incumbent innovations drive growth and generate income inequality.
In particular, we derive a double-Pareto distribution of income distribution, which is used as
an approximation of an observed distribution. It consists of what we call the Left and Right
distributions connected at mode. A double-Pareto distribution has two Pareto exponents, eachfor the Left and Right distributions.4 Indeed, there are studies which provide evidence in supportof such approximation (see below).
An advantage of this approximation-based approach is that we can derive iso-Gini loci, iso-top
income share loci and iso-bottom income share loci in a tractable way in the space of those two
Pareto exponents. These tools make it possible to examine how the top/bottom income sharesare related to the Gini coefficient in an intuitive way. They also allow us to explore economic
mechanisms working behind the X relationship in a simple way. Admittedly, any approximation, including ours, causes loss of information of the underlying phenomenon. To examine what3

In an earlier version of the paper, correlation coefficients of bottom 1% and 5% were included. However, manyof the corresponding income shares are revised to zero as of writing (including the USA) as the dataset is updated.
4
In the literature, a double-Pareto distribution would typically arise if income follows a geometric brownian
motion with Poisson “death”, matched by “birth” of entrepreneurs entering at a single point of a given profit. We
depart from this perspective in that no geometric brownian motion is assumed.

5

information is retained/lost, we use the disaggregated data of 100% national income in the US
developed by Piketty, Saez and Zucman (2018). The result shows that the trends of inequality
indices, required to analyze the X inequality indices, are well preserved though their levels seem
affected. In this sense, a double-Pareto distribution seems suited to approximate an observed
distribution.
Our second contribution is related to identification/quantification of factors behind the X
inequality relationship in calibration analysis. The model is calibrated to the U.S. economy, using
innovation-related data. Our result shows that a declining business dynamism, captured by afall in new firm entry rate as well as decreasing R&D productivity levels, is a major contributor
to the X relationship. Falling corporate income taxes were also found important in line with
Nallareddy, Rouen and Serrato (2018). Business dynamism is a driver of income growth via
creating new products/jobs and reallocating resources from obsolete production units to more
efficient ones. Its declining trend since the 1980s in the U.S. is a focus of several studies (e.g.
see Decker, Haltiwanger, Jarmin and Miranda (2014), Decker, Haltiwanger, Jarmin and Miranda(2016b) and Akcigit and Ates (2019b)). It is feared that a falling business dynamism generates
less opportunities to climb income ladders and less vibrant social mobility, exacerbating inequality(see Fikri, Lettieri and Reyes (2017) and Furman and Orszag (2018)). Our result indeed confirms
such concern.
As our third contribution, we introduce a new approach of modeling incumbent innovations
as a driver of income inequality. The top 1% (and even smaller percent) incomes follows a Pareto
distribution, and one of its important characteristics is a heavy tail. If an innovation-driven
growth model is to capture this property, profits of some firms stretch to near infinity, despitethat the total profits are finite. A challenge is to introduce such extremely large profits in anotherwise standard R&D model based on the homothetic utility/production functions. Acemoglu
and Cao (2015) tackles this problem by developing a “lab-equipment” model. In their model,
however, entrants earn a greater profit (which could be very large) than a previous incumbent
upon entry, though imitators always start at a profit level below the average. In contrast, in ourmodel entrants start from low profit, located in the Left distribution, and they can earn a huge
profit only after a series of successful incumbent innovations.5 Jones and Kim (2018) sidestep
the issue by assuming that profits are proportional to human capital that follows a stochastic
growth process. More importantly, the study limits the role of innovations to causing firm exits,
complemented with the assumption that incumbent firms do not conduct R&D. We take a stepfurther by introducing entrant as well as incumbent innovations in shaping a Pareto distribution.
Incumbent innovations drive an exponential growth of income, enabling successful entrepreneurs
to earn extremely high incomes, whereas entrant firms cause creative destruction, which keep thecollapse of income distribution in check. Our approach is inspired by a pioneering work of Kletteand Kortum (2004), as explained below. Note that Jones and Kim (2018) and our work are bothconsistent with the observation of Smith, Yagan, Zidar and Zwick (2019) that a major source of
top income is “pass-through” entrepreneurial profits, which accrue as returns to human capital,
rather than capital income. Our approach of taking profits as an important source of increasing
inequality is also supported by Barkai (2020). The study provides evidence that a large increasein pure profits contributed to a declining income shares of labour and capital in the U.S.
5

In terms of modeling approach, our model can be best viewed as complementing Acemoglu and Cao (2015)
because we assume labour as an input for R&D. In models with R&D workers, the CES functions are often used tomodel expansion of variety goods (see Romer (1990)), and the Cobb-Douglas utility/production functions are often
assumed for quality improvement of goods (see Aghion and Howitt (1992) and Grossman and Helpman (1991)).
The CES function is used for quality improvement in Li (2001) for the first time, developed further by Li (2003)
and used by others including Dinopoulos and Segerstrom (2010).

6

In the model, there is a continuum of measure J > 1 of final output, produced using aunit continuum of intermediate goods. A monopoly firm run by an entrepreneur earn profits
from producing intermediate goods. To model a Pareto distribution, profits of some firms must
increase to near infinity. This poses a difficulty in building an otherwise standard R&D modelwith the homothetic production function. We solve the issue, resorting to a valuable insight ofKlette and Kortum (2004) that the number of intermediate goods can be treated as countable, i.e.
1, 2, 3, · · · in a continuum of product space. Countability implies that there are infinitely many
products that are potentially produced by monopoly. Because of this property, some firms canearn disproportionately large profits even in the [J × 1] intermediate product space.
Turning to R&D activities, there are entrant and incumbent innovations, both of which improve quality of intermediate goods. In particular, entrant innovations render existing goods
obsolete. Upon successful innovation, entrepreneurs enter a given intermediate product industry
where they start producing some products. After entry, they engage in further R&D as incumbents. As long as entrant innovations do not arrive in their industries, their profits continue toincrease without limit. This is the expanding force that stretches the income distribution in the
direction of infinity. However, they exit the market and their products become obsolete if they
are hit by entrant innovation. Some of those goods are replaced by new products and others
become available as competitive goods. This is the contracting force which prevents the income
distribution from collapsing in steady state.
A double-Pareto distribution has two parameters, which we call the Left and Right exponents.
They are endogenously determined and depends on Poisson rates of entrant and incumbent innovations. In turn, those Poisson rates are determined by incentives for R&D, entrant and incumbent,
as in a standard Schumpeterian model. Based on iso-Gini, iso-top pT % income share and isobottom pB % income share loci (e.g. pT = 1 and pB = 40), we identify the areas where the X
relationship emerges in the space spanned by the two Pareto exponents. Using those loci and theresulting equilibrium conditions, comparative statics analysis can be easily conducted, and theyshow how the Gini coefficient and the top/bottom income shares respond to parameter changes.
In addition, our model can accommodate contrasting results of Jones and Kim (2018) and Aghion,
Akcigit, Bergeaud, Blundell and Hémous (2019) regarding entrant innovations. The former predicts that entrant innovations reduce top income inequality because they destroy monopoly rents
and induce exists of incumbent firms. According to the latter study, on the other hand, entrant
innovations can increase top income inequality.6 In our model, the both case can arise, depending
upon parameters, at least on the theoretical level.
There are studies on a double-Pareto distribution. Reed (2001) argues that size distributionof some economic variables, including income, exhibits a double-Pareto distribution. In addition,using U.S. data drawn from the Current Population Survey (2000–2009) and the Panel Study ofIncome Dynamics (1968–1993), Toda (2012) establishes that personal labour income conditioned
on education experiences follows a double-Pareto distribution. Toda (2011) also demonstrates
that U.S. male wage in 1970-1993 appears to follow a double-Pareto distribution once its trend
is removed. In addition, Toda and Walsh (2015) show that cross-sectional U.S. consumption(quarterly data in 1979-2004) obeys the power law in both the upper and lower tails. As far as
the lower and upper tails of income are concerned, Reed (2003) and Reed and Wu (2008) argue
that the lower and upper tails of incomes exhibit a Pareto distribution, though the middle range
is best captured by log-normal distribution. In an early study, Champernowne (1953) considers
that the lower tail follows a Pareto distribution.
6

On the other hand, Aghion et al. (2019) and Aghion, Akcigit, Hyytinen and Toivanen (2017) show that
innovation is inclusive in the sense that it promotes social mobility.

7Turning to the literature, Jones and Kim (2018) is closely related to our study, as mentioned
above. Their analysis is based on a Schumpeterian growth model mainly to examine how innovation affects the top 1% income share. Aghion et al. (2019) also develop a model where innovation
plays a role in shaping top 1 % income inequality. Another related study is Acemoglu and Cao(2015) which extends a standard Schumpeterian model by introducing entrant and incumbent
R&D. They show that the firm size measured by sales follows a Pareto distribution. In contrast
to those studies, our model uses a double-Pareto distribution as an approximation of the entire
income distribution and explores the X inequality relationship. In addition to those studies in the
Schumpeterian framework, there are competitive models which account for a Pareto distributionof income. An important contribution is made by Aoki and Nirei (2017) and Nirei (2009). Gabaix
and Landier (2008) can also cited in this vein, given that they develop a competitive assignment
model. Gabaix, Lasry, Lions and Moll (2016) extends a random growth model to account forfast rise in top income.7 Our model is also related to Klette and Kortum (2004) because theirinsight plays a crucial role in generating a Pareto distribution. Their model is Schumpeterian withincumbent firms expanding the portfolio of products through innovation. Entrant innovation also
exists but is not drastic enough to cause outright exit of incumbent firms, and assumptions are
such that firm distribution follows a logarithmic rather than Pareto distribution. The model iswidely used in research. For example, Lentz and Mortensen (2008) use it to explore the link
between growth and resource reallocation. Akcigit and Kerr (2018) is another study of entrant
and incumbent innovation. They show that the firm distribution matters for long-run growth.
More recently, Peters (2020) shows that entry mitigates misallocation of resources in a growing
economy.
The structure of the paper is as follows. Section 2 gives a basic structure of the model,
taking entrant and incumbent innovation as given. It shows the emergence of a double-Pareto
distribution of profit income. In Section 3, we derive iso-Gini and iso-income share contours.
Section 4 endogenizes entrant and incumbent innovations. We conduct comparative statics in
Section 5. Calibration analysis is developed in Section 6 to identify contributing factors behind
the X inequality relationship in the U.S. Section 7 concludes.

2

A Schumpeterian Profit Distribution with Pareto Tails

The purpose of this section is to demonstrate the emergence of a profit distribution with Pareto
tails in our Schumpeterian model in the simplest possible setting. For this end, the model isdeveloped here, taking the incentive structure of production and R&D activities as given. This
allows us to highlight key mechanisms of the model.

2.1

The Basic Model Settings

Consumers are risk-neutral with no saving. Her instantaneous utility is given by
1

RJ

U = e J 0 ln Yj dj ,

J ≥1

(1)

where Yj is differentiated final output j. We assume that Yj is competitively produced with acontinuum of intermediate goods yji according toln Yj =

Z 1

ln qji yji di,qji = λkji ,λ > 1,

kji = 0, 1, 2, ...

0

7There are studies on a Pareto distribution of wealth. For example, see Benhabib, Bisin and Zhu (2011).

8

(2)

1
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Figure 2: If an entrepreneur succeeds in entrant R&D, she starts with hi
number of monopoly goods in intermediate good industry i. After entry
the entrepreneur generates further innovation as an incumbent.

where qji is the quality level of yji . Intermediate product firms conduct R&D to increase kji ,and the highest quality products are always used for final output production. Quality innovations
allow monopoly firms to produce some intermediate goods and earn profits. In this section, we
use π to denote (net) profit per intermediate product and take π as given.
Consider an intermediate goods industry i in Figure 2. yji is a product i used to produce
final output j, and it is assumed to be specific to a product j. That is, yji is a different productfrom yj ′ i , j ′ ̸= j, and their quality improvement requires separate successful innovations. In anintermediate product industry i, a single monopoly firm operates along with competitive firms. We
also assume that a monopoly firm, run by an entrepreneur, produces multiple products (not all)in i. A firm initially produces a continuum of hi < J products when it enters an industry i after
successful entrant R&D. All other products in i are competitively produced (more explanation on
the entry/exit process through creative destruction later). After entry, a firm conducts R&D tofurther increase the quality of competitive products in i to become their sole producer. Let ususe ni to denote that number of products the firm produces in i. It is equivalent to
ni = hi + mi

(3)

which consists of hi and mi , the latter of which is materialized through incumbent R&D. We alsouse ci to denote the remaining goods in i that are competitively produced.
To introduce a Pareto distribution of profits in this otherwise standard Schumpeterian model,
we use an insight of Klette and Kortum (2004) that mi can be treated as countable, i.e. mi =0, 1, 2, 3, · · · in a continuum of product space in i. Countability of mi has two implications. First,
there are infinitely many products that are potentially produced by monopoly, no matter how large
it is. Second, “most” of products in i are competitively produced. This applies to all intermediate
goods industries i ∈ [0, 1]. The state of a firm in industry i is completely characterized byni . Indeed, some monopoly firms are lucky enough to produce an exceptionally large number of
products, earning overly huge profits. This is one of the prominent features necessary to generatea Pareto distribution.
Let N denote the number of monopoly productsRin the economy. R Similarly, the number of11competitive products in i is given by C. Then, N = 0 ni di and C = 0 ci di hold. Given that a9single monopoly firm produces multiple products in each intermediate industry, N is equivalent
to the average number of products produced by monopoly firms.8 We also require
J = C + N.

(4)

Note that ni can be exceptionally large to generate a Pareto distribution and this feature isaccommodated in (4) because the average of ni is finite, as will be established.
The economy is characterized by a turnover of monopoly firms through entry/exit, causedby creative destruction of intermediate products. Consider potential entrant firms conducting
R&D. We assume that an entrant R&D success follows a Poisson process with an arrival rateof gE , which is taken as given in this section. It is undirected in the sense that an industry is
randomly chosen from i ∈ [0, 1] to implement successful innovation. This assumption simplifies
analysis, but it also captures in a simple way an unpredictable nature of R&D outcomes.9 To
introduce a drastic nature of creative destruction, we assume that all of the previous incumbent
products in i are rendered obsolete by entrant innovation in the same intermediate industry.10
The “death” of firms due to entrant R&D is the contracting force of the income distribution, which
prevents its collapse in steady state. A successful entrant in turn increases the quality level ofa continuum of hi products, which are randomly allocated to her.11 We proceed in two steps
regarding the assumption of hi . First, we assume that the value of hi is assumed to be constantfor all intermediate goods industries, though its location in Figure 2 is random. In this case, aprofit income follows a Pareto distribution. In the second step, the Pareto distribution in the firstcase is interpreted as the right part of the entire distribution. The left part arises once the valueof hi is randomly distributed in addition to randomness of its location in the figure.
As mentioned above, after entry incumbent firms engage in R&D to improve quality of competitive goods in their own industries. Incumbent R&D in industry i makes it possible to expand
the portfolio of the firm’s products stochastically with the Poisson arrival rate of gI per product.
In this section, we take gI as given. The “per product” assumption plays a crucial role in generating a Pareto distribution in the right tail. To illustrate this point, consider an incumbent firmwith ni products. The arrival rate of incumbent innovation is now given by12
gI ni .

(5)

Its salient feature is that the more products are improved in quality, the higher the arrival rateof an additional innovation. This is the expanding force of the profit distribution.
There are two things worth mention regarding the assumption (5). First, it means that therate of innovation is different among firms. Initially, incumbent innovation occurs at a lower rate
because ni is low. But as more and more innovations are generated, income growth accelerates.
8

An incumbent in i will be indifferent between incumbent and entrant R&D. We assume that incumbents invest
in R&D in her own industry only for simplicity.
9
A well-known example of this type of uncertainty is a microwave oven, which was invented from radar technology
for military purposes. The Internet and GPS are also byproducts of military R&D expenditure. Viagra is an
example of a commercial product which was originally created for different purposes. A similar assumption is usedin Kortum (1997) and Acemoglu, Akcigit, Alp, Bloom and Kerr (2018).
10
The large creative destruction effect is reported by Guvenen, Ozkan and Song (2014). Using U.S. data, theyshow that the distribution of unfavorable shocks to the rich is left-skewed, meaning that the richer is more likely
to be hit by shocks. Akcigit and Kerr (2018) and Aghion et al. (2019) also provide evidence for entrants’ drastic
innovation, using patent data.
11
That is, the location of a continuum hi in Figure 2 is random. Hence, some of the previous incumbent products
may be included in the initial product portfolio of an entrant.
12
As will be explained in more detail, ni in (5) corresponds to the number of R&D projects rather than a positive
externality.

10

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|{z}

z

Figure 3: The Right distribution of entrepreneurial income. An arrow (1)indicates entrepreneurs earning more profits and moving rightward in the
distribution.

This result is consistent with the finding of Piketty et al. (2018) who show that the averageannual growth of income is increasing in income percentiles in the U.S. in 1980-2014 with a growrate accelerating above the top 1%.13 Second, as we will establish, the number of products ni is
distributed according to a double-Pareto distribution in equilibrium, and hence so is gI ni . This
is in line with Guvenen, Karahan, Ozkan and Song (2015) which show that growth of earnings isdouble-Pareto distributed, using a large U.S. panel data set. Third, the assumption also captures
heterogeneity of firm growth among, stressed by Luttmer (2011) and Gabaix et al. (2016).

2.2

Step I: Pareto Distribution of Profit Income

Taking π as given, define
zi = n i π(6)

as the total (net) profit earned by an entrepreneur in an intermediate good industry i.
Assume that the value of hi is fixed at h, i.e.
z = hπ∀i.

(7)All entrant firms start from the initial profit z. After entry, firms engage in R&D to increase therange of products they produce according to (5). Whenever innovation occurs, the total profit
increases by π, and its expected increase during ∆t is given by∆zi = π {1 × ni gI ∆t + 0 × ni (1 − gI ∆t)} = zi gI ∆t.

(8)

It shows that total profit geometrically grows and can be very large. This section derives the
distribution of zi . In Figure 3, (8) corresponds to the rightward movement of an entrepreneur, as
indicated by arrow (1). On the other hand, there is always a possibility that entrant innovation
occurs, causing exits of incumbents, which is captured by arrow (2) in the figure.
13

See Figure II on p.579 of Piketty et al. (2018).

11Given these assumptions, we denote the cumulative distribution function of z by FR (z, t)which depends on time t. Define its counter cumulative distribution as
C (z, t) = FR (∞, t) − FR (z, t) .

(9)

Now, consider how C (n, t) changes during a small time interval ∆t:14
C (z, t + ∆t) − C (z, t) = [C (z − ∆z, t) − C (z, t)] − gE ∆tC (z, t) .

(10)

On the LHS is the total change in C (n, t) which is decomposed into two terms on the RHS. The
first term captures an inflow of firms into C (z, t) due to an increase in z through incumbent
R&D, captured by the shared area. The second term is a flow of existing firms due to entrant
innovations. Rearranging the equation using (8), and then letting ∆t → 0 givesdC (z, t)dC (z, t)= zgI −
− gE C (z, t)dtdz
In steady state, C (z, t) is constant. Therefore, solving the resulting differential equation using(9), we end up with z −ζ ζFR (z) = FR (∞) 1 −,fR (z) = FR (∞) ζ z −ζ−1
(11)
z
z
where
ζ≡

gE
> 1.
gI

(12)

fR (z) gives the number of entrepreneurs earning z. We have FR (∞) = 1 for a constant h.15
In particular, it is a Pareto distribution with the Pareto exponent ζ, which is assumed to begreater than one because it is required for a finite mean of z. This demonstrates that the powerlaw exponent is determined by the two key variables in our Schumpeterian model. A higher
Poisson rate gE raises the exponent ζ, meaning that the right tail gets thiner. This is intuitive
because 1/gE is the average period of earning profits, which means that monopoly rents are lost
more frequently for a higher gE . On the other hand, a higher growth of incumbent profits via
gI reduces the Pareto exponent, making the right tail thicker. This is because entrepreneurs
monopolize more products for a given period of time, moving faster rightward in Figure 3.

2.3

Step II: The Entire Distribution of Profit Income

The Pareto distribution in the previous section is derived under the assumption that the initial
profit z is the same for all entrant firms. Indeed, there is no reason why it should be the case, andit seems more natural to assume that the entry level of profits differ. It may be due to uncertainty
of R&D activities in general or it may be caused by the timing of launching new products, regional
characteristics and even business cycles. Due to those uncertain factors, the distribution of profits
would extend below z. Some entrepreneurs are lucky enough to start near z, while unlucky ones
are far off z.16
To capture this observation, we introduce an additional uncertainty into R&D by randomizing
initial profit levels of entrant firms. More specifically, dropping the subscript i for simplicity,
14

Derivation here is based on Jones and Kim (2018).
Remember that there is a single monopoly firm in each intermediate goods industry i ∈ [0, 1].
16Arguably the assumption is more realistic because many self-employed can also be found in the bottom partof the income distribution.
15

12

we assume that the value of h is randomly drawn for entrant firms upon successful innovationaccording to
FH (h; Θ) ,fH (h; Θ) ,h ∈ h, h ,h≥0(13)where Θ is a set of parameters of the assumed distribution. FH (.) and fH (.) are the cumulative
distribution and density functions of h. Note that z defined in (7) is now the maximum starting profit for entrant firms. In what follows, we derive the distribution of z = hπ for z ≤ z.
Conveniently, the distribution (11) is still valid for z < z.17
Making use of (6) and (13) and by changing the variables, let us rewrite fH (h) in terms of z
as
z 
fH
;Θ
π
ϕ (z; Θ) =,z ∈ (z, z]
πwhere z = hπ and z = hπ. Then,Z z
ϕ (s; Θ) ds,Φ (z; Θ) = 1(14)Φ (z; Θ) =z

is the probability that the initial profit for entrant firms is equal to z or less.
Now, use FL (z, t) to denote the cumulative distribution function of z for z < z. We call FL (z)the Left distribution because it is relevant to the range of z ∈ (z, z]. Similarly, FR (z) is termed
the Right distribution for z ∈ (z, ∞), and it is the same as (11).18 To derive the exact expressionof FL (z), consider how it changes during time interval ∆t:
(a)

(b)

z
}|
{ z
}|
{
FL (z, t + ∆t) − FL (z, t) = gE ∆tFR (∞, t) Φ (z, t; Θ) + gE ∆t [FL (z̃, t) − FL (z, t)] Φ (z, t; Θ)− gE ∆tFL (z, t) [1 − Φ (z, t; Θ)] + [FL (z − ∆z, t) − FL (z, t)]
|
{z
} |
{z
}
(c)

(d)

(15)
This equation is best explained by using Figure 4 which shows the flows of firms.19 Entrant firms
always start at z ≤ z, and the exact entry profit is randomly determined by (13). After entry,
firms move rightward in the distribution due to their own incumbent R&D as long as they are
not hit by entrant innovation. Otherwise, they exit the market. Note that such exits can happen
anywhere in the distribution of z ∈ (z, ∞). Also note that the total number of monopoly firms
must be such that
1 = FL (z) + FR (∞) .
(16)
Let us explore sources of changes in FL (z). First consider the term (a) of (15). Due to entrant
innovations, the number of exiting firms coming from FR (∞) during ∆t is gE ∆tFR (∞, t), outof which a fraction Φ (z, t) flow into FL (z). Similarly, the term (b) represents an inflow of firms
from [FL (z, t) − FL (z, t)]. There are two sources of firm outflows. One is captured by the term(c). Exiting firms are replaced with entrants which start between z and z. Another source isincumbent R&D, which moves firms rightward in the distribution out of FL (z). The term (d)represents this effect, which is captured by the shaded area in Figure 4.
As before, rearrangement and letting ∆t → 0 yielddFL (z) ζζdFL (z, t)=+ FL (z) − Φ (z; Θ)
dt
dz
z
z

17

Although z is not included, the validity of (11) does not change.
Its derivation using (10) does not depend on the assumption of a constant h.
19
In the figure, fL (z; Θ) > 0 is also possible.
18

13

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Figure 4: The Left distribution of entrepreneurial income. The arrows
indicate exit of firms and entry of new firms. z is the boarder line net
profit between the Left and Right distributions.

where (8) and (16) are used. The last term captures the effect of “birth” of entrant firms. Utilizing
condition of the LHS being zero in steady state, one can confirm that the solution to the above
differential equation isZ z
B (z; Θ)FL (z; Θ) = ζsζ−1 Φ (s; Θ) ds
(17)
,
B
(z;
Θ)
=
zζ
z
where Θ is made explicit in FL (z). Naturally, the Left distribution FL (z; Θ) depends on Φ (z; Θ),i.e. the distribution of initial profits of entrant firms. Its associated density isζ[Φ (z; Θ) − FL (z; Θ)] .
(18)z
Having derived (18), the question arises: how is it related to (11)? The answer is that thedensity is continuous at z in the following sense:fL (z; Θ) =fL (z; Θ) = fR (z)(19)

Indeed, it is easy to confirm this equality, using the second equation of (11), (16) and (18). In
addition, note that FL (z; Θ) and FR (z) must satisfy the condition that an inflow of firms intoFL (z) must be matched by an outflow out of it in steady state. The former is given by gE FR (∞)
during dt because all entrants must start at or below z. Making use of this flow condition,
Appendix A derives an outflow of firms crossing the borderline z from FL (z) to FR (∞), andshows that equating those flows givesZ z
1  z ζ
+ 1 fL (z; Θ) dz.
(20)
1=
gI ζ zzThis is the condition which relates ζ and gI to the distribution parameters Θ. It is an equilibrium
condition which consistently links the Left and Right distributions of profit income. Note that Θ
can consist of multiple parameters (e.g. mean and standard deviation for log-normal distribution,right-truncated at z).
14

2.4

Double-Pareto Distribution

The exact shape of the left distribution FL (z; Θ) depends on Φ (z; Θ), and hence the assumed
function of fH (h; Θ). To fix our idea, let us consider
fH (h) =ξhξ−1,
ξ > 1, h = 0,(21)ξhin what follows. This is a Pareto distribution of h with Θ ≡ ξ. We assume ξ > 1 to ensure thatthe density is strictly increasing. It is strictly concave for 1 < ξ < 2, linear for ξ = 2 and convex
for ξ > 2. Therefore, entrant entrepreneurs are more likely start with lower profit income as ξ
falls. In this sense, a lower ξ exacerbates inequality in the left tail.20
Under this assumption, (20) is reduced to
ξ=

1
− ζ.
gI

(22)

This condition endogenously determines the value of ξ for given gI and ζ, which is consistent withflows of firms/entrepreneurs in the entire distribution of profit income. In particular, the Pareto
exponents are negatively related, and its implications will be discussed below. Using (12) and(22), ξ can be expressed in terms of Poisson rates of innovation
ξ=1 − gEgI

When (22) holds, z is now distributed according toζ  z ξ


F
(z)
=
 L
ξ+ζ z
 ζ
F (z) =ξzFL (z) + FR (z) = 1 −
ξ+ζ z

(23)

0<z≤z

(24)

z<z<∞

This is a double-Pareto distribution where z exactly obeys the Pareto law in both tails.21 F (z)collapses to a right-tailed Pareto distribution FR (z) for ξ → ∞. Note that FL (z) = gE andFR (∞) = 1 − gE . This implies that the proportion gE of all entrepreneurs are located in theLeft distribution and others are on the other side. The result is intuitive because a higher gEmeans that creative destruction occurs more often so that more firms tend be found in the Left
distribution.

2.5

The Number of Monopoly Industries

In this section, we consider the determination of N which is the number of monopoly products inthe economy. We derive it by examining the number of intermediate products flowing into andout of N . The approach will be turn out to be useful for later analysis.22
20
21This is reflected in the Gini coefficient calculated for fH (h) alone, which is 1/ (2ξ + 1).
The associated density function is given by

ξζ
z ξ−1



· ξ
0<z<z
ξ
+
ζ
z
f (z) =ζ ξζ · zz≤z<∞ξ + ζ z ζ+1

22

Alternatively, we can directly calculate N . Recall that ni , the number of monopoly products in a given
intermediate industry i, is related to the total profit z through (6). This relationship allows us to rewrite thedensity functions in footnote 21 in terms of n. This method also allows us to derive (27).

15When an entrepreneur succeeds in entrant innovation in industry i, all incumbent products
becomeR i obsolete in the industry and their number is ni gE dt during time dt. Integrating it over igives 0 ni gE dtdi = N gE dt which is the number of intermediate products flowing out of N . On
the other hand, an entrant creates h number of products in i. Given that h is random, its average
is
Z h

hξ
hfH (h) dh =≡ ĥ ξ; h .
(25)
ξ+1
0

Note that ĥ ξ; h may include goods produced by the previous incumbent, and we count them as
an inflow here. Therefore, ĥ ξ; h gE dt is an inflow of goods due to entrant innovation during dt.
In addition, new intermediate goods are created via incumbent R&D with an Raverage flow of ni gI dt
1
products being generated during dt in industry i. Integrating it over i gives 0 (ni gI dt) di = N gI .
Equating inflows and outflows givesN gE dt = ĥgE + N gI dt
(26)
⇓

N=


hξζ
≡ N ξ, ζ; h(ξ + 1) (ζ − 1)(27)

A higher h̄ raises N because h̄ determines the maximum number of monopoly products for entrant
firms. To develop an intuitive explanation of how ξ and ζ affect N , recall that N is equivalent
to the average number of products produced by monopoly firms. Consider ζ which is negatively
related to N . A higher ζ, caused by a higher gE for a given gI (see (12)), means that a turnover offirms is relatively high, and hence the number of products per monopoly firm falls. On the other
hand, N rises as ξ increases. An intuitive account for it can be easily developed, using (23). It
shows that a higher ξ arises due to a lower gE for a given gI , implying a lower turnover of firms
in contrast to ζ. A finite value of N requires ζ > 1.

3

Inequality Measures

Using the double-Pareto distribution, we next demonstrate that the Gini coefficient and the
top/bottom income shares can be expressed in terms of the two Pareto exponents.

3.1

Iso-Gini Contours

It is straightforward, though tedious, to show that the Gini coefficient of (24) is given byZ ∞2 ξ 2 + ξζ + ζ 2 + ξ − ζG=F (z) (1 − F (z)) =.
(2ξ + 1) (ξ + ζ) (2ζ − 1)0(28)

One can confirm that G is decreasing both in ξ and ζ . Figure 5 draws an iso-Gini locus, which areconvex to (1, 1).23 Inequality measured by G falls as we move northeastward. To interpret its slope,note that the Gini of the Right distribution would be 1/ (2ζ − 1) if FR (z) alone was considered
independently and calculated separately, and similarly 1/ (2ξ + 1) for FL (z) alone. This showsthat a higher ζ and ξ reduces the Gini within each side, and increasing ζ and reducing ξ is akinto shifting inequality from the Right to the Left distribution. Following this interpretation, theslope of an iso-Gini curve is the marginal rate of substitution between inequalities in the Left and23

See Appendix B for proof.

16

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Figure 5: The four areas divided by iso-SB and iso-ST curves. Startingfrom A0, the X relationship holds in Areas 1 and 4.
Right distributions. A unit increase ζ (falling inequality in the Right distribution) requires a fallin ξ (increasing inequality in the Left distribution) by the amount equivalent to the slope of an
iso-Gini curve for a given level of the Gini coefficient. A downward-sloping iso-Gini locus implies
that a fall in either ξ or ζ, taking the other Pareto exponent constant, necessarily increases theGini coefficient.

3.2

Top/Bottom Income Shares

The top/bottom income shares can also be easily calculated in our framework. For this, define
100p̄ with p̄ = F (z̄) as the percentile for z̄, the threshold income between the Left and Right
distributions. First consider the bottom 100pB % income share for p < p̄, which we use SB todenote. Appendix C shows that it is defined by
1+ 1
SB = pB ξ



1
1
ξ ξ
1−
1+
,
ζ
ζ

for pB ≤ p̄,

∂SB
> 0,
∂ξ

∂SB
> 0.
∂ζ

(29)

This equation allows us to draw an iso-SB curve in Figure 5, which shows different combinationsof (ξ, ζ) for a given (SB , pB ). SB increases in ζ because its higher value means a thiner right tails.
BB is also increasing in ξ despite that the left tail gets thiner with it. Its reason is more involved,
but it is basically driven by the fact that net profit at the 100pB percentile increases with a higher
ξ.24 The signs of the derivatives in (29) imply that, holding pB constant, SB gets higher/lower in
the northeast/southwest area. This property will be exploited to examine how the income sharechanges in response to parameter changes.
In the appendix, we also derive the top 100(1 − pT )% income share (e.g. 1 − pT = 0.01), ST
24

The cumulative income up to 100pB is given by gI ξζprofit at 100pB percentile. It is increasing in ξ.

1
R zL (pB )  z ξ
ξ
ξ+ζ
dz
where
z
(p
)
=
z
p
is the net
L
B
B
ζ
0
z

17

for pT > p̄, which is defined by25

ST = (1 − pT )

1− ζ1

1
ξ
1 ,

ζ ζ
1+
ξ
1+

for pT ≥ p̄,

∂ST
< 0,
∂ξ

∂ST
< 0.
∂ζ

(30)

This defines iso-ST loci in Figure 5, giving different combinations of (ξ, ζ) for a given (ST , 1 − pT ).
An intuition for the signs of the derivatives are similar to the one given for (29). The derivativesin (30) confirm that ST is larger/smaller in the southwest/northeast area for a constant pT .
Appendix C also demonstrates that an iso-ST curve is steeper than an iso-SB curve, giving rise
to a unique intersection point A0, as illustrated in Figure 5. The two curves divide the (ξ, ζ) space
into four areas. To interpret them, consider an economy located at A0. The following list gives
what happens as the economy moves from A0 to four regions, holding pB and 1 − pT constant:Area 1: SB falls, and ST increases.
Area 2: SB increases, and ST increases (i.e. the income share of the middle income range betweenpB and pT falls).
Area 3: SB falls, and ST falls (i.e. the income share of the middle income range between pB andpT rises).
Area 4: SB increases, and ST falls.
Those four cases allow us to explore how income shares respond to parameter changes.
The next task is to locate an iso-Gini curve passing through A0. Appendix D shows that if
pB and pT are sufficiently different from p̄, an iso-Gini is steeper than an iso-SB curve, but lessso than an iso-ST curve, as illustrated in Figure 5. A word “sufficiently” does not mean pB → 0
or pT → 1. In fact, an iso-Gini curve is sandwiched between two iso-income share curves for alarge range of values of pB and pT . For example, Toda (2012) obtains estimates of ζ = 2.34 andξ = 1.15 using U.S. data from the Current Population Survey (2000-2009). He also shows thata calculated Gini coefficient based on those values is close to an actual value. Using the same
numbers, we have p̄ = 0.67 and the sandwiched case arises for pB ≤ 0.65 and 1 − pT ≤ 1 − 0.67,
which practically covers an almost entire range of values. In addition, our calibration analysis
below suggests that the sandwiched case is a norm.26
The X inequality relationship between the Gini coefficient and the top/bottom income sharein Table 1 corresponds to Areas 1 and 4 in Figure 5. Now consider again an economy at A0
in the figure. Further suppose that the economy moves to a random point in (ξ, ζ) space. It isclear from the figure that the economy is more likely to end up in Areas 1 and 4 than other ares
because the iso-SB and iso-ST curves are downward-sloping, i.e. the X relationship is more likely
to occur than otherwise. Having said this, however, the economy does not move randomly, butsystematically according to economic incentives. To explore this, we need to endogenize R&D
activity.
ζ −1
ξ+1
and ST =for p = p̄.
ξ+ζ
ξ+ζ
26
Focussing upon the sandwiched case also allows us to avoid taxonomic analysis.
25

One can easily confirm SB =18

Panel (a)Panel (b)log(FL(x)) or log(1 FR(x))

0

1970
1975
1980
1985
1990
1995
2000
2005
2010
2015

2
4
6
8
10
12
14
16

10

5

0

log xx0
(

5

1.80

1.10

1.75

1.08

1.70

1.06

1.65

1.04

1.60

1.02

1.55
1980

10

1990

1995

2000

2005

2010

2015

1.50

for the 0-35th percentile
for the 65-100th percentile (right axis))

FL(x) = xx0 is used for x/x0 < 0 and
1 FR(x) = xx0 is used for x/x0 > 0 where
x0 is income at the 60th percentile.
(

1985

)

(

)

Figure 6: Both panels use an income variable called “ptinc” (pre-tax national income; non-negativevalues only) with a frequency varaible “dweght” in Piketty et al. (2018).

3.3

Data and a Double-Pareto Approximation

In general, an observed income distribution does not follow a particular distribution, and different distributions are proposed to model it. Examples include exponential, lognormal, gamma,
Levy-stable, double-Pareto-lognormal, and they are considered as an approximation of a true
distribution at best.27 Similarly, we use a double-Pareto as an approximation of an underlying
income distribution. But before we move on, we consider the extent to which the double-Pareto
approximation is useful in understanding the X inequality relationship. For this, we use data of
Piketty et al. (2018). Its advantage is that it allows us to calculate summary statistics based on
disaggregated data of 100% national income in the U.S.
We first examine some properties relevant to our analysis. Panel (a) of Figure 6 shows a log-log
plot of the data for five-year intervals in 1970-2015. The threshold is set to the 55th percentile
and a tent map shape is clearly visible for all years.28 This visual inspection suggests that adouble-Pareto distribution is in the ballpark.
Having said this, however, approximation means loss of information by definition (more on
this later). In addition, a question remains on how to calculate the Left and Right tail Pareto
exponents. A possibility may be to choose a given threshold percentile which divides the entire
distribution into the Left and Right parts and calculate exponents, respectively. However, thisapproach does not necessarily generate the best fit. Instead, we search different percentiles tocalculate ξˆ and ζ̂ separately and choose the ones which meet the following conditions: (i) ξˆ and ζ̂are both greater than 1, and (ii) prediction errors measured by the coefficient of variation of root
mean square deviation is minimized in five percentile intervals. According to these criterion, we
found the 35th percentile as the upper threshold for the Left distribution and the 65th percentile
as the lower threshold for the Right distribution in the 1981-2016 period. Panel (b) shows the
maximum likelihood estimates of ξ and ζ based on those percentiles. A clear downward trend isnoticeable for ζ̂. This means that the Gini coefficient tends to increase according to (28). On the27
28

See Toda (2012) for some references.
The tent map shape does not dramatically change even if the 60th or 50th percentile is used.

19other hand, ξˆ has a positive trend. This tends to decrease the Gini coefficient according to (28).
An increasing trend of the Gini coefficient in the U.S. means that the former effect dominates thelatter.
Given those estimates, we compare the inequality indices calculated using data and predictedby a double-Pareto distribution with estimates of ξˆ and ζ̂. Panel (a) of Figure 7 (next page)shows the Gini coefficient. The predicted values closely follow the actual index, though slightdeviation occurs in more recent years. As far as the Gini coefficient is concerned, a double-Pareto
approximation works strikingly well.29 Panels (b) and (c) show the top 1% and 5% income shares.
It is immediately clear that predicted values again closely move together with the data, though
there are differences in levels. Level differences are even clearer for the bottom 30% and 40%
income shares in Panels (d) and (e). They are due to information loss caused by a double-Pareto
approximation. What is remarkable, however, is that the tendency of changes, i.e. the slope, ispreserved for all of those indices. Indeed, the null hypothesis that the slopes of a linear trendof data and predicted values are the same cannot be rejected even at the 10% level for the Gini
coefficient and the bottom 30% and 40% income shares. Regarding the top 1% and 5% income
shares, although a similar null hypothesis can be rejected at 1% level, the figures suggest thatthe data and the approximated series move closely together. Because the purpose of our analysis
is not to explain levels, but changes or trend of the inequality indices, we take a Double-Pareto
distribution as a reasonable approximation of the underlying income distribution for our purposes.

4

Endogenous Growth

The previous sections regarded gE and gI as exogenous and focused on how innovation affects
inequality. In what follows, we endogenize those variables, introducing the reverse channels by
which inequality affects innovation incentives.

4.1

Consumers

As mentioned above, consumers are risk-neutral, hence the interest rate r is equal to the rate oftime preferences. We assume that the price index associated with (1) is one.30 Given these, thedemand for Yj is
E
Yj =
(31)
JPj
where E is consumption expenditure and Pj is the price of Yj .

4.2

Demand for Intermediate Products

Consumption goods j is competitively produced according to (2). Profit maximization requiresthat yji pji = Yj PJ holds. Hence, a demand function of intermediate product yji is given by
yji =

E
Jpji

(32)

where pji is the price of yji .
This confirms that the effect of a falling ζ̂ dominates that of an increasing ξˆ in Panel (b) of Figure 6. Also notethat the Gini coefficient calculated using the data of Piketty et al. (2018) is somewhat higher than those reported
in other sources, summarized in UNU-WIDER (2020).
1 RJ
30
One can show that the price index associated with U is e J 0 ln Pj dj = 1.
2920

Panel (a)Gini Coefficient
Data
Double Pareto Prediction

0.59
0.58
0.57
0.56
0.55
0.54
0.53
0.52
1980

1985

1990

1995

Panel (b)Top 1% Income Share (%)

2005

36

18

34

2015

Data
Double Pareto Prediction

38

20

2010

Panel (c)Top 5% Income Share (%)

40

Data
Double Pareto Prediction

22

2000

32

16

30

14

28

12
1980

1985

1990

1995

2000

2005

2010

26

2015

1980

Panel (d)Bottom 30% Income Share (%)

6.4

1990

1995

2000

2005

2010

2015

Panel (e)Bottom 40% Income Share (%)

Data
Double Pareto Prediction

6.2

1985

Data
Double Pareto Prediction

11.0

6.0

10.5

5.8

10.0

5.6
5.4

9.5

5.2

9.0

5.0

8.5

4.8
1980

1985

1990

1995

2000

2005

2010

2015

1980

1985

1990

1995

2000

2005

2010

2015

Figure 7: The plots labelled Data are calculated using the same variables in Figure 6. Thoselabelled Double Pareto Prediction use ξˆ and ζ̂ in Figure 6 to calculate (28), (29) and (30).

21

4.3

Profits

One unit of intermediate goods is produced with one worker. Intermediate goods are produced
competitively or by monopoly firms. In the latter case, like other Schumpeterian models, firms
charge the price with the quality step λ as a constant markup over the marginal cost, i.e. pji = λw
E1
where w is wage. Therefore, profit per product is Λ where Λ ≡ 1 − .
J
λ

4.4

R&D Technology

To enter an intermediate good industry i as a monopoly, an entrepreneur has to be successful
δE
in R&D first. Entrant innovation occurs with a Poisson arrival rate of δ E ≡ 1−µ for eachRE
entrepreneur. RE is the total number of entrepreneurs who engage in entrant R&D, and its
presence in the denominator captures the negative congestion externality. Because of free entry,
there are potentially many researchers, and each of them takes RE as given. The Poisson rate forthe economy as a whole is gE = δ E RE orµ,gE = δE RE

δE > 0,0 < µ ≤ 1.

(33)

It is equivalent to gE in Section 2.
Recall that entrant R&D is undirected in the sense that an intermediate industry where
innovation is implemented is randomly chosen from all industries i ∈ [0, 1] after innovation. Oncean industry i is chosen, then all goods produced by the previous incumbent monopoly firm in i,are rendered obsolete. Then, a range of goods with a measure hi < J is randomly selected foran entrant to start with, and their quality levels increases by a factor λ. An entrepreneur earns
profits hi π at the time of entry. Any other goods in i are now competitively produced. hi goods
may include those produced by the previous incumbent.
After entry, an entrepreneur turns to incumbent R&D to increase profits further. If successful,
a competitive product in i is randomly picked and its quality rises by a factor λ, increasing theentrepreneur’s profit by π. We assume that incumbent R&D takes the form of multiple projects,
and a single project runs and is financed out of profit arising from each intermediate goods
production. That is, the number of R&D projects is equivalent to the number of products thatfirm produces. Specifically, innovation for each project follows a Poisson process with an arrival
rate ofγ
gIi = δI RIi,
δI > 0, 0 < γ < 1(34)where RIi is the number of workers in each project of a firm operating in industry i.
Now consider the expected change in ni due to incumbent R&D. Given multiple R&D projects,
it changes according to
dni = 1 × ni gIi dt + 0 × ni (1 − gIi dit) = ni gIi dt.

(35)

Consider the first term of the first equality. A firm runs ni multiple projects, and each generates
an arrival rate gIi . Therefore, ni gIi dt is equivalent to a flow of innovations during dt, each ofwhich improves the quality of a good. The second term represents the case where projects fail.
(35) shows that ni grows exponentially on average.

22

4.5

R&D Decisions

Let us consider an R&D decision facing an incumbent firm with ni products. Let Vi denote thevalue of the incumbent firm. Given (35), define the value of that firm:Vi (ni ) = max ni πi dt + E (1 − ρdt) (1 − gE dt) Vi (ni (t + dt)) + Vi (t + dt)|ni =n̄i
(36)
RIi

where

E
− (1 − sI ) wRIi
(37)
J
is after-tax profit net of R&D expenditure, τ is the corporate tax rate and sI is the rate of subsidy
to incumbent R&D. The second term on the RHS is interpreted as follows. Vi (ni (t + dt)) is thevalue if an additional innovation occurs, and it is realized if no entrant innovation occurs withthe probability of (1 − gE dt) during dt. The remaining term (1 − ρdt) discounts the future value.
The last term captures capital gains due to growth of E and w which is realized irrespective ofinnovation. Appendix F shows that the optimal RIi is defined byπi ≡ (1 − τ ) Λ

RIi =



γδI
V
·
1 − sI w

where
V ≡

 1

1−γ

≡ RI

∀ i ∈ [0, 1](38)π
ρ + gE − gI − gw

(39)v = ĥ (ξ) V(40)

is the value of an incumbent firm per product or V ≡ Vi (ni ) /ni and gw is the growth rate of wage
which captures capital gains. The presence of gE in V captures the risk of losing profits, and gI
represents an additional gain from incumbent innovation. Because R&D employment per productis the same for i ∈ [0, 1] (see (38)), we have πi = π and gIi = gI . The latter is equivalent to gI in
Section 2. RI in (38) is increasing in δI , sI and V /w as expected.
Turning to entrant R&D decisions, recall that hi is a random variable, and so is the value of an
entrant firm because it depends on hi . We therefore distinguish between its ex ante and ex postvalues. An ex ante or ex post value of innovation is the one before or after the result of uncertain
R&D, i.e. the value of hi becomes known. In fact, the ex post value per product is equivalent to
V in (39), hence the ex post firm value is given by hi V . Next, let us use v to denote the ex ante
value of entrant innovation. Using the average of hi in (25), v is given bybecause V is the same for all intermediate good industries. Free entry is assumed, and it leads to
δE
1−µ
RE

v = (1 − sE ) w(41)

where sE is the rate of subsidy for entrant R&D.
To explore how R&D incentives respond to research-related parameters, let us use (38), (40)and (41) to get"# 1RE
1 − sI δE ĥ (ξ) 1−γ
=
·
·
(42)
RI
1 − sE δIγ
where we assume µ = γ, i.e. the extent of the diminishing returns of entrant and incumbent
R&D is the same for simplicity. This assumption is maintained in what follows.31 The ratio of
31

Assuming µ ̸= γ complicates a equilibrium condition without generating an additional insight.

23

entrant-to-incumbent R&D workers RE /RI is increasing in (1 − sI ) δE / (1 − sE ) δI . An intuition
goes as follows. An increase in δE or sE encourages entrant R&D, but discourages incumbent
innovations because the risk of losing all profits and existing the market rises. Regarding a higherδI or sI , it indeed promotes incumbent and entrant innovations because the value of a monopoly
product V increases. However, incumbent R&D incentives are larger than entrants’ to the extent
that RE /RI falls. (42) also shows that the relative R&D employment is increasing in ĥ (ξ), theaverage value of h. This is because a higher ĥ (ξ) raises the expected return from entrant R&D,
leading to a greater employment in entrant R&D.

4.6

Labour Market

There are four sources that require workers. First, the number of entrepreneurs earning profits is
1. This is because there is a single monopoly firm run by an entrepreneur in each of intermediate
goods industry i ∈ [0, 1]. Second, workers are used for R&D. Those who engage in entrant R&D is
1
RE = (gE /δE ) µ from (33). In addition, (22) and (23) allow us to write gE = ζ/ (ξ + ζ). Therefore,
1
RE = (ζ/δE (ξ + ζ)) µ ≡ RE (ξ, ζ) is equivalent to entrepreneurs trying to enter the intermediate
goods industry. Incumbent R&D workers per product is similarly calculated from (34) and (22)1
as RI = (1/δI (ξ + ζ)) γ ≡ RI (ξ, ζ). The total number of workers used for incumbent research is
RI N where N is given in (27). The remaining workers are used for manufacturing, which employs
CZ/J + N Z/Jλ = (J − ΛN ) Z/J where Z ≡ E/w. Workers are fully employed for
L = 1 + RE (ξ, ζ) + RI (ξ, ζ) N (ξ, ζ) + [J − ΛN (ξ, ζ)]

5

Steady State Equilibrium

5.1

GrowthZ
J(43)

Entrant and incumbent innovations improve quality of intermediate products. This manifests itself
in the form of wage growth and utility growth.32 Indeed, Appendix E shows that the following
holds in steady state
Q̇
ẇ
U̇
gQ ≡
=
=
(44)
Q
w
U
RJ R1
where ln (Q) = J1 0 0 kji didj · ln (λ) and Q is the overall quality index for the whole economy.
Its change during dt is given byZ Z 
1 J 1 the number of quality
d ln (Q) =× [(kji + 1) − kji ] didj · ln (λ)
improvement during dt
J 0 0Note that the number of innovations, entrant and incumbent, is equivalent to the right-hand sideof (26). Therefore, we have

32

Entrant
Incumbent
Contribution
Contributionz }| {z }| {
gQ = (
gE ĥ (ξ)+ gI N (ξ, ζ) ) ln (λ) =See Grossman and Helpman (1991) for example.

24ζ
N (ξ, ζ) ln (λ)
ξ+ζ

(45)

using (22), (12) and (27). The first equality decomposes growth into entrant and incumbent
contributions. The former depends only on the Pareto exponent of the Left distribution ξ because
entrants start from there.

5.2

Equilibrium Conditions

To solve the model, let us first use (12), (33) and (34) to rewrite (42) as
ζ=A



ξ
ξ+1

 γ

1−γ

A≡

,



1 − sI h·
1 − sE γ

γ 
 1−γ

δE
δI

 1

1−γ

.

(46)

We call it the R&D-incentive condition because it captures the optimal R&D decisions of entrant
and incumbent firms. A positive relationship between ζ and ξ is due the fact that the ratio of
entrant-to-incumbent R&D workers RE /RI in (42) is positively related to ξ, as explained above.
The second condition is based on the ex ante value of successful entrant innovation. Using(37), (39), (40), (41), (43) and (45), one can derive what we call the firm-value condition:Γ (ξ, ζ) =δEĥ (ξ)·Π (ξ, ζ)1 − sE RE (ξ, ζ)1−γ

(47)

where
L − 1 − RE (ξ, ζ) − RI (ξ, ζ) N (ξ, ζ),J − ΛN (ξ, ζ)(ζ − 1 + γ) − ζN (ξ, ζ) (ln λ)Γ (ξ, ζ) ≡ ρ +.
ξ+ζΠ (ξ, ζ) ≡ (1 − τ ) Λ

(48)
(49)

Although Π (ξ, ζ) and Γ (ξ, ζ) look complicated, they have clear interpretations. Π (ξ, ζ) is theafter-tax (gross) profit and Γ (ξ, ζ) is the effective discount rate, both expressed in terms of thetwo Pareto exponents.33(46) and (47) is the system of two equations with two unknowns (ξ, ζ). Note that gE and gI inequilibrium can be recovered once ξ and ζ are determined, using (22) and (23). Before considering
comparative static analysis, we distinguish two cases, focusing upon a unique equilibrium.34 The
first case is illustrated in Figure 8 where the R&D-incentive condition is steeper than the firmvalue condition at equilibrium.35 In the second case (not shown), the relative slopes are reversed.
In what follows, we focus on the first case because its equilibrium is stable in the following sense.
The R&D-incentive condition is based on the optimal R&D decisions. Incumbents optimallychose RI , and RE is determined via free entry. In particular, when firms consider whether to
conduct entrant R&D, they take the ex ante value v = ĥ (ξ) V in (40) as given. That is, theR&D-incentive condition (46) determines ζ, taking ξ as given. On the other hand, the firm-value
condition (47) concerns the ex ante value of entrant innovation, which essentially determines the33Π (ξ, ζ) is equivalent to (1 − τ )Eexpressed in terms of ξ and ζ. Γ (ξ, ζ) is equivalent to the denominator of
Jw

(39).
34
Implicitly differentiating the labour market condition (43), we can show that the partial derivative of Z w.r.t.
ξ is negative using the fact that the net profit (37) is positive. The partial derivative of Z w.r.t. ζ, on the other
hand, turns out ambiguous, because a higher ζ raises RE and manufacturing employment, but reduces RI . In what
follows, we assume ∂Z≥ 0. This assumption implies that a change in manufacturing employment in particular∂ζ
dominates and seems reasonable given the fact that R&D employment is about 1% in total employment in data
(e.g. see calibration analysis below).
35
The R&D-incentives condition always satisfies ζ > 1 for ξ = 1.

25

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Incentive
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Condition

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1

gE
gE

A1
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Figure 8: Steady state equilibrium as an intersection point between the
R&D-incentive condition and the firm-value condition. Equilibrium is stable when the former is steeper than the latter. The line from the origin
passing through the equilibrium point is equivalent to (1 − gE ) /gE . An
iso-gI curve is defined by (22).

Pareto exponent ξ of the Left distribution where entrant firms start, taking ζ as given. Viewed
this way, equilibrium illustrated in Figure 8 is stable. In addition, calibration analysis below showsthat the stable case is a norm.

5.3

Comparative Statics

Focusing on the stable case in Figure 8, the following summarizes comparative statics results:
Result 1: Following an increase in λ, L or a fall in J, ρ, τ ,• ξ and ζ decrease.
• gE and gI increase.
• the Gini coefficient increases, the bottom pB income share falls, andthe top pT income share increases.
Result 2: Following an increase in δI or sI ,• ξ and ζ decrease.
• gE and gI increase.
• the Gini coefficient increases, the bottom pB income share falls, andthe top pT income share increases.
Result 3: Suppose that λ is not too large such that Γ (ξ, ζ) − ρ ≥ 0.
Following an increase in δE or sE ,• ξ decreases and ζ increases.
• gE and gI change ambiguously.
26• changes in the Gini coefficient increases, the bottom pB income share falls, andthe top pT income share are ambiguous.
Result 4: Suppose sI = sE initially. Following a simultaneous increase in sI and sE ,• ξ and ζ decrease.
• gE and gI increase.
• the Gini coefficient increases, the bottom pB income share falls, andthe top pT income share increases.
To develop an intuition, note that ξ/ζ = (1 − gE ) /gE which is equivalent to the slope of aline from the origin to A0 in Figure 8. Also note that (22) defines an iso-gI contour with theslope of −1. As we move southwestward in the figure, gI gets higher, and it becomes lower in
the area above the iso-gI locus. Now, consider Result 1. A fall in τ , for example, means thatan after-tax profit per product is larger. This effect manifests itself in a downward shift of thefirm-value condition, moving an equilibrium from A0 to A1 . It causes the ratio ξ/ζ to fall, and A1
is located below the iso-gI curve, given that the R&D-incentives condition unaffected. Intuitively,
the tax reduction induces both types of R&D, but its impact on incumbent R&D is greater thanentrant R&D because the effect is realized immediately for incumbents, but in future for entrants
conditional on an R&D success. Such difference results in a fall in ζ. The result also reveals theinequality-worsening effects of a lower corporate tax, and it is also empirically supported (e.g. see
Nallareddy et al. (2018)). Other parameters are similarly interpreted.
Turning to Result 2, as δI gets larger, the firm-value condition shifts down and the R&Dincentive condition moves left. As a result, equilibrium moves from A0 to A2 . This is because
a higher incumbent R&D productivity boosts an incentive for incumbent R&D. In fact, it also
induces entrant R&D because a higher δI increases the ex ante value of entrant innovation. Those
two effects reduce ξ. In addition, the effect on gI is strong to the extent that ζ falls and inequality
worsens for the same reason explained above. Regarding the subsidy rate of incumbent R&D sI ,its higher value affects the R&D-incentive condition only, moving equilibrium to A3 in Figure 8.
gE is positively affected for the same reason as in a higher δI .
Result 3 summarizes the effects of entrant R&D productivity improvement and a higher rateof entrant subsidy. They affect the Pareto exponents differently, hence changes in the inequality
indices are ambiguous. Intuitively, a greater δE and sE makes entrant R&D attractive, resulting
in a higher gE . On the other hand, it increases the risk of losing profits for incumbent firms. This
discourages incumbent R&D. Those effects lead to opposite changes in ξ and ζ. This result leads
us to an interesting observation regarding entry. Jones and Kim (2018) show that entry of new
firms tends to reduce inequality via creative destruction, while entrants can increase inequality
in Aghion et al. (2019) because of higher markups they enjoy. Aghion et al. (2019) also reported
that entrant and incumbent innovations both are positively correlated with top 1% income share.
These findings suggest that firm entry can increase or decrease inequality. Our model is in linewith both of those findings.
In Result 4, a simultaneous increase in the rate of subsidies to incumbent and entrant R&D
is considered. By assumption, the R&D-incentive condition is unaffected, while the firm-value
condition shifts downward. A new equilibrium is given by a point like A1 . It is the combinationof the effects caused by a higher sE and sI in Results 2 and 3. This result qualitatively implies
that more generous R&D subsidies may be behind the X inequality relationship.
According to these results, most parameter changes, ceteris paribus, moves the economy to
Areas 1 and 4 in Figure 5. In this sense, our model predicts that those parameter shifts may haveplayed a role in generating the X inequality relationship. While these are useful insights, they do27Externally Set Parameters

Internally Set Parameters

ρ

0.07

τ

0.30 → 0.20 (changes linearly)

J

1.249

γ

0.35

sE

0.05 → 0.20 (changes linearly)

L

10.0

sI

δE , δIh, λPanels (c)-(f)of Figure 9

0.05 → 0.20 (changes linearly)

Table 2: Calibrated parameter values.
not inform us about the extent to which each factor contributed to the X inequality relationshipobserved in many countries. To tackle this issue, we next resort to calibration analysis.

6

Calibration

6.1

Strategy

The main purpose of this calibration exercise is to identify underlying factors of the Xinequality
relationship in the U.S. in recent decades. In particular, we quantify their contributions based on
data. Our strategy has two steps.
In the first step, six parameters are set externally, and others are disciplined subject to data.
In particular, we set the latter parameters on the basis of three types of data: (i) the entry rate offirms, (ii) the share of R&D workers in working population, and (iii) TFP growth. Given the seriesof ξˆ and ζ̂ in Panel (b) of Figure 6, parameters are matched with (i) and (ii) for the 1981-2016
period, and (iii) for the average in the period. By so doing, we basically allow parameter values
to change to be consistent with ξˆ and ζ̂. In the second step, we let all parameters to change as in
the first step, except for a single parameter which is held fixed at the 1981 level. Shutting down
the effect of a parameter makes it possible to identify the extent to which it contributed to the
X inequality relationship. We conduct this exercise for six parameters of interest to quantify theindividual contribution of parameters to changes in inequality indices.

6.2

Calibrated Values and the Model Fit

Six parameters in Table 2 are externally set. The subjective rate of time preference ρ is set
to 0.07 to roughly mimic the long-run annual rate of return from the stock market. γ is theparameter which determines the degree of diminishing marginal product of R&D workers forentrant and incumbent firms. This parameter plays an important role in characterizing the natureof equilibrium. Kortum (1993) reports point estimates between 0.1 and 0.6. In a more recent
attempt, Acemoglu, Akcigit, Hanley and Kerr (2016) runs a first-difference regression, reporting
0.286-0.455 with the the average of 0.35. They also conduct robustness checks, e.g. by restricting
dataset, and obtain similar values. Acemoglu and Akcigit (2012) use those values for the analysis
of IPR and innovation, and Acemoglu et al. (2018) use 0.5. Given those studies, we set γ = 0.35,
and the result does not dramatically change as long as γ ≤ 0.5. We set the working population
L = 10.0 for the following reason. In our model there are always a measure one of entrepreneurs
earning positive monopoly profits. and those profits can be lower than wage. In this sense,
entrepreneurs in our model are more like self-employed in data. The US Bureau of Labor Statistics
compiles data of self-employed, incorporated and unincorporated both starting in 2000. Its ratio
to the total employment is stable with the average of 10.7% in the 2000-2016 period, implying aroughly one in 10 are self-employed. L = 10.0 is used in line with this number.

28

Regarding a corporate profit tax rate and R&D subsidy rate, we borrow the values that
Akcigit and Ates (2019b) use. They provide a brief historical account for changes in those rates.
Setting the corporate tax at 30% and the subsidy rate at 5% in 1981 in their calibration, theauthors examine declining business dynamism in the US by changing those rates to 20% in 2010,
respectively. Although they use a sophisticated approach of changing those rates over the period,
we adopt a simpler approach of linearly changing them. Incumbent and entrant subsidy rates areequalized. The remaining parameters are internally set in the following way.
Panel (a) of Figure 9 shows the entry rate of establishments in the U.S, constructed using theBusiness Dynamic Statistics compiled by the Census Bureau. It is the ratio of new establishments
to the total number of active establishments in a given year. Its long-run trend is negative,
although it increased in early 1980 and before a steep dive due to the financial crisis in 2008. In
Panel (b), the share of R&D workers is plotted, using the data from the OECD Main Science andTechnology Indicators. It is defined as the ratio of the full-time equivalent number of researchers
to the total employment. In contrast to the rate of firm entry, it steadily increases over theperiod. Finally, we also use the TFP growth rates, adjusted for capital utilization and labour
efforts, which are reported in Fernald (2014).
Using those values and given the (ξ, ζ) series in Panel (b) of Figure 5, we set up the system
of four equations to determine four parameters δI , δE , h and λ. The first equation is the rate offirm entry, which is given by
DataER =gE ĥ (ξ)ζ
ĥ (ξ)
=
·
.
J
ξ+ζ
J

(50)

where DataER is Panel (a) of Figure 9. At each moment, gE number of innovations occur acrossi ∈ [0, 1], and each innovation creates ĥ (ξ) number of products. We take those products asestablishments in data. gE ĥ (ξ) is divided by J to make it consistent with the definition of thedata on the LHS, which corresponds to a series in Panel (a) of Figure 9. Rewriting the firstequality using (12) and (22), we can use (50) to pin down the value of h.
Let DataRD denote a series in Panel (b) of Figure 9. Then, the share of R&D workers satisfies
the following condition
DataRD × L = RE (ξ, ζ) + RI (ξ, ζ) N (ξ, ζ) .

(51)

The remaining two conditions are the R&D-incentive condition (46) and the firm-value condition(47) which we use to make parameter values data-consistent. Making use of those, we simultaneously determine the values of δI , δE , h and λ over the 1981-2016 period for a given J. Finally,
given these parameter values and using gQ in (45), we set J = 1.249 to match the average annual
TFP growth rate over the period, which is 0.859 from the data. This gives us recalculated valuesof δI , δE , h and λ.
The results are shown in Panels (c)-(f) of Figure 9. A noticeable feature is that R&D productivity levels, entrant and incumbent both, steadily fell. Importantly, the rate of reduction inδE is 19.9% which is greater than 16.7% for incumbents. This has the following implication forinequality. (42) shows that its RHS falls, tending to reduce the ratio of entrant to incumbent
R&D employment. This translates into a reduction of the Right exponent ζ, making the righttail thicker. In fact, this result captures a falling trend of ζ in Panel (b) of Figure 6. A similar
observation can be made for the Left distribution. Its Pareto exponent is given in (23), which isincreasing in the figure.
h is the maximum initial number of products for entrants, and it fell by 16.2%, comparing the1981 and 2016 values in Panels (e) of Figure 9. This follows the 20.6% reduction of the firm entry
29Panel (b)The Share of R&D Workers (%)

Panel (a)Entry Rate of Firms in US
0.14
0.9

0.13

0.8

0.12
0.11

0.7

0.10

0.6
1980

1985

1990

1995

2000

2005

2010

1980

2015

1985

1990

1995

2000

2005

2010

2015

The entry rate is defined as the ratio of the number tothe number of new establishments to the total numberof establishments in a given year.
Data: the Business Dynamic Statistics, the Census Bureau.

The ratio of the full-time equivalent number of
researchers to the total employment.
Data: Main Science and Technology Indicators (2021)

Panel (c)Entrant R&D Productivity

Panel (d)Incumbent R&D Productivity

1.95
1.90
1.85
1.80
1.75
1.70
1.65
1.60
1.55

E

I

1.20
1.15
1.10
1.05
1.00

1980

1985

1990

1995

2000

2005

2010

0.95

2015

1980

Panel (e)No of Entrant's Products

0.575

1990

1995

2000

2005

2010

2015

Panel (f)Size of Innovation
h

0.550

1.030

0.525

1.028

0.500

1.026

0.475
0.450

1.024

0.425

1.022

0.400

1.020

0.375

1985

1980

1985

1990

1995

2000

2005

2010

2015

1980

1985

1990

1995

2000

2005

2010

2015

Figure 9: Panels (a) and (b) plot data, while calibrated values are shown in Panels (c)-(f).

30ModelData

Source(1)

TFP Growth

0.86%

0.86%

Fernald (2014)(2)

Rate of Firm Entry

11.70%

11.70%

Business Dynamic Statistics,
the US Census Bureau(3)

Share of R&D Workers

0.75%

0.75%

OECD Main Science andTechnology Indicators(4)

Size of Innovation

1.023

1.075

(5)

Incumbent Contribution
to TFP Growth61.25%

75.17%(6)

Entrant Contribution
to TFP Growth38.75%

24.83%

Garcia-Macia, Hsieh and Klenow (2019)

Table 3: The “Model” column shows the 1981-2016 averages. (1)-(3) in the “Data”
columns are the average values in the 1981-2016 period, and those in (4)-(6) give theaverage of the three periods, 1983-1993, 1993-2003, 2003-2013.

rate in data. An implication is that the income distribution becomes skew to the right. This
tends to increase inequality, as will be discussed. The size of quality step λ is shown in Panel (f).
Table 3 summarizes the model fit on the basis of the average values. Note that parametervalues are chosen so that the model fits the data for (1)-(3), while (4)-(6) compare the model
prediction with values reportetd in Garcia-Macia et al. (2019). The size of innovation λ is slightly
lower than the value reported, but it falls in the range considered “plausible” by Stokey (1995).36
λ is also the monopoly price markup over marginal cost. Its increasing trend is consistent with thefact that the markup increased in the US in recent decades, though the level of markup predictedby our model is small.37 (5) and (6) give the contribution of incumbent and entrant innovations to
TFP growth. Though the model under- (or over-)predicts the incumbent (or entrant) contribution,
those values are roughly in line with the data. In addition, Garcia-Macia et al. (2019) report thatincumbent contribution increased while entrants’ fell in the 1983-2013 period, and this trend is
captured by our model.38 Despite the parsimonious and stylized nature of the model, the fit of themodel seems broadly reasonable. Figure 10 (next page) illustrates an equilibrium in 1981 basedon calibrated parameter values. It shifts northeastward, generating the X inequality relationship,
i.e. a higher Gini coefficient, a lower bottom income share and a higher top income share.

6.3

Quantifying Factors for X Inequality Relationship

Calibrated parameter values used in Table 2 are data-consistent based on the Pareto exponents inPanel (b) of Figure 6. Put differently, we can reproduce those series of ξˆ and ζ̂. More importantly,
we can also reproduce Double-Pareto Prediction series of the Gini coefficient and top/bottom
income shares in Panels (a)-(e) of Figure 7, using the R&D-incentive and firm-value conditions
with those calibrated parameter values. Viewed from the model’s perspective, therefore, changes
36

Stokey (1995) considers [1.02, 1.6] as a plausible range. Acemoglu and Cao (2015) use 1.1 and 1.2 for simulation,
which are also in the range.
37
Evidence cited in Akcigit and Ates (2019a) shows that the markup increases from 20% to 50% between 1980
and 2010.
38
According to Garcia-Macia et al. (2019), entrant contribution is 32.3% in 1983-1993 and fell to 19.8% in
2003-2013. In our model, the corresponding percents is 41.6% and 36.3%.

31

(Left Exponent)

2.0

Equilibrium in 1981 & 2016

Firm-Value Condition in 1981
R&D-Incentive Condition in 1981
iso-Gini
iso-ST (1%)
iso-SB (40%)
Firm-Value Condition in 2016
R&D-Incentive Condition in 2016

1.8
1.6
1.4
1.2
1.0
1.0

1.2

1.4

1.6

1.8

(Right Exponent)

2.0

Figure 10: Based on calibrated values of the parameters, a movementof equilibrium from 1981 to 2016 is depicted with the R&D-incentive andfirm-value conditions. Contours for a given Gini Coefficient, bottom 40%
income shares and top 10% income shares for 1981 are also shown.

in ξ, ζ and the inequality indices are the results of changing all parameters at the same time.
Given this observation, we quantify the contribution of each parameter to the X relationship,using a method similar to the one employed in Akcigit and Ates (2019b). We conduct counterfactual experiments by holding one parameter at the 1981 level at a time, while other parameterschange as documented in Table 2. Inevitably, the inequality indices deviate from the original
series, and such deviation allows us to measure the contribution of a parameter held fixed. We
repeat this process for δE , δI , h, λ, τ and sI = sE . To quantify deviation, we use the following
measures:
yend

1 X
Dy − Dykdk
D2016 − D2016
y=1981
Ω1 =
,
Ω2 =
(52)
D2016 − D1981
Dyend − D1981
D refers to the Gini coefficient, the top 10% income share or the bottom 40% income share, andk is a variable fixed at the 1981 level.39 For example, D2016 is the Gini coefficient in 2016 andkD2016
is the Gini coefficient in 2016 with a variable k is fixed at the 1981 level. d is the numberof years used in the numerator in Ω2 . Note that Ω1 measures deviation in 2016, while Ω2 uses
the average of deviation as a measure of the contribution of a variable k.40 Also note that thelarger the value of Ω1 and Ω2 , the greater the contribution made by a variable k. If Ω1 or Ω2 isnegative, it means a negative contribution being made by a variable k.
Consider entrant R&D productivity δE . It is best explained using Panel (a) of Figure 11. The
Gini coefficient, the bottom 40% income share and the top 1% income share are shown, and series
labelled “Double-Pareto Prediction” and “Data” are equivalent to those in Panels (a), (b) and (e) of
Figure 7. Series labelled “δE fixed” shows what would happen if the parameter was held constantat the 1981 level. Consider the left graph. For a constant δE , the Gini coefficient falls rather thanrises. It means that the effect of a reduction of δE is so strong that if it is removed, then the Gini
coefficient follows a clear negative trend. In this sense, a falling δE made a significant contribution
39
40

A similar index is used in Akcigit and Ates (2019b).
In (52), yend is the end year which may differ for the reason mentioned in Footnote 41.

32

Panel (a): E Fixed at 1981 Level
Gini Coefficient

12.5

Bottom 40% Income (%)

Top 1% Income (%)

0.58

12.0

22

0.56

11.5

20

11.0

18

0.54

10.5

0.52

16

10.0

0.50

9.5

14

0.48

9.0

12

0.46
1980 1985 1990 1995 2000 2005 2010 2015

8.5
1980 1985 1990 1995 2000 2005 2010 2015

101980 1985 1990 1995 2000 2005 2010 2015

E fixed

Double-Pareto Approximation

Data

Panel (b): I Fixed at 1981 Level
Gini Coefficient

12

0.9

10

0.8

8

Bottom 40% Income (%)

80
70
60

6

0.7

50
40

4
0.6

30

2

0.5
1980

1985

1990

1995

2000

0
1980

2005

I fixedTop 1% Income (%)

90

20
1985

1990

1995

2000

2005

10
1980

Double-Pareto Approximation

1985

1990

1995

2000

2005

Data

Panel (c): h Fixed at 1981 Level
0.60

Gini Coefficient

Bottom 40% Income (%)
11.5

0.58

22

11.0

20

10.5

0.56

18

10.0

0.54
0.52
1980 1985 1990 1995 2000 2005 2010 2015

Top 1% Income (%)

9.5

16

9.0

14

8.5
1980 1985 1990 1995 2000 2005 2010 2015121980 1985 1990 1995 2000 2005 2010 2015

h fixed

Double-Pareto Approximation
Figure 11:

33

Data

Panel (a): Fixed at 1981 Level
Gini Coefficient

Bottom 40% Income (%)

0.58

13

0.56

12

0.54

11

0.52

10

0.50

9

1980 1985 1990 1995 2000 2005 2010 2015

22
20
18
16
14

1980 1985 1990 1995 2000 2005 2010 2015

fixed

Top 1% Income (%)

121980 1985 1990 1995 2000 2005 2010 2015

Double-Pareto Approximation

Data

Panel (b): Fixed at 1981 Level
Gini Coefficient

Bottom 40% Income (%)

0.59

Top 1% Income (%)

0.58

11.0

22

0.57

10.5

20

10.0

18

0.54

9.5

16

0.53

9.0

14

8.5
1980 1985 1990 1995 2000 2005 2010 2015121980 1985 1990 1995 2000 2005 2010 2015

0.56
0.55

0.52
1980 1985 1990 1995 2000 2005 2010 2015

fixed

Double-Pareto Approximation

Data

Panel (c): sE and sI Fixed at 1981 Level
Gini Coefficient

Bottom 40% Income (%)

0.59

11.5

0.58

11.0

0.57

22
20

10.5

0.56

Top 1% Income (%)

18

0.55

10.0

0.54

9.5

16

0.53

9.0

14

8.5
1980 1985 1990 1995 2000 2005 2010 2015121980 1985 1990 1995 2000 2005 2010 2015

0.52
0.51
1980 1985 1990 1995 2000 2005 2010 2015

sI, sE fixed

Double-Pareto Approximation
Figure 12:

34

Data

Table 4:
Measure: Ω1

D: Double-Pareto Approximated Series
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

δE

δI

h

λ

τ

sI = sE

δE and hτ andsI = sE(7)/(8)Gini Coefficient

1.66

−3.00

0.67

0.69

0.43

0.55

1.79

0.92

1.94

Top 1% Share

1.64

−3.43

0.68

0.43

0.28

0.35

1.83

0.57

3.24

Bottom 40% Share

1.43

−3.58

0.69

1.32

0.81

1.04

1.28

1.78

0.72

Table 5:
Measure: Ω2

D: Double-Pareto Approximated Series
(1)

(2)

(3)

(4)

(5)

(6)

(7)

(8)

(9)

δE

δI

h

λ

τ

sI = sE

δE and hτ andsI = sE(7)/(8)Gini Coefficient

0.91

−1.75

0.20

0.27

0.22

0.27

0.93

0.92

1.01

Top 1% Share

0.88

−2.07

0.20

0.16

0.13

0.16

0.93

0.57

1.64

Bottom 40% Share

0.80

−2.08

0.21

0.54

0.43

0.52

0.73

1.78

0.41

to an increase in the Gini coefficient. A similar pattern arises in the right graph of the top 1%
income share. It would have fallen below 10% in 2010 if entrant R&D productivity had been left
unchanged in 1981. The middle graph shows the bottom 40% income share. A δE -fixed series
is trend-less or has a slightly positive trend, while the Data and Double-Pareto Prediction series
fall. It confirms that a fall in δE had large impacts on different aspects of inequality.
Turning to Panel (b), it shows the case of fixing incumbent R&D productivity δI . In sharp
contrast to δE , the trends of the δI -fixed series are all reversed.41 For example, consider the Gini
coefficient. If δI was fixed at the 1981 level, it would have increased as shown in the left graph. Itmeans that a decreasing incumbent R&D productivity mitigated inequality measured by the Gini
coefficient. The top 1% share in right graph is similarly interpreted. In the case of the bottom
40% income share in the middle graph, it would have been as low as 4% (ignoring an observation
with less than 1% in 2003) with a δI fixed at the 1981 level. These imply that the worsening of
inequality is mitigated due to a declining incumbent productivity.
An intuition for these results of entrant and incumbent R&D productivity levels is simple. If
δE is kept at the 1981 level, changes in gE become minimal, while a falling δI tends to reducegI . As a result, the left and right Pareto exponents ξ = (1 − gE ) /gI and ζ = gE /gI both tend toincrease. In Figure 5, this means that an economy moves northeastward from A0 for a constantδE . Fixing δI is the opposite case where equilibrium moves southwestward.
To quantify the contrasting results of fixing δE and δI , let us turn to Columns (1) and (2) ofTable 4. It uses the end-year deviation Ω1 as a measure of contribution with the Double-Pareto
approximated series used for D in (52). The numbers in Column (1) are all positive, while negative
41

Data of years after 2008 are all dropped from the figure because they make either ξ less than one or the Gini
coefficient greater than one.

35

in Column (2). The same pattern remains in Table 12 with the average cumulative measure Ω2 .
These results concerning the relative roles of incumbent/entrant firms are in line Garcia-Macia
et al. (2019) which find a dwindling role of entrant innovation in TFP growth in the 1983-2013
period.
The same quantifying approach is applied to h, λ, τ and sE = sI . Panel (c) of Figure 11 showsthat the h-fixed series are trend-less, though they are more volatile compared with the δE -fixed
series. It means that a falling h has a positive impact on the inequality indices, though its effectis less than δE , as confirmed in Column (3) of Tables 4 and 5. Panel (a) of Figure 12 shows thecase of the quality step λ. It exhibits a volatile pattern similar to h, and its impacts are also
comparable to h, as Column (4) of the tables confirm. In Panels (b) and (c) of the figure, theτ -fixed and sE = sI fixed series follow more steady patterns. Visual inspection of the graphs
indicates their significant impacts, which are confirmed in Columns (5) and (6) of Tables 4 and 5.

6.4

Declining Business Dynamism and Policy Changes

Having considered the impacts of each parameter separately, there are two issues that we consider
next. First, how are those parameter changes interact in generating the X inequality relationship?
Do they reinforce or contract each other? Second, the following difference seems to have emerged.
The effects of δE and h on the three inequality indices, documented in Tables 4 and 5 are similar
in magnitude, whereas λ, τ and sI = sE affected the bottom 40% income share more because themagnitude of their impacts on the bottom share is about twice as large as the top 1% income
share and the Gini coefficient. How do we interpret these results? To address those questions, wegroup those parameters (except λ) into two. One group consists of δE and h capturing an aspectof a declining business dynamism in the U.S., and another group of τ and sI = sE consists ofchanging fiscal policy measures.
A declining business dynamism is characterized by a falling pace of startups and new businesses with an increasing share of older firms. As Acemoglu et al. (2018) argue, it would leadto adverse impacts on growth and productivity because it means a slower pace of reallocation of
resources from less efficient to more efficient businesses. To the extent that new firms’ innovations,involving job creation and destruction, are important to productivity growth, a declining business dynamism, observed in the U.S. at least since 1980, is a serious concern to policy makers.42
Evidently, data show that an incentive for new firms to enter the market declined. In particular,
according to Decker et al. (2014), a declining business dynamism is observed in almost all sectors and all geographic regions, though variations exist.43 Whatever factors working behind thephenomenon, it is captured by a falling δE and h in our model. To assess the contribution of adeclining business dynamism to the X inequality relationship, let us apply the method used above.
That is, we fix those two parameters at the 1981 level and change others. The results are shownin Column (7) of Tables 4 and 5. The magnitude of the impacts are certainly large. However,compared with δE , an increase in the magnitude is not particularly dramatic. In addition, themagnitude even slightly fell for the bottom 40% income share. What it suggests is that δE andh have a relatively large “substitutability” in explaining the X inequality relationship, especially42

Startup firms account for about 20 percent of total job creation (see Decker et al. (2014)).
As factors that are not sector-specific and region-specific, Decker et al. (2014) suggest regulation increasing
adjustment costs (e.g.Gutiérrez, Jones and Philippon (2019)) and technological progress plus globalization favoring
big businesses. Decker, Haltiwanger, Jarmin and Miranda (2016a) refer to network externalities which work in favorof big firms, and Akcigit and Ates (2019b) argue that a slower knowledge diffusion from frontier firms to lagging
firms is a possible cause. Astebro, Braguinsky and Ding (2020) report that an increasing burden of knowledge in
R&D and management discouraged startups, again favoring big firms, while population growth slowdown is citedas an important factor in Peters and Walsh (2019).
43

36

for the bottom income share.
Let us turn to the fiscal policy τ and sI = sE . Akcigit and Ates (2019b) consider changes in
the fiscal policy as a possible cause for a declining business dynamism. According to the study,
the U.S. went through major tax system overhauls in the 1980s with a substantial reduction ofa statutory corporate tax rate. They also showed that an effective tax rate, which takes into
account various tax benefits and determines actual tax bills, also dramatically fell. Akcigit andAtes (2019b) also explains an increasing intervention in supporting R&D in the period. The US
government began a federal R&D tax credit in 1981, and in the next year state-level support
started in Minnesota and spread to other states. Major recipients were incumbent firms because
taxable profits were needed for the tax credit. In order to quantify their contribution to the X
inequality relationship, τ and sI = sE are held fixed at the 1981 level, letting other parameterschange. Consider Column (8) of Table 4 first. The magnitude increases nearly in a linear way inthe sense that summing the numbers in (5) and (6) approximately gives the magnitude in Column(8). In Table 5, on the other hand, the result is more dramatic because of the cumulative natureof the index Ω2 . The number in (8) is nearly twice as large as the sum of (5) and (6). In thissense, those policy measures are “complimentary” and their changes reinforce the effects of theother .
Given the above discussion, two results stand out. The effect of a declining business dynamism
seems to have generated a larger impact on the Gini coefficient than the fiscal policy changes,
though their impacts are comparable when the cumulative index Ω2 is used in Table 5. Second,
Column (9) shows the ratio of (7) over (8). It indicates that the top income share is more affectedby a declining business dynamism, and the bottom income share by the policy changes. In thissense, the two factors operated on different aspects of inequality to a different degree.

7

Conclusion

Inequality can be measured in different ways. The Gini coefficient and the top/bottom income
shares are often used in the literature. The Gini coefficient is a summary measure of the entire
distribution and the income shares show how the chosen part of the distribution changes relative
to the whole distribution. Although they show different aspects of inequality, data show thatthey are systematically related. That is, the Gini coefficient is negatively related to the bottom
pB % income share and positively to the top pT % income share, giving rise to what we call the
X inequality relationship. It is certainly intuitive that they are related in an observed way, but
equations remain. How do we explain it? What economic forces are working behind?
We explore these issues by constructing a Schumpeterian growth model which gives rise to adouble-Pareto distribution of income as a result of entrant and incumbent innovations. A doublePareto income distribution allows us to develop iso-Gini loci and iso-income share schedules ina tractable way. In equilibrium, the rates of incumbent and entrant innovations determine theLeft and Right Pareto exponents, which in turn characterize a market equilibrium. Comparative
statics analysis shows that changes of most parameters generate the X inequality relationship.
The results imply that incumbent and entrant innovations play an important role in generating
the X inequality relationship.
We also used the model to quantify the underlying factors behind the relationship in the
U.S. via calibration. Making use of innovation-related data to pin down parameter values, weconsider the impact of each parameter on equality indices. We found that the largest impact wascaused by deterioration of entrant R&D productivity. Calibration also shows a fall in incumbent
R&D productivity, which was found to mitigate inequality. These contrasting results highlight37

the important roles played by different types of innovations behind the X inequality relationship.
In addition, we also grouped parameters into two; one capturing fiscal policy changes and theother for a declining business dynamism. Both are certainly important in understanding the X
inequality relationship. But the latter seems to be have a particularly important implication,
because some studies (e.g. Fikri et al. (2017) and Furman and Orszag (2018)) point out that adeclining business dynamism is behind an increasing inequality in the U.S.
Our calibration analysis focuses upon the U.S. only. But, the X inequality relationship holds
in other countries. Furthermore, Calvino, Criscuolo and Verlhac (2020) provides evidence ofa declining business dynamism being “pervasive” in many countries. Our result indicates thepossibility that those two phenomena are related in those economies as well.

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Appendix A

Derivation of (22)ξζFootnote 21 gives fL (z) = CL z ζ−1 where CL = (ξ+ζ)zξ . It is the number of entrepreneurs at z in the left
distribution. Define Tz as the time period just required for them to reach z, starting from z. Then their
number after Tz falls tofL (z) e−gE Tz(53)

because some exit due to entrant innovations. Now consider an entrant which innovates at tz with h. Her
profit at t ≥ tz is given by z = egI (t−tz ) hz where egI (t−tz ) h ≤ 1. After Tz , it increases to z = z, at whiche g I Tz h = h⇒

Tz = −

ln zz
.
gI

(54)

Substituting (54) into (53) yields CzζL z ξ+ζ−1 , which is the number of entrepreneurs who reach z startingfrom z. Integrating it from 0 to z yields the flow of entrepreneurs reaching z
Z z
0

CL ξ+ζ−1
ξζ
z
dz =
2.
ζ
z
(ξ + ζ)40

Equating it to gE FR (z) and using (16) yields (22).

Appendix B

Derivation of (28) and Iso-Gini Contours

Using (24), first calculate the total net profit which is also equal to the average net profit
Z ∞ξζ
Ztotal =zf (z) dz =z(ξ + 1) (ζ − 1)0(55)

where f (z) is given in footnote 21. Then, the Gini coefficient G is defined byZ ∞Ztotal G =F (z) [1 − F (z)] dz = GL (z) + GR (z)
0

where


ζ  z ξζ  z ξ
1−
dz
z
ξ+ζ z
0 ξ+ζ
ξζ
2ξ + 1 + ζ=z
2 · (ξ + 1) (2ξ + 1)
(ξ + ζ)GL (z) =

Z z

and
Z ∞
GR (z) =1−ξ  z −ζξ  z −ζ
dz
ξ+ζ z
ξ+ζ z
z
ξζ
ξ + 2ζ − 1=z2 ·(ξ + ζ) (ζ − 1) (2ζ − 1)

after tedious rearrangement. Now, making use of GL (z̄) and GR (z̄), the Gini coefficient is re-expressedas (28) (again after tedious rearrangement).
To calculate the slope of an iso-Gini contour, note that∂G2ζ (ζ − 1) (2ζ + 4ξ + 1)=−
2
2 < 0,∂ξ(ξ + ζ) (2ζ − 1) (2ξ + 1)

∂G2ξ (ξ + 1) (4ζ + 2ξ − 1)
=−
< 0.
2
2
∂ζ
(ξ + ζ) (2ζ − 1) (2ξ + 1)
These allow us to derive the following:ξ (ξ + 1) (2ξ + 1) (4ζ + 2ξ − 1)dξ=−<0dζ G=Gζ (ζ − 1) (2ζ − 1) (2ζ + 4ξ + 1)
To show convexity of an iso-Gini curve, define b ≡ ζξ so thatdξ(ξ + 1) (2ξ + 1) [4ξ + b (2ξ − 1)]

 .
= −b2
dζ G=G(ξ − b) (2ξ − b) 2 ξ + 4ξ + 1
b


dξ
∂
One can easily show ∂b
− dζ

G=G



> 0, establishing the desired result.

41

(56)

Appendix C

Derivation of (29) and (30)ζFirst note p̄ = F (z̄) = ξ+ζand the total income Ztotal is defined in (55). Now define the bottom percentilepB ≤ p̄ such that pB = F (z (pB )) where z (pB ) is net profit at pB . This definition givesz (pB ) = z



ξ+ζ
pB
ζ

 ξ1

.

Using this result, calculate the cumulative income up to z (pB )Z (pB ) =Z z(pB )zf (z) dz =0ξζ(ξ + ζ) (ξ + 1)



ξ+ζ
pB
ζ

1+ ξ1z.

(pB )Then, the bottom 100pB % income share is defined by SB = ZZLtotal, which gives (29). It is straightforward
to calculate the slope of SBξ(ξ+1)∂SBdξζ(ζ−1)∂ζ= − ∂SB = −ζdζ Bottom
1+ 1+
L (p , ξ, ζ)
∂ξ
B

ξ

(57)

B

1
> 0 because
where LB (pB , ξ, ζ) = log p 1+
( ζξ )






ξ
ζ
ξ
ξ
pB 1 +< p̄ 1 +
=
1+
= 1.
ζ
ζ
ξ+ζ
ζ

Convexity can also be shown, but omitted.
Next, define the top percentile 1 − pT for pT ≥ p̄ such that pT = F (z (pT )) where z (pT ) is net profit
at pT . This definition givesz (pT ) = z



− ζ1

.

ξ+ζ
−
ξ+1



ξ+ζ
(1 − pT )ξCalculate the cumulative income up to z (pT )Z (pT ) =Z z(pT )0ξζ
zf (z) dz =z(ξ + ζ) (ζ − 1)(ξ+ζ
(1 − pT )ξ1− ζ1 ).

T)
Then, the top 100 (1 − pT ) % income share is defined by ST = 1 − Z(pZtotal , which gives (30). Its slope is





∂ST
dξ
ξ (ξ + 1)ξ∂ζ= − ∂ST = −1+ 1+
LT (1 − pB , ξ, ζ)dζ Topζ (ζ − 1)ζ∂ξwhere LT (1 − pT , ξ, ζ) = log (1−p )1 1+ ζ > 0 because
T (
ξ)






ζ
ζ
ξ
ζ
(1 − pT ) 1 +< (1 − p̄) 1 +
=
1+
= 1.
ξ
ξ
ξ+ζ
ξ
Convexity can also be shown, but omitted. Comparing (57) and (58) confirms
p <p

p >p

dξ B
dξ T
<
dζ Bottomdζ Top

42

.

(58)

Appendix D

Relative Slopes of Iso-Gini, Iso-SB and Iso-ST Curves

Using (57), one can easily confirm that
∂
∂pB

p <p

dξ B
dζ Bottom

!

> 0.

(59)

It means that an iso-SB curve pivots anti-clockwise around a given (ξ, ζ) with a lower pB . Similarly, using
(58),
!
p >p
dξ T
∂
>0
(60)
∂pT
dζ Bottom
which implies that an iso-ST contour pivots clockwise around a given (ξ, ζ) as pT becomes larger. Also
note that
p =p
p =p
dξ B
dξ T
ξ (ξ + 1)==−.
dζ Bottomdζ Topζ (ζ − 1)from (57) and (58).
Now, using (56)p =p4ξ (ξ + 1) (ξ + ζ)
dξ B
dξ
=
(ζ + 1 − ξ)−dζ G=Gdζ Bottom
ζ (ζ − 1) (2ζ − 1) (2ζ + 4ξ + 1)
This shows that there are two possible cases:p =pCase 1:dξdξ B≥dζ G=Gdζ Bottom

Case 2:dξdξ B<dζ G=Gdζ Bottom

for ζ + 1 ≥ ξ

p =pfor ζ + 1 < ξFirst consider Case 1. (60) means that an increase in pT makes an iso-ST curve pivots clockwise. Hence,
if pT increases sufficiently and pB falls only slightly, then
p <p

p >p

dξ B
dξ
dξ B
<
<
dζ Bottomdζ G=Gdζ Bottom(61)

arises. Turning to Case 2, if pB decreases sufficiently and pT rises only slightly, then (61) holds.

Appendix E

Growth Rate

First, consider utility maximization with the Lagrangian equation
"
Z
L=e

1
J

RJ

ln Yj dj
0

J

+µ E−

#

Pj Yj dj .

0U
The F.O.C is JY= µPj . Using this and the budget constraint gives (31). Substituting this back into (1)
j
gives
E
U=
(62)
JPU

where

1

RJ

PU = e J 0 ln Pj dj = 1

43

(63)

is the price index which we normalize to one.
Next consider profit maximization of final output producers:
Z 1
R1
pij yij di.
ΠY j = Pj e 0 ln qij yij di −
0

P

R1

The F.O.C. is yijj e 0 ln qij yij di = pij , which gives Pj Yj = pij yij and hence (32). Plug this into (2) with thehighest quality levels to obtain the price index in final output industry j:
Z 1
Z 1
ln Pj =
ln pij di −ln qij di.
00

Substitute this into (63) and rewrite the resulting equation with pji = λw for monopoly products andpji = w for competitive goods, we obtainw N
1 = e J ln λ
(64)
Q
Q̇
Q̇
U̇
It means Q= ẇ
w . This together with (62) and (63) also means Q = U .

Appendix F

Endogenizing gI

Rewrite (36) as
0 = maxni (t)
ni (t) (1 − τ ) Λ E
J − (1 − sI ) wRIi (t)γ+Vi′ (ni (t)) δI RIi (t) ni (t) − (ρ + gE ) Vi (n (t)) + V̇i (t)The F.O.C. is
RIi (t) =Vi′ (ni (t)) γδI(1 − sI ) w

1
 1−γ

Assume V = Vi (ni ) /ni . Then, the F.O.C. is reduced to (38). Using (38), rewrite (36) to give (39).

44